**Mechanobiology of Fracture Healing: Basic Principles and Applications in Orthodontics and Orthopaedics**

Antonio Boccaccio and Carmine Pappalettere *Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari Italy* 

### **1. Introduction**

20 Theoretical Biomechanics

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The Chapter describes how mechanobiological models can be utilized to predict the spatial and temporal patterns of the tissues differentiating within a fracture site during the healing process. It will be structured in four main Sections. Firstly, the basic principles of mechanobiology, the main theories and the principal models utilized to simulate the cellular processes involved in fracture healing will be illustrated. Second, two examples will be given showing how a mechano-regulation model, - where the bone callus is modeled as a biphasic poroelastic material and the stimulus regulating tissue differentiation is hypothesized to be a function of the strain and fluid flow-, can be utilized to assess bone regeneration in an ostetomized mandible submitted to distraction osteogenesis and in a fractured lumbar vertebra. Finally, the main limitations of the model utilized and, in general, of mechanobiological algorithms as well as the future perspectives will be outlined. Fracture healing is a physiological process that initiates immediately after the fracture event and occurs by following two different modalities: by primary fracture healing or by secondary fracture healing. Primary healing involves a direct attempt by the cortex to reestablish itself once it has become interrupted. When stabilisation is not adequate to permit primary healing, the abundant capillaries required for bone repair are constantly ruptured and secondary healing takes place. Secondary healing involves responses within the periosteum and external soft tissues and subsequent formation of an external callus. Secondary fracture healing occurs in the following stages. Blood emanates from the ruptured vessels and a haemorrhage quickly fills the fracture gap space. Macrophages remove the dead tissue and generate initial granulation tissue for the migration of undifferentiated mesenchymal stem cells (MSCs), originating an initial stabilizing callus. These cells proliferate and migrate from the surrounding soft tissue (Einhorn, 1998, McKibbin, 1978) (Fig. 1a). Then, stem cells disperse into the fracture callus, divide (mitosis) and simultaneously migrate within the fracture site (Fig. 1b). In the next stage, mesenchymal cells may differentiate into chondrocytes, osteoblasts or fibroblasts, depending on the biological and mechanical conditions (Fig. 1c). These differentiated cells begin to synthesize the extracellular matrix of their corresponding tissue (Doblaré et al., 2004) (Fig. 1d). Intramembranous woven bone is produced by direct differentiation of the stem cells into osteoblasts. Endochondral ossification occurs when chondrocytes are replaced by osteoblasts.

Mechanobiology of Fracture Healing: Basic

given by:

assumes the form:

exponentially with time.

as the cellular dispersal, the proliferation, the apoptosis, etc.

Principles and Applications in Orthodontics and Orthopaedics 23

potential to differentiate along various lineages of mesenchymal origin in response to appropriate stimuli (Chen et al., 2007). Bone marrow is the most important source for MSCs (Simmons, 1985, Brighton and Hunt, 1991, Glowacki, 1998). However, MSCs have been also identified in different other tissues such as adipose, periosteum, trabecular bone, synovium, skeletal muscle, dental pulp and periodontal ligament (Barry and Murphy, 2004, Ballini et al., 2007, Ballini et al., 2010). Quiescent MSCs become mobilised during repair and remodelling through regulation by external chemical and physical signals that control their activation, proliferation, migration, differentiation and survival i.e. their fate (Byrne, 2008). One key aspect in mechanobiology of MSCs is the modelling of the cellular processes such

Concerning the process of cellular dispersal, it has been suggested that the movement of stem cells can be thought of as an assemblage of particles, with each particle moving around in a random way (Murray, 1989). In a number of studies (Lacroix and Prendergast, 2002, Geris et al., 2004, Andreykiv et al., 2005), a diffusion equation has been used to simulate the movement of cells through regenerating tissues. If *c* is the concentration of stem cells in a given volume and *D* the diffusion coefficient, the derivative of *c* with respect to the time is

> *<sup>c</sup>* <sup>2</sup> *D c t* = ∇ **<sup>d</sup>**

However such a modelling of cellular dispersal presents the limitation that the diffusion coefficient assumes a value that does not depend on the cell phenotype or the tissue through which the cell is moving. Furthermore, this approach implicitly assumes that cells attempt to achieve a homogenous population density within the area of analysis. Lacroix et al. (2002) developed further the diffusion equation (1) by including the processes of cellular mitosis and apoptosis (programmed cell death). Therefore, the rate of change in cell concentration

> <sup>2</sup> ( ) *<sup>c</sup> <sup>c</sup> D c cs c kc*

=∇ + − **<sup>d</sup>**

The first term on the right-hand side of equation (2) describes cell migration by simple linear diffusion; the second term describes cell mitosis, where *cc(x,t)* is the chemical concentration of a mitosis-inducing factor; *s(cc)* is a function describing the mitosis rate per cell; and *k* is a constant describing the cell death or removal rate (Sherratt et al., 1992). Since the mesenchymal stem cells can differentiate into cells of different phenotypes *i* (i.e. fibroblasts, chondrocytes and osteoblasts) that produce different tissues *j* (i.e. fibrous tissue, cartilage and bone), a logical progression of the idea proposed by Lacroix et al., (Lacroix et al., 2002), would be that the diffusion coefficient *D* would depend on the cell phenotype *i* and the tissue type *j* through which the cell is moving. This modelling has been adopted in Kelly and Prendergast (Kelly and Prendergast, 2005, 2006). Boccaccio et al. (Boccaccio et al., 2007, 2008a), modelled the cellular dispersal by using the diffusion equation (1) however, they accounted for the fact that MSCs not only require time to differentiate, but that the differentiated cell types require time to synthesise and remodel new tissue. To this purpose, based on the results of Richardson et al. (Richardson et al., 1992) who observed an exponential increase in stiffness during tibial fracture healing, they assumed that the Young's modulus of all tissues within the fracture callus increases

*t*

**<sup>d</sup>** (1)

**<sup>d</sup>** (2)

### **2. Mechanobiology: Basic principles**

Comparing patterns of differentiation during tissue repair to predictions of the mechanical environment within the mesenchymal tissue has led to the development of a number of hypothesis for mechano-regulated tissue differentiation. Theories on the relationship between mechanics and biology were originally proposed in relation to fracture healing. These theories later evolved into 'mechanobiological algorithms'; a finite set of rules that govern the effects of mechanical loading on stem cells and tissues. Mechanobiology merges the older science of mechanics with the newer and emerging disciplines of molecular biology and genetics.

Fig. 1. Let Ω be an arbitrary fracture domain loaded and constrained over part of the surface. Immediately after the fracture event the mesenchymal stem cells (MSCs) reside outside the domain in the surrounding soft tissue (a). Then, stem cells disperse into the domain, divide (mitosis) and simultaneously migrate within the domain (b). Depending on the biological and mechanical conditions MSCs differentiate into fibroblasts, chondrocytes and osteoblasts (c). These differentiated cells begin to synthesize the extracellular matrix of their corresponding tissue (d).

At the centre of mechanobiology is the cellular process of mechano-transduction, or the way by which the cells sense and respond to mechanical forces or, in general to biophysical stimuli. Experimental and analytical models are often integrated in mechanobiology to gain a deeper understanding of the cells' response to mechanical factors. Experiments provide insights and measurements, which can then be interpreted within the context of analytical frameworks. Analytical simulations permit investigation of possible explanations that require in vivo validation and will suggest further experimental investigations (van der Meulen and Huiskes, 2002).

### **2.2 Mechanobiology of mesenchymal stem cells**

Mesenchymal stem cells (MSCs) are nonhematopoietic progenitor cells found in adult tissues. They posses an extensive proliferative ability in an uncommitted state and hold the

Comparing patterns of differentiation during tissue repair to predictions of the mechanical environment within the mesenchymal tissue has led to the development of a number of hypothesis for mechano-regulated tissue differentiation. Theories on the relationship between mechanics and biology were originally proposed in relation to fracture healing. These theories later evolved into 'mechanobiological algorithms'; a finite set of rules that govern the effects of mechanical loading on stem cells and tissues. Mechanobiology merges the older science of

Fig. 1. Let Ω be an arbitrary fracture domain loaded and constrained over part of the surface. Immediately after the fracture event the mesenchymal stem cells (MSCs) reside outside the domain in the surrounding soft tissue (a). Then, stem cells disperse into the domain, divide (mitosis) and simultaneously migrate within the domain (b). Depending on the biological and mechanical conditions MSCs differentiate into fibroblasts, chondrocytes and osteoblasts

a) b)

c) d)

At the centre of mechanobiology is the cellular process of mechano-transduction, or the way by which the cells sense and respond to mechanical forces or, in general to biophysical stimuli. Experimental and analytical models are often integrated in mechanobiology to gain a deeper understanding of the cells' response to mechanical factors. Experiments provide insights and measurements, which can then be interpreted within the context of analytical frameworks. Analytical simulations permit investigation of possible explanations that require in vivo validation and will suggest further experimental investigations (van der

Mesenchymal stem cells (MSCs) are nonhematopoietic progenitor cells found in adult tissues. They posses an extensive proliferative ability in an uncommitted state and hold the

(c). These differentiated cells begin to synthesize the extracellular matrix of their

mechanics with the newer and emerging disciplines of molecular biology and genetics.

**2. Mechanobiology: Basic principles**

corresponding tissue (d).

Meulen and Huiskes, 2002).

**2.2 Mechanobiology of mesenchymal stem cells** 

potential to differentiate along various lineages of mesenchymal origin in response to appropriate stimuli (Chen et al., 2007). Bone marrow is the most important source for MSCs (Simmons, 1985, Brighton and Hunt, 1991, Glowacki, 1998). However, MSCs have been also identified in different other tissues such as adipose, periosteum, trabecular bone, synovium, skeletal muscle, dental pulp and periodontal ligament (Barry and Murphy, 2004, Ballini et al., 2007, Ballini et al., 2010). Quiescent MSCs become mobilised during repair and remodelling through regulation by external chemical and physical signals that control their activation, proliferation, migration, differentiation and survival i.e. their fate (Byrne, 2008). One key aspect in mechanobiology of MSCs is the modelling of the cellular processes such as the cellular dispersal, the proliferation, the apoptosis, etc.

Concerning the process of cellular dispersal, it has been suggested that the movement of stem cells can be thought of as an assemblage of particles, with each particle moving around in a random way (Murray, 1989). In a number of studies (Lacroix and Prendergast, 2002, Geris et al., 2004, Andreykiv et al., 2005), a diffusion equation has been used to simulate the movement of cells through regenerating tissues. If *c* is the concentration of stem cells in a given volume and *D* the diffusion coefficient, the derivative of *c* with respect to the time is given by:

$$\frac{\mathbf{dc}}{\mathbf{dt}} = D\nabla^2 \mathbf{c} \tag{1}$$

However such a modelling of cellular dispersal presents the limitation that the diffusion coefficient assumes a value that does not depend on the cell phenotype or the tissue through which the cell is moving. Furthermore, this approach implicitly assumes that cells attempt to achieve a homogenous population density within the area of analysis. Lacroix et al. (2002) developed further the diffusion equation (1) by including the processes of cellular mitosis and apoptosis (programmed cell death). Therefore, the rate of change in cell concentration assumes the form:

$$\frac{\mathbf{dc}}{\mathbf{dt}} = D\nabla^2 \mathbf{c} + c\mathbf{s}(\mathbf{c}\_c) - k\mathbf{c} \tag{2}$$

The first term on the right-hand side of equation (2) describes cell migration by simple linear diffusion; the second term describes cell mitosis, where *cc(x,t)* is the chemical concentration of a mitosis-inducing factor; *s(cc)* is a function describing the mitosis rate per cell; and *k* is a constant describing the cell death or removal rate (Sherratt et al., 1992). Since the mesenchymal stem cells can differentiate into cells of different phenotypes *i* (i.e. fibroblasts, chondrocytes and osteoblasts) that produce different tissues *j* (i.e. fibrous tissue, cartilage and bone), a logical progression of the idea proposed by Lacroix et al., (Lacroix et al., 2002), would be that the diffusion coefficient *D* would depend on the cell phenotype *i* and the tissue type *j* through which the cell is moving. This modelling has been adopted in Kelly and Prendergast (Kelly and Prendergast, 2005, 2006). Boccaccio et al. (Boccaccio et al., 2007, 2008a), modelled the cellular dispersal by using the diffusion equation (1) however, they accounted for the fact that MSCs not only require time to differentiate, but that the differentiated cell types require time to synthesise and remodel new tissue. To this purpose, based on the results of Richardson et al. (Richardson et al., 1992) who observed an exponential increase in stiffness during tibial fracture healing, they assumed that the Young's modulus of all tissues within the fracture callus increases exponentially with time.

Mechanobiology of Fracture Healing: Basic

environment (Carter et al., 1998)

Principles and Applications in Orthodontics and Orthopaedics 25

relating tissue differentiation to mechanical loading. They proposed that local stress or strain history influences tissue differentiation over time (Carter et al., 1988). These ideas were later developed further and a more general mechano-regulation theory was proposed (Carter et al., 1998) (Fig. 3). They postulated that: (i) compressive hydrostatic stress history guides the formation of cartilaginous matrix constituents; (ii) tensile strain history guides connective tissue cells in their production and turnover of fibrous matrix constituents; (iii) fibrocartilage is formed when a tissue loading history consists of a combination of high levels of hydrostatic compressive stress and high levels of tensile strain; (iv) direct bone formation is permitted, in regions exposed to neither significant compressive hydrostatic stress nor significant tensile strain, provided there is an adequate blood supply; (v) preosseous tissue can be diverted down a chondrogenic pathway in regions of low oxygen tension. The mechano-regulation theory of Claes and Heigele (Claes and Heigele, 1999) was initially presented in quantitative terms, and although the resulting concept is similar to that of Carter et al. (Carter et al., 1998), they based their mechano-regulation theory on the observation that bone formation occurs mainly near calcified surfaces and that both intramembranous and endochondral ossification exist in fracture healing. Depending on local strain and hydrostatic pressure different cellular reactions and tissue differentiation

processes were predicted to occur (Claes et al., 1998; Claes and Heigele, 1999).

Fig. 3. Schematic of the mechano-regulation model developed by Carter and colleagues representing the role of the hydrostatic stress history and the maximum principle tensile strain history on the differentiation of mesenchymal stem cells in a well-vascularised

Prendergast and Huiskes (Prendergast and Huiskes, 1995) and Prendergast et al. (Prendergast et al., 1997), created a poroelastic finite element model of a bone-implant interface to analyse the mechanical environment on differentiating cells. They found that the biophysical stimuli experienced by the regenerating tissue at the implant interface are not only generated by the tissue matrix, but also to a large extent by the drag forces from the interstitial flow. Based on this study, a new mechano-regulation theory was developed taking into consideration that connective tissues are poroelastic and comprise both fluid and solid. They proposed a mechano-regulatory pathway composed of two biophysical stimuli; octahedral strain of the solid phase and interstitial fluid velocity relative to the solid. Fluid

In reality, diffusion is not the mechanism of stem cell dispersal; cells disperse by crawling or proliferation or are transported in a moving fluid (Prendergast et al., 2009). In order to better simulate the cellular processes involved during the fracture healing process, Pérez and Prendergast (Pérez and Prendergast , 2007) developed a 'random-walk' model to describe cell proliferation and migration, with and without a preferred direction. In this approach, a regular lattice of points is superimposed on the fracture domain. Each lattice point is either empty, or occupied by a stem cell. Cell movement can be simulated by moving a cell from one lattice point to another; cell proliferation, by dividing a cell so that the daughter cell takes up a neighbouring lattice point; cell apoptosis, by removing a cell at a lattice point.

Fig. 2. Diagram showing the mechano-regulation model developed by Pauwels (Pauwels, 1960). The combination of two biophysical stimuli, shear strain and hydrostatic pressure, will act on the mesenchymal cell pool leading to either hyaline cartilage, fibrocartilage or fibrous tissue as represented on the perimeter of the quadrant. The larger arrows indicate that, as time passes, ossification of these soft tissues occurs, provided that the soft tissue has stabilized the environment. Reprinted from Bone, Vol. 19, Issue 2, Weinans H, Prendergast PJ, Tissue adaptation as a dynamical process far from equilibrium, Pages No. 143-149, Copyright (1996) with permission from Elsevier.

### **2.3 Principal mechano-regulation models**

Pauwels, (Pauwels, 1960), who was the first to propose the hypothesis of a mechano-regulated tissue differentiation, suggested that the distortional shear stress is a specific stimulus for the development of collagenous fibres and that hydrostatic compressive stress is a specific stimulus for cartilage formation. When a soft tissue has stabilized the environment, differentiation of MSCs into osteoblasts is favoured leading to the formation of bone (Fig. 2). Based on a qualitative analysis of clinical results of fracture healing, Perren (Perren, 1979) proposed that tissue differentiation is controlled by the tolerance of various tissues to strain. The basis of this theory, - normally known as 'the interfragmentary strain theory' - is that a tissue that ruptures or fails at a certain strain level cannot be formed in a region experiencing strains greater than this level. Based on the framework of Pauwels (Pauwels, 1960), Carter et al. (Carter et al., 1988, Carter and Wong, 1988) expanded the concepts

In reality, diffusion is not the mechanism of stem cell dispersal; cells disperse by crawling or proliferation or are transported in a moving fluid (Prendergast et al., 2009). In order to better simulate the cellular processes involved during the fracture healing process, Pérez and Prendergast (Pérez and Prendergast , 2007) developed a 'random-walk' model to describe cell proliferation and migration, with and without a preferred direction. In this approach, a regular lattice of points is superimposed on the fracture domain. Each lattice point is either empty, or occupied by a stem cell. Cell movement can be simulated by moving a cell from one lattice point to another; cell proliferation, by dividing a cell so that the daughter cell takes up a neighbouring lattice point; cell apoptosis, by removing a cell at a lattice point.

Fig. 2. Diagram showing the mechano-regulation model developed by Pauwels (Pauwels, 1960). The combination of two biophysical stimuli, shear strain and hydrostatic pressure, will act on the mesenchymal cell pool leading to either hyaline cartilage, fibrocartilage or fibrous tissue as represented on the perimeter of the quadrant. The larger arrows indicate that, as time passes, ossification of these soft tissues occurs, provided that the soft tissue has stabilized the environment. Reprinted from Bone, Vol. 19, Issue 2, Weinans H, Prendergast PJ, Tissue adaptation as a dynamical process far from equilibrium, Pages No. 143-149,

Pauwels, (Pauwels, 1960), who was the first to propose the hypothesis of a mechano-regulated tissue differentiation, suggested that the distortional shear stress is a specific stimulus for the development of collagenous fibres and that hydrostatic compressive stress is a specific stimulus for cartilage formation. When a soft tissue has stabilized the environment, differentiation of MSCs into osteoblasts is favoured leading to the formation of bone (Fig. 2). Based on a qualitative analysis of clinical results of fracture healing, Perren (Perren, 1979) proposed that tissue differentiation is controlled by the tolerance of various tissues to strain. The basis of this theory, - normally known as 'the interfragmentary strain theory' - is that a tissue that ruptures or fails at a certain strain level cannot be formed in a region experiencing strains greater than this level. Based on the framework of Pauwels (Pauwels, 1960), Carter et al. (Carter et al., 1988, Carter and Wong, 1988) expanded the concepts

Copyright (1996) with permission from Elsevier.

**2.3 Principal mechano-regulation models** 

relating tissue differentiation to mechanical loading. They proposed that local stress or strain history influences tissue differentiation over time (Carter et al., 1988). These ideas were later developed further and a more general mechano-regulation theory was proposed (Carter et al., 1998) (Fig. 3). They postulated that: (i) compressive hydrostatic stress history guides the formation of cartilaginous matrix constituents; (ii) tensile strain history guides connective tissue cells in their production and turnover of fibrous matrix constituents; (iii) fibrocartilage is formed when a tissue loading history consists of a combination of high levels of hydrostatic compressive stress and high levels of tensile strain; (iv) direct bone formation is permitted, in regions exposed to neither significant compressive hydrostatic stress nor significant tensile strain, provided there is an adequate blood supply; (v) preosseous tissue can be diverted down a chondrogenic pathway in regions of low oxygen tension. The mechano-regulation theory of Claes and Heigele (Claes and Heigele, 1999) was initially presented in quantitative terms, and although the resulting concept is similar to that of Carter et al. (Carter et al., 1998), they based their mechano-regulation theory on the observation that bone formation occurs mainly near calcified surfaces and that both intramembranous and endochondral ossification exist in fracture healing. Depending on local strain and hydrostatic pressure different cellular reactions and tissue differentiation processes were predicted to occur (Claes et al., 1998; Claes and Heigele, 1999).

Fig. 3. Schematic of the mechano-regulation model developed by Carter and colleagues representing the role of the hydrostatic stress history and the maximum principle tensile strain history on the differentiation of mesenchymal stem cells in a well-vascularised environment (Carter et al., 1998)

Prendergast and Huiskes (Prendergast and Huiskes, 1995) and Prendergast et al. (Prendergast et al., 1997), created a poroelastic finite element model of a bone-implant interface to analyse the mechanical environment on differentiating cells. They found that the biophysical stimuli experienced by the regenerating tissue at the implant interface are not only generated by the tissue matrix, but also to a large extent by the drag forces from the interstitial flow. Based on this study, a new mechano-regulation theory was developed taking into consideration that connective tissues are poroelastic and comprise both fluid and solid. They proposed a mechano-regulatory pathway composed of two biophysical stimuli; octahedral strain of the solid phase and interstitial fluid velocity relative to the solid. Fluid

Mechanobiology of Fracture Healing: Basic

**2.4 Mechanobiology: Domains of applicability** 

healing process in cancellous bone.

engineering (Boccaccio et al., 2011a).

Applications of mechanobiology can be found in three main areas:

Principles and Applications in Orthodontics and Orthopaedics 27

i. In the development of new clinical therapies, for example in bone fracture healing, or osteoporosis. Different studies are reported in literature in which mechanobiological models are utilized to pursue this aim: Lacroix and Prendergast (Lacroix and Prendergast, 2002) predicted the patterns of tissue differentiation during fracture healing of long bones; Shefelbine et al. (Shefelbine et al., 2005) simulated the fracture

ii. In the improvement of implant design. With implants such as prostheses, cells migrate up to the implant surface and begin to synthesis matrix, but if the micromotion is too high bone will not form to stabilise the implant – instead a soft tissue layer will form (Huiskes, 1993, Prendergast, 2006). A number of articles can be found in literature where mechanobiological models are utilized to predict the patterns of tissue differentiation at the tissue-implant interface: Andreykiv et al. (Andreykiv et al., 2005) simulated the bone ingrowth on the surface of a glenoid component; Moreo et al., (Moreo et al., 2009a,b) developed a mechano-regulation algorithm that models the main biological interactions occurring at the surface of endosseous implants and is able to

reproduce most of the biological features of the osseointegration phenomenon. iii. In bone tissue engineering and regenerative medicine. Appropriate biophysical stimuli are needed in bone scaffolds, in addition to nutrients and appropriate levels of oxygen supply, to favour an appropriate tissue differentiation process (Martin et al., 2004, Prendergast et al., 2009). A number of studies (Byrne et al., 2007, Milan et al., 2009; Olivares et al., 2009, Sanz-Herrera et al., 2009) are reported in literature that through a combined use of finite element method and mechano-regulation algorithms described the possible patterns of the tissues differentiating within biomimetic scaffolds for tissue

In this Chapter we will focus on the first domain of applicability (i) and, specifically, two examples will be illustrated that show how mechanobiology can be used to predict the patterns of tissue differentiation in a human mandible osteotomized and submitted to distraction osteogenesis as well the regrowth and the remodelling process of the cancellous bone in a vertebral fracture. Predictions were conducted by implementing the mechano-

regulation model of Prendergast and colleagues (Prendergast et al., 1997).

**3. Mechanobiology of mandibular symphyseal distraction osteogenesis** 

Mandibular Symphyseal Distraction Osteogenesis (MSDOG) is a common clinical procedure aimed to modify the geometrical shape of the mandible for correcting problems of dental overcrowding and arch shrinkage. Such problems are usually solved by tooth extraction or expansion protocols. However, these clinical procedures are unstable and tend to relapse towards the original dimension (McNamara and Brudon, 1993). Mandibular distraction osteogenesis may solve transverse mandibular deficiency problems. With this clinical procedure the mandibular geometry is definitively changed so that the risk of a relapsing towards the original dimension is avoided. In spite of consolidated clinical use, the process of tissue differentiation and bone regrowth in an osteotomized mandible remains poorly understood. Clinically, MSDOG can be divided into four stages: firstly the mandible is osteotomized and then instrumented with a distraction appliance; secondly a seven to ten day latency period is waited after the surgical operation in order to allow the formation of a

flow and substrate strain in the tissue, are used as a basis for the stimulus *S* for cell differentiation as follows:

$$S = \frac{\gamma}{a} + \frac{v}{b} \tag{3}$$

where γ is the octahedral shear strain, *v* is the interstitial fluid flow velocity, *a*=3.75% and b=3μms-1 are empirical constants. High stimulus levels (*S*>3) promote the differentiation of mesenchymal cells into fibroblasts, intermediate levels (1<*S*<3) stimulate the differentiation into chondrocytes, and low levels of these stimuli (*S*<1) promote the differentiation into osteoblasts. Simulation of the time-course of tissue differentiation was presented by Huiskes et al., (Huiskes et al., 1997) (Fig. 4). The solid line shows what would occur in an environment where a high shear persists (i.e. maintenance of fibrous tissue and inhibition of ossification) whereas the dashed line shows what would occur if the presence of the soft tissue could progressively reduce the micromotions (i.e ossification would occur). Recently, the mechano-regulation model of Prendergast et al., (Prendergast et al., 1997) has been further developed to include factors such as angiogenesis (Checa and Prendergast, 2009), and the role of the mechanical environment on the collagen architecture in regenerating soft tissues (Nagel and Kelly, 2010). Gómez-Benito et al. (Gómez-Benito et al., 2005), presented a mathematical model to simulate the effect of mechanical stimuli on most of the cellular processes that occur during fracture healing, namely proliferation, migration and differentiation. They simulated the process of bone healing as a process driven by a mechanical stimulus, Ψ(*x*,*t*) assumed to be the second invariant of the deviatoric strain tensor.

Fig. 4. Schematic of the mechano-regulation model proposed by Prendergast et al. (Prendergast et al., 1997). The solid line shows what would occur in an environment where a high shear persists (i.e. maintenance of fibrous tissue and inhibition of ossification) whereas the dashed line shows what would occur if the presence of the soft tissue could progressively reduce the micromotions (i.e ossification would occur).

The models above reviewed are based on theories of mechano-transduction, the way in which cells sense and respond to mechanical forces or displacements. Other bio-regulatory theories are reported in literature that put in relationship biochemical factors with the spatial and temporal patterns of tissue differentiation observed during the healing process of a fractured bone (Bailón-Plaza and van der Meulen, 2001, Geris et al., 2008).

### **2.4 Mechanobiology: Domains of applicability**

26 Theoretical Biomechanics

flow and substrate strain in the tissue, are used as a basis for the stimulus *S* for cell

*<sup>v</sup> <sup>S</sup> a b*

where γ is the octahedral shear strain, *v* is the interstitial fluid flow velocity, *a*=3.75% and b=3μms-1 are empirical constants. High stimulus levels (*S*>3) promote the differentiation of mesenchymal cells into fibroblasts, intermediate levels (1<*S*<3) stimulate the differentiation into chondrocytes, and low levels of these stimuli (*S*<1) promote the differentiation into osteoblasts. Simulation of the time-course of tissue differentiation was presented by Huiskes et al., (Huiskes et al., 1997) (Fig. 4). The solid line shows what would occur in an environment where a high shear persists (i.e. maintenance of fibrous tissue and inhibition of ossification) whereas the dashed line shows what would occur if the presence of the soft tissue could progressively reduce the micromotions (i.e ossification would occur). Recently, the mechano-regulation model of Prendergast et al., (Prendergast et al., 1997) has been further developed to include factors such as angiogenesis (Checa and Prendergast, 2009), and the role of the mechanical environment on the collagen architecture in regenerating soft tissues (Nagel and Kelly, 2010). Gómez-Benito et al. (Gómez-Benito et al., 2005), presented a mathematical model to simulate the effect of mechanical stimuli on most of the cellular processes that occur during fracture healing, namely proliferation, migration and differentiation. They simulated the process of bone healing as a process driven by a mechanical stimulus, Ψ(*x*,*t*) assumed to be the second

Fig. 4. Schematic of the mechano-regulation model proposed by Prendergast et al.

the dashed line shows what would occur if the presence of the soft tissue could

of a fractured bone (Bailón-Plaza and van der Meulen, 2001, Geris et al., 2008).

progressively reduce the micromotions (i.e ossification would occur).

(Prendergast et al., 1997). The solid line shows what would occur in an environment where a high shear persists (i.e. maintenance of fibrous tissue and inhibition of ossification) whereas

The models above reviewed are based on theories of mechano-transduction, the way in which cells sense and respond to mechanical forces or displacements. Other bio-regulatory theories are reported in literature that put in relationship biochemical factors with the spatial and temporal patterns of tissue differentiation observed during the healing process

<sup>γ</sup> = + (3)

differentiation as follows:

invariant of the deviatoric strain tensor.

Applications of mechanobiology can be found in three main areas:


In this Chapter we will focus on the first domain of applicability (i) and, specifically, two examples will be illustrated that show how mechanobiology can be used to predict the patterns of tissue differentiation in a human mandible osteotomized and submitted to distraction osteogenesis as well the regrowth and the remodelling process of the cancellous bone in a vertebral fracture. Predictions were conducted by implementing the mechanoregulation model of Prendergast and colleagues (Prendergast et al., 1997).

### **3. Mechanobiology of mandibular symphyseal distraction osteogenesis**

Mandibular Symphyseal Distraction Osteogenesis (MSDOG) is a common clinical procedure aimed to modify the geometrical shape of the mandible for correcting problems of dental overcrowding and arch shrinkage. Such problems are usually solved by tooth extraction or expansion protocols. However, these clinical procedures are unstable and tend to relapse towards the original dimension (McNamara and Brudon, 1993). Mandibular distraction osteogenesis may solve transverse mandibular deficiency problems. With this clinical procedure the mandibular geometry is definitively changed so that the risk of a relapsing towards the original dimension is avoided. In spite of consolidated clinical use, the process of tissue differentiation and bone regrowth in an osteotomized mandible remains poorly understood. Clinically, MSDOG can be divided into four stages: firstly the mandible is osteotomized and then instrumented with a distraction appliance; secondly a seven to ten day latency period is waited after the surgical operation in order to allow the formation of a

Mechanobiology of Fracture Healing: Basic

(in yellow).

2 3

1

the deformable portion of the distractor device (Fig. 5c).

and Prendergast, 2002, Kelly and Prendergast, 2005).

**3.2 Boundary and loading conditions** 

Principles and Applications in Orthodontics and Orthopaedics 29

device. Since the stiffness of the mandibular bone is orders of magnitude greater than the callus, we modelled the portion of bone and of the device far from the osteotomized region as a rigid body. Conversely, the portion of the bone, of bone callus and of the device near to the middle sagittal plane was modelled with 3D deformable elements. With this strategy we reduced the computational cost of the analysis without introducing significant alterations with respect to the anatomo-physiological behaviour of the mandibular district. The finite element model consists of about 12000 3-node un-deformable triangular elements (see Fig. 5b) and about 5400 8-node hexahedral elements for meshing the osteotomized region and

Fig. 6. (a) FEM model of the osteotomized region. Different regions and materials included in the model: (b) cortical bone (in light blue), (c) cancellous bone (in red), (d) fracture callus

a) b) c) d)

Following Meyer et al.*,* (Meyer et al., 2004), a 2 mm thick gap between the two mandibular *ramus* was created. The gap, surrounded by cortical and cancellous bone (Fig. 6) was hypothesized to be initially occupied by granulation tissue. The callus and bone tissue forming the portion of mandible near to the middle sagittal plane were modelled as biphasic poroelastic materials. Following Schwartz-Dabney and Dechow, (Schwartz-Dabney and Dechow, 2003) the cortical bone was modelled as an orthotropic material. The material properties used for all the other tissues are the same as used in previous models (Lacroix

The FEM model is subjected to three boundary conditions applied simultaneously (Fig. 5b). Boundary condition (i) simulates the temporomandibular joint. The condyles are represented by two reference points at the locations of articulation. These reference points are connected to the mandible arms through coupling constraints. The behaviour of the temporomandibular joint disc is modelled by constraining these reference points to three fixed points by means of spring elements aligned to the coordinate system. The mandible hence can rotate about an axis defined by the line connecting the two condyles and translate along the coordinate directions. Boundary condition (ii) models the mastication. The action of the most important muscles involved in the mastication process, was simulated. Force intensity and direction are those used in a previous study (Boccaccio et al., 2006). Boundary condition (iii) simulates the unilateral occlusion on one tooth on the right mandibular arm.

good quality bone callus; thirdly the distraction device is progressively expanded with a well defined rate for seven-ten day time period; the final stage is the maturation period during which the patient is maintained in rigid external fixation. At the end of this period, more space is available on the inferior arch so that the teeth which are initially in intimate contact, can be repositioned (through orthodontic treatments) in the correct locations.

The second stage is crucial for successful MSDOG. If the latency period is too short, a weak and insufficient callus will form, and without a good callus not enough new bone may form and complications may arise such as fibrous union, non-union, tooth loss and periodontal defects (Conley and Legan, 2003). On the other hand, too long a latency period may substantially increase the risk of premature bone union, which can hinder the subsequent expansion process. Furthermore, the duration of latency period depends strictly on the aging of patient (Conley and Legan, 2003). In the case of young children, the accelerated healing process allows clinical protocols with shorter latency period to be adopted while, in the case of elder patients, as the healing process progresses slowly, longer latency periods are required. The distraction period (i.e., the third stage) is also critical. Too fast a rate of expansion of the appliance can lead to poor bone quality within the distraction gap, partial union, fibrous union or atrophic non-union. Conversely, too slow a rate can lead to premature consolidation hence hindering the distraction process (Conley and Legan, 2003). Such issues have been investigated by developing a mechano-regulation model of a human mandible osteotomized and submitted to distraction osteogenesis.

### **3.1 Finite element model**

The 3D model of a human mandible has been reconstructed from CT scan data and the processing of the CT files was made by means of the *Mimics*® *Version 7.2* software (Materialise Inc.) (Fig. 5(a-c)). The model also includes an orthodontic distractor tooth-borne

Fig. 5. (a) Epoxy resin model of the osteotomized mandible with a tooth-borne device; (b) mandible-distractor orthodontic device FEM model; (c) details of the osteotomized region and of the tooth-borne device

good quality bone callus; thirdly the distraction device is progressively expanded with a well defined rate for seven-ten day time period; the final stage is the maturation period during which the patient is maintained in rigid external fixation. At the end of this period, more space is available on the inferior arch so that the teeth which are initially in intimate contact, can be repositioned (through orthodontic treatments) in the correct locations. The second stage is crucial for successful MSDOG. If the latency period is too short, a weak and insufficient callus will form, and without a good callus not enough new bone may form and complications may arise such as fibrous union, non-union, tooth loss and periodontal defects (Conley and Legan, 2003). On the other hand, too long a latency period may substantially increase the risk of premature bone union, which can hinder the subsequent expansion process. Furthermore, the duration of latency period depends strictly on the aging of patient (Conley and Legan, 2003). In the case of young children, the accelerated healing process allows clinical protocols with shorter latency period to be adopted while, in the case of elder patients, as the healing process progresses slowly, longer latency periods are required. The distraction period (i.e., the third stage) is also critical. Too fast a rate of expansion of the appliance can lead to poor bone quality within the distraction gap, partial union, fibrous union or atrophic non-union. Conversely, too slow a rate can lead to premature consolidation hence hindering the distraction process (Conley and Legan, 2003). Such issues have been investigated by developing a mechano-regulation model of a human

The 3D model of a human mandible has been reconstructed from CT scan data and the processing of the CT files was made by means of the *Mimics*® *Version 7.2* software (Materialise Inc.) (Fig. 5(a-c)). The model also includes an orthodontic distractor tooth-borne

a) b)

c)

Fig. 5. (a) Epoxy resin model of the osteotomized mandible with a tooth-borne device; (b) mandible-distractor orthodontic device FEM model; (c) details of the osteotomized region

mandible osteotomized and submitted to distraction osteogenesis.

**3.1 Finite element model** 

and of the tooth-borne device

device. Since the stiffness of the mandibular bone is orders of magnitude greater than the callus, we modelled the portion of bone and of the device far from the osteotomized region as a rigid body. Conversely, the portion of the bone, of bone callus and of the device near to the middle sagittal plane was modelled with 3D deformable elements. With this strategy we reduced the computational cost of the analysis without introducing significant alterations with respect to the anatomo-physiological behaviour of the mandibular district. The finite element model consists of about 12000 3-node un-deformable triangular elements (see Fig. 5b) and about 5400 8-node hexahedral elements for meshing the osteotomized region and the deformable portion of the distractor device (Fig. 5c).

Fig. 6. (a) FEM model of the osteotomized region. Different regions and materials included in the model: (b) cortical bone (in light blue), (c) cancellous bone (in red), (d) fracture callus (in yellow).

Following Meyer et al.*,* (Meyer et al., 2004), a 2 mm thick gap between the two mandibular *ramus* was created. The gap, surrounded by cortical and cancellous bone (Fig. 6) was hypothesized to be initially occupied by granulation tissue. The callus and bone tissue forming the portion of mandible near to the middle sagittal plane were modelled as biphasic poroelastic materials. Following Schwartz-Dabney and Dechow, (Schwartz-Dabney and Dechow, 2003) the cortical bone was modelled as an orthotropic material. The material properties used for all the other tissues are the same as used in previous models (Lacroix and Prendergast, 2002, Kelly and Prendergast, 2005).

### **3.2 Boundary and loading conditions**

The FEM model is subjected to three boundary conditions applied simultaneously (Fig. 5b). Boundary condition (i) simulates the temporomandibular joint. The condyles are represented by two reference points at the locations of articulation. These reference points are connected to the mandible arms through coupling constraints. The behaviour of the temporomandibular joint disc is modelled by constraining these reference points to three fixed points by means of spring elements aligned to the coordinate system. The mandible hence can rotate about an axis defined by the line connecting the two condyles and translate along the coordinate directions. Boundary condition (ii) models the mastication. The action of the most important muscles involved in the mastication process, was simulated. Force intensity and direction are those used in a previous study (Boccaccio et al., 2006). Boundary condition (iii) simulates the unilateral occlusion on one tooth on the right mandibular arm.

Mechanobiology of Fracture Healing: Basic

latency period.

days.

**patients** 

Principles and Applications in Orthodontics and Orthopaedics 31

where *cmax* is the highest concentration of cells which may occupy any one element domain, *Egranulation* is the Young's modulus of the granulation tissue. This rule of mixtures has been successfully adopted by Lacroix and Prendergast (Lacroix and Prendergast, 2002), to describe the delay between the time when stimulus first acts on the cells and the process of differentiation into a new phenotype. The algorithm was written in FORTRAN environment and each iteration corresponds to 4.8 hours, that is, the diffusion equation (1) computes the change of cells concentration occurring every 4.8 hours. Therefore, with 5 iterations a time period of 1 day is covered. This algorithm has been used to predict the patterns of the tissues differentiating within the bone callus of the osteotomized mandible during a ten day

The analyses carried out revealed that the Young's modulus of the bone, cartilage and fibrous tissue decreases towards the centre of the callus as we move away from the adjacent bone marrow (Fig. 8a). In order to investigate if premature bone bridging between the two sides of the fracture callus could hinder the subsequent distraction process, the amounts of new bone within the callus with a predicted Young's modulus greater than 0.7 MPa were isolated (see Fig. 8b which illustrates the process of bone formation in the frontal plane 1-3). After eight days, portions of bone tissue linking the left with the right side of the callus are predicted to form. The presence of bone bridges will hinder the distraction process. This suggests that is better to apply clinical protocols with a latency period not longer than seven-eight days so that the risk of a premature bone union is avoided. This is in agreement with Conley and Legan, (Conley and Legan, 2003) who suggest a latency period of seven

**3.4 Determination of the optimal duration of the latency period for differently aged** 

differentiation were predicted for each case (Fig. 9).

The mechano-regulation model schematically illustrated in Figure 7 has been further developed to investigate how the optimal duration of the latency period changes for differently aged patients (Boccaccio et al., 2008b). Three different cases were considered: young (up to 20 years old), adult (about 55 years old) and elder (more than 70 years old) patients. Based on the histological analyses conducted by Chen et al., (Chen et al., 2005), Baxter et al., (Baxter et al., 2004), Mendes et al. (Mendes et al., 2002), and Park et al. (Park et al., 2005), the diffusion coefficient *D* was hypothesized to be a function of the patient's age*.* Let *telder*, *tadult* and *tyoung* be the time periods required by MSCs for recovering completely the bone callus domain respectively for the elder, adult and young patients. Following Chen et al. (Chen et al., 2005) who measured the proliferation of MSCs in an *in vitro* culture at different time intervals (4, 8, 12 16 and 20 days) the diffusion coefficient *D* for the differently aged patients has been set: *telder* =3 weeks, *tadult* =2 weeks and *tyoung*=1 week. Patterns of tissue

A bony bridge between the left and right sides of the fracture callus forms after 5, 8 and 9 days for the young, adult and elder patients, respectively (Fig. 9). Such results lead us to conclude that the optimal duration of the latency period is: 5-6 days for the young patient, 7- 8 and 9-10 days for the adult and the elder patients, respectively. These evaluations are in agreement with literature. For instance, Conley & Legan (Conley and Legan, 2003) suggest a

max max ( 1) *granulation c c c*

*E iter E E c c*

( ) max

<sup>−</sup> + = ⋅+ ⋅ (4)

The occlusion is modelled by constraining, with simple-supports preventing *u*3 displacements (see direction 3 in Fig. 6), the second premolar.

Fig. 7. Schematic of the implemented mechano-regulation algorithm

### **3.3 Determination of the optimal duration of the latency period**

The mechano-regulation model of Prendergast and colleagues (Prendergast et al., 1997), combined with the above described finite element model was utilized to investigate the tissue differentiation process after osteotomy and to carry out an investigation on the optimal duration of the latency period and on its effects on the bone regeneration process. The spreading of mesenchymal stem cells throughout the bone callus was simulated with the diffusion equation (1) (Boccaccio et al., 2008a). The diffusion coefficient *D* was set so that the complete cell coverage in the callus is achieved two weeks after the osteotomy. As MSCs disperse from the bone marrow throughout the callus, they will differentiate into different cell phenotypes based on the value of a biophysical stimulus *S* computed with the equation (3). After the calculation of the new tissue phenotype and of the number of the MSCs invading the bone callus domain, the algorithm evaluates the mechanical properties for every element based on the exponential law described above (Section 2.2) and a simple rule of mixtures. The diffusion equation, the formulation for the calculation of the stimulus, the exponential law as well as the rule of mixtures were implemented into an algorithm a graphical summary of which is depicted in Fig. 7. If Ē is the Young's modulus for a given element averaged over the previous 10 iterations and *c* is the concentration of cells invading the domain in the current iteration *iter*, then the Young's modulus for that element and for next iteration *iter+1* can be computed as follows:

The occlusion is modelled by constraining, with simple-supports preventing *u*3-

displacements (see direction 3 in Fig. 6), the second premolar.

Fig. 7. Schematic of the implemented mechano-regulation algorithm

**3.3 Determination of the optimal duration of the latency period** 

The mechano-regulation model of Prendergast and colleagues (Prendergast et al., 1997), combined with the above described finite element model was utilized to investigate the tissue differentiation process after osteotomy and to carry out an investigation on the optimal duration of the latency period and on its effects on the bone regeneration process. The spreading of mesenchymal stem cells throughout the bone callus was simulated with the diffusion equation (1) (Boccaccio et al., 2008a). The diffusion coefficient *D* was set so that the complete cell coverage in the callus is achieved two weeks after the osteotomy. As MSCs disperse from the bone marrow throughout the callus, they will differentiate into different cell phenotypes based on the value of a biophysical stimulus *S* computed with the equation (3). After the calculation of the new tissue phenotype and of the number of the MSCs invading the bone callus domain, the algorithm evaluates the mechanical properties for every element based on the exponential law described above (Section 2.2) and a simple rule of mixtures. The diffusion equation, the formulation for the calculation of the stimulus, the exponential law as well as the rule of mixtures were implemented into an algorithm a graphical summary of which is depicted in Fig. 7. If Ē is the Young's modulus for a given element averaged over the previous 10 iterations and *c* is the concentration of cells invading the domain in the current iteration *iter*, then the Young's

modulus for that element and for next iteration *iter+1* can be computed as follows:

$$E(iter+1) = \frac{c}{c\_{\text{max}}} \cdot \overline{E} + \frac{\left(c\_{\text{max}} - c\right)}{c\_{\text{max}}} \cdot E\_{gravitational} \tag{4}$$

where *cmax* is the highest concentration of cells which may occupy any one element domain, *Egranulation* is the Young's modulus of the granulation tissue. This rule of mixtures has been successfully adopted by Lacroix and Prendergast (Lacroix and Prendergast, 2002), to describe the delay between the time when stimulus first acts on the cells and the process of differentiation into a new phenotype. The algorithm was written in FORTRAN environment and each iteration corresponds to 4.8 hours, that is, the diffusion equation (1) computes the change of cells concentration occurring every 4.8 hours. Therefore, with 5 iterations a time period of 1 day is covered. This algorithm has been used to predict the patterns of the tissues differentiating within the bone callus of the osteotomized mandible during a ten day latency period.

The analyses carried out revealed that the Young's modulus of the bone, cartilage and fibrous tissue decreases towards the centre of the callus as we move away from the adjacent bone marrow (Fig. 8a). In order to investigate if premature bone bridging between the two sides of the fracture callus could hinder the subsequent distraction process, the amounts of new bone within the callus with a predicted Young's modulus greater than 0.7 MPa were isolated (see Fig. 8b which illustrates the process of bone formation in the frontal plane 1-3). After eight days, portions of bone tissue linking the left with the right side of the callus are predicted to form. The presence of bone bridges will hinder the distraction process. This suggests that is better to apply clinical protocols with a latency period not longer than seven-eight days so that the risk of a premature bone union is avoided. This is in agreement with Conley and Legan, (Conley and Legan, 2003) who suggest a latency period of seven days.

### **3.4 Determination of the optimal duration of the latency period for differently aged patients**

The mechano-regulation model schematically illustrated in Figure 7 has been further developed to investigate how the optimal duration of the latency period changes for differently aged patients (Boccaccio et al., 2008b). Three different cases were considered: young (up to 20 years old), adult (about 55 years old) and elder (more than 70 years old) patients. Based on the histological analyses conducted by Chen et al., (Chen et al., 2005), Baxter et al., (Baxter et al., 2004), Mendes et al. (Mendes et al., 2002), and Park et al. (Park et al., 2005), the diffusion coefficient *D* was hypothesized to be a function of the patient's age*.* Let *telder*, *tadult* and *tyoung* be the time periods required by MSCs for recovering completely the bone callus domain respectively for the elder, adult and young patients. Following Chen et al. (Chen et al., 2005) who measured the proliferation of MSCs in an *in vitro* culture at different time intervals (4, 8, 12 16 and 20 days) the diffusion coefficient *D* for the differently aged patients has been set: *telder* =3 weeks, *tadult* =2 weeks and *tyoung*=1 week. Patterns of tissue differentiation were predicted for each case (Fig. 9).

A bony bridge between the left and right sides of the fracture callus forms after 5, 8 and 9 days for the young, adult and elder patients, respectively (Fig. 9). Such results lead us to conclude that the optimal duration of the latency period is: 5-6 days for the young patient, 7- 8 and 9-10 days for the adult and the elder patients, respectively. These evaluations are in agreement with literature. For instance, Conley & Legan (Conley and Legan, 2003) suggest a

Mechanobiology of Fracture Healing: Basic

predicted and compared (Fig.10).

Principles and Applications in Orthodontics and Orthopaedics 33

Fig. 9. Bone regeneration process for the young, adult and elder patients

**3.5 Influence of expansion rates on mandibular distraction osteogenesis** 

Another important issue investigated with the mechano-regulation model of Prendergast and colleagues (Prendergast et al., 1997) and the above described finite element model, is the influence of expansion rates on the distraction process. The algorithm shown in Fig. 7 has been expanded to include the modelling of the distraction of the orthodontic appliance. Two different protocols of expansion were investigated: the first one where the device is distracted by 0.6mm/day for a period of ten days, the second where the device is distracted at 1.2 mm/day for five days. The final result with the aforementioned protocols is a widening of mandibular arch by about 6 mm corresponding to the space occupied by a lower incisor. The total number of iterations was set in order to cover a time period of 43 days, consisting of a one week latency period and a 37 day distraction and maturation period. In the case of the 0.6 mm/day distraction rate, this latter period included ten days of distraction and a 27 day maturation period, while the 1.2 mm/day distraction rate consisted of five days of distraction and a 32 day maturation period. In each iteration which simulates a distraction process, a structural FE analysis was run in order to model just the expansion of the appliance. In this analysis a given displacement was imposed to the arms of the distractor. Further details about the implemented algorithm are reported in Boccaccio et al. (Boccaccio et al., 2007). The spatial and temporal changes in tissue differentiation produced with these two different protocols were

The computational analyses revealed that the lower expansion rate of 0.6 mm/day leads to greater amounts of bone tissue 'bridging' the left and right sides of the callus (Fig. 10). It is possible that excessive bone quantities in this area may hinder the process of distraction

latency period not longer than 7 days. Mattik et al. (Mattik et al., 2001), reported three clinical cases of young patients 18, 19 and 28 years old, for whom a latency period of 5 days was adopted. Lazar et al. (Lazar et al., 2003) submitted a 62-years old patient to mandibular distraction osteogenesis. The first distraction was given to the mandible following a delayed latency period of 10 days.

Fig. 8. (a) 3D visualization of tissue differentiation in the bone callus**;** (b) 3D visualization of the bone regeneration process in the frontal plane 1-3

latency period not longer than 7 days. Mattik et al. (Mattik et al., 2001), reported three clinical cases of young patients 18, 19 and 28 years old, for whom a latency period of 5 days was adopted. Lazar et al. (Lazar et al., 2003) submitted a 62-years old patient to mandibular distraction osteogenesis. The first distraction was given to the mandible following a delayed

Fig. 8. (a) 3D visualization of tissue differentiation in the bone callus**;** (b) 3D visualization of

the bone regeneration process in the frontal plane 1-3

latency period of 10 days.

a)

b)

Fig. 9. Bone regeneration process for the young, adult and elder patients

### **3.5 Influence of expansion rates on mandibular distraction osteogenesis**

Another important issue investigated with the mechano-regulation model of Prendergast and colleagues (Prendergast et al., 1997) and the above described finite element model, is the influence of expansion rates on the distraction process. The algorithm shown in Fig. 7 has been expanded to include the modelling of the distraction of the orthodontic appliance. Two different protocols of expansion were investigated: the first one where the device is distracted by 0.6mm/day for a period of ten days, the second where the device is distracted at 1.2 mm/day for five days. The final result with the aforementioned protocols is a widening of mandibular arch by about 6 mm corresponding to the space occupied by a lower incisor. The total number of iterations was set in order to cover a time period of 43 days, consisting of a one week latency period and a 37 day distraction and maturation period. In the case of the 0.6 mm/day distraction rate, this latter period included ten days of distraction and a 27 day maturation period, while the 1.2 mm/day distraction rate consisted of five days of distraction and a 32 day maturation period. In each iteration which simulates a distraction process, a structural FE analysis was run in order to model just the expansion of the appliance. In this analysis a given displacement was imposed to the arms of the distractor. Further details about the implemented algorithm are reported in Boccaccio et al. (Boccaccio et al., 2007). The spatial and temporal changes in tissue differentiation produced with these two different protocols were predicted and compared (Fig.10).

The computational analyses revealed that the lower expansion rate of 0.6 mm/day leads to greater amounts of bone tissue 'bridging' the left and right sides of the callus (Fig. 10). It is possible that excessive bone quantities in this area may hinder the process of distraction

Mechanobiology of Fracture Healing: Basic

at the level of individual trabeculae.

Principles and Applications in Orthodontics and Orthopaedics 35

The objective of this study was to investigate if biophysical stimuli play a role in regulating the process of tissue differentiation and bone remodeling in a fractured vertebral body. Our hypothesis is that the mechano-regulation model for tissue differentiation proposed by Prendergast et al. (Prendergast et al., 1997) and that has previously been used to predict the time-course of fracture repair in long and flat bones (Lacroix and Prendergast, 2002, Boccaccio et al., 2007) can be used to predict trabecular bone healing in fractured vertebrae

To determine the magnitude of such stimuli at the level of individual trabeculae, a multiscale finite element approach has been adopted. A macro-scale finite element model (Fig. 11) of the spinal segment L3-L4-L5, including a mild wedge fracture in the body of the L4 vertebra, is used to determine the boundary conditions acting on a micro-scale finite element model of a portion of fractured trabecular bone. The micro-scale model, in turn, is utilized to predict the local patterns of tissue differentiation within the fracture site and then how the equivalent mechanical properties of the macro-scale model change with time.

TIE CONSTRAINT

Fig. 11. (a) Finite element models of the spinal segment L3-L4-L5; (b) the body of fractured

CONSTRAINT EQUATION

b)

The L3, L5 vertebrae and the posterior processes of the L4 were modelled as un-deformable rigid bodies. Deformable elements have been utilized to model the body of the L4 vertebra as well as the intervertebal discs. Between the intervertebral disc and the vertebral bodies a 'tie' constraint was applied (Fig. 11a). Constraint equations were used to attach the posterior processes to the body of the L4 vertebra (Fig. 11a). Each intervertebral disc includes the cartilagineous endplates, the nucleus polposus, the annulus fibrosus with the collagen fibres. The effects of the flavum, intertransverse, interspinous and supraspinous ligaments have been included in the model. Further details about the geometry of the intervertebral discs and the modelling of the ligaments are reported in Boccaccio et al. (Boccaccio et al., 2008c). The height of the fractured vertebra decreases by the 20% from the posterior processes towards the anterior side (Fig. 11 (b)). In the Genant grading (Genant et al., 1993), such a fracture is classified as a mild wedge fracture. The body of L4 includes the cortical

vertebra L4; its height decreases towards the anterior side.

a)

because a premature bone union can occur. Therefore a faster distraction rate of 1.2 mm/day is preferable in cases where there is an increased risk of bone union.

Fig. 10. Bone regeneration process for a distraction rate of (a) 0.6 mm/day and (b) 1.2 mm/day

### **4. Mechanobiology of fracture repair in vertebral bodies**

Vertebral fractures commonly occur in elderly people with osteoporosis. For example, in the United States vertebral fractures account for nearly half of all osteoporotic fractures (Cummings et al., 2002). With age the structure of cancellous bone within vertebral bodies transforms from that characterized by predominately plate-like trabeculae to rod-like trabeculae. This change leads to an age-related decrease in trabecular bone mass (Amling et al., 1996). The reduced bone mass observed in vertebral bodies, particularly with osteoporosis, is generally accompanied by greater amounts of microcallus formations around injured trabeculae (Hansson and Roos, 1981). This weakening of the tissue means that spine fractures may occur after minimal trauma (Einhorn, 2005).

because a premature bone union can occur. Therefore a faster distraction rate of 1.2

a)

b)

mm/day is preferable in cases where there is an increased risk of bone union.

Fig. 10. Bone regeneration process for a distraction rate of (a) 0.6 mm/day and (b) 1.2

Vertebral fractures commonly occur in elderly people with osteoporosis. For example, in the United States vertebral fractures account for nearly half of all osteoporotic fractures (Cummings et al., 2002). With age the structure of cancellous bone within vertebral bodies transforms from that characterized by predominately plate-like trabeculae to rod-like trabeculae. This change leads to an age-related decrease in trabecular bone mass (Amling et al., 1996). The reduced bone mass observed in vertebral bodies, particularly with osteoporosis, is generally accompanied by greater amounts of microcallus formations around injured trabeculae (Hansson and Roos, 1981). This weakening of the tissue means

**4. Mechanobiology of fracture repair in vertebral bodies** 

that spine fractures may occur after minimal trauma (Einhorn, 2005).

mm/day

The objective of this study was to investigate if biophysical stimuli play a role in regulating the process of tissue differentiation and bone remodeling in a fractured vertebral body. Our hypothesis is that the mechano-regulation model for tissue differentiation proposed by Prendergast et al. (Prendergast et al., 1997) and that has previously been used to predict the time-course of fracture repair in long and flat bones (Lacroix and Prendergast, 2002, Boccaccio et al., 2007) can be used to predict trabecular bone healing in fractured vertebrae at the level of individual trabeculae.

To determine the magnitude of such stimuli at the level of individual trabeculae, a multiscale finite element approach has been adopted. A macro-scale finite element model (Fig. 11) of the spinal segment L3-L4-L5, including a mild wedge fracture in the body of the L4 vertebra, is used to determine the boundary conditions acting on a micro-scale finite element model of a portion of fractured trabecular bone. The micro-scale model, in turn, is utilized to predict the local patterns of tissue differentiation within the fracture site and then how the equivalent mechanical properties of the macro-scale model change with time.

Fig. 11. (a) Finite element models of the spinal segment L3-L4-L5; (b) the body of fractured vertebra L4; its height decreases towards the anterior side.

The L3, L5 vertebrae and the posterior processes of the L4 were modelled as un-deformable rigid bodies. Deformable elements have been utilized to model the body of the L4 vertebra as well as the intervertebal discs. Between the intervertebral disc and the vertebral bodies a 'tie' constraint was applied (Fig. 11a). Constraint equations were used to attach the posterior processes to the body of the L4 vertebra (Fig. 11a). Each intervertebral disc includes the cartilagineous endplates, the nucleus polposus, the annulus fibrosus with the collagen fibres. The effects of the flavum, intertransverse, interspinous and supraspinous ligaments have been included in the model. Further details about the geometry of the intervertebral discs and the modelling of the ligaments are reported in Boccaccio et al. (Boccaccio et al., 2008c). The height of the fractured vertebra decreases by the 20% from the posterior processes towards the anterior side (Fig. 11 (b)). In the Genant grading (Genant et al., 1993), such a fracture is classified as a mild wedge fracture. The body of L4 includes the cortical

Mechanobiology of Fracture Healing: Basic

Boccaccio et al., (Boccaccio et al., 2011b).

vertebra.

Principles and Applications in Orthodontics and Orthopaedics 37

Further details about the micro-scale finite element model of trabecular bone are reported in

A multi-scale approach was adopted. The equations describing tissue differentiation were implemented into an algorithm, a graphical summary of which is depicted in Fig. 14. The time period investigated corresponds to the first 100 days after the fracture event. The macro-scale model of the spinal segment was utilized to determine the elastic and poroelastic boundary conditions acting on eight different micro-scale models which were

Fig. 14. Schematic of the algorithm utilized to model the fracture repair process in the L4

hypothesized to represent different regions in the fractured cancellous bone located in the neighbourhood of the points P1,…,P8 (Fig. 12(c)). In order to evaluate the above mentioned

shell of 0.5 mm thickness (Mizrahi et al., 1993), the cancellous bone and the fracture gap (Fig. 12(a-c)). The cancellous and the cortical bone have been modelled as biphasic poroelastic materials possessing transversely isotropic elastic properties. The distribution of the cancellous bone Young's modulus was assumed to be heterogeneous. An anisotropy ratio of 7/10, -i.e. the Young's modulus in the transversal plane is 7/10 the Young's modulus along the cranio-caudal direction-, was assumed (Eberlein et al., 2001). Further details about the spatial distribution of the Young's modulus of the cancellous bone are reported elsewhere (Boccaccio et al., 2008c).

Fig. 12. The body of the fractured vertebra L4 includes the cortical shell (in blue, (a), (b)) the cancellous bone (in red, (b)) and the fracture gap (in green, (c)). The points P1, P2, …, P8 within the fracture gap in correspondence of which the analysis of the fracture repair process was carried out are indicated.

The micro-scale model of the trabecular bone was similar in geometry to that used by Shefelbine et al. (Shefelbine et al., 2005) (Fig. 13). A diastasis of 0.5 mm was simulated, with the trabeculae bordering the gap idealized as prismatic domains 0.1 mm thick. The space between fractured trabeculae was hypothesized to be occupied by granulation tissue. Both, the trabecular bone and the granulation tissue were modelled as biphasic poroelastic materials.

Fig. 13. Geometry (a) and section (b) of the micro-scale model. In red are represented the trabeculae spicules, in blue sky the granulation tissue.

shell of 0.5 mm thickness (Mizrahi et al., 1993), the cancellous bone and the fracture gap (Fig. 12(a-c)). The cancellous and the cortical bone have been modelled as biphasic poroelastic materials possessing transversely isotropic elastic properties. The distribution of the cancellous bone Young's modulus was assumed to be heterogeneous. An anisotropy ratio of 7/10, -i.e. the Young's modulus in the transversal plane is 7/10 the Young's modulus along the cranio-caudal direction-, was assumed (Eberlein et al., 2001). Further details about the spatial distribution of the Young's modulus of the cancellous bone are reported elsewhere

Fig. 12. The body of the fractured vertebra L4 includes the cortical shell (in blue, (a), (b)) the cancellous bone (in red, (b)) and the fracture gap (in green, (c)). The points P1, P2, …, P8 within the fracture gap in correspondence of which the analysis of the fracture repair

b) c)

a)

The micro-scale model of the trabecular bone was similar in geometry to that used by Shefelbine et al. (Shefelbine et al., 2005) (Fig. 13). A diastasis of 0.5 mm was simulated, with the trabeculae bordering the gap idealized as prismatic domains 0.1 mm thick. The space between fractured trabeculae was hypothesized to be occupied by granulation tissue. Both, the trabecular bone and the granulation tissue were modelled as biphasic poroelastic

Fig. 13. Geometry (a) and section (b) of the micro-scale model. In red are represented the

a) b)

trabeculae spicules, in blue sky the granulation tissue.

(Boccaccio et al., 2008c).

process was carried out are indicated.

materials.

Further details about the micro-scale finite element model of trabecular bone are reported in Boccaccio et al., (Boccaccio et al., 2011b).

A multi-scale approach was adopted. The equations describing tissue differentiation were implemented into an algorithm, a graphical summary of which is depicted in Fig. 14. The time period investigated corresponds to the first 100 days after the fracture event. The macro-scale model of the spinal segment was utilized to determine the elastic and poroelastic boundary conditions acting on eight different micro-scale models which were

Fig. 14. Schematic of the algorithm utilized to model the fracture repair process in the L4 vertebra.

hypothesized to represent different regions in the fractured cancellous bone located in the neighbourhood of the points P1,…,P8 (Fig. 12(c)). In order to evaluate the above mentioned

Mechanobiology of Fracture Healing: Basic

of the bone architecture (Amling et al., 1996).

the cancellous bone in a fractured lumbar vertebra.

**5. Discussion** 

Principles and Applications in Orthodontics and Orthopaedics 39

The analyses carried out predicted that the space between the fractured trabeculae is mostly occupied by fibrous tissue in the first days after the fracture event (Fig. 15) During the first 30-35 days after the fracture event the amount of fibrous tissue decreases significantly and disappears completely after six weeks. Small amounts of cartilage appear during the first week, and approximately 40% of the space between the fractured trabeculae is occupied by cartilage after one month. This cartilaginous tissue is completely replaced by bone after two months (Fig. 15). Small amounts of bone are predicted after the first two weeks and after the second month the space is entirely occupied by bone. Bone deposition is predicted to initiate at the fractured trabecular ends. The bone remodeling process appears to start after the second month and reaches equilibrium at the end of the third month. The remodeled

The spatial and temporal patterns of tissue differentiation predicted by this model are in general agreement with those observed experimentally. Diamond et al., (Diamond et al., 2007) describe 4 stages of fracture healing process in the vertebral body, with significant overlap between the various stages of healing. Chondrogenesis was evident in the second stage of the fracture healing process, which followed the initial granulation tissue stage. The appearance of cartilaginous tissue in the days following the fracture event is also predicted by the model (Fig. 15). This cartilaginous tissue is predicted to be gradually replaced by woven bone. The model then predicts a peak in bone formation (see 56th day, Fig. 15). Hyperosteoidosis/Osteosclerosis (excessive formation of osteoid) is also observed experimentally at comparable time-points (Diamond et al., 2007). Finally remodeling of the cancellous bone architecture is predicted. Complete new trabeculae are predicted to form due to bridging of the microcallus between the remnant trabeculae, leading to restructuring

In this Chapter the basic principles of mechanobiology are described as well as the principal theories and the main models utilized to simulate the cellular processes involved in fracture healing. Two examples have been illustrated that show how mechanobiology can be used to predict the patterns of tissue differentiation in a human mandible osteotomized and submitted to distraction osteogenesis as well the regrowth and the remodelling process of

Different are the limitations of the above outlined mechano-regulation theories. The main criticism raised against the models of Pauwels (Pauwels, 1960), Carter and colleagues (Carter et al., 1988, Carter and Wong, 1988) and Claes and Heigele (Claes and Heigele, 1999) is that there are several reasons that interstitial fluid flow could be a more realistic mechanical variable for feedback information to the cells during tissue differentiation than hydrostatic pressure (Owan et a., 1997, Jacobs et al., 1998). The interfragmentary strain theory, although has the advantage of being simple to be used since interfragmentary movement can be easily monitored, presents the limitation that it models the fracture as a one dimensional entity thus ignoring the three dimensional complexity of the callus. The model of Prendergast et al. (Prendergast et al., 1997) although takes into account the interstitial fluid flow neglects osmotic effects and charged-density flows in the tissue (Mow et al., 1999). Several mechano-regulation algorithms proposed to control tissue differentiation during bone healing have been shown to accurately predict temporal and spatial tissue distributions during normal fracture healing. As these algorithms are different

trabeculae are aligned with those bordering the fractured region (Fig. 15).

boundary conditions, an axial compression of 1000 N -which is the typical load acting on the lumbar vertebrae of a 70 Kg subject in the erect standing position- was applied in the centre of mass of the L3 vertebra and ramped over a time period of 1 s (which can be considered the time in which a subject assumes the erect position). The nodes located on the inferior surface of the L5 vertebra have been clamped. To switch from the macro to the micro-scale model, proper localization rules have been utilized. The micro-scale model, in turn, served to predict the local patterns of the tissues differentiating during the fracture repair process. A compression test was simulated on the micro-scale model reproducing the same elastic and poroelastic boundary conditions as those determined from the macro-scale model. The results obtained from this finite element analysis were used to determine the biophysical stimulus acting in each element of the micro-scale model and then to implement the mechano-regulation model of Prendergast et al. (Prendergast et al., 1997). The change of the tissue phenotype leaded to a change of the equivalent mechanical properties possessed by the fracture gap of the macro-scale model. By adopting homogenization techniques the information obtained from the micro-scale model regarding the material properties was upscaled to the macro-scale level. The new material properties were implemented into the macro-scale model and a new iteration initiated. Further details regarding the algorithm are reported in a previous study (Boccaccio et al., 2011b).

Fig. 15. Patterns of the tissues differentiating within the fractured vertebra during the fracture healing process.

The analyses carried out predicted that the space between the fractured trabeculae is mostly occupied by fibrous tissue in the first days after the fracture event (Fig. 15) During the first 30-35 days after the fracture event the amount of fibrous tissue decreases significantly and disappears completely after six weeks. Small amounts of cartilage appear during the first week, and approximately 40% of the space between the fractured trabeculae is occupied by cartilage after one month. This cartilaginous tissue is completely replaced by bone after two months (Fig. 15). Small amounts of bone are predicted after the first two weeks and after the second month the space is entirely occupied by bone. Bone deposition is predicted to initiate at the fractured trabecular ends. The bone remodeling process appears to start after the second month and reaches equilibrium at the end of the third month. The remodeled trabeculae are aligned with those bordering the fractured region (Fig. 15).

The spatial and temporal patterns of tissue differentiation predicted by this model are in general agreement with those observed experimentally. Diamond et al., (Diamond et al., 2007) describe 4 stages of fracture healing process in the vertebral body, with significant overlap between the various stages of healing. Chondrogenesis was evident in the second stage of the fracture healing process, which followed the initial granulation tissue stage. The appearance of cartilaginous tissue in the days following the fracture event is also predicted by the model (Fig. 15). This cartilaginous tissue is predicted to be gradually replaced by woven bone. The model then predicts a peak in bone formation (see 56th day, Fig. 15).

Hyperosteoidosis/Osteosclerosis (excessive formation of osteoid) is also observed experimentally at comparable time-points (Diamond et al., 2007). Finally remodeling of the cancellous bone architecture is predicted. Complete new trabeculae are predicted to form due to bridging of the microcallus between the remnant trabeculae, leading to restructuring of the bone architecture (Amling et al., 1996).

## **5. Discussion**

38 Theoretical Biomechanics

boundary conditions, an axial compression of 1000 N -which is the typical load acting on the lumbar vertebrae of a 70 Kg subject in the erect standing position- was applied in the centre of mass of the L3 vertebra and ramped over a time period of 1 s (which can be considered the time in which a subject assumes the erect position). The nodes located on the inferior surface of the L5 vertebra have been clamped. To switch from the macro to the micro-scale model, proper localization rules have been utilized. The micro-scale model, in turn, served to predict the local patterns of the tissues differentiating during the fracture repair process. A compression test was simulated on the micro-scale model reproducing the same elastic and poroelastic boundary conditions as those determined from the macro-scale model. The results obtained from this finite element analysis were used to determine the biophysical stimulus acting in each element of the micro-scale model and then to implement the mechano-regulation model of Prendergast et al. (Prendergast et al., 1997). The change of the tissue phenotype leaded to a change of the equivalent mechanical properties possessed by the fracture gap of the macro-scale model. By adopting homogenization techniques the information obtained from the micro-scale model regarding the material properties was upscaled to the macro-scale level. The new material properties were implemented into the macro-scale model and a new iteration initiated. Further details regarding the algorithm are

Fig. 15. Patterns of the tissues differentiating within the fractured vertebra during the

reported in a previous study (Boccaccio et al., 2011b).

fracture healing process.

In this Chapter the basic principles of mechanobiology are described as well as the principal theories and the main models utilized to simulate the cellular processes involved in fracture healing. Two examples have been illustrated that show how mechanobiology can be used to predict the patterns of tissue differentiation in a human mandible osteotomized and submitted to distraction osteogenesis as well the regrowth and the remodelling process of the cancellous bone in a fractured lumbar vertebra.

Different are the limitations of the above outlined mechano-regulation theories. The main criticism raised against the models of Pauwels (Pauwels, 1960), Carter and colleagues (Carter et al., 1988, Carter and Wong, 1988) and Claes and Heigele (Claes and Heigele, 1999) is that there are several reasons that interstitial fluid flow could be a more realistic mechanical variable for feedback information to the cells during tissue differentiation than hydrostatic pressure (Owan et a., 1997, Jacobs et al., 1998). The interfragmentary strain theory, although has the advantage of being simple to be used since interfragmentary movement can be easily monitored, presents the limitation that it models the fracture as a one dimensional entity thus ignoring the three dimensional complexity of the callus. The model of Prendergast et al. (Prendergast et al., 1997) although takes into account the interstitial fluid flow neglects osmotic effects and charged-density flows in the tissue (Mow et al., 1999). Several mechano-regulation algorithms proposed to control tissue differentiation during bone healing have been shown to accurately predict temporal and spatial tissue distributions during normal fracture healing. As these algorithms are different

Mechanobiology of Fracture Healing: Basic

fracture healing, both, in the early and in the final stages.

Principles and Applications in Orthodontics and Orthopaedics 41

during the latency period which is not observed histologically. Explicitly including in the model the time taken for mineralization etc following differentiation of progenitor cells into osteoblasts may result in model predictions more comparable to *in vivo* findings. Another important limitation of the mechano-regulation model used to assess the bone regeneration process in a human mandible osteotomized and distracted as well as in a fractured vertebra is represented by the utilization of the exponential law. Such a law was introduced to account for the fact that mesenchymal cells not only require time to differentiate but, that differentiated cells require some time also for synthesising and remodelling new tissue. In reality, the exponential law should be utilized only to model the early stages of the fracture healing process; when this process is close to the end, saturation phenomena (e.g. the mineralization process) occur within the fracture callus and therefore, at this point, the exponential law should be replaced with another law that allows to better describe these conclusive processes. As a first approximation, we utilized the exponential law to model the early stages of the fracture healing (Boccaccio et al., 2011b) while, toward the end of the process, we replaced the exponential law with a linear constant law. Further research should be carried out on the mathematical function that better describes the entire process of

Many experiments on skeletal failure and repair have been performed in the last century aimed to determine the influence of biological, mechanical, hormonal factors on the healing process. Despite this effort, there are still many unanswered questions. This indicates the complexity of the biological problems and has stimulated the development of computational models that can analyze the influence of all factors and make predictions under different boundary and loading conditions. These models must also be validated with experimental analyses. However, in many cases the computational models cannot be validated directly because of the difficulties in performing some measurements in vivo. Despite this, indirect validations can be performed if the conclusions of the computer simulations are similar to the experimental or clinical results. Once the mechano-regulation model has been validated, it can be conveniently utilized to assess the regeneration process within the fracture site in the case in which different boundary and loading conditions act on it. For instance, the mechanoregulation model of fracture repair in vertebral bodies illustrated above can be used to predict the spatial and temporal patterns of repair during altered loading conditions, which may prove beneficial in developing rehabilitation regimes following vertebral fractures. Also, this same model can be utilized to evaluate how the patterns of tissue differentiation change if the fractured vertebra is supported with minimally invasive percutaneous fixation devices (Palmisani et al., 2009). Furthermore, the mechanobiological model of the human mandible osteotomized and distracted can be utilized to estimate how the bony tissue regeneration process within the fracture callus change for different mandibular distraction devices such as

those analyzed in previous studies (Boccaccio et al., 2008d; Boccaccio et al., 2011c).

Future perspectives include the development of computer power. A more robust integration is required, in future, between biology, mechanics and materials science. This should lead to the development of mechano-regulation models that more accurately describe physiological processes such as fracture healing, tissue genesis etc. A very promising research area for mechanobiology is in the field of tissue engineering. Mechanobiology could be an efficient and cheap tool to determine optimal parameters governing scaffold performance. Future perspectives of numerical simulations of biomaterial scaffolds for tissue engineering rely also on the development of new methods to account for the multi-scale dimension of the problems. At the micro-scale level one can analyse cell/biomaterial interactions and how

in nature and biophysical parameters, it raises the question of which reflects the actual mechanobiological processes the best. Isaksson et al. (Isaksson et al., 2006), addressed this issue by corroborating the mechano-regulatory algorithms of Carter and colleagues (Carter et al., 1988, Carter and Wong, 1988), Claes and Heigele (Claes and Heigele, 1999, Claes et al., 1998) and Prendergast and coworkers (Prendergast et al., 1997), with more extensive in vivo bone healing data from animal experiments. They compared the patterns of tissue differentiation predicted by the different models and the patterns observed in vivo in an ovine tibia model. They concluded that none of the algorithms predicted patterns of healing entirely similar to those observed experimentally. However, patterns predicted by the algorithm based on deviatoric strain and fluid velocity (i.e. the model of Prendergast et al., (Prendergast et al., 1997)) was closest to experimental results.

Another important limitation for computational mechanobiology is represented by the fact that the mechano-regulatory algorithms include empirical constants, the values of which must be determined by comparison to a biological reality. For example, in the mechanoregulation rule developed by Prendergast et al. (Prendergast et al., 1997) the constants *b* and *a* (see equation (3)) do not have a specific physical meaning and can be determined by following the 'trial and error' method outlined in van der Melulen and Huiskes (van der Melulen and Huiskes, 2002): "Computational mechano-biologists hypothesize a potential rule and determine if the outcome of this hypothesis produces realistic tissue structures and morphologies, hence 'trial-and-error'. If the results correspond well, they might be an explanation for the mechanism being modelled. This method of research is common practice and productive in physics, less common in biology (Huiskes, 1995); although 'theoretical biology' is based on this type of approach". Certainly, physicists can use this approach (the computational gedanken experiment) because there are so few rules in physics and the predictions are amenable to exact quantitative testing. In biology the phenomena to be observed and analysed are much more complicated than in physics, so cut-and-try theoretical experimentation could not be really useful. Further research should be carried out on the efficiency and the correctness of this philosophy of biological research.

Concerning the mechano-regulation model of fracture repair in the body of the L4 vertebra, the most important limitation is that the implemented algorithm does not include a damaged tissue region that would allow tissue to fracture and new callus to form in regions experiencing high levels of biophysical stimulation (e.g. strain). Therefore this study has only considered the original injury event. In reality, histological analyses (Diamond et al., 2007) revealed that a typical feature of vertebral fractures is the overlap between the different tissues corresponding to the different temporal stages of the fracture healing process. This can be justified with the argument that in vertebral bodies the fracture stabilization that permits orderly repair in long bones is not possible due to repetitive injury. As far as concerns the model of the human mandible osteotomized and distracted, the most important limitation is that after the first simulation the numerical predictions state that the callus consists of a mixture of bone, cartilage and fibrous tissue (Figures 8-9), where it might be more accurate to state that in the first few days after the osteotomy the callus consists of progenitor cells subjected to a stimulus that if maintained will result in these cells differentiating into either fibroblasts, chondrocytes or osteoblasts, depending on the magnitude of the stimulus. This may explain why histological findings (Loboa et al., 2005) from animal model studies differ from the initial model predictions as progenitor cells take time to differentiate and produce a tissue phenotype that is recognised by appropriate histological staining. For example, significant bone and soft tissue formation is predicted

in nature and biophysical parameters, it raises the question of which reflects the actual mechanobiological processes the best. Isaksson et al. (Isaksson et al., 2006), addressed this issue by corroborating the mechano-regulatory algorithms of Carter and colleagues (Carter et al., 1988, Carter and Wong, 1988), Claes and Heigele (Claes and Heigele, 1999, Claes et al., 1998) and Prendergast and coworkers (Prendergast et al., 1997), with more extensive in vivo bone healing data from animal experiments. They compared the patterns of tissue differentiation predicted by the different models and the patterns observed in vivo in an ovine tibia model. They concluded that none of the algorithms predicted patterns of healing entirely similar to those observed experimentally. However, patterns predicted by the algorithm based on deviatoric strain and fluid velocity (i.e. the model of Prendergast et al.,

Another important limitation for computational mechanobiology is represented by the fact that the mechano-regulatory algorithms include empirical constants, the values of which must be determined by comparison to a biological reality. For example, in the mechanoregulation rule developed by Prendergast et al. (Prendergast et al., 1997) the constants *b* and *a* (see equation (3)) do not have a specific physical meaning and can be determined by following the 'trial and error' method outlined in van der Melulen and Huiskes (van der Melulen and Huiskes, 2002): "Computational mechano-biologists hypothesize a potential rule and determine if the outcome of this hypothesis produces realistic tissue structures and morphologies, hence 'trial-and-error'. If the results correspond well, they might be an explanation for the mechanism being modelled. This method of research is common practice and productive in physics, less common in biology (Huiskes, 1995); although 'theoretical biology' is based on this type of approach". Certainly, physicists can use this approach (the computational gedanken experiment) because there are so few rules in physics and the predictions are amenable to exact quantitative testing. In biology the phenomena to be observed and analysed are much more complicated than in physics, so cut-and-try theoretical experimentation could not be really useful. Further research should be carried

out on the efficiency and the correctness of this philosophy of biological research.

Concerning the mechano-regulation model of fracture repair in the body of the L4 vertebra, the most important limitation is that the implemented algorithm does not include a damaged tissue region that would allow tissue to fracture and new callus to form in regions experiencing high levels of biophysical stimulation (e.g. strain). Therefore this study has only considered the original injury event. In reality, histological analyses (Diamond et al., 2007) revealed that a typical feature of vertebral fractures is the overlap between the different tissues corresponding to the different temporal stages of the fracture healing process. This can be justified with the argument that in vertebral bodies the fracture stabilization that permits orderly repair in long bones is not possible due to repetitive injury. As far as concerns the model of the human mandible osteotomized and distracted, the most important limitation is that after the first simulation the numerical predictions state that the callus consists of a mixture of bone, cartilage and fibrous tissue (Figures 8-9), where it might be more accurate to state that in the first few days after the osteotomy the callus consists of progenitor cells subjected to a stimulus that if maintained will result in these cells differentiating into either fibroblasts, chondrocytes or osteoblasts, depending on the magnitude of the stimulus. This may explain why histological findings (Loboa et al., 2005) from animal model studies differ from the initial model predictions as progenitor cells take time to differentiate and produce a tissue phenotype that is recognised by appropriate histological staining. For example, significant bone and soft tissue formation is predicted

(Prendergast et al., 1997)) was closest to experimental results.

during the latency period which is not observed histologically. Explicitly including in the model the time taken for mineralization etc following differentiation of progenitor cells into osteoblasts may result in model predictions more comparable to *in vivo* findings. Another important limitation of the mechano-regulation model used to assess the bone regeneration process in a human mandible osteotomized and distracted as well as in a fractured vertebra is represented by the utilization of the exponential law. Such a law was introduced to account for the fact that mesenchymal cells not only require time to differentiate but, that differentiated cells require some time also for synthesising and remodelling new tissue. In reality, the exponential law should be utilized only to model the early stages of the fracture healing process; when this process is close to the end, saturation phenomena (e.g. the mineralization process) occur within the fracture callus and therefore, at this point, the exponential law should be replaced with another law that allows to better describe these conclusive processes. As a first approximation, we utilized the exponential law to model the early stages of the fracture healing (Boccaccio et al., 2011b) while, toward the end of the process, we replaced the exponential law with a linear constant law. Further research should be carried out on the mathematical function that better describes the entire process of fracture healing, both, in the early and in the final stages.

Many experiments on skeletal failure and repair have been performed in the last century aimed to determine the influence of biological, mechanical, hormonal factors on the healing process. Despite this effort, there are still many unanswered questions. This indicates the complexity of the biological problems and has stimulated the development of computational models that can analyze the influence of all factors and make predictions under different boundary and loading conditions. These models must also be validated with experimental analyses. However, in many cases the computational models cannot be validated directly because of the difficulties in performing some measurements in vivo. Despite this, indirect validations can be performed if the conclusions of the computer simulations are similar to the experimental or clinical results. Once the mechano-regulation model has been validated, it can be conveniently utilized to assess the regeneration process within the fracture site in the case in which different boundary and loading conditions act on it. For instance, the mechanoregulation model of fracture repair in vertebral bodies illustrated above can be used to predict the spatial and temporal patterns of repair during altered loading conditions, which may prove beneficial in developing rehabilitation regimes following vertebral fractures. Also, this same model can be utilized to evaluate how the patterns of tissue differentiation change if the fractured vertebra is supported with minimally invasive percutaneous fixation devices (Palmisani et al., 2009). Furthermore, the mechanobiological model of the human mandible osteotomized and distracted can be utilized to estimate how the bony tissue regeneration process within the fracture callus change for different mandibular distraction devices such as those analyzed in previous studies (Boccaccio et al., 2008d; Boccaccio et al., 2011c).

Future perspectives include the development of computer power. A more robust integration is required, in future, between biology, mechanics and materials science. This should lead to the development of mechano-regulation models that more accurately describe physiological processes such as fracture healing, tissue genesis etc. A very promising research area for mechanobiology is in the field of tissue engineering. Mechanobiology could be an efficient and cheap tool to determine optimal parameters governing scaffold performance. Future perspectives of numerical simulations of biomaterial scaffolds for tissue engineering rely also on the development of new methods to account for the multi-scale dimension of the problems. At the micro-scale level one can analyse cell/biomaterial interactions and how

Mechanobiology of Fracture Healing: Basic

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these interactions influence the tissue differentiation process, at the macro-scale level one can evaluate the 'average' boundary and loading conditions acting on the scaffold implanted in the specific anatomical site.

### **6. Conclusions**

This Chapter presented the principal mechano-regulation theories recently developed to simulate the tissue differentiation and the main cellular processes involved in fracture healing. Two examples have then been given illustrating how a mechano-regulation algorithm - where the bone callus is modeled as a biphasic poroelastic material and the stimulus regulating tissue differentiation is hypothesized to be a function of the strain and fluid flow -, can be utilized to assess the bone regeneration process both, in a human mandible submitted to distraction osteogenesis and in a fractured lumbar vertebra. The principal limitations of mechanobiological algorithms as well as the future research lines in the field have been finally outlined.

### **7. References**


these interactions influence the tissue differentiation process, at the macro-scale level one can evaluate the 'average' boundary and loading conditions acting on the scaffold

This Chapter presented the principal mechano-regulation theories recently developed to simulate the tissue differentiation and the main cellular processes involved in fracture healing. Two examples have then been given illustrating how a mechano-regulation algorithm - where the bone callus is modeled as a biphasic poroelastic material and the stimulus regulating tissue differentiation is hypothesized to be a function of the strain and fluid flow -, can be utilized to assess the bone regeneration process both, in a human mandible submitted to distraction osteogenesis and in a fractured lumbar vertebra. The principal limitations of mechanobiological algorithms as well as the future research lines in

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**6. Conclusions**

**7. References** 


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**0**

**3**

<sup>1</sup>*Brazil* <sup>2</sup>*Denmark*

**Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass**

<sup>1</sup>*Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte* <sup>2</sup>*University of Aalborg, Department of Biotechnology, Chemistry, and Environmental*

It is generally accepted that bone morphology may be influenced by functional bone strains or stresses (McMahon, 1973, 1975a, b; Alexander, 1977; Bertram & Biewener 1990, 1992; Biewener, 1990; Christiansen, 1999, 2002a, b). However, the problem as to which specific mechanical characteristics are most relevant remains open (e.g., Rubin & Lanyon, 1984; Fritton et al., 2000; Biewener, 2000, 2005). Aiming to establish correlations between structural proportions and posture of mammalian limbs coping with body's locomotory functions, including support of mass in the gravitational field, scaling studies of limb long bones in terrestrial mammals have been subject to long standing debate and controversy (Biewener, 1982, 1983, 1989, 1990, 2000, 2005; Biewener et al., 1983; Biewener & Taylor, 1986; Selker & Carter, 1989; Bertram & Biewener, 1990, 1992; Christiansen, 1997, 1998, 1999, 2002a, b, 2007; Fariña et al., 1997; Carrano, 1998, 1999, 2001; Currey, 2003; Kokshenev, 2003; Kokshenev et al., 2003). The exploration of basic concepts of stability of ideal and non-ideal solid cylinders loaded in non-critical, transient and near critical mechanical regimes, mapped to arbitrary loaded curved limb long bones, resulted in a number of mechanical patterns of similarity in long bones adjusted to their design (Kokshenev, 2007). Established under fairly general assumptions, the proposed scaling rules (for peak longitudinal-bone and transverse-bone elastic forces and momenta, compressive and shear strains, corresponding to axial and non-axial bending and torsional components of tensorial stress) congruent with bone allometry explained the two basic patterns of functional stresses *in vivo* revealed in the limb bones of fast running terrestrial

The theoretically established patterns of bone design (Kokshenev, 2007) have been also tested (Kokshenev & Christiansen, 2010) by the surprisingly varied differential scaling of the limb long bones in Asian (*Elephas maximus*) and African (*Loxodonta africana*) elephants. These terrestrial giants have more upright limb bones to vertical, notably much more upright propodials (humerus and femur), which are held at a distinctly greater angle compared to the ground than is the case in other large, quadrupedal mammals. Studies of their locomotor mechanics have also indicated differences from other terrestrial mammals, in that fast locomotion is ambling with no suspended phase in the stride, but with duty factors exceeding 0.5 (Alexander et al., 1979; Hutchinson et al., 2006). The theoretical predictions

**1. Introduction**

mammals by Rubin & Lanyon (1982, 1984).

Valery B. Kokshenev1 and Per Christiansen2

*Engineering, Sohngaardsholmsvej 57, DK-9000 Aalborg*


## **Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass**

Valery B. Kokshenev1 and Per Christiansen2

<sup>1</sup>*Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte* <sup>2</sup>*University of Aalborg, Department of Biotechnology, Chemistry, and Environmental Engineering, Sohngaardsholmsvej 57, DK-9000 Aalborg* <sup>1</sup>*Brazil*

<sup>2</sup>*Denmark*

### **1. Introduction**

48 Theoretical Biomechanics

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It is generally accepted that bone morphology may be influenced by functional bone strains or stresses (McMahon, 1973, 1975a, b; Alexander, 1977; Bertram & Biewener 1990, 1992; Biewener, 1990; Christiansen, 1999, 2002a, b). However, the problem as to which specific mechanical characteristics are most relevant remains open (e.g., Rubin & Lanyon, 1984; Fritton et al., 2000; Biewener, 2000, 2005). Aiming to establish correlations between structural proportions and posture of mammalian limbs coping with body's locomotory functions, including support of mass in the gravitational field, scaling studies of limb long bones in terrestrial mammals have been subject to long standing debate and controversy (Biewener, 1982, 1983, 1989, 1990, 2000, 2005; Biewener et al., 1983; Biewener & Taylor, 1986; Selker & Carter, 1989; Bertram & Biewener, 1990, 1992; Christiansen, 1997, 1998, 1999, 2002a, b, 2007; Fariña et al., 1997; Carrano, 1998, 1999, 2001; Currey, 2003; Kokshenev, 2003; Kokshenev et al., 2003). The exploration of basic concepts of stability of ideal and non-ideal solid cylinders loaded in non-critical, transient and near critical mechanical regimes, mapped to arbitrary loaded curved limb long bones, resulted in a number of mechanical patterns of similarity in long bones adjusted to their design (Kokshenev, 2007). Established under fairly general assumptions, the proposed scaling rules (for peak longitudinal-bone and transverse-bone elastic forces and momenta, compressive and shear strains, corresponding to axial and non-axial bending and torsional components of tensorial stress) congruent with bone allometry explained the two basic patterns of functional stresses *in vivo* revealed in the limb bones of fast running terrestrial mammals by Rubin & Lanyon (1982, 1984).

The theoretically established patterns of bone design (Kokshenev, 2007) have been also tested (Kokshenev & Christiansen, 2010) by the surprisingly varied differential scaling of the limb long bones in Asian (*Elephas maximus*) and African (*Loxodonta africana*) elephants. These terrestrial giants have more upright limb bones to vertical, notably much more upright propodials (humerus and femur), which are held at a distinctly greater angle compared to the ground than is the case in other large, quadrupedal mammals. Studies of their locomotor mechanics have also indicated differences from other terrestrial mammals, in that fast locomotion is ambling with no suspended phase in the stride, but with duty factors exceeding 0.5 (Alexander et al., 1979; Hutchinson et al., 2006). The theoretical predictions

biomechanics (see reviews by Hutchinson, 2006; and Alexander, 2006), as well as inferred physiology and growth rates (e.g., Ji et al., 1998; Horner et al., 1999; Rensberger & Watabe, 2000; Sander, 2000; Erickson, 2004; Padian et al., 2004). However, locomotor complexity cannot be captured in a simple model since the number of required parameters increases rapidly as a model becomes more complex (Hutchinson & Gatesy, 2006). One issue of great importance in biomechanics is therefore to more accurately employ generalizations following from dynamic similarity concepts suggested a few dimensionless universal (body mass independent) numbers (Alexander & Jayes, 1983; Gatesy & Biewener, 1991) or a few number of universal scaling (exponents in scaling) rules (Garland, 1983; Heglund & Taylor, 1988; Biewener, 1990; Farley et al.,1993; Bejan & Marden, 2006; Kokshenev, 2010), which potentially make a bridge between locomotory trends in large modern elephants, the largest extant, and the largest extinct proboscideans (e.g., Kokshenev & Christiansen, 2010). Although the experimental exploration of dynamic similarity determinants was repeatedly questioned, e.g., in the context of dimensionless body safety factors (Biewener, 1982, 1990, 2000, 2005) or Froude numbers (Hutchinson et al., 2006), the empirical studies of locomotor evolution in extinct animals (e.g., archosaurs reviewed by Hutchinson, 2006) evidence for that evolution in running and turning abilities in extinct giants, accomplished by transformations in poses and postures of animals, expose a certain kind of adaptive similarity driven by body mass. Developing here the application of dynamic stress similarity generally supported by some available dynamic data on reconstructed extinct giants, we demonstrate how the empirically established traits of locomotory evolution in extinct terrestrial giants can be explained as

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 51

The principal sources of data on *maximal body masses M*(exp) max inferred from remains of some of the largest non-avian dinosaurs and mammals are listed in Table 1. Other published data are

The *in vivo* studies by Rubin & Lanyon (1982, 1984) on the individual-bone functional peak

The axial and bending components of the compressive *tibial* stress where scaled to body mass (ranged from 7.3 *kg* to 2500 *kg* ) and analyzed by the least-squares liner regression method

*axial <sup>M</sup>*<sup>−</sup>0.05, with *<sup>σ</sup>*(exp)

similarity (shown below Table 2 in Rubin & Lanyon, 1984) can be read here as

*bone* (*M*, *<sup>V</sup>*max) = *<sup>σ</sup>*(exp)

*bone* (*M*, *V*), resulted from gauge implantation in the long limb bones, revealed the dynamic stress (or strain) similarity in animals of different body mass *M* running at a certain *speed V*. For the special studied case of biped and quadruped animals (from a turkey to an elephant) running at their *fastest speeds V*max, the corresponding pattern of dynamic stress

*axial* (*M*) + *<sup>σ</sup>*(exp)

*bend* (*M*) = *<sup>C</sup>*(exp)

*bend* (*M*). (1)

*bend <sup>M</sup>*0.08, (2)

affected by maximal body mass through limb-bone adaptations.

**2. Materials and methods**

**2.1.1 Maximal body mass**

cited at appropriate points.

**2.1.2 Dynamic strain and stress similarities**

*<sup>σ</sup>*(exp)

*<sup>σ</sup>*(max)

*axial* (*M*) = *<sup>C</sup>*(exp)

**2.1 Materials**

stress *<sup>σ</sup>*(*peak*)

resulted in

by Kokshenev (2007) were compared with data from a phylogenetically wide sample of proboscideans from Christiansen (2007). Consequently, some salient features distinguishing limb postures characteristic of locomotion in proboscideans and mammals were established. The limb bone allometry in Asian elephants and the Elephantidae was shown to be congruent with adaptation to bending-torsion proportions of stress induced by muscular forces, likewise in other mammals, whereas limb bones in African elephants appear adapted for coping with the dominating axial compressive forces of gravity. Since the hindlimb bones in extant and most likely extinct elephants were shown to be more compliant than forelimb bones, the limb compliance gradient of limb locomotory function, contrasting in sign to other mammals, was shown to constitute a new important dynamic constraint preventing elephants from achieving a full-body aerial phase in fast gaits of locomotion.

Considering the athletism of non-avian dinosaurs and adopting the assumption of *dynamic similarity*, Alexander (1976) analyzed inferred bending stress in their limb bones, which led him to the hypothesis (Alexander, 1981, 1985a) that animals appear to be built with universal (equal bodyweight-independent) *limb safety factors* (bone strengths related to corresponding peak bone stresses). However, systematic studies of functional limb bone stress by Rubin & Lanyon (1982, 1984) clearly demonstrated that the dimensionless factor of limb safety may not be chosen as the determinant of dynamic similarity revealed across body mass through the speed and frequency in fast running bipeds and quadrupeds. Instead, a new kind of *dynamic strain similarity* (physically equivalent to dynamic stress similarity) observed in limb bones of walking, trotting, and galloping animals experimentally established by many researchers (Rubin & Lanyon, 1982, 1984; Rubin & Lanyon, 1982; Biewener et al., 1983; Biewener & Taylor, 1986) has been only recently understood in light of more general *mechanical elastic similarity* (Kokshenev, 2007). It has been in particular demonstrated that the consistency between the elastic strain similarity by Rubin & Lanyon (1982, 1984) and the *elastic static stress similarity* by McMahon (1975a) exists, since both are underlaid by the same patterns of elastic forces emerging in solids, that can be revealed if the external gravitational forces are substituted by predominating functional muscular forces (Kokshenev, 2003; Kokshenev et al., 2003). In this chapter, we try to establish a link between the mechanical elastic similarity in limb bones of different-sized animals (McMahon, 1973, 1975a, b; Rubin & Lanyon, 1982, 1984; Kokshenev, 2003, 2007) and the seminal dynamic similarity in locomotion of animals (Alexander, 1976, 1989; Alexander & Jayes, 1983), arising in turn from the more general mechanical similarity of uniform classical systems (Kokshenev, 2011a, b).

Following Schmidt-Neielsen (1984), it has been widely recognized that body mass is often a major locomotory factor scaling muscle force output establishing limits for body ability in animals (Alexander, 1985b; Hutchinson & Garcia, 2002; Biewener, 2005, Hutchinson et al., 2005; Marden, 2005), through their maximal speed (Garland, 1983; Jones & Lindstedt, 1993; Sellers & Manning, 2007) and maximal size (Hokkanen, 1986a; Biewener, 1989; Kokshenev, 2007). Reconstructing body mass and locomotion in extinct animals (Alexander, 1989, 1991, 1998, 2006; Fariña & Blanco, 1996; Fariña et al., 1997; Carrano, 1998; 2001; Carrano & Biewener, 1999; Farlow et al., 2000; Wilson & Carrano, 1999; Hutchinson & Gatesy, 2006, Sellers & Manning, 2007), the biomechanical modeling also includes their locomotor habits (e.g., Fariña, 1995; Paul & Christiansen, 2000; Christiansen & Paul 2001; Blanco & Jones, 2005; Fariña et al., 2005). The evolutionary history of dinosaurs and mammals provide evidence for convergent similarities of skeletal design (e.g., near parasagittal limb postures and hinge-like joints), locomotor kinematics (Alexander, 1991, 1998; Farlow et al., 2000; Carrano, 1998, 1999; 2001; Paul & Christiansen, 2000; Hutchinson et al., 2006), and biomechanics (see reviews by Hutchinson, 2006; and Alexander, 2006), as well as inferred physiology and growth rates (e.g., Ji et al., 1998; Horner et al., 1999; Rensberger & Watabe, 2000; Sander, 2000; Erickson, 2004; Padian et al., 2004). However, locomotor complexity cannot be captured in a simple model since the number of required parameters increases rapidly as a model becomes more complex (Hutchinson & Gatesy, 2006). One issue of great importance in biomechanics is therefore to more accurately employ generalizations following from dynamic similarity concepts suggested a few dimensionless universal (body mass independent) numbers (Alexander & Jayes, 1983; Gatesy & Biewener, 1991) or a few number of universal scaling (exponents in scaling) rules (Garland, 1983; Heglund & Taylor, 1988; Biewener, 1990; Farley et al.,1993; Bejan & Marden, 2006; Kokshenev, 2010), which potentially make a bridge between locomotory trends in large modern elephants, the largest extant, and the largest extinct proboscideans (e.g., Kokshenev & Christiansen, 2010). Although the experimental exploration of dynamic similarity determinants was repeatedly questioned, e.g., in the context of dimensionless body safety factors (Biewener, 1982, 1990, 2000, 2005) or Froude numbers (Hutchinson et al., 2006), the empirical studies of locomotor evolution in extinct animals (e.g., archosaurs reviewed by Hutchinson, 2006) evidence for that evolution in running and turning abilities in extinct giants, accomplished by transformations in poses and postures of animals, expose a certain kind of adaptive similarity driven by body mass. Developing here the application of dynamic stress similarity generally supported by some available dynamic data on reconstructed extinct giants, we demonstrate how the empirically established traits of locomotory evolution in extinct terrestrial giants can be explained as affected by maximal body mass through limb-bone adaptations.

### **2. Materials and methods**

### **2.1 Materials**

2 Will-be-set-by-IN-TECH

by Kokshenev (2007) were compared with data from a phylogenetically wide sample of proboscideans from Christiansen (2007). Consequently, some salient features distinguishing limb postures characteristic of locomotion in proboscideans and mammals were established. The limb bone allometry in Asian elephants and the Elephantidae was shown to be congruent with adaptation to bending-torsion proportions of stress induced by muscular forces, likewise in other mammals, whereas limb bones in African elephants appear adapted for coping with the dominating axial compressive forces of gravity. Since the hindlimb bones in extant and most likely extinct elephants were shown to be more compliant than forelimb bones, the limb compliance gradient of limb locomotory function, contrasting in sign to other mammals, was shown to constitute a new important dynamic constraint preventing elephants from achieving

Considering the athletism of non-avian dinosaurs and adopting the assumption of *dynamic similarity*, Alexander (1976) analyzed inferred bending stress in their limb bones, which led him to the hypothesis (Alexander, 1981, 1985a) that animals appear to be built with universal (equal bodyweight-independent) *limb safety factors* (bone strengths related to corresponding peak bone stresses). However, systematic studies of functional limb bone stress by Rubin & Lanyon (1982, 1984) clearly demonstrated that the dimensionless factor of limb safety may not be chosen as the determinant of dynamic similarity revealed across body mass through the speed and frequency in fast running bipeds and quadrupeds. Instead, a new kind of *dynamic strain similarity* (physically equivalent to dynamic stress similarity) observed in limb bones of walking, trotting, and galloping animals experimentally established by many researchers (Rubin & Lanyon, 1982, 1984; Rubin & Lanyon, 1982; Biewener et al., 1983; Biewener & Taylor, 1986) has been only recently understood in light of more general *mechanical elastic similarity* (Kokshenev, 2007). It has been in particular demonstrated that the consistency between the elastic strain similarity by Rubin & Lanyon (1982, 1984) and the *elastic static stress similarity* by McMahon (1975a) exists, since both are underlaid by the same patterns of elastic forces emerging in solids, that can be revealed if the external gravitational forces are substituted by predominating functional muscular forces (Kokshenev, 2003; Kokshenev et al., 2003). In this chapter, we try to establish a link between the mechanical elastic similarity in limb bones of different-sized animals (McMahon, 1973, 1975a, b; Rubin & Lanyon, 1982, 1984; Kokshenev, 2003, 2007) and the seminal dynamic similarity in locomotion of animals (Alexander, 1976, 1989; Alexander & Jayes, 1983), arising in turn from the more general mechanical similarity of

Following Schmidt-Neielsen (1984), it has been widely recognized that body mass is often a major locomotory factor scaling muscle force output establishing limits for body ability in animals (Alexander, 1985b; Hutchinson & Garcia, 2002; Biewener, 2005, Hutchinson et al., 2005; Marden, 2005), through their maximal speed (Garland, 1983; Jones & Lindstedt, 1993; Sellers & Manning, 2007) and maximal size (Hokkanen, 1986a; Biewener, 1989; Kokshenev, 2007). Reconstructing body mass and locomotion in extinct animals (Alexander, 1989, 1991, 1998, 2006; Fariña & Blanco, 1996; Fariña et al., 1997; Carrano, 1998; 2001; Carrano & Biewener, 1999; Farlow et al., 2000; Wilson & Carrano, 1999; Hutchinson & Gatesy, 2006, Sellers & Manning, 2007), the biomechanical modeling also includes their locomotor habits (e.g., Fariña, 1995; Paul & Christiansen, 2000; Christiansen & Paul 2001; Blanco & Jones, 2005; Fariña et al., 2005). The evolutionary history of dinosaurs and mammals provide evidence for convergent similarities of skeletal design (e.g., near parasagittal limb postures and hinge-like joints), locomotor kinematics (Alexander, 1991, 1998; Farlow et al., 2000; Carrano, 1998, 1999; 2001; Paul & Christiansen, 2000; Hutchinson et al., 2006), and

a full-body aerial phase in fast gaits of locomotion.

uniform classical systems (Kokshenev, 2011a, b).

### **2.1.1 Maximal body mass**

The principal sources of data on *maximal body masses M*(exp) max inferred from remains of some of the largest non-avian dinosaurs and mammals are listed in Table 1. Other published data are cited at appropriate points.

### **2.1.2 Dynamic strain and stress similarities**

The *in vivo* studies by Rubin & Lanyon (1982, 1984) on the individual-bone functional peak stress *<sup>σ</sup>*(*peak*) *bone* (*M*, *V*), resulted from gauge implantation in the long limb bones, revealed the dynamic stress (or strain) similarity in animals of different body mass *M* running at a certain *speed V*. For the special studied case of biped and quadruped animals (from a turkey to an elephant) running at their *fastest speeds V*max, the corresponding pattern of dynamic stress similarity (shown below Table 2 in Rubin & Lanyon, 1984) can be read here as

$$
\sigma\_{bone}^{(\max)}(M, V\_{\max}) = \sigma\_{axial}^{(\exp)}(M) + \sigma\_{bend}^{(\exp)}(M). \tag{1}
$$

The axial and bending components of the compressive *tibial* stress where scaled to body mass (ranged from 7.3 *kg* to 2500 *kg* ) and analyzed by the least-squares liner regression method resulted in

$$
\sigma\_{\text{axial}}^{(\text{exp})}(M) = \mathbb{C}\_{\text{axial}}^{(\text{exp})} M^{-0.05}, \text{with } \sigma\_{\text{bend}}^{(\text{exp})}(M) = \mathbb{C}\_{\text{bend}}^{(\text{exp})} M^{0.08}, \tag{2}
$$

Mammals Proboscidea Elephantidae

(*mam*) < 1.

*tors* , with *α* + *β* + *τ* = 1, (5)

*limb* = 0.78 ± 0.02 (6)

Limb bones *N λ*(exp) *r N λ*(exp) *r N λ*(exp) *r* Humerus 189 0.7631 0.9738 16 *1.134* 0.831 7 0.912 0.990 Radius 189 0.7530 0.9957 10 *1.078* 0.878 6 0.813 0.853 Ulna 189 0.849 0.9600 14 0.929 0.866 6 0.727 0.888 Femur 189 0.8431 0.9763 14 0.802 0.816 7 0.747 0.966 Tibia 189 0.7641 0.9499 11 0.772 0.857 6 0.751 0.925 Limb bone 189 **0.795** 0.971 13 **0.943** 0.850 6 **0.790** 0.924 Table 2. Statistical data on the slenderness of individual and effective limb bones in animals reproduced from Table 1 in Kokshenev & Christiansen (2010). The mean slenderness exponents are presented by the slopes *λ*(exp) derived in *N* species through the least-squares (LS) regression with the *correlation coefficient r*. The LS characterization of the effective limb bone corresponding to the overall-bone mean data is introduced as the standard mean of all data on five bones. The *bold numbers* are the data used below. The *italic numbers* indicate

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 53

functions of body mass are not available, the effective limb bone for extinct species can be introduced through the statistical data on the *slenderness exponents λ* (= *dLog*10*L*/*dLog*10*C*) obtained regardless of the body mass data. Some employed below examples of extinct

In fast gaits of locomotion, all types of elastic strains (and thus stresses), including axial uniform compressive strains, non-uniform tensile strains and transverse non-uniform shear strains, are generally involved in the hindlimbs of terrestrial animals. Considering the cases of the fastest speed dynamic regimes discussed in Eq. (1), we employ the principle of superposition for the simple axial, and complex bending and shear-strain (torsional) elastic

emerging in an effective limb bone. The peak stress appears as a combination of the partial functional axial, bending, and torsional stresses weighted respectively by relative probabilities *α*, *β*, and *τ*. These mass-independent probabilities play the role of indicators of the dynamic stress similarity, likewise the limb duty factors and swept angles discussed in the theory of

Unlike the individual-bone data discussed in Eq. (1), we introduce an *effective mammalian limb bone* unifying six long bones (humerus, radius, ulna, femur, tibia, and fibula), through their geometric similarity determined by the bone-diameter and bone-length *allometric exponents*

established (Kokshenev, 2007, Table 1) via the LRS data from mostly running extant mammals (79 species, 98 specimens, Table 2 in Christiansen, 1999). One can see that the data on

*bend* <sup>+</sup> *τσ*(max)

*limb* <sup>=</sup> 0.28 <sup>±</sup> 0.02, with *<sup>λ</sup>*(*pred*)

*axial* <sup>+</sup> *βσ*(max)

(exp)

some the slope data contrasting to systematic mammalian data *<sup>λ</sup>*(exp)

Mammalia are listed in Table 2.

*<sup>σ</sup>*(max)

*d* (exp)

fields, resulted in the total compressive stress

*limb* (*M*, *<sup>V</sup>*max) = *ασ*(max)

*limb* = 0.36 ± 0.02 and *l*

dynamic similarity in animal locomotion (Kokshenev, 2011a, b).

**2.2 Theory**

**2.2.1 Stress similarity**


Table 1. Some data on maximal body masses in terrestrial giants. The limb duty limb factor is provided as an expected characteristics indicating that animals could run with a suspended phase in the stride (with *βduty* < 0.5) or would able to progress with a walking gait only (with *βduty* > 0.5).

where

$$\mathcal{C}\_{\text{axial}}^{(\text{exp})} = 11 MPa \cdot (\text{kg})^{0.05} \text{ and } \mathcal{C}\_{\text{bend}}^{(\text{exp})} = 28 MPa \cdot (\text{kg})^{-0.08}.\tag{3}$$

Testing the dynamic stress similarity hypothesis by Rubin & Lanyon (1984), the continuous-speed dynamic similarity between different-sized animals, moving in a certain *gait*, was studied by Biewener & Taylor (1986) through the gait-dependent individual-bone stress function *<sup>σ</sup>*(max) *bone* (*M*, *V*), also analyzed by Rubin & Lanyon (1982). Experimental studies of the peak stress measured in the midshaft of long bones in fast walking, moderately running (or trotting) animals, and fast running (or galloping) animals with smoothly changing speed, resulted in that *<sup>σ</sup>*(max) *bone* (*M*, *V*) is a linear piecewise function of speed, which domains are limited by the gait-dependent "speed-equivalent" points *V*(max) *trans* , as suggested by Biewener & Taylor (1986).

When the allometric data from individual bones in running mammals are generalized to *effective limb bone*(e.g., Kokshenev, 2003), the corresponding peak functional stress is suggested here in the form

$$
\sigma\_{\text{limb}}^{(\text{max})}(M, V) = \sigma\_{\text{axial}}^{(\text{max})}(M) + \sigma\_{\text{bond}}^{(\text{max})}(M) \frac{V}{V\_{\text{trans}}^{(\text{max})}},\tag{4}
$$

where the *transient* speeds *V*(max) *trans* are characteristic points of the crossover-gait (walk-to-run) dynamic states or transient-mode (trot-to-gallop) dynamic states, discussed in the context of dynamic similarity theory (Kokshenev, 2011a, b). The reliably of the suggested gait-dependent pattern of dynamic stress similarity in an effective bone shown in Eq. (4) can be inferred from the *in vivo* compressive stress data exemplified by those from three walking, trotting, and galloping adult goats analyzed by Biewener & Taylor (1986, Figs. 2B and 4), which generally unify the experimental data from the radius and tibia in a goat with those in a dog (Rubin & Lanyon, 1982) and a horse (Rubin & Lanyon, 1982; Biewener et al., 1983).

### **2.1.3 Limb bone design**

Aiming to make a bridge between limb bone design in extant and extinct animals listed in Table 1, for which systematic data on the bone length (*L* ) and bone circumference (*C* ) as


Table 2. Statistical data on the slenderness of individual and effective limb bones in animals reproduced from Table 1 in Kokshenev & Christiansen (2010). The mean slenderness exponents are presented by the slopes *λ*(exp) derived in *N* species through the least-squares (LS) regression with the *correlation coefficient r*. The LS characterization of the effective limb bone corresponding to the overall-bone mean data is introduced as the standard mean of all data on five bones. The *bold numbers* are the data used below. The *italic numbers* indicate some the slope data contrasting to systematic mammalian data *<sup>λ</sup>*(exp) (*mam*) < 1.

functions of body mass are not available, the effective limb bone for extinct species can be introduced through the statistical data on the *slenderness exponents λ* (= *dLog*10*L*/*dLog*10*C*) obtained regardless of the body mass data. Some employed below examples of extinct Mammalia are listed in Table 2.

### **2.2 Theory**

4 Will-be-set-by-IN-TECH

Taxon Largest species *<sup>M</sup>*(exp) max , *kg* References *<sup>β</sup>*(*pred*)

Dinosauria,Sauropoda, *Argentinosaurus* 70000 Mazetta et al., 2004 > 0.5 Dinosauria, Sauropoda *Supersaurus* 38000 Lovelace et al., 2007 > 0.5 Dinosauria, Sauropoda *Giraffatitan* 38000 Taylor, 2009 > 0.5 Dinosauria, Theropoda *Giganotosaurus* 9000 Mazetta et al., 2004 < 0.5 Dinosauria, Theropoda *Tyrannosaurus* 8000 Mazetta et al., 2004 < 0.5 Dinosauria, Ornithopoda *Shantungosaurus* 16000 Horner et al., 2004 < 0.5 Mammalia, Proboscidea *Mammuthus trogontherii* 20000 Christiansen, 2004 ≈ 0.5 Mammalia, Proboscidea *Deinotherium giganteum* 18000 Christiansen, 2004 ≈ 0.5 Mammalia, Perissodactyla *Indricotherium* 19000 Fortelius & Kappe-

Table 1. Some data on maximal body masses in terrestrial giants. The limb duty limb factor is provided as an expected characteristics indicating that animals could run with a suspended phase in the stride (with *βduty* < 0.5) or would able to progress with a walking gait only

Testing the dynamic stress similarity hypothesis by Rubin & Lanyon (1984), the continuous-speed dynamic similarity between different-sized animals, moving in a certain *gait*, was studied by Biewener & Taylor (1986) through the gait-dependent individual-bone

of the peak stress measured in the midshaft of long bones in fast walking, moderately running (or trotting) animals, and fast running (or galloping) animals with smoothly changing speed,

When the allometric data from individual bones in running mammals are generalized to *effective limb bone*(e.g., Kokshenev, 2003), the corresponding peak functional stress is suggested

dynamic states or transient-mode (trot-to-gallop) dynamic states, discussed in the context of dynamic similarity theory (Kokshenev, 2011a, b). The reliably of the suggested gait-dependent pattern of dynamic stress similarity in an effective bone shown in Eq. (4) can be inferred from the *in vivo* compressive stress data exemplified by those from three walking, trotting, and galloping adult goats analyzed by Biewener & Taylor (1986, Figs. 2B and 4), which generally unify the experimental data from the radius and tibia in a goat with those in a dog (Rubin &

Aiming to make a bridge between limb bone design in extant and extinct animals listed in Table 1, for which systematic data on the bone length (*L* ) and bone circumference (*C* ) as

*axial* (*M*) + *<sup>σ</sup>*(max)

*bone* (*M*, *V*), also analyzed by Rubin & Lanyon (1982). Experimental studies

*bone* (*M*, *V*) is a linear piecewise function of speed, which domains are

*bend* (*M*) *<sup>V</sup>*

*trans* are characteristic points of the crossover-gait (walk-to-run)

*V*(max) *trans*

*axial* <sup>=</sup> <sup>11</sup>*MPa* · (*kg*)0.05 and *<sup>C</sup>*(exp)

limited by the gait-dependent "speed-equivalent" points *V*(max)

*limb* (*M*, *<sup>V</sup>*) = *<sup>σ</sup>*(max)

Lanyon, 1982) and a horse (Rubin & Lanyon, 1982; Biewener et al., 1983).

*<sup>σ</sup>*(max)

where the *transient* speeds *V*(max)

**2.1.3 Limb bone design**

(with *βduty* > 0.5).

stress function *<sup>σ</sup>*(max)

resulted in that *<sup>σ</sup>*(max)

& Taylor (1986).

here in the form

*<sup>C</sup>*(exp)

where

*duty*

< 0.5

lmann, 1993

*bend* <sup>=</sup> <sup>28</sup>*MPa* · (*kg*)<sup>−</sup>0.08. (3)

*trans* , as suggested by Biewener

, (4)

### **2.2.1 Stress similarity**

In fast gaits of locomotion, all types of elastic strains (and thus stresses), including axial uniform compressive strains, non-uniform tensile strains and transverse non-uniform shear strains, are generally involved in the hindlimbs of terrestrial animals. Considering the cases of the fastest speed dynamic regimes discussed in Eq. (1), we employ the principle of superposition for the simple axial, and complex bending and shear-strain (torsional) elastic fields, resulted in the total compressive stress

$$
\sigma\_{\text{limb}}^{(\text{max})}(M, V\_{\text{max}}) = \mathfrak{a}\sigma\_{\text{axial}}^{(\text{max})} + \beta \sigma\_{\text{bond}}^{(\text{max})} + \tau \sigma\_{\text{tors}}^{(\text{max})}, \text{with } \mathfrak{a} + \beta + \tau = 1,\tag{5}
$$

emerging in an effective limb bone. The peak stress appears as a combination of the partial functional axial, bending, and torsional stresses weighted respectively by relative probabilities *α*, *β*, and *τ*. These mass-independent probabilities play the role of indicators of the dynamic stress similarity, likewise the limb duty factors and swept angles discussed in the theory of dynamic similarity in animal locomotion (Kokshenev, 2011a, b).

Unlike the individual-bone data discussed in Eq. (1), we introduce an *effective mammalian limb bone* unifying six long bones (humerus, radius, ulna, femur, tibia, and fibula), through their geometric similarity determined by the bone-diameter and bone-length *allometric exponents*

$$d\_{limb}^{(\text{exp})} = 0.36 \pm 0.02 \text{ and } l\_{limb}^{(\text{exp})} = 0.28 \pm 0.02, \text{ with } \lambda\_{limb}^{(\text{prod})} = 0.78 \pm 0.02\tag{6}$$

established (Kokshenev, 2007, Table 1) via the LRS data from mostly running extant mammals (79 species, 98 specimens, Table 2 in Christiansen, 1999). One can see that the data on

When compared with Eq. (3), the experimental data *<sup>C</sup>*(exp)

(Kokshenev, 2007, Fig. 1).

**2.2.2 Critical body mass**

*mobil* (*M*) = *<sup>C</sup>*(exp)

*mobil <sup>M</sup>*0.08 <sup>=</sup> *<sup>s</sup>*

introduced on the basis of Eqs. (5) and (10). Here *s*

The *in vitro* data on the mean individual-bone strength *s*

long bones damaged under pure axial compression, i.e., *s*

*bend* = 56.1 *MPa*, *s*

⎞ ⎠

1 0.08

Alexander (1981) and Biewener (1990), pure bending and torsion, i.e., *s*

result in limiting estimates for the critical masses (Kokshenev, 2007)

(max)

to complex bending-torsion loading and *<sup>C</sup>*(exp)

*s* (max)

> ⎛ ⎝ *s* (max) *bend C*(max) *bend*

*M*(*crit*) *bend* =

gait anyway.

**2.2.3 Body safety**

(max) *mobil* �

constraint

*<sup>σ</sup>*(max)

*s* (*mean*)

adopted without change, whereas the data for the body mobility function *<sup>C</sup>*(exp)

*M <sup>M</sup>*(*crit*) max �0.08

modified by the statistical analysis extended by *in vitro* data on the tibial failure in torsion

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 55

Taking into consideration that instead of the axial-bone stress the bending and torsional functional stresses limit the body mobility of animals involved in strenuous activity, the upper limit of body mass, or the *critical body mass M*(*crit*) max , follows from the relevant functional-stress

*tors* ≈ 60 *MPa* reported by Taylor et al. (2003). Given that generally the ±10 % domain is adopted for the statistically established mean bone strengths, the maximal amplitude data

*tors* = 66 *MPa*, and *s*

= 9.3 *tons* and *M*(*crit*)

These two critical body masses determine two kinds of similarity patterns of well distinct *body postures* in giants, whose body mobility is supposedly limited by critical pure bending and pure torsional stresses emerging in their limbs at maximal speeds of critical locomotion. Using empirical data in Table 1, the patterns of *most erect posture* and *most sprawling posture* in giants, may be respectively presented by the obligate biped *Giganotosaurus*, likely capable for relatively fast running and the largest terrestrial quadruped *Argentinosaurus*, an animal so huge that a true run would likely have severely injured the limb bones via critical torsional stresses. Indeed, sauropod anatomy speaks vehemently against anything other than a walk

Bearing in mind that the limb safety during strenuous activity of terrestrial animals establishes the risk level of skeleton damage (Alexander, 1981; Biewener, 1982, 1989, 1990), a generalized

, providing *<sup>M</sup>*(*crit*) max <sup>=</sup>

(*mean*)

(*mean*)

(max)

� *<sup>s</sup>* (max) *tors C*(max) *tors*

(max)

*mobil* is shown in Eq. (11).

*tors* =

*supp* for the body support function is

⎛ ⎝ *s* (max) *mobil <sup>C</sup>*(exp) *mobil*

*mobil* is the strength of the bone subjected

*f unct* are known for a compact

*axial* ≈ 200 *MPa* employed by

*axial* = 220 *MPa* (13)

� 1 0.08

(*mean*)

⎞ ⎠

*bend* ≈ 51 *MPa* and

= 71 *tons*. (14)

1 0.08

, (12)

*mobil* is slightly

*slenderness exponent <sup>λ</sup>*(*pred*) *limb* = *l* (exp) *limb* /*d* (exp) *limb* shown in Eq. (6) match well those established from a wider spectrum of extant mammals (189 species, 612 specimens, see Table 3 below). The theory of stress similarity, arising from the more general mechanical elastic similarity, provides three patterns of elastic similarity functional stresses discussed in Eq. (5), namely,

$$
\sigma\_{func}^{(\max)}(M) = \mathbb{C}\_{func}^{(\max)} M^{\mu},\tag{7}
$$

establishing the stress similarity through the corresponding scaling exponents

$$
\mu\_{\text{bend}} = \mu\_{\text{tors}} = d - l,\\
\text{with } d + l = \frac{2}{3},\\
\text{and } \mu\_{\text{axial}} = -\frac{2}{3}\mu\_{\text{bend}}.\tag{8}
$$

in limbs of adult, or large species of different-sized animals (Kokshenev, 2007, Fig. 2). Leaving for a while the scaling parameters *C*(max) *f unc* in Eq. (7), requiring a special consideration beyond the scaling theory, we stress that the bone-diameter *d* and the bone-length *l* scaling exponents are also external parameters of the dynamic similarity theory.

When the most simplest isometric similarity (*d* (*isom*) <sup>0</sup> = *l* (*isom*) <sup>0</sup> = 1/3) is adopted for the limb design, evidently excluding bone curvature effects in bone stress (e.g., Bertram & Biewener, 1992), the only one type of functional stress associated with the oversimplified *body support function* can be distinguished through the weight-independent isometrically universal stresses

$$
\sigma\_{bend}^{(\text{isom})} \sim \sigma\_{\text{tors}}^{(\text{isom})} \sim \sigma\_{\text{axial}}^{(\text{isom})} \propto M^0,\tag{9}
$$

straightforwardly following from Eqs. (7) and (8). In contrast, when the mammalian-limb realistic design discussed in Eq. (6) is adopted, the scaling rules for *stress similarity functions* determined by the scaling exponents

$$
\mu\_{bend}^{(pred)} = \mu\_{tors}^{(pred)} = 0.08 \text{ and } \mu\_{axial}^{(pred)} = -0.05 \tag{10}
$$

become well theoretically observable through the axial-bone and bending-bone stress similarity discussed in Eq. (2). Thereby, it has been repeatedly demonstrated that the dynamic strain similarity experimentally established in limbs of fast running animals (Rubin & Lanyon, 1982, 1984) is grounded by the mechanical elastic similarity established for effective limb bones (Kokshenev, 2007). In other words, we have shown that the effective limb bone from running mammals shown in Eq. (6) is designed as adapted to the primary locomotory functions, presented by the *body mobility* and support functions patterned in Eq. (10).

The experimental data by Rubin & Lanyon (1982) for the axial and bending stress discussed in Eqs. (1) and (2) have been obtained at special experimental conditions of a special choice of the local-midshaft coordinate systems consequently excluding shear strains. In the context of the limb stress similarity suggested in Eqs. (5) and (7) these conditions are viewed as to be conventionally introduced by the two sets of stress indicators (*α*, *β*, *τ*) presented by (1, 0, 0) and (0, 1, 0), respectively. Restoring the torsional stress *<sup>σ</sup>*(max) *tors* in all stress-similarity equations through the corresponding scaling exponent in Eq. (10), we also adopt in Eq. (5) the scaling factors

$$\mathbb{C}^{(\text{max})}\_{\text{bend}} = \mathbb{C}^{(\text{max})}\_{\text{tors}} = \mathbb{C}^{(\text{exp})}\_{\text{mobil}} = 27 MPa \cdot (\text{kg})^{-0.08} \text{ and } \mathbb{C}^{(\text{max})}\_{\text{axial}} = \mathbb{C}^{(\text{exp})}\_{\text{sup}} = 11 MPa \cdot (\text{kg})^{0.05}. \tag{11}$$

When compared with Eq. (3), the experimental data *<sup>C</sup>*(exp) *supp* for the body support function is adopted without change, whereas the data for the body mobility function *<sup>C</sup>*(exp) *mobil* is slightly modified by the statistical analysis extended by *in vitro* data on the tibial failure in torsion (Kokshenev, 2007, Fig. 1).

### **2.2.2 Critical body mass**

6 Will-be-set-by-IN-TECH

from a wider spectrum of extant mammals (189 species, 612 specimens, see Table 3 below). The theory of stress similarity, arising from the more general mechanical elastic similarity, provides three patterns of elastic similarity functional stresses discussed in Eq. (5), namely,

*f unc* (*M*) = *<sup>C</sup>*(max)

in limbs of adult, or large species of different-sized animals (Kokshenev, 2007, Fig. 2). Leaving

the scaling theory, we stress that the bone-diameter *d* and the bone-length *l* scaling exponents

design, evidently excluding bone curvature effects in bone stress (e.g., Bertram & Biewener, 1992), the only one type of functional stress associated with the oversimplified *body support function* can be distinguished through the weight-independent isometrically universal stresses

*tors* <sup>∼</sup> *<sup>σ</sup>*(*isom*)

straightforwardly following from Eqs. (7) and (8). In contrast, when the mammalian-limb realistic design discussed in Eq. (6) is adopted, the scaling rules for *stress similarity functions*

*tors* <sup>=</sup> 0.08 and *<sup>μ</sup>*(*pred*)

become well theoretically observable through the axial-bone and bending-bone stress similarity discussed in Eq. (2). Thereby, it has been repeatedly demonstrated that the dynamic strain similarity experimentally established in limbs of fast running animals (Rubin & Lanyon, 1982, 1984) is grounded by the mechanical elastic similarity established for effective limb bones (Kokshenev, 2007). In other words, we have shown that the effective limb bone from running mammals shown in Eq. (6) is designed as adapted to the primary locomotory

The experimental data by Rubin & Lanyon (1982) for the axial and bending stress discussed in Eqs. (1) and (2) have been obtained at special experimental conditions of a special choice of the local-midshaft coordinate systems consequently excluding shear strains. In the context of the limb stress similarity suggested in Eqs. (5) and (7) these conditions are viewed as to be conventionally introduced by the two sets of stress indicators (*α*, *β*, *τ*) presented by (1, 0, 0)

through the corresponding scaling exponent in Eq. (10), we also adopt in Eq. (5) the scaling

functions, presented by the *body mobility* and support functions patterned in Eq. (10).

*mobil* <sup>=</sup> <sup>27</sup>*MPa* · (*kg*)−0.08 and *<sup>C</sup>*(max)

(*isom*) <sup>0</sup> = *l*

3

*limb* shown in Eq. (6) match well those established

, and *<sup>μ</sup>axial* <sup>=</sup> <sup>−</sup><sup>2</sup>

(*isom*)

*f unc* in Eq. (7), requiring a special consideration beyond

*f unc <sup>M</sup>μ*, (7)

3

<sup>0</sup> = 1/3) is adopted for the limb

*axial* <sup>∝</sup> *<sup>M</sup>*0, (9)

*axial* = −0.05 (10)

*tors* in all stress-similarity equations

*supp* <sup>=</sup> <sup>11</sup>*MPa* · (*kg*)0.05. (11)

*axial* <sup>=</sup> *<sup>C</sup>*(exp)

*μbend*, (8)

(exp)

*slenderness exponent <sup>λ</sup>*(*pred*)

*limb* = *l*

for a while the scaling parameters *C*(max)

determined by the scaling exponents

factors

*C*(max)

*bend* <sup>=</sup> *<sup>C</sup>*(max)

*tors* <sup>=</sup> *<sup>C</sup>*(exp)

When the most simplest isometric similarity (*d*

(exp) *limb* /*d*

*<sup>σ</sup>*(max)

*<sup>μ</sup>bend* <sup>=</sup> *<sup>μ</sup>tors* <sup>=</sup> *<sup>d</sup>* <sup>−</sup> *<sup>l</sup>*, with *<sup>d</sup>* <sup>+</sup> *<sup>l</sup>* <sup>=</sup> <sup>2</sup>

are also external parameters of the dynamic similarity theory.

*<sup>σ</sup>*(*isom*)

*bend* <sup>=</sup> *<sup>μ</sup>*(*pred*)

*<sup>μ</sup>*(*pred*)

and (0, 1, 0), respectively. Restoring the torsional stress *<sup>σ</sup>*(max)

*bend* <sup>∼</sup> *<sup>σ</sup>*(*isom*)

establishing the stress similarity through the corresponding scaling exponents

Taking into consideration that instead of the axial-bone stress the bending and torsional functional stresses limit the body mobility of animals involved in strenuous activity, the upper limit of body mass, or the *critical body mass M*(*crit*) max , follows from the relevant functional-stress constraint

$$\sigma\_{mobil}^{(\max)}(M) = \mathcal{C}\_{mobil}^{(\exp)}M^{0.08} = \operatorname\*{s}\_{mobil}^{(\max)} \left(\frac{M}{M\_{\max}^{(crit)}}\right)^{0.08}, \text{ providing } M\_{\max}^{(crit)} = \left(\frac{\operatorname\*{s}\_{mobil}^{(\max)}}{\mathcal{C}\_{mobil}^{(\exp)}}\right)^{\frac{1}{0.08}},\tag{12}$$

introduced on the basis of Eqs. (5) and (10). Here *s* (max) *mobil* is the strength of the bone subjected to complex bending-torsion loading and *<sup>C</sup>*(exp) *mobil* is shown in Eq. (11).

The *in vitro* data on the mean individual-bone strength *s* (*mean*) *f unct* are known for a compact long bones damaged under pure axial compression, i.e., *s* (*mean*) *axial* ≈ 200 *MPa* employed by Alexander (1981) and Biewener (1990), pure bending and torsion, i.e., *s* (*mean*) *bend* ≈ 51 *MPa* and *s* (*mean*) *tors* ≈ 60 *MPa* reported by Taylor et al. (2003). Given that generally the ±10 % domain is adopted for the statistically established mean bone strengths, the maximal amplitude data

$$s\_{bend}^{(\text{max})} = 56.1 \text{ MPa}, s\_{tors}^{(\text{max})} = 66 \text{ MPa}, \text{and} \ s\_{axial}^{(\text{max})} = 220 \text{ MPa} \tag{13}$$

result in limiting estimates for the critical masses (Kokshenev, 2007)

$$M\_{bend}^{(crit)} = \left(\frac{s\_{bend}^{(max)}}{\mathcal{C}\_{bend}^{(max)}}\right)^{\text{dir}} = 9.3 \text{ tons} \text{ and } M\_{lors}^{(crit)} = \left(\frac{s\_{lors}^{(max)}}{\mathcal{C}\_{lors}^{(max)}}\right)^{\text{dir}} = 71 \text{ tons}.\tag{14}$$

These two critical body masses determine two kinds of similarity patterns of well distinct *body postures* in giants, whose body mobility is supposedly limited by critical pure bending and pure torsional stresses emerging in their limbs at maximal speeds of critical locomotion. Using empirical data in Table 1, the patterns of *most erect posture* and *most sprawling posture* in giants, may be respectively presented by the obligate biped *Giganotosaurus*, likely capable for relatively fast running and the largest terrestrial quadruped *Argentinosaurus*, an animal so huge that a true run would likely have severely injured the limb bones via critical torsional stresses. Indeed, sauropod anatomy speaks vehemently against anything other than a walk gait anyway.

### **2.2.3 Body safety**

Bearing in mind that the limb safety during strenuous activity of terrestrial animals establishes the risk level of skeleton damage (Alexander, 1981; Biewener, 1982, 1989, 1990), a generalized

as follows from Eq. (16) taken at the critical condition *S*(*crit*)

bone stress (Kokshenev, 2007).

**3. Results**

**3.1 Critical mass**

(17) are estimated through

the critical condition *S*(*crit*)

determined by the critical mass

*M*(*crit*)

*M*(*crit*)

*<sup>μ</sup>*(*pred*) *axial* <sup>=</sup> <sup>−</sup><sup>4</sup>

9 1 − *λ*

*mobil*(*β*) = 137.1

*mobil*(*β*) = 137.1

The domain shown in Eq. (18) for the body mobility function conventionally indicates the condition for realization of the near-critical transient states of continuous dynamic similarity, associated with locally equilibrated universal transformations of near-critical compressive strains into non-critical global shear strains, eventually decreasing the resulted compressive

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 57

Several aspects concerning the problem of evolution of locomotory trends in extinct animals are discussed in view of generalizations suggested by the dynamic stress similarity approach.

In order to establish the role of competing axial, bending, and torsional stresses in the problem of the maximal mass of terrestrial giants, we analyze the domains of observation of the

is associated with maximal body mass *<sup>M</sup>*(exp) max known for the largest giants (Table 1). Since in most cases of bone specimens for extinct species the allometric exponents *d* and *l* are not available, we employ statistical data on the slenderness exponent *λ*(exp) for the effective limb bone (Table 2). Instead of Eq. (10), the theoretical predictions for the scaling exponents in Eq.

corresponding probabilities *<sup>α</sup>*, *<sup>β</sup>*, and *<sup>τ</sup>* through Eq. (17) where the critical mass *<sup>M</sup>*(*crit*)

<sup>1</sup> <sup>+</sup> *<sup>λ</sup>* and *<sup>μ</sup>*(*pred*)

hereafter designated as [0;1], the resulted observable domains are shown in Table 3.

established with the help of the definitive equation for the exponent *λ* = *l*/*d* excluding *d* and *l* from those shown in Eq. (8). Allowing the whole domains for all kinds of stresses to be exploited by giants during near-critical locomotion, e.g., for bending stress 0 ≤ *β* ≤ 1,

Fig. 1 illustrates the analysis in Table 3 indicating domains of exploration of the axial, bending and torsional stresses by limb bones in the largest mammalian species running or walking at

critical mass nor maximal near-critical mass of terrestrial giants is affected by axial stress. In other words, instead of body safety function lying beyond the critical locomotion domain, as shown in Eq. (16), the body mobility function establishes the critical behavior (see Fig. 2)

(<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)56.1 <sup>+</sup> *<sup>β</sup>*<sup>66</sup> <sup>1</sup>

represented from Eq. (19) in the mammalian limb-bone approximation discussed in Eq. (6).

(<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)56.1 <sup>+</sup> *<sup>β</sup>*<sup>66</sup> <sup>1</sup>

<sup>1</sup> The provided analysis is treated as semi-qualitative since, unlike the scaling exponents (Table 3), the data on the scaling factors, discussed in Eq. (10), are still taken in the mammalian-limb approximation.

*bend* <sup>=</sup> *<sup>μ</sup>*(*pred*)

*tors* <sup>=</sup> <sup>2</sup> 3 1 − *λ*

*body* <sup>=</sup> 1. The semi-qualitative analysis<sup>1</sup> indicates that neither maximal

0.08

0.08

*mobil*(*M*, *V*max) = 1, with *α* = 0.

*body* (*α*, *β*, *τ*)

<sup>1</sup> <sup>+</sup> *<sup>λ</sup>* (20)

, with 0 ≤ *β* ≤ 1, (21)

, with 0 ≤ *β* ≤ 1, (22)

body-skeleton *safety function Sbody*(*M*, *V*) based on the additivity of functional stresses *<sup>σ</sup>*(max) *f unct* (*M*, *V*) could be suggested as

$$\mathcal{S}\_{body}(M, V) = \left(\sum\_{f:\text{functions}} \frac{\sigma\_{funct}^{(\text{max})}(M, V)}{s\_{funct}^{(\text{max})}}\right)^{-1}.\tag{15}$$

The dynamic stress similarity experimentally and theoretically represented in, respectively, Eqs. (1), (4), (11) and (5), (7), (8), suggests the following piecewise functional form, namely

$$S\_{body}^{-1}(M, V) = \frac{\alpha}{s\_{axial}^{(\text{max})}} \mathcal{C}\_{supp}^{(\text{exp})} M^{\mu\_{\text{axial}}} + \left(\frac{\beta}{s\_{bedo}^{(\text{max})}} + \frac{\tau}{s\_{tors}^{(\text{max})}}\right) \mathcal{C}\_{molil}^{(\text{exp})} M^{\mu\_{\text{fell}}} \frac{V}{V\_{\text{max}}}, \text{ with } S\_{body} \ge 2,\tag{16}$$

for the body safety function activated in slow (walk) and fast (walk and run) gaits of animals moving within a certain speed domain limited by *V*max. The shown domain (analyzed in Figs. 1 and 3 in Kokshenev, 2007) conventionally establishes a condition for realization of the mechanically equilibrated, *non-critical* dynamic similarity states in slow and moderately fast gaits of locomotion, as exemplified by the non-critical axial stress (Kokshenev, 2007, Table 2, Eq. (2)).

Coming back to the problem of the maximal mass of terrestrial giants mentioned in the Introduction, a question arises whether the body support function may establish the critical body mass *M*(*crit*) *body* determined by critical-point condition *<sup>S</sup>*(*crit*) *body* (*M*(*crit*) *body* , *V*max) = 1, which in fact violate the domain of validation shown in Eq. (16)? Searching for the answer, let analyze the relevant model solution *M*(*crit*) *body* (*α*, *β*, *τ*) provided by

$$\frac{\mathfrak{a}}{s\_{\text{axial}}^{(\text{max})}} \mathbf{C}\_{\text{supp}}^{(\text{exp})} M\_{\text{body}}^{\mu\_{\text{axial}}} + \left( \frac{\mathfrak{f}}{s\_{\text{bend}}^{(\text{max})}} + \frac{\tau}{s\_{\text{tors}}^{(\text{exp})}} \right) \mathbf{C}\_{\text{model}}^{(\text{exp})} M\_{\text{body}}^{\mu\_{\text{bend}}} = 1, \text{ with } a + \boldsymbol{\beta} + \tau = 1,\tag{17}$$

following from Eq. (16). In Eq. (14), the two particular solutions *M*(*crit*) *body* (0, 1, 0) = 9.3 *tons* and *M*(*crit*) *body* (0, 0, 1) = 71 *tons* were discussed as critical masses attributed, respectively, to the largest theropod and the largest sauropod, obtained in the effective limb-bone approximation (shown in Eqs. (10) and (11)). One can see that the obtained limiting masses introduce the validation domain for the *body mobility function* in giants, namely

$$S\_{mobil}(M, V) = \left(\frac{M\_{\text{max}}^{(crit)}}{M}\right)^{\mu\_{\text{leul}}} \frac{V\_{\text{max}}}{V}, \text{ with } S\_{mobil} \ge 1. \tag{18}$$

The critical mass discussed in Eq. (12) is now specified by

$$M\_{\text{max}}^{(crit)}(\beta) = \left(\frac{\mathcal{C}\_{mobil}^{(\text{exp})-1} s\_{bend}^{(\text{max})} s\_{tors}^{(\text{max})}}{\beta s\_{tors}^{(\text{max})} + \tau s\_{bend}^{(\text{max})}}\right)^{\mu\_{bend}^{-1}}, \text{with } \tau = 1 - \beta,\tag{19}$$

as follows from Eq. (16) taken at the critical condition *S*(*crit*) *mobil*(*M*, *V*max) = 1, with *α* = 0. The domain shown in Eq. (18) for the body mobility function conventionally indicates the condition for realization of the near-critical transient states of continuous dynamic similarity, associated with locally equilibrated universal transformations of near-critical compressive strains into non-critical global shear strains, eventually decreasing the resulted compressive bone stress (Kokshenev, 2007).

### **3. Results**

8 Will-be-set-by-IN-TECH

body-skeleton *safety function Sbody*(*M*, *V*) based on the additivity of functional stresses

The dynamic stress similarity experimentally and theoretically represented in, respectively, Eqs. (1), (4), (11) and (5), (7), (8), suggests the following piecewise functional form, namely

+

for the body safety function activated in slow (walk) and fast (walk and run) gaits of animals moving within a certain speed domain limited by *V*max. The shown domain (analyzed in Figs. 1 and 3 in Kokshenev, 2007) conventionally establishes a condition for realization of the mechanically equilibrated, *non-critical* dynamic similarity states in slow and moderately fast gaits of locomotion, as exemplified by the non-critical axial stress (Kokshenev, 2007, Table 2,

Coming back to the problem of the maximal mass of terrestrial giants mentioned in the Introduction, a question arises whether the body support function may establish the critical

fact violate the domain of validation shown in Eq. (16)? Searching for the answer, let analyze

⎞ <sup>⎠</sup> *<sup>C</sup>*(exp)

*mobil <sup>M</sup>μbend*

*V*max

⎞ ⎠ *μ*−<sup>1</sup> *bend*

(max) *tors*

(max) *bend*

*body* (0, 0, 1) = 71 *tons* were discussed as critical masses attributed, respectively, to the largest theropod and the largest sauropod, obtained in the effective limb-bone approximation (shown in Eqs. (10) and (11)). One can see that the obtained limiting masses introduce the

�*μbend*

(max) *bend s*

*body* (*α*, *β*, *τ*) provided by

*τ s* (max) *tors*

*<sup>σ</sup>*(max) *f unct* (*M*, *V*)

*τ s* (max) *tors*

⎞ <sup>⎠</sup> *<sup>C</sup>*(exp)

*s* (max) *f unct*

⎞ ⎠ −1

*mobil <sup>M</sup>μbend <sup>V</sup>*

*body* (*M*(*crit*)

*V*max

. (15)

, with *Sbody* ≥ 2,

*body* , *V*max) = 1, which in

*body* (0, 1, 0) = 9.3 *tons*

*body* = 1, with *α* + *β* + *τ* = 1, (17)

*<sup>V</sup>* , with *Smobil* <sup>≥</sup> 1. (18)

, with *τ* = 1 − *β*, (19)

(16)

⎛ <sup>⎝</sup> ∑ *f unctions*

⎛ ⎝

*body* determined by critical-point condition *<sup>S</sup>*(*crit*)

*β s* (max) *bend*

*<sup>σ</sup>*(max)

*S*−<sup>1</sup>

Eq. (2)).

body mass *M*(*crit*)

*α s* (max) *axial*

and *M*(*crit*)

*body*(*M*, *<sup>V</sup>*) = *<sup>α</sup>*

*s* (max) *axial*

the relevant model solution *M*(*crit*)

*<sup>C</sup>*(exp) *supp <sup>M</sup>μaxial body* +

*<sup>C</sup>*(exp)

⎛ ⎝

*β s* (max) *bend*

validation domain for the *body mobility function* in giants, namely

*Smobil*(*M*, *V*) =

*<sup>M</sup>*(*crit*) max (*β*) =

The critical mass discussed in Eq. (12) is now specified by

⎛ ⎝ +

following from Eq. (16). In Eq. (14), the two particular solutions *M*(*crit*)

� *<sup>M</sup>*(*crit*) max *M*

*C*(exp)−<sup>1</sup> *mobil s*

> *βs* (max) *tors* + *τs*

*f unct* (*M*, *V*) could be suggested as

*Sbody*(*M*, *V*) =

*supp Mμaxial* +

Several aspects concerning the problem of evolution of locomotory trends in extinct animals are discussed in view of generalizations suggested by the dynamic stress similarity approach.

### **3.1 Critical mass**

In order to establish the role of competing axial, bending, and torsional stresses in the problem of the maximal mass of terrestrial giants, we analyze the domains of observation of the corresponding probabilities *<sup>α</sup>*, *<sup>β</sup>*, and *<sup>τ</sup>* through Eq. (17) where the critical mass *<sup>M</sup>*(*crit*) *body* (*α*, *β*, *τ*) is associated with maximal body mass *<sup>M</sup>*(exp) max known for the largest giants (Table 1). Since in most cases of bone specimens for extinct species the allometric exponents *d* and *l* are not available, we employ statistical data on the slenderness exponent *λ*(exp) for the effective limb bone (Table 2). Instead of Eq. (10), the theoretical predictions for the scaling exponents in Eq. (17) are estimated through

$$
\mu\_{\text{axial}}^{(pred)} = -\frac{4}{9} \frac{1-\lambda}{1+\lambda} \text{ and } \mu\_{\text{bend}}^{(pred)} = \mu\_{\text{tors}}^{(pred)} = \frac{2}{3} \frac{1-\lambda}{1+\lambda} \tag{20}
$$

established with the help of the definitive equation for the exponent *λ* = *l*/*d* excluding *d* and *l* from those shown in Eq. (8). Allowing the whole domains for all kinds of stresses to be exploited by giants during near-critical locomotion, e.g., for bending stress 0 ≤ *β* ≤ 1, hereafter designated as [0;1], the resulted observable domains are shown in Table 3.

Fig. 1 illustrates the analysis in Table 3 indicating domains of exploration of the axial, bending and torsional stresses by limb bones in the largest mammalian species running or walking at the critical condition *S*(*crit*) *body* <sup>=</sup> 1. The semi-qualitative analysis<sup>1</sup> indicates that neither maximal critical mass nor maximal near-critical mass of terrestrial giants is affected by axial stress. In other words, instead of body safety function lying beyond the critical locomotion domain, as shown in Eq. (16), the body mobility function establishes the critical behavior (see Fig. 2) determined by the critical mass

$$M\_{mobil}^{(crit)}(\beta) = \left[\frac{137.1}{(1-\beta)56.1+\beta66}\right]^{\frac{1}{608}}, \text{ with } 0 \le \beta \le 1,\tag{21}$$

represented from Eq. (19) in the mammalian limb-bone approximation discussed in Eq. (6).

$$M\_{mobil}^{(crit)}(\beta) = \left[\frac{137.1}{(1-\beta)56.1+\beta66}\right]^{\frac{1}{0.08}}, \text{ with } 0 \le \beta \le 1,\tag{22}$$

<sup>1</sup> The provided analysis is treated as semi-qualitative since, unlike the scaling exponents (Table 3), the data on the scaling factors, discussed in Eq. (10), are still taken in the mammalian-limb approximation.

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 0.5

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 59

0 0.2 0.4 0.6 0.8 1

Fig. 1. The comparative role of limb adaptation in the largest mammal and elephant (extinct species) through the elastic reaction to muscle loading forces (via turning-bending stress, Fig. 1A) and ground-reaction forces (mostly via axial stress, Fig. 1B). The *solid* and *dashed lines*

the gravitational field) as a relative squared-speed characteristics of animal locomotion (Alexander & Jayes, 1983; Gatesy& Biewener, 1991), one obtains the scaling relations

dynamic similarities in a walk and a run gaits in giants, moving in the near critical mobility regime. When the walk-to-run continuous transition, *in vivo* established for modern elephants

is mapped onto the walk-to-run crossover in evolution of trends in extinct largest giants indicated by transient bending-torsion stress similarity at *β* = 0.5, the model similarity scaling equations for near critical level of walking and running quadrupedal giants are suggested as

*<sup>b</sup>* <sup>∼</sup> *<sup>L</sup>*(*isom*) *<sup>m</sup>* <sup>∼</sup> *<sup>M</sup>*1/3

E

*critic* <sup>=</sup> *<sup>V</sup>*<sup>2</sup>

*<sup>b</sup>* , adopted for model Froude numbers in the simplest

*<sup>w</sup>*−*<sup>r</sup>* <sup>≈</sup> 1 and duty factors *<sup>β</sup>*(exp)

*walk* <sup>∼</sup> *<sup>M</sup>*−<sup>1</sup> max(*β*) and *Fr*(mod) *run* <sup>∼</sup> *<sup>M</sup>*−2/3 max (*β*) distinguishing

*walk* <sup>=</sup> 24.7*M*−<sup>1</sup> max(*β*) and *Fr*(*quadr*) *run* <sup>=</sup> 8.48*M*−2/3 max (*β*), (23)

*crit*/*gLb*, where *g* is

*duty* ≈ 0.5

max(*β*). Hence, one obtaines the

Bending stress indicator,

,

0.3

0

*<sup>b</sup>* and *Fr*(mod) *run* <sup>∼</sup> *<sup>L</sup>*−<sup>2</sup>

by Hutchinson et al. (2006) at Froude numbers *Fr*(exp)

(here critical mass is taken in *tons*) with the help of Eq.(22).

isometric geometry approximation *L*(*isom*)

*Fr*(*quadr*)

critical-mass scaling rules *Fr*(mod)

*Fr*(mod)

*walk* <sup>∼</sup> *<sup>L</sup>*−<sup>3</sup>

0.2

B

correspond to employing and non-employing domains (Table 3).

Choosing the *model* critical-state *Froude number* (*Fr*(mod)

0.4

0.6

0.8

1

A

0.1

0.1

0.3

0.5


Table 3. Maximal body mass and limb bone design determine the stress-similarity domains through the corresponding stress indicators. The domains of the axial (*α*), bending (*β*), and torsion (*τ*) stresses explored by the body safety function (shown in squared brackets) are obtained through Eq. (17) where the scaling exponents *<sup>μ</sup>*(*pred*) *axial* and *<sup>μ</sup>*(*pred*) *bend* <sup>=</sup>*μ*(*pred*) *tors* are

estimated through Eq. (20). The data on the critical mass *<sup>M</sup>*(exp) max for *Mammuthus trogontherii* is used for the largest taxon in both Mammalia, Proboscidea, and Elephantidae (Table 1). The largest masses for extant Asian (*Elephas maximus*) and African (*Loxodonta africana*) elephants are for exceptionally large bulls (Wood, 1976; Nowak, 1991; McFarlan, 1992; Blashford-Snell & Lenska, 1996). The LS regression data on the slenderness exponents *λ*(exp) of long bones from *N* species are taken from Table 2 and from *n* specimens for elephants are taken from Table 2 in Kokshenev & Christiansen (2010). *Notation*: *n.e.s.* - not employing stress.


Table 4. Dynamic characterization of the slow and fast striated muscles (of resting length *Lm*) contracting at optimum-speed (maximum-efficiency), maximum-speed (maximum-power), and near critical (low-safety) dynamic conditions (reproduced from Table 4 in Kokshenev, 2009).

represented from Eq. (19) in the mammalian limb-bone approximation discussed in Eq. (6). In Fig. 3 we extend the analysis of locomotory trends in Fig. 2 to other large giants, modeled by near critical safety factors lying in narrow domain 1.0 < *Smobil* ≤ 1.1, corresponding to the 10% domain established for the mean bone strengths, as discussed in Eq. (13).

### **3.2 Froude numbers**

Modeling the animal locomotion by contractions of locomotory muscles activated at natural frequencies, the data on optimal speeds *<sup>V</sup>*(exp) *opt* ∼ *<sup>M</sup>*1/6 of bipeds (Gatesy& Biewener, 1991) and quadrupeds (Heglund & Taylor, 1988) running in fast locomotor modes can be explained by activation of the fast locomotor muscles of length, linearly scaling with *limb length* (or hip height) *L*(*isom*) *<sup>b</sup>* <sup>∼</sup> *<sup>M</sup>*1/3 (Kokshenev, 2009, 2010) presented in the isometric-similarity approximation discussed in Eq. (9). Modeling the low-safety locomotion of large giants by critical-velocity muscle contractions (see Table 4), let us adopt the scaling relations *V*(mod) *walk* ∼ *L*−<sup>1</sup> *<sup>b</sup>* and *<sup>V</sup>*(mod) *run* <sup>∝</sup> *<sup>L</sup>*−1/2 *<sup>b</sup>* for critically walking or running giants moving due to activation of slow and fast limb muscles, respectively.

10 Will-be-set-by-IN-TECH

Mammals *N* = 189 0.795 -0.051 0.076 20000 [0; 0.03] [0.85; 0] [0.15; 0.97] Proboscidean *N* = 13 0.943 -0.013 0.020 20000 *n.e.s.* [1; 0.95] [0.95; 1] Elephantidae *N* = 6 0.790 -0.520 0.078 20000 *n.e.s.* [1; 0.30] [0.40; 1] Asian elephant *n* = 106 0.851 -0.036 0.054 8000 *n.e.s.* [1; 0.35] [0.45; 1] African elephant *n* = 56 0.648 -0.094 0.141 10000 [0.35; 0.45] [0.65; 0] [0; 0.55] Table 3. Maximal body mass and limb bone design determine the stress-similarity domains through the corresponding stress indicators. The domains of the axial (*α*), bending (*β*), and torsion (*τ*) stresses explored by the body safety function (shown in squared brackets) are

estimated through Eq. (20). The data on the critical mass *<sup>M</sup>*(exp) max for *Mammuthus trogontherii* is used for the largest taxon in both Mammalia, Proboscidea, and Elephantidae (Table 1). The largest masses for extant Asian (*Elephas maximus*) and African (*Loxodonta africana*) elephants are for exceptionally large bulls (Wood, 1976; Nowak, 1991; McFarlan, 1992; Blashford-Snell & Lenska, 1996). The LS regression data on the slenderness exponents *λ*(exp) of long bones from *N* species are taken from Table 2 and from *n* specimens for elephants are taken from Table 2 in Kokshenev & Christiansen (2010). *Notation*: *n.e.s.* - not employing stress.

Dynamic conditions Optimum speed Maximum power Near critical regime Locomotory muscle type slow fast slow fast slow fast Contraction frequency *<sup>L</sup>*−<sup>1</sup> *<sup>m</sup> <sup>L</sup>*−1/2 *<sup>m</sup> <sup>L</sup>*−3/2 *<sup>m</sup> <sup>L</sup>*−<sup>1</sup> *<sup>m</sup> <sup>L</sup>*−<sup>2</sup> *<sup>m</sup> <sup>L</sup>*−3/2 *<sup>m</sup>*

Table 4. Dynamic characterization of the slow and fast striated muscles (of resting length *Lm*) contracting at optimum-speed (maximum-efficiency), maximum-speed (maximum-power), and near critical (low-safety) dynamic conditions (reproduced from Table 4 in Kokshenev,

represented from Eq. (19) in the mammalian limb-bone approximation discussed in Eq. (6). In Fig. 3 we extend the analysis of locomotory trends in Fig. 2 to other large giants, modeled by near critical safety factors lying in narrow domain 1.0 < *Smobil* ≤ 1.1, corresponding to the

Modeling the animal locomotion by contractions of locomotory muscles activated at natural

and quadrupeds (Heglund & Taylor, 1988) running in fast locomotor modes can be explained by activation of the fast locomotor muscles of length, linearly scaling with *limb length* (or

approximation discussed in Eq. (9). Modeling the low-safety locomotion of large giants by critical-velocity muscle contractions (see Table 4), let us adopt the scaling relations *V*(mod)

*<sup>b</sup>* <sup>∼</sup> *<sup>M</sup>*1/3 (Kokshenev, 2009, 2010) presented in the isometric-similarity

*<sup>b</sup>* for critically walking or running giants moving due to activation of

*<sup>m</sup> <sup>L</sup>*−1/2 *<sup>m</sup> <sup>L</sup>*<sup>0</sup>

*<sup>m</sup> <sup>L</sup>*1/2

10% domain established for the mean bone strengths, as discussed in Eq. (13).

*bend <sup>M</sup>*(exp) max , *kg <sup>α</sup> <sup>τ</sup> <sup>β</sup>*

*axial* and *<sup>μ</sup>*(*pred*)

*bend* <sup>=</sup>*μ*(*pred*)

*<sup>m</sup> <sup>L</sup>*−<sup>1</sup> *<sup>m</sup> <sup>L</sup>*−1/2 *<sup>m</sup>*

*opt* ∼ *<sup>M</sup>*1/6 of bipeds (Gatesy& Biewener, 1991)

*walk* ∼

*tors* are

*axial <sup>μ</sup>*(*pred*)

Largest giants in *<sup>N</sup>*, *<sup>n</sup> <sup>λ</sup>*(exp) *<sup>μ</sup>*(*pred*)

Contraction velocity *L*<sup>0</sup>

frequencies, the data on optimal speeds *<sup>V</sup>*(exp)

2009).

**3.2 Froude numbers**

hip height) *L*(*isom*)

*<sup>b</sup>* and *<sup>V</sup>*(mod) *run* <sup>∝</sup> *<sup>L</sup>*−1/2

slow and fast limb muscles, respectively.

*L*−<sup>1</sup>

obtained through Eq. (17) where the scaling exponents *<sup>μ</sup>*(*pred*)

Fig. 1. The comparative role of limb adaptation in the largest mammal and elephant (extinct species) through the elastic reaction to muscle loading forces (via turning-bending stress, Fig. 1A) and ground-reaction forces (mostly via axial stress, Fig. 1B). The *solid* and *dashed lines* correspond to employing and non-employing domains (Table 3).

Choosing the *model* critical-state *Froude number* (*Fr*(mod) *critic* <sup>=</sup> *<sup>V</sup>*<sup>2</sup> *crit*/*gLb*, where *g* is the gravitational field) as a relative squared-speed characteristics of animal locomotion (Alexander & Jayes, 1983; Gatesy& Biewener, 1991), one obtains the scaling relations *Fr*(mod) *walk* <sup>∼</sup> *<sup>L</sup>*−<sup>3</sup> *<sup>b</sup>* and *Fr*(mod) *run* <sup>∼</sup> *<sup>L</sup>*−<sup>2</sup> *<sup>b</sup>* , adopted for model Froude numbers in the simplest isometric geometry approximation *L*(*isom*) *<sup>b</sup>* <sup>∼</sup> *<sup>L</sup>*(*isom*) *<sup>m</sup>* <sup>∼</sup> *<sup>M</sup>*1/3 max(*β*). Hence, one obtaines the critical-mass scaling rules *Fr*(mod) *walk* <sup>∼</sup> *<sup>M</sup>*−<sup>1</sup> max(*β*) and *Fr*(mod) *run* <sup>∼</sup> *<sup>M</sup>*−2/3 max (*β*) distinguishing dynamic similarities in a walk and a run gaits in giants, moving in the near critical mobility regime. When the walk-to-run continuous transition, *in vivo* established for modern elephants by Hutchinson et al. (2006) at Froude numbers *Fr*(exp) *<sup>w</sup>*−*<sup>r</sup>* <sup>≈</sup> 1 and duty factors *<sup>β</sup>*(exp) *duty* ≈ 0.5 is mapped onto the walk-to-run crossover in evolution of trends in extinct largest giants indicated by transient bending-torsion stress similarity at *β* = 0.5, the model similarity scaling equations for near critical level of walking and running quadrupedal giants are suggested as

$$Fr\_{walk}^{(quadr)} = 24.7 M\_{\text{max}}^{-1}(\beta) \text{ and } Fr\_{run}^{(quadr)} = 8.48 M\_{\text{max}}^{-2/3}(\beta), \tag{23}$$

(here critical mass is taken in *tons*) with the help of Eq.(22).

1 1.02 1.04 1.06 1.08 1.1

*F*(mod) *mobil*

*Apatosaurus*

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 61

*Elephas recki*

*Tarbosaurus*

*Rebbachisaurus Barosaurus*

*Edmontosaurus*

*Loxodonta africana Elephas antiquus*

*Diplodocus*

*Mammuthus columbi*

*Daspletosaurus*

*Olorotitan*

*Chubutisaurus*

*Elephas maximus*

*Mg* <sup>=</sup> 2.08 · *<sup>M</sup>*<sup>−</sup>0.20, with *Smobil* <sup>≥</sup> 1,

(24)

*Mapusaurus*

Near critical safety factor, *Smobil*

Fig. 3. Maximal body masses for terrestrial giants modeled by the near-critical locomotor behavior. *Solid lines* are drawn by Eq.(18) taken at maximal stride speeds. The *points* are data

Our study develops Alexander's hypothesis on that the locomotor dynamic similarity in non-avian dinosaurs can be inferred from the bending stress in their limbs (Alexander, 1976). We have demonstrated how the knowledge on elastic stress (or strain) similarity reliably established in limbs of different-sized running mammals may provide rationalizations of locomotory trials empirically suggested for extinct terrestrial giants. Allowing the generalized mammalian limb bone to be arbitrary loaded, the body functions of giants of different taxa subjected to cyclic loading during locomotor activity are viewed in terms of a few dynamic similarity patters provided by elastic theory of solids. Our theoretical study suggests that even though the dynamic similarity is underlaid by a certain set of *elastic force patterns*, there

The elastic patterns of forces emerging in the effective mammalian limb bone loaded in distinct (globally equilibrated non-critical stationary states and near-critical transient states

from Table 1 and other available sources. The star corresponds to 6-*ton* modeled

*Mg* <sup>=</sup> 1.89 · *<sup>M</sup>*<sup>−</sup>0.20,

*Tyrannosaurus* (Hutchinson et al., 2007; Sellers & Manning, 2007).

are several ways in which they may be realized and thus described.

*F*(mod) *bend*

1

namely

*F*(mod) *tors*

discussed in Fig. 5.

**4. Discussion**

*Mg* <sup>=</sup> 0.87 · *<sup>M</sup>*<sup>−</sup>0.20,

10

Maximal body mass,

*Mmax* (*ton*)

100

*Argentinosaurus*

*Supersaurus*

*Mammuthus trogontherii*

*Mammuthus sungari*

*Seismosaurus*

*Carcharodontosaurus*

*Antarctosaurus*

*Paralititan*

*Acrocanthosaurus*

*Tyrannosaurus*

*Giganotosaurus Shantungosaurus*

Fig. 2. Analysis of locomotor evolution of the largest terrestrial giants scaled by the limb bending parameter. The *solid line* is critical mass drawn by Eq. (22). The *points* correspond to giants in Table 1. The *arrows* indicate the biomechanical trends described by competitive empirically revealed traits (Hutchinson, 2006, Table 1): (1), more erect posture; (2), less erect posture; (4), quadrupedalism; (5), striding bipedalism; (6), body-size increase; (7), body-size decrease; (8), more upright pose; (9), more crouched pose; (22), poor bipedal running ability; (23), good bipedal running ability; (24), poor turning ability; (25); improved turning ability; (30), more cursorial limb proportions; (31), less cursorial limb proportions.

### **3.3 Turning and bending mobilities**

Being remarkably similar, both limb-bone longitudinal bending elastic force *Fbend* and transverse torsion force *Ftors* scale with body mass as *M*3*d*−*<sup>l</sup>* , within the domain of similar dynamic states including transient near-critical states (Kokshenev, 2007, Eq. (17), Table 2). The force scaling equations, taken in the mammalian limb-bone approximation, i.e., with 3*d* − *l* = 1.20, are completed by the scaling factors approximated by muscle-reaction forces from the largest 6-*ton Tyrannosaurus* modeled by Hutchinson et al. (2007).2 Treating the total force exerted by leg muscles over the center of body mass as the vectorial sum of vertical and horizontal forces, resulted in the total muscle force output, the corresponding skeletal body reaction force, limiting body mobility is found as the root mean square of the corresponding elastic forces emerging in the effective limb bones, i.e., *F*(mod) *mobil* = (*F*(mod)<sup>2</sup> *bend* <sup>+</sup> *<sup>F</sup>*(mod)<sup>2</sup> *tors* )1/2. When extrapolated to other largest extinct giants, the relative forces provide the model estimates,

<sup>2</sup> Specifically, for the case of *Ftors* we employ the data (Hutchinson et al., 2007, Table 6, Model 3) on maximal (medial and lateral) rotation muscle moments and moment arms of turning force exerted about center of mass of the trunk. The case of *F*(mod) *bend* is adjusted with the maximal vertical ground reaction force that the limb could support (Hutchinson et al., 2007, Table 9, Model 23).

Fig. 3. Maximal body masses for terrestrial giants modeled by the near-critical locomotor behavior. *Solid lines* are drawn by Eq.(18) taken at maximal stride speeds. The *points* are data from Table 1 and other available sources. The star corresponds to 6-*ton* modeled *Tyrannosaurus* (Hutchinson et al., 2007; Sellers & Manning, 2007).

namely

12 Will-be-set-by-IN-TECH

body-size increase (6)

*Giraffatitan Supersaurus*

*M. trogontherii*

(30), more cursorial limb proportions; (31), less cursorial limb proportions.

transverse torsion force *Ftors* scale with body mass as *M*3*d*−*<sup>l</sup>*

elastic forces emerging in the effective limb bones, i.e., *F*(mod)

about center of mass of the trunk. The case of *F*(mod)

(2,4,8, 22,24,31)

22

b d i

Diplodocoidea

 Titanosauriformes

W !!E

0 0.2 0.4 0.6 0.8 1 1.2

Perissodactyla

*Shantungosaurus Indricotherium*

Proboscidea

E

*mobil* = (*F*(mod)<sup>2</sup>

*Tyrannosaurus*

, within the domain of similar

*bend* <sup>+</sup> *<sup>F</sup>*(mod)<sup>2</sup>

*bend* is adjusted with the maximal vertical ground

*tors* )1/2. When

Carnosauria

Coelurosauria

*Giganotosaurus*

(7,9,25,30)

(7,9,30)

(1,5,9,23,25,30) 9,23,25,30)

Theropoda (30)

body-size decrease (7)

Ornithopoda

Limb bending stress indicator,

Being remarkably similar, both limb-bone longitudinal bending elastic force *Fbend* and

dynamic states including transient near-critical states (Kokshenev, 2007, Eq. (17), Table 2). The force scaling equations, taken in the mammalian limb-bone approximation, i.e., with 3*d* − *l* = 1.20, are completed by the scaling factors approximated by muscle-reaction forces from the largest 6-*ton Tyrannosaurus* modeled by Hutchinson et al. (2007).2 Treating the total force exerted by leg muscles over the center of body mass as the vectorial sum of vertical and horizontal forces, resulted in the total muscle force output, the corresponding skeletal body reaction force, limiting body mobility is found as the root mean square of the corresponding

extrapolated to other largest extinct giants, the relative forces provide the model estimates,

<sup>2</sup> Specifically, for the case of *Ftors* we employ the data (Hutchinson et al., 2007, Table 6, Model 3) on maximal (medial and lateral) rotation muscle moments and moment arms of turning force exerted

reaction force that the limb could support (Hutchinson et al., 2007, Table 9, Model 23).

Fig. 2. Analysis of locomotor evolution of the largest terrestrial giants scaled by the limb bending parameter. The *solid line* is critical mass drawn by Eq. (22). The *points* correspond to giants in Table 1. The *arrows* indicate the biomechanical trends described by competitive empirically revealed traits (Hutchinson, 2006, Table 1): (1), more erect posture; (2), less erect posture; (4), quadrupedalism; (5), striding bipedalism; (6), body-size increase; (7), body-size decrease; (8), more upright pose; (9), more crouched pose; (22), poor bipedal running ability; (23), good bipedal running ability; (24), poor turning ability; (25); improved turning ability;

*Deinotherium*

Elephantidae

0

10

20

Titanosauria

**3.3 Turning and bending mobilities**

Critical

30

 body mass,

40

50

*Mmax*

(*ton*)

60

70

*Argentinosaurus*

Sauropoda*,* 

80

$$\frac{F\_{\text{tors}}^{(\text{mod})}}{M\text{g}} = 0.87 \cdot M^{-0.20},\\\frac{F\_{\text{bond}}^{(\text{mod})}}{M\text{g}} = 1.89 \cdot M^{-0.20},\\\frac{F\_{\text{model}}^{(\text{mod})}}{M\text{g}} = 2.08 \cdot M^{-0.20}, \text{with } S\_{\text{model}} \ge 1,\tag{24}$$

discussed in Fig. 5.

### **4. Discussion**

Our study develops Alexander's hypothesis on that the locomotor dynamic similarity in non-avian dinosaurs can be inferred from the bending stress in their limbs (Alexander, 1976). We have demonstrated how the knowledge on elastic stress (or strain) similarity reliably established in limbs of different-sized running mammals may provide rationalizations of locomotory trials empirically suggested for extinct terrestrial giants. Allowing the generalized mammalian limb bone to be arbitrary loaded, the body functions of giants of different taxa subjected to cyclic loading during locomotor activity are viewed in terms of a few dynamic similarity patters provided by elastic theory of solids. Our theoretical study suggests that even though the dynamic similarity is underlaid by a certain set of *elastic force patterns*, there are several ways in which they may be realized and thus described.

The elastic patterns of forces emerging in the effective mammalian limb bone loaded in distinct (globally equilibrated non-critical stationary states and near-critical transient states

5 15 25 35 45 55 65 75 85

Fig. 5. Locomotor mobilities expected for extinct giants. The curves for relative limb bone elastic forces are drawn via Eq. (24) and notations correspond to Figs. 2 and 4. The *stars* are

the data on 3D *Tyrannosaurus* experimentally modeled by Hutchinson et al. (2007).

Turning mobility poor turning ability (24)

Sauropoda

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 63

*Giraffatitan*

*Supersaurus*

Proboscidea

Maximal body mass, *Mmax* (*ton*)

the corresponding oversimplified bending stress estimated under assumption of the almost isometric mammalian bones, as shown in Eq. (9). In this case we have ignored small by finite scaling bone-mass effects (Prange et al., 1979), which significance in limb bone allometry has previously been discussed (Kokshenev et al., 2003) and statistically evaluated (Kokshenev, 2007). In the second case, the desirable observation of the seemingly universal functional limb stress implies that the safety factor domain reliably established in running mammals as lying between 2 and 4 should indicate almost invariable safety function approximated by the number 3 ± 1. In the current research, the dynamic similarity features in giants moving in near-critical dynamic regimes are revealed through the stress-similarity indicators of the underlying patterns of axial, bending and torsional elastic forces. As is common in dynamic similarity theories, the patterns of scaling rules and related dimensionless numbers (Kokshenev, 2011a) or functions (Kokshenev, 2011b) determine the conditions of observation

Our recent study of similarities and dissimilarities in locomotory trends of extant and extinct proboscideans, indicated by the slenderness characteristics of design of the effective limb bones (Kokshenev & Christiansen, 2010), has revealed new features of their locomotory functions. The Asian elephants (*E. maximus*) are dynamically similar to the family Elephantidae and have been found to be distinct from the African elephants (*L. africana*). Moreover, the Elephantidae as group including both kinds of extant elephants, clearly exposed a negative limb duty gradient (with respect to other mammals, Fig. 6) in the body mobility function, closely related to the proximal-distal gradient (Biewener et al., 2006) in body ability function. Qualitatively, the hind-to-fore gradient in limb locomotor activity

*Paralititan*

*Argentinosaurus*

*Antarctosaurus*

0.1

of dynamic similarity states.

*Shantungosaurus*

*Apatosaurus*

*Shantungosaurus*

*Elephas maximus* Ornithopoda

*Loxodonta africana Elephas antiquus*

*Elephas recki Mammuthus columbi*

*Deinotherium*

poor mob. walkers

*Mammuthus sungari*

*Seismosaurus*

*Chubutisaurus*

*Mammuthus trogontherii* 

Theropoda

*Tyrannosaurus*

*M. trogontherii*

good turn. ability (25) E >0.5 W>0.5

*Giganotosaurus*

*Rebbachisaurus*

*Diplodocus*

*Barosaurus*

*Indricotherium*

0.2

Coelurosauria

Carnosauria

*Lambeosaurus*

*Edmontosaurus*

0.3

0.4

Fig. 4. Froude numbers expected from the largest terrestrial giants during near-critical locomotion. The *solid curves* are drawn through Eqs. (23) and Eq. (22); The *points* (Table 1) and notations for taxa and traits correspond to those in Fig. 1. The *star* shows the experimental data *Fr*(exp) *run* <sup>=</sup> 2.1 for the fast running 6-*ton Tyrannosaurus* modeled by Sellers & Manning (2007).

of ) dynamic similarity regimes (Kokshenev, 2007) have been revealed earlier through the bone design observed in different-sized animals through the allometric data on linear bone dimensions (Kokshenev et al., 2003; Kokshenev, 2007) and slenderness (Kokshenev, 2003; Kokshenev & Christiansen, 2010). All theoretical studies of the data were critically discussed within the context of McMahon's pioneering criteria of elastic similarity (McMahon, 1973, 1975a, b). The observation of elastic force patterns through the locomotory functions evaluated on the basis of available data on the maximal masses of extinct giants is one of the objectives of the presented approach. The critical-state similarity regime is treated as theoretically established if the observable characteristics are shown to be driven by one of the predominating critical-force patterns.

Even though animals are built from the same bone tissue and muscle tissue materials, the safety factors of limbs in running animals are not universally equal (Kokshenev, 2007, Fig. 3), as was suggested by earlier estimates (Alexander, 1981, 1985a) of the isolated dynamic stress states and systematical observations of limb safety factors (Biewener, 1982, 1983, 1989, 1990, 2000, 2005). On the other hand, in non-critical dynamic regimes of locomotion the body safety function discussed in Eq. (16) could also conventionally be observed, either theoretically or experimentally, as "almost" universal. The first kind of observations could be made through 14 Will-be-set-by-IN-TECH

0

0

E

experimental data *Fr*(exp) *run* <sup>=</sup> 2.1 for the fast running 6-*ton Tyrannosaurus* modeled by Sellers

of ) dynamic similarity regimes (Kokshenev, 2007) have been revealed earlier through the bone design observed in different-sized animals through the allometric data on linear bone dimensions (Kokshenev et al., 2003; Kokshenev, 2007) and slenderness (Kokshenev, 2003; Kokshenev & Christiansen, 2010). All theoretical studies of the data were critically discussed within the context of McMahon's pioneering criteria of elastic similarity (McMahon, 1973, 1975a, b). The observation of elastic force patterns through the locomotory functions evaluated on the basis of available data on the maximal masses of extinct giants is one of the objectives of the presented approach. The critical-state similarity regime is treated as theoretically established if the observable characteristics are shown to be driven by one of

Even though animals are built from the same bone tissue and muscle tissue materials, the safety factors of limbs in running animals are not universally equal (Kokshenev, 2007, Fig. 3), as was suggested by earlier estimates (Alexander, 1981, 1985a) of the isolated dynamic stress states and systematical observations of limb safety factors (Biewener, 1982, 1983, 1989, 1990, 2000, 2005). On the other hand, in non-critical dynamic regimes of locomotion the body safety function discussed in Eq. (16) could also conventionally be observed, either theoretically or experimentally, as "almost" universal. The first kind of observations could be made through

Fig. 4. Froude numbers expected from the largest terrestrial giants during near-critical locomotion. The *solid curves* are drawn through Eqs. (23) and Eq. (22); The *points* (Table 1)

and notations for taxa and traits correspond to those in Fig. 1. The *star* shows the

0.5

bipedalism (5)

1

1.5

2

Carnosauria *Tyrannosaurus*

A B

good runners (5, 23)

poor runners (22)

Hadrosauridae

Non-elephantid

(1,7,9,25,30)

2.5

0 20 40 60

*Shantungosaurus*

*Giganotosaurus*

Theropoda

(1,7,9,25,30)

*Indricotherium*

(2,6,8 24,31)

Critical body mass, *Mmax* (*ton*)

quadrupedalism (4)

*Deinotherium M. trogontherii*

Proboscidea

Ornithopoda

*Giraffatitan Supersaurus*

*Argentinosaurus*

Sauropoda

0 0.2 0.4 0.6 0.8 1 1.2

Limb bending stress indicator,

Diplodocoidea Titanosauriformes

Elephantidae

walkers

the predominating critical-force patterns.

Titanosauria

variable posture (3)

Hyracodontidae

(2,6,8,24,31)

0

& Manning (2007).

0.5

1

Froude

1.5

number, *Frcrit*

2

2.5

Fig. 5. Locomotor mobilities expected for extinct giants. The curves for relative limb bone elastic forces are drawn via Eq. (24) and notations correspond to Figs. 2 and 4. The *stars* are the data on 3D *Tyrannosaurus* experimentally modeled by Hutchinson et al. (2007).

the corresponding oversimplified bending stress estimated under assumption of the almost isometric mammalian bones, as shown in Eq. (9). In this case we have ignored small by finite scaling bone-mass effects (Prange et al., 1979), which significance in limb bone allometry has previously been discussed (Kokshenev et al., 2003) and statistically evaluated (Kokshenev, 2007). In the second case, the desirable observation of the seemingly universal functional limb stress implies that the safety factor domain reliably established in running mammals as lying between 2 and 4 should indicate almost invariable safety function approximated by the number 3 ± 1. In the current research, the dynamic similarity features in giants moving in near-critical dynamic regimes are revealed through the stress-similarity indicators of the underlying patterns of axial, bending and torsional elastic forces. As is common in dynamic similarity theories, the patterns of scaling rules and related dimensionless numbers (Kokshenev, 2011a) or functions (Kokshenev, 2011b) determine the conditions of observation of dynamic similarity states.

Our recent study of similarities and dissimilarities in locomotory trends of extant and extinct proboscideans, indicated by the slenderness characteristics of design of the effective limb bones (Kokshenev & Christiansen, 2010), has revealed new features of their locomotory functions. The Asian elephants (*E. maximus*) are dynamically similar to the family Elephantidae and have been found to be distinct from the African elephants (*L. africana*). Moreover, the Elephantidae as group including both kinds of extant elephants, clearly exposed a negative limb duty gradient (with respect to other mammals, Fig. 6) in the body mobility function, closely related to the proximal-distal gradient (Biewener et al., 2006) in body ability function. Qualitatively, the hind-to-fore gradient in limb locomotor activity

provoking critical bending deformations under loading conditions of the absence of non-axial external forces (e.g., Kokshenev et al., 2003; Kokshenev, 2003). McMahon therefore proposed that terrestrial animals would optimize their skeleton support function so that were similarly in danger of elastic mechanical failure by buckling under gravity (e.g., Christiansen, 1999). The corresponding to buckling mechanism loading, the simplest model for terrestrial giants, as an animal standing on one leg and subjected to increasing with weight gravitational force, was proposed by Hokkanen (1986a). Although it has been clearly demonstrated that the maximal body mass of terrestrial giants is not limited by axial critical forces, associated with buckling mechanism of damage of naturally curved bones (e.g., Currey, 1967; Hokkanen, 1986a, b; Selker & Carter, 1989), the corresponding scaling pattern of the Euler's force (e.g., Kokshenev et al., 2003, Eq. (1)) emerging under aforesaid dynamic conditions may survive and even predominate at loads far from critical. Moreover, as one may infer from the reliable observation of the effective pillar-type bone, the pattern of critical buckling force most likely controls the limb elastic stress of animals, in which limb muscles function as to resist non-axial external forces, through the proximal-distal functional gradients, evolutionary resulting in the limb bone design mostly adapted to the axial-bone peak loading. Hence, when one considers competitive patterns of functional elastic stresses in a certain effective bone of unknown design, as discussed on general basis in Eq. (5), equally with the term of non-critical axial

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 65

*axial* , which negative allometry (*μaxial* <sup>&</sup>lt; 0) was predicted by *<sup>σ</sup>*(*pred*)

(Kokshenev, 2007, Eq. (5), Table 3), one should not exclude the possibility of axial stress

Following the ESM (Fig. 6), the individual limb bones in *L. africana* are mostly influenced by gravitational and ground reaction forces, generating buckling-type dynamic elastic stress

functional axial, bending and torsional stresses occurring in the limbs of the largest extinct giants at similar critical states modeled by the maximal body masses of Asian and African elephants, Elephantidae, and mammals as whole (Table 3, Fig. 1). Unlike the case of Asian elephants and running mammals, the axial predominating stress becomes observable through the maximal body mass of African elephants (Fig. 1A). As follows from Fig. 6, more reliable observation of the stress similarity in African elephants, as well as bovids (McMahon, 1975b),

similarity model. Since the exponent for critical axial stress is expected to be twice as high as that for critical bending stress, we have thereby shown that in near-critical dynamic regimes the buckling force effects appear to dominate over bending-torsional force effects. Since the

cylinders, the critical (yield-state) bending strains (Kokshenev et al., 2003; Eq. (4)) dynamically transform into transient-state pure bending elastic strains (Kokshenev, 2007; Eq. (15)), as has

The analysis of elastic peak forces developed in ideal solid cylinders indicated that various conditions of application of external loads affect solely scaling factors, leaving unchanged scaling exponents determining elastic force patterns (e.g., Kokshenev et al., 2003, Eq. (1)) employed above in rationalization of the functional stress in Eq. (5) when applied to extinct animals. Unlike ideal cylinders, the individual long skeletal bones are mostly curved along the

*axial* <sup>∝</sup> *<sup>M</sup>*2(*d*−*l*) (Kokshenev et al. 2003, Eq. (4)) underlying the pattern of

*axial* . This finding is supported by our study of the competing

*axial* <sup>=</sup> <sup>−</sup>0.05 and *<sup>μ</sup>*(*pred*)

<sup>0</sup>*bend* = 1/8, predicted by McMahon's elastic

<sup>0</sup> = 2/3, are not ideal long

*axial* <sup>∝</sup> *<sup>M</sup>*2/3−2*<sup>d</sup>*

*bend* = 0.08 in Eq.

stress *<sup>σ</sup>*(max)

predicted as *<sup>σ</sup>*(*ESM*)

of the peak amplitude *<sup>σ</sup>*(*ESM*)

(16) are substituted by *<sup>μ</sup>*(*ESM*)

been explained earlier by Currey (1967).

might be obtained when the scaling exponents *<sup>μ</sup>*(*pred*)

<sup>0</sup>*axial* <sup>=</sup> 1/4 and *<sup>μ</sup>*(*ESM*)

effective pillar-type limb bones, scaled to body mass as *<sup>λ</sup>*(*ESM*)

pillar-type bones.

Fig. 6. The observation of elastic similarity across different taxa via bone slenderness exponents (Table 2) for the effective forelimb bone (humerus, radius, and ulna) and hindlimb bone (femur and tibia). The *arrows* indicate deviations in the trends of adaptation of forelimb and hindlimb bones associated with gradients in the corresponding mechanical functions. The *bar* shows statistical error. *Notations*: GSM and ESM indicate the predictions of the geometric similarity model and elastic similarity model of McMahon (1973, 1975a); SSMM indicates the static stress similarity model (McMahon, 1975a, SSM) modified by muscle forces (Kokshenev, 2003). For further details, see Fig. 4 in Kokshenev & Christiansen (2010).

caused by the difference in bone proportions of hind and fore limbs (Fig. 6, mammals ) explains the differing *postures* in mammals as group as well as elephants (Kokshenev & Christiansen, 2010, Fig. 5). The negative gradient of limb functions for the Elephantidae (including the extant species, but mainly comprising of extinct species) suggests a trend for forelimbs to be designed more isometrically, and also for the forelimbs of *E. maximus* to be mechanically stiffer than the hindlimbs. The newly established dynamic constraint, contrasting in sign to that of other mammals and mechanically extending the hindlimb ground contact duration, explains why elephants are prevented from achieving a full-body aerial phase during fast gaits of locomotion.

The design remarkably observed (Table 6) for the effective limb bone in the African elephant (5 limb bones, 56 specimens, Table 3) is surprisingly similar to that established in adult ungulates (5 limb bones, 118 specimens, McMahon, 1975b) mostly from the family Bovidae (see also analysis by Selker & Carter, 1989). The distinct from other effective bones, say, *pillar-type limb bone*, is statistically established via the slenderness exponent *<sup>λ</sup>*(exp) <sup>≈</sup> *<sup>λ</sup>*(*ESM*) <sup>0</sup> = 2/3 predicted by the elastic similarity model (ESM, Fig. 6). The pillar-like limb bone pattern underlaid by the *critical buckling force* (McMahon, 1973, 1975a) is well distinguished from the mammalian-type limb bone, determined experimentally in Eq. (6) and predicted by SSMM through *<sup>λ</sup>*(*SSMM*) <sup>0</sup> = 7/9 (Kokshenev & Christiansen, 2010, footnote 1). The buckling force was introduced in the theory of elasticity of solids by Euler as the critical axial elastic force 16 Will-be-set-by-IN-TECH

Forelimb

Elephantidae

Taxonomic category

exponents (Table 2) for the effective forelimb bone (humerus, radius, and ulna) and hindlimb bone (femur and tibia). The *arrows* indicate deviations in the trends of adaptation of forelimb and hindlimb bones associated with gradients in the corresponding mechanical functions. The *bar* shows statistical error. *Notations*: GSM and ESM indicate the predictions of the geometric similarity model and elastic similarity model of McMahon (1973, 1975a); SSMM indicates the static stress similarity model (McMahon, 1975a, SSM) modified by muscle forces (Kokshenev, 2003). For further details, see Fig. 4 in Kokshenev & Christiansen (2010).

caused by the difference in bone proportions of hind and fore limbs (Fig. 6, mammals ) explains the differing *postures* in mammals as group as well as elephants (Kokshenev & Christiansen, 2010, Fig. 5). The negative gradient of limb functions for the Elephantidae (including the extant species, but mainly comprising of extinct species) suggests a trend for forelimbs to be designed more isometrically, and also for the forelimbs of *E. maximus* to be mechanically stiffer than the hindlimbs. The newly established dynamic constraint, contrasting in sign to that of other mammals and mechanically extending the hindlimb ground contact duration, explains why elephants are prevented from achieving a full-body aerial

The design remarkably observed (Table 6) for the effective limb bone in the African elephant (5 limb bones, 56 specimens, Table 3) is surprisingly similar to that established in adult ungulates (5 limb bones, 118 specimens, McMahon, 1975b) mostly from the family Bovidae (see also analysis by Selker & Carter, 1989). The distinct from other effective bones, say, *pillar-type*

predicted by the elastic similarity model (ESM, Fig. 6). The pillar-like limb bone pattern underlaid by the *critical buckling force* (McMahon, 1973, 1975a) is well distinguished from the mammalian-type limb bone, determined experimentally in Eq. (6) and predicted by SSMM

was introduced in the theory of elasticity of solids by Euler as the critical axial elastic force

<sup>0</sup> = 7/9 (Kokshenev & Christiansen, 2010, footnote 1). The buckling force

*limb bone*, is statistically established via the slenderness exponent *<sup>λ</sup>*(exp) <sup>≈</sup> *<sup>λ</sup>*(*ESM*)

Fig. 6. The observation of elastic similarity across different taxa via bone slenderness

*Elephas maximus*

*Loxodonta*

 *africana*

SSMM compliant limbs

GSM

more stiff

<sup>0</sup> = 2/3

ESM

stiff limbs

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

phase during fast gaits of locomotion.

through *<sup>λ</sup>*(*SSMM*)

Limb bone slenderness exponent

, O

Mammals

LS data

Hindlimb

provoking critical bending deformations under loading conditions of the absence of non-axial external forces (e.g., Kokshenev et al., 2003; Kokshenev, 2003). McMahon therefore proposed that terrestrial animals would optimize their skeleton support function so that were similarly in danger of elastic mechanical failure by buckling under gravity (e.g., Christiansen, 1999). The corresponding to buckling mechanism loading, the simplest model for terrestrial giants, as an animal standing on one leg and subjected to increasing with weight gravitational force, was proposed by Hokkanen (1986a). Although it has been clearly demonstrated that the maximal body mass of terrestrial giants is not limited by axial critical forces, associated with buckling mechanism of damage of naturally curved bones (e.g., Currey, 1967; Hokkanen, 1986a, b; Selker & Carter, 1989), the corresponding scaling pattern of the Euler's force (e.g., Kokshenev et al., 2003, Eq. (1)) emerging under aforesaid dynamic conditions may survive and even predominate at loads far from critical. Moreover, as one may infer from the reliable observation of the effective pillar-type bone, the pattern of critical buckling force most likely controls the limb elastic stress of animals, in which limb muscles function as to resist non-axial external forces, through the proximal-distal functional gradients, evolutionary resulting in the limb bone design mostly adapted to the axial-bone peak loading. Hence, when one considers competitive patterns of functional elastic stresses in a certain effective bone of unknown design, as discussed on general basis in Eq. (5), equally with the term of non-critical axial stress *<sup>σ</sup>*(max) *axial* , which negative allometry (*μaxial* <sup>&</sup>lt; 0) was predicted by *<sup>σ</sup>*(*pred*) *axial* <sup>∝</sup> *<sup>M</sup>*2/3−2*<sup>d</sup>* (Kokshenev, 2007, Eq. (5), Table 3), one should not exclude the possibility of axial stress predicted as *<sup>σ</sup>*(*ESM*) *axial* <sup>∝</sup> *<sup>M</sup>*2(*d*−*l*) (Kokshenev et al. 2003, Eq. (4)) underlying the pattern of pillar-type bones.

Following the ESM (Fig. 6), the individual limb bones in *L. africana* are mostly influenced by gravitational and ground reaction forces, generating buckling-type dynamic elastic stress of the peak amplitude *<sup>σ</sup>*(*ESM*) *axial* . This finding is supported by our study of the competing functional axial, bending and torsional stresses occurring in the limbs of the largest extinct giants at similar critical states modeled by the maximal body masses of Asian and African elephants, Elephantidae, and mammals as whole (Table 3, Fig. 1). Unlike the case of Asian elephants and running mammals, the axial predominating stress becomes observable through the maximal body mass of African elephants (Fig. 1A). As follows from Fig. 6, more reliable observation of the stress similarity in African elephants, as well as bovids (McMahon, 1975b), might be obtained when the scaling exponents *<sup>μ</sup>*(*pred*) *axial* <sup>=</sup> <sup>−</sup>0.05 and *<sup>μ</sup>*(*pred*) *bend* = 0.08 in Eq. (16) are substituted by *<sup>μ</sup>*(*ESM*) <sup>0</sup>*axial* <sup>=</sup> 1/4 and *<sup>μ</sup>*(*ESM*) <sup>0</sup>*bend* = 1/8, predicted by McMahon's elastic similarity model. Since the exponent for critical axial stress is expected to be twice as high as that for critical bending stress, we have thereby shown that in near-critical dynamic regimes the buckling force effects appear to dominate over bending-torsional force effects. Since the effective pillar-type limb bones, scaled to body mass as *<sup>λ</sup>*(*ESM*) <sup>0</sup> = 2/3, are not ideal long cylinders, the critical (yield-state) bending strains (Kokshenev et al., 2003; Eq. (4)) dynamically transform into transient-state pure bending elastic strains (Kokshenev, 2007; Eq. (15)), as has been explained earlier by Currey (1967).

The analysis of elastic peak forces developed in ideal solid cylinders indicated that various conditions of application of external loads affect solely scaling factors, leaving unchanged scaling exponents determining elastic force patterns (e.g., Kokshenev et al., 2003, Eq. (1)) employed above in rationalization of the functional stress in Eq. (5) when applied to extinct animals. Unlike ideal cylinders, the individual long skeletal bones are mostly curved along the

scaled by the bending stress set up in the limbs of animals moving at critically low limb safety factors. Our analysis, indicating probabilities of bending (*β*) and torsional (*τ*) limb stresses as seen in various maximal body masses, is expected to establish a bridge between body size traits 6 and 7 and other biomechanical traits established for dinosaurs and reviewed by Hutchinson (2006). The anatomical data from bipedal (trait 5) theropods (long legs, long tibia relative to femur, long metatarsus, and very large limb muscles) suggest that they had well developed anatomical adaptations for running modes, as shown through the expected duty factor in Table 1. These generally agree with the observation in Fig. 2 of Sauropoda in, say, *run-mode domain β* > 0.5 > *τ* (we recall that *τ* = 1 − *β*) indicated by limbs adapted for peak bending effects rather than peak torsional effects. Anatomy leaves no doubt whatsoever that all sauropods (trait 4) were capable of progression with a walking gait only, which generally determines the *walk-mode* domain *τ* > 0.5 > *β*. The established trends linked to locomotor characterization of the limbs of giants, developing with body mass (trait 6) are supported by changes in competing biomechanical traits: from striding bipedalism (trait 5), related to good bipedal running ability (trait 23) to quadrupedalism (trait 4) through decreasing bipedal running ability (22). Likewise, the trend of changes in posture, pose, and limb proportions with body mass (varying from corresponding traits 1, 9, and 30 to traits 2, 8, and 31) could also be elucidated through the limb stress indicators.<sup>3</sup> A crossover between giants found in run-mode and walk-mode domains during the evolution of competitive locomotory traits is indicated by the transient indicators *τ* ≈ *β* ≈ 0.5, corresponding approximately to the Elephantidae (Fig. 2), whose limbs are expected to be biomechanically adapted to the transient

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 67

In Fig. 3, the analysis of locomotor traits from Fig. 2 for the largest extinct giants is extended to smaller giants, belonging to the same anatomical type and large-scale taxon (e.g., family, order). When the critical (and near-critical) body mass criterion of the similarity in evolution of locomotory trends is adopted, the largest representatives of the taxa Titanosauria, Diplodocoidea, Titanosauriformes, Elephantidae, non-elephantid, proboscideans (a paraphyletic group of morphologically comparable taxa), Perissodactyla, Hadrosauridae, Carnosauria, and Coelurosauria are treated as *locomotory patterns of similarity* presented by biomechanically similar animals having limbs adaptively designed to explore the bending and torsion stresses in the strict narrow domains indicated (Fig. 2) by the corresponding probabilities *β* and *τ*. For example, the dynamic similarity pattern of *Mammuthus trogontherii* (Fig. 2) is now extended by the Elephantidae, including the largest extant 10-*ton* African and 8-*ton* Asian elephants (Fig. 3). Being analyzed by the body safety function in Eq. (16), where *τ* �= 1 − *β*, these two compared elephants also differ in the *mean*

*Af rican* <sup>&</sup>lt; 0.5, for African elephant, and *<sup>τ</sup>*(*mean*)

for Asian elephant (Table 3, Fig. 1B). These different limb bone traits imply that even though both elephants generally fall into the transient walk-to-run evolutionary domain indicated by

rather than walking animals. In contrast, African elephants fall into the domain of walkers

<sup>3</sup> In this context, the set of traits (6, 8, 31) suggested for the Carnosauria by Hutchinson et al. (2006) should be likely substituted by the set (7, 9, 30), if this group is indeed characteristic of the non-avian

*Asian* <sup>≈</sup> *<sup>β</sup>*(*mean*)

*Asian* > 0.5, reminiscent of running mammals

*Asian* > 0.5,

(walk-to-run) dynamic similarity states.

*Af rican* <sup>≈</sup> *<sup>β</sup>*(*mean*)

*Af rican* < 0.5.

*eleph* , Asian elephants, showing *<sup>β</sup>*(*mean*)

*stress indicators <sup>τ</sup>*(*mean*)

*eleph* <sup>≈</sup> *<sup>β</sup>*(*mean*)

indicated by *<sup>β</sup>*(*mean*)

Theropoda as whole.

*<sup>τ</sup>*(*mean*)

longitudinal axis in all but the largest extant mammals (Bertram & Biewener, 1988, 1992), and even in the largest, elephant-sized theropod dinosaurs (e.g., Farlow et al., 1995; Christiansen, 1998). Hence, external loads produce a complex non-axial compression in long bones, causing different components (axial and non-axial) of reactive (elastic) bending and torsional stresses. Moreover, since the bone cross section is also not an ideal circle, near maximal bending stress can be avoided via the transient non-critical dynamic regimes by deviation of the bone from the sagittal plane, thereby decreasing bending stress through torsional stress (Kokshenev, 2007). Switching from bending deformation to out-off-plane torsion, such a mechanism reduces the risk of limb damage, naturally increasing bone safety factors. In this study, the stress in the non-uniformly loaded limbs of animals is modeled by an arbitrary composition of axial, bending, and torsional stresses, as shown in Eq. (5).

Unlike the case of the ESM underlaid by Euler's force, no explicit bending force pattern for McMahon's static stress similarity model (SSM, McMahon, 1975a) was established. Instead, in the *in vitro* experimental studies of long bone strength the *critical transverse-bone force* has been discussed as the *perpendicular* bending force that a bone can withstand without breaking (e.g., Hokkanen, 1986b, Eq. (11)) under either three-point or cantilever loading (Selker & Carter, 1989; Eq. (4)). In spite of that it was widely adopted that most fractures in limbs of living animals are due to longitudinal-bone and transverse-bone torsional forces (e.g., Carter et al., 1980; Rubin & Lanyon, 1982; Biewener et al., 1983; Biewener & Taylor, 1986; Selker & Carter, 1989) such generalizations resulting in transverse-bending, longitudinal-bending, and transverse-torsional elastic force patterns unified by the SSM have recently been established and tested (Kokshenev, 2007, Eq. (17) and Table 3). In this study, we have repeatedly demonstrated in Eq. (10) that the dynamic elastic stress similarity patterns (Kokshenev, 2007, Eq. (15)) can be observed through the effective (mammalian) limb bone design, which appears to be common to both extant and extinct large animals (Fig. 6). It is well established that the peak muscle contractions involved in locomotion are primarily responsible for bending and torsional stresses in bones (Carter et al., 1980; Biewener, 1982). Consequently, the definitive scaling equations of the SSM of long bones (McMahon, 1975a), completely ignoring predominating muscular forces, required a modification as indicated by the bone stress index (Selker & Carter, 1989; Table 2). When experimentally indicated modifications have been provided through the gravitational forces substituted by external bending and torsional muscle's forces, the *modified* McMahon's model (SSMM) had turned to be reliable, i.e., observable through the bone allometry data from extant mammals (Kokshenev et al., 2003, Fig. 1, dashed area 2; Kokshenev, 2003, Fig. 2) and extinct animals (Fig. 6, SSMM). According to the SSMM, in the Elephantidae and in particular in *E. maximus*, the external bone off-axial muscular forces, causing a complex bending-torsion elastic bone stress during fast locomotion, provide a relatively high level of limb compliance conducted by the limb bones. These qualitative SSMM predictions are well supported by the analysis illustrated in Fig. 1B. Asian elephants are involved in torsional effects with the probability (0.35 ≤ *τ* ≤ 1) that on average is twice as high as that (0 ≤ *τ* ≤ 0.65) of the largest extant African elephants (Table 3). Relatively large domains of exploration of bending-torsional stress, completely excluding the axial stress domain common to mammals, Proboscideans and Elephantidae (Table 3). These distinct dynamically features of limb bone functions accomplished by different predominating critical stresses result in distinct bone designs, as illustrated by the running-mammal and pillar-like effective limb bones.

In Fig. 2, the maximal masses of some extinct giants, from the largest theropod (9-*ton Giganotosaurus* and 8-*ton Tyrannosaurus*) to the largest sauropod (70-*ton Argentinosaurus*), are 18 Will-be-set-by-IN-TECH

longitudinal axis in all but the largest extant mammals (Bertram & Biewener, 1988, 1992), and even in the largest, elephant-sized theropod dinosaurs (e.g., Farlow et al., 1995; Christiansen, 1998). Hence, external loads produce a complex non-axial compression in long bones, causing different components (axial and non-axial) of reactive (elastic) bending and torsional stresses. Moreover, since the bone cross section is also not an ideal circle, near maximal bending stress can be avoided via the transient non-critical dynamic regimes by deviation of the bone from the sagittal plane, thereby decreasing bending stress through torsional stress (Kokshenev, 2007). Switching from bending deformation to out-off-plane torsion, such a mechanism reduces the risk of limb damage, naturally increasing bone safety factors. In this study, the stress in the non-uniformly loaded limbs of animals is modeled by an arbitrary composition

Unlike the case of the ESM underlaid by Euler's force, no explicit bending force pattern for McMahon's static stress similarity model (SSM, McMahon, 1975a) was established. Instead, in the *in vitro* experimental studies of long bone strength the *critical transverse-bone force* has been discussed as the *perpendicular* bending force that a bone can withstand without breaking (e.g., Hokkanen, 1986b, Eq. (11)) under either three-point or cantilever loading (Selker & Carter, 1989; Eq. (4)). In spite of that it was widely adopted that most fractures in limbs of living animals are due to longitudinal-bone and transverse-bone torsional forces (e.g., Carter et al., 1980; Rubin & Lanyon, 1982; Biewener et al., 1983; Biewener & Taylor, 1986; Selker & Carter, 1989) such generalizations resulting in transverse-bending, longitudinal-bending, and transverse-torsional elastic force patterns unified by the SSM have recently been established and tested (Kokshenev, 2007, Eq. (17) and Table 3). In this study, we have repeatedly demonstrated in Eq. (10) that the dynamic elastic stress similarity patterns (Kokshenev, 2007, Eq. (15)) can be observed through the effective (mammalian) limb bone design, which appears to be common to both extant and extinct large animals (Fig. 6). It is well established that the peak muscle contractions involved in locomotion are primarily responsible for bending and torsional stresses in bones (Carter et al., 1980; Biewener, 1982). Consequently, the definitive scaling equations of the SSM of long bones (McMahon, 1975a), completely ignoring predominating muscular forces, required a modification as indicated by the bone stress index (Selker & Carter, 1989; Table 2). When experimentally indicated modifications have been provided through the gravitational forces substituted by external bending and torsional muscle's forces, the *modified* McMahon's model (SSMM) had turned to be reliable, i.e., observable through the bone allometry data from extant mammals (Kokshenev et al., 2003, Fig. 1, dashed area 2; Kokshenev, 2003, Fig. 2) and extinct animals (Fig. 6, SSMM). According to the SSMM, in the Elephantidae and in particular in *E. maximus*, the external bone off-axial muscular forces, causing a complex bending-torsion elastic bone stress during fast locomotion, provide a relatively high level of limb compliance conducted by the limb bones. These qualitative SSMM predictions are well supported by the analysis illustrated in Fig. 1B. Asian elephants are involved in torsional effects with the probability (0.35 ≤ *τ* ≤ 1) that on average is twice as high as that (0 ≤ *τ* ≤ 0.65) of the largest extant African elephants (Table 3). Relatively large domains of exploration of bending-torsional stress, completely excluding the axial stress domain common to mammals, Proboscideans and Elephantidae (Table 3). These distinct dynamically features of limb bone functions accomplished by different predominating critical stresses result in distinct bone designs, as illustrated by the running-mammal and

In Fig. 2, the maximal masses of some extinct giants, from the largest theropod (9-*ton Giganotosaurus* and 8-*ton Tyrannosaurus*) to the largest sauropod (70-*ton Argentinosaurus*), are

of axial, bending, and torsional stresses, as shown in Eq. (5).

pillar-like effective limb bones.

scaled by the bending stress set up in the limbs of animals moving at critically low limb safety factors. Our analysis, indicating probabilities of bending (*β*) and torsional (*τ*) limb stresses as seen in various maximal body masses, is expected to establish a bridge between body size traits 6 and 7 and other biomechanical traits established for dinosaurs and reviewed by Hutchinson (2006). The anatomical data from bipedal (trait 5) theropods (long legs, long tibia relative to femur, long metatarsus, and very large limb muscles) suggest that they had well developed anatomical adaptations for running modes, as shown through the expected duty factor in Table 1. These generally agree with the observation in Fig. 2 of Sauropoda in, say, *run-mode domain β* > 0.5 > *τ* (we recall that *τ* = 1 − *β*) indicated by limbs adapted for peak bending effects rather than peak torsional effects. Anatomy leaves no doubt whatsoever that all sauropods (trait 4) were capable of progression with a walking gait only, which generally determines the *walk-mode* domain *τ* > 0.5 > *β*. The established trends linked to locomotor characterization of the limbs of giants, developing with body mass (trait 6) are supported by changes in competing biomechanical traits: from striding bipedalism (trait 5), related to good bipedal running ability (trait 23) to quadrupedalism (trait 4) through decreasing bipedal running ability (22). Likewise, the trend of changes in posture, pose, and limb proportions with body mass (varying from corresponding traits 1, 9, and 30 to traits 2, 8, and 31) could also be elucidated through the limb stress indicators.<sup>3</sup> A crossover between giants found in run-mode and walk-mode domains during the evolution of competitive locomotory traits is indicated by the transient indicators *τ* ≈ *β* ≈ 0.5, corresponding approximately to the Elephantidae (Fig. 2), whose limbs are expected to be biomechanically adapted to the transient (walk-to-run) dynamic similarity states.

In Fig. 3, the analysis of locomotor traits from Fig. 2 for the largest extinct giants is extended to smaller giants, belonging to the same anatomical type and large-scale taxon (e.g., family, order). When the critical (and near-critical) body mass criterion of the similarity in evolution of locomotory trends is adopted, the largest representatives of the taxa Titanosauria, Diplodocoidea, Titanosauriformes, Elephantidae, non-elephantid, proboscideans (a paraphyletic group of morphologically comparable taxa), Perissodactyla, Hadrosauridae, Carnosauria, and Coelurosauria are treated as *locomotory patterns of similarity* presented by biomechanically similar animals having limbs adaptively designed to explore the bending and torsion stresses in the strict narrow domains indicated (Fig. 2) by the corresponding probabilities *β* and *τ*. For example, the dynamic similarity pattern of *Mammuthus trogontherii* (Fig. 2) is now extended by the Elephantidae, including the largest extant 10-*ton* African and 8-*ton* Asian elephants (Fig. 3). Being analyzed by the body safety function in Eq. (16), where *τ* �= 1 − *β*, these two compared elephants also differ in the *mean stress indicators <sup>τ</sup>*(*mean*) *Af rican* <sup>≈</sup> *<sup>β</sup>*(*mean*) *Af rican* <sup>&</sup>lt; 0.5, for African elephant, and *<sup>τ</sup>*(*mean*) *Asian* <sup>≈</sup> *<sup>β</sup>*(*mean*) *Asian* > 0.5, for Asian elephant (Table 3, Fig. 1B). These different limb bone traits imply that even though both elephants generally fall into the transient walk-to-run evolutionary domain indicated by *<sup>τ</sup>*(*mean*) *eleph* <sup>≈</sup> *<sup>β</sup>*(*mean*) *eleph* , Asian elephants, showing *<sup>β</sup>*(*mean*) *Asian* > 0.5, reminiscent of running mammals rather than walking animals. In contrast, African elephants fall into the domain of walkers indicated by *<sup>β</sup>*(*mean*) *Af rican* < 0.5.

<sup>3</sup> In this context, the set of traits (6, 8, 31) suggested for the Carnosauria by Hutchinson et al. (2006) should be likely substituted by the set (7, 9, 30), if this group is indeed characteristic of the non-avian Theropoda as whole.

*runners* capable of showing relatively high numbers *Fr*(mod) *run* <sup>&</sup>gt; 1.5, since they have limbs mostly subjected to and, thus, adapted to near-critical bending stress of the domain, say, 0.8 *β* < 1, and (ii) *poor runners* attributed to Froude numbers lying in the transient-state domain 1 <sup>&</sup>lt; *Fr*(mod) *<sup>w</sup>*-*<sup>r</sup>* <sup>≤</sup> 1.5 and using moderate bending and moderate torsional stress indicated by 0.5 < *β* 0.8 . The walk-mode domain (0 < *β* ≤ 0.5) establishes the locomotory

Evolution of Locomotor Trends in Extinct Terrestrial Giants Affected by Body Mass 69

evolution in giants of gait-dependent *Froude function Fr*(*M*) is found to be in good agreement

6-*ton Tyrannosaurus* showing relatively high running speed 8 *m*/*s* (Sellers & Manning, 2007). One may also expect that more reliable estimates for critical speeds discussed above could be

It generally follows from Fig. 4 that with increasing body mass the evolution of locomotory patterns from *good runners* through *poor runners* to *walkers* is controlled by decreasing bending stress and increasing torsional elastic stress in the limbs. Based on the corresponding indicators *β* and *τ*, we are broadly able to characterize the accompanied evolution in postures and limb proportions, changing in striding running bipeds from (i) erect posture (trait 1) and more cursorial limbs (trait 30) to (iii) sprawling, or less erect posture (trait 2) and less cursorial limb proportions (trait 31), attributed to quadrupeds. The intermediate biped-to-quadruped locomotory evolution indicated by near equal proportions of moderate bending and torsional stress in limbs, may roughly be approached by elephantid and non-elephantid proboscideans, having rather short and compact bodies and fairly long limbs; these animals were undoubtedly not fast running with a suspended phase and therefore are found (Fig. 4) to be poor runners (trait 22). The largest land animal, *Argentinosaurus*, certainly could not run at all, but grouping in the walk-mode domain, it may be broadly regarded as

While locomotor ability is provided by the relative body force output, body mobility is generally associated with the corresponding reaction-body elastic force, provided in most part by limb bone elastic forces. The evolution of body mobility, resulting from turning and bending limb mobilities, moderately decreasing with body mass (Fig. 5), is concomitant with a suggested evolution from striding bipedalism (trait 5) to quadrupedalism (trait 4). The walking and running mobilities are distinguished through the walk-mode and run-mode domains of the bone-stress similarity indicators. The animals considered by Hutchinson (2006) to possess good running abilities (trait 23) as well as improved turning abilities (trait 25) are presented (Figs. 4 and 5) by a single pattern of good runners, which relatively high mobility is due to bending effects exceeding torsional effects in the limb bones. The animals characterized by poor turning ability (trait 24) and poor bending ability are treated as walkers

constrained by relatively high-level torsional elastic stress developed in long bones.

As can be inferred from Fig. 5, smaller quadrupedal giants from the Proboscidea, poorly running elephants at relatively high safety factors, may expose the same body mobility as the slow running largest bipedal giants of the Theropoda, operating close to critical levels. The body mobility of extinct elephants, moving at near critical levels, e.g., presented by *Deinotherium giganteum* (Fig. 5), is one-fourth of that of running African and Asiatic elephants

derived from the predicted Froude numbers (Fig. 4B) by re-scaling method.

*Tyrus* = 2.10 (shown by star in Fig. 4B) for the experimentally modeled

*walk* ≤ 1 is generally expected. The suggested scenario of

pattern of (iii) *walkers*, for which *Fr*(mod)

with the data *Fr*(exp)

having been a good walker.

(Kokshenev, 2011b, Fig. 2).

The top-speed dynamic regimes, broadly related to maximal body mobility, should not be expected for large animals (Garland, 1983; Christiansen, 2002b) especially if they move at near critical conditions. Within the framework of the current approach to critical locomotion, the top speed of the fastest land mammal, the cheetah (Sharp, 1997), conventionally separating small and large fast-running animals (Jones & Lindstedt, 1993) and possibly determining a crossover in mammalian scaling trends of the appendicular skeleton (Christiansen, 1999), corresponds to the non-critical domain of body safety function discussed in Eq. (16) and the data *Smobil* = 3/2 (Kokshenev, 2007) for the body mobility function discussed in Eq. (18). The non-critical linear dynamic similarity as seen in optimal locomotion by stride speeds *V*(mod) *opt* <sup>∼</sup> *<sup>M</sup>*1/6and natural frequencies *<sup>T</sup>*(exp)−<sup>1</sup> *opt* ∼ *<sup>M</sup>*1/6 in running bipeds (Gatesy & Biewener, 1991) and trotting and galloping quadrupeds (Heglund & Taylor, 1988; Farley et al., 1993) can be explained by the optimal-speed regime of muscle activation shown in Table 4. As such, dynamic similarity was shown (Kokshenev, 2009, 2010, 2011a) to be determined by minimum mechanical muscle action controlled during contractions by the condition of *linear* dynamic-length changes with muscle length *Lm* (Table 4). The non-critical *bilinear* dynamic regime (maximum-power regime, Table 4) was proven to be responsible for muscle design adaptation to the primary locomotory functions (Kokshenev, 2008), whereas the higher non-linear regime of contraction of fast and slow muscles is associated here with near critical loading dynamic conditions (1 <sup>&</sup>lt; *Smobil* <sup>≤</sup> 1.1, Fig. 3) resulting in model speeds *<sup>V</sup>*(mod) *walk* ∼ *<sup>M</sup>*−1/3 *max* and *<sup>V</sup>*(mod) *run* <sup>∼</sup> *<sup>M</sup>*−1/6 *max* , which follow from Table 4. When the near critical regime speeds are compared with maximal speeds (limited by 2 *m*/*s* for large quadrupedal sauropods and by 4 *m*/*s* for bipedal dinosaurs) obtained by Tulborn (1990) using graphs of relative stride length against Froude number (Alexander, 1976), the estimates *V*(mod) *walk* <sup>=</sup> 8.3*M*−1/3 *max* and *<sup>V</sup>*(mod) *run* <sup>=</sup> 8.1*M*−1/6 *max* (masses are taken in *tons*) for the relatively slow-walking *Argentinosaurus* and slow-running *Giganotosaurus* (Table 1) are suggested. However, these estimates do not corroborate the idea of a continuos-speed evolution in gaits during a crossover from striding bipedalism (trait 5) to quadrupedalism (trait 4), as generally expected near the crossover-gait duty factor of around one half (Table 1) at a transient critical mass of around 25 *tons* (Fig. 2). Indeed, the estimates suggested for speeds within the scenario of a one-step continuous transition are not consistent, since they do not match when taken at the transient critical mass. We therefore propose a *two-step scenario* for evolution of locomotory functions: from (i) good bipedal running ability (trait 23), via (ii) decreasing bipedal running ability (trait 22), to (iii) quadrupedalism (trait 4).

Instead of speeds, the study of locomotor dynamic similarity in different-sized animals through Froude numbers is more appropriate (Alexander, 1976; Alexander & Jayes, 1983, Gatesy & Biewener, 1991). Most mammals appear to change their gait from walking to running discontinuously (abruptly) at a duty factor close to one half and Froude numbers below one (Ahlborn & Blake, 2002; Alexander & Jayes, 1983, Gatesy & Biewener, 1991). Extant elephants, however, exhibit a *continuous* walk-to-run transition at magnitudes *Fr*(exp) *<sup>w</sup>*-*<sup>r</sup>* <sup>≈</sup> <sup>1</sup> (Hutchinson et al., 2006; for comparative analysis of quadrupeds see Kokshenev, 2011b, Fig. 1). Mapping this transient-state similarity point onto the transient critical mass discussed in the context of similarity in locomotor evolution in extinct giants, Eq. (23) describes the crossover from striding bipedalism (trait 5) to quadrupedalism (trait 4). Then, when introducing the two-step evolution scenario via locomotor traits 23→ 22→ 24 (Fig. 4) in the run-mode domain (0.5 < *β* ≤ 1), we determine two locomotory patterns: (i) *good* 20 Will-be-set-by-IN-TECH

The top-speed dynamic regimes, broadly related to maximal body mobility, should not be expected for large animals (Garland, 1983; Christiansen, 2002b) especially if they move at near critical conditions. Within the framework of the current approach to critical locomotion, the top speed of the fastest land mammal, the cheetah (Sharp, 1997), conventionally separating small and large fast-running animals (Jones & Lindstedt, 1993) and possibly determining a crossover in mammalian scaling trends of the appendicular skeleton (Christiansen, 1999), corresponds to the non-critical domain of body safety function discussed in Eq. (16) and the data *Smobil* = 3/2 (Kokshenev, 2007) for the body mobility function discussed in Eq. (18). The non-critical linear dynamic similarity as seen in optimal locomotion by stride speeds

Biewener, 1991) and trotting and galloping quadrupeds (Heglund & Taylor, 1988; Farley et al., 1993) can be explained by the optimal-speed regime of muscle activation shown in Table 4. As such, dynamic similarity was shown (Kokshenev, 2009, 2010, 2011a) to be determined by minimum mechanical muscle action controlled during contractions by the condition of *linear* dynamic-length changes with muscle length *Lm* (Table 4). The non-critical *bilinear* dynamic regime (maximum-power regime, Table 4) was proven to be responsible for muscle design adaptation to the primary locomotory functions (Kokshenev, 2008), whereas the higher non-linear regime of contraction of fast and slow muscles is associated here with near critical loading dynamic conditions (1 <sup>&</sup>lt; *Smobil* <sup>≤</sup> 1.1, Fig. 3) resulting in model speeds *<sup>V</sup>*(mod)

*<sup>M</sup>*−1/3 *max* and *<sup>V</sup>*(mod) *run* <sup>∼</sup> *<sup>M</sup>*−1/6 *max* , which follow from Table 4. When the near critical regime speeds are compared with maximal speeds (limited by 2 *m*/*s* for large quadrupedal sauropods and by 4 *m*/*s* for bipedal dinosaurs) obtained by Tulborn (1990) using graphs of relative stride

*<sup>V</sup>*(mod) *run* <sup>=</sup> 8.1*M*−1/6 *max* (masses are taken in *tons*) for the relatively slow-walking *Argentinosaurus* and slow-running *Giganotosaurus* (Table 1) are suggested. However, these estimates do not corroborate the idea of a continuos-speed evolution in gaits during a crossover from striding bipedalism (trait 5) to quadrupedalism (trait 4), as generally expected near the crossover-gait duty factor of around one half (Table 1) at a transient critical mass of around 25 *tons* (Fig. 2). Indeed, the estimates suggested for speeds within the scenario of a one-step continuous transition are not consistent, since they do not match when taken at the transient critical mass. We therefore propose a *two-step scenario* for evolution of locomotory functions: from (i) good bipedal running ability (trait 23), via (ii) decreasing bipedal running ability (trait 22), to (iii)

Instead of speeds, the study of locomotor dynamic similarity in different-sized animals through Froude numbers is more appropriate (Alexander, 1976; Alexander & Jayes, 1983, Gatesy & Biewener, 1991). Most mammals appear to change their gait from walking to running discontinuously (abruptly) at a duty factor close to one half and Froude numbers below one (Ahlborn & Blake, 2002; Alexander & Jayes, 1983, Gatesy & Biewener, 1991). Extant elephants, however, exhibit a *continuous* walk-to-run transition at magnitudes *Fr*(exp) *<sup>w</sup>*-*<sup>r</sup>* <sup>≈</sup> <sup>1</sup> (Hutchinson et al., 2006; for comparative analysis of quadrupeds see Kokshenev, 2011b, Fig. 1). Mapping this transient-state similarity point onto the transient critical mass discussed in the context of similarity in locomotor evolution in extinct giants, Eq. (23) describes the crossover from striding bipedalism (trait 5) to quadrupedalism (trait 4). Then, when introducing the two-step evolution scenario via locomotor traits 23→ 22→ 24 (Fig. 4) in the run-mode domain (0.5 < *β* ≤ 1), we determine two locomotory patterns: (i) *good*

length against Froude number (Alexander, 1976), the estimates *V*(mod)

*opt* ∼ *<sup>M</sup>*1/6 in running bipeds (Gatesy &

*walk* ∼

*walk* <sup>=</sup> 8.3*M*−1/3 *max* and

*V*(mod)

quadrupedalism (trait 4).

*opt* <sup>∼</sup> *<sup>M</sup>*1/6and natural frequencies *<sup>T</sup>*(exp)−<sup>1</sup>

*runners* capable of showing relatively high numbers *Fr*(mod) *run* <sup>&</sup>gt; 1.5, since they have limbs mostly subjected to and, thus, adapted to near-critical bending stress of the domain, say, 0.8 *β* < 1, and (ii) *poor runners* attributed to Froude numbers lying in the transient-state domain 1 <sup>&</sup>lt; *Fr*(mod) *<sup>w</sup>*-*<sup>r</sup>* <sup>≤</sup> 1.5 and using moderate bending and moderate torsional stress indicated by 0.5 < *β* 0.8 . The walk-mode domain (0 < *β* ≤ 0.5) establishes the locomotory pattern of (iii) *walkers*, for which *Fr*(mod) *walk* ≤ 1 is generally expected. The suggested scenario of evolution in giants of gait-dependent *Froude function Fr*(*M*) is found to be in good agreement with the data *Fr*(exp) *Tyrus* = 2.10 (shown by star in Fig. 4B) for the experimentally modeled 6-*ton Tyrannosaurus* showing relatively high running speed 8 *m*/*s* (Sellers & Manning, 2007). One may also expect that more reliable estimates for critical speeds discussed above could be derived from the predicted Froude numbers (Fig. 4B) by re-scaling method.

It generally follows from Fig. 4 that with increasing body mass the evolution of locomotory patterns from *good runners* through *poor runners* to *walkers* is controlled by decreasing bending stress and increasing torsional elastic stress in the limbs. Based on the corresponding indicators *β* and *τ*, we are broadly able to characterize the accompanied evolution in postures and limb proportions, changing in striding running bipeds from (i) erect posture (trait 1) and more cursorial limbs (trait 30) to (iii) sprawling, or less erect posture (trait 2) and less cursorial limb proportions (trait 31), attributed to quadrupeds. The intermediate biped-to-quadruped locomotory evolution indicated by near equal proportions of moderate bending and torsional stress in limbs, may roughly be approached by elephantid and non-elephantid proboscideans, having rather short and compact bodies and fairly long limbs; these animals were undoubtedly not fast running with a suspended phase and therefore are found (Fig. 4) to be poor runners (trait 22). The largest land animal, *Argentinosaurus*, certainly could not run at all, but grouping in the walk-mode domain, it may be broadly regarded as having been a good walker.

While locomotor ability is provided by the relative body force output, body mobility is generally associated with the corresponding reaction-body elastic force, provided in most part by limb bone elastic forces. The evolution of body mobility, resulting from turning and bending limb mobilities, moderately decreasing with body mass (Fig. 5), is concomitant with a suggested evolution from striding bipedalism (trait 5) to quadrupedalism (trait 4). The walking and running mobilities are distinguished through the walk-mode and run-mode domains of the bone-stress similarity indicators. The animals considered by Hutchinson (2006) to possess good running abilities (trait 23) as well as improved turning abilities (trait 25) are presented (Figs. 4 and 5) by a single pattern of good runners, which relatively high mobility is due to bending effects exceeding torsional effects in the limb bones. The animals characterized by poor turning ability (trait 24) and poor bending ability are treated as walkers constrained by relatively high-level torsional elastic stress developed in long bones.

As can be inferred from Fig. 5, smaller quadrupedal giants from the Proboscidea, poorly running elephants at relatively high safety factors, may expose the same body mobility as the slow running largest bipedal giants of the Theropoda, operating close to critical levels. The body mobility of extinct elephants, moving at near critical levels, e.g., presented by *Deinotherium giganteum* (Fig. 5), is one-fourth of that of running African and Asiatic elephants (Kokshenev, 2011b, Fig. 2).

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### **6. Acknowledgments**

One of the authors (V.B.K.) acknowledges financial support by the national agency FAPEMIG.

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