**4. The practicality of computer-aided mathematical model generation**

I will now turn to the question of whether model generation is practical. Numerous examples of my work in brain research provide evidence that formal generation of mathematical models has an attractive place in knowledge discovery. I will highlight oxygen exchange and blood flow in the aging brain to show how mathematical models can serve as an instrument of knowledge discovery. The mathematical modeling paradigm we propose for the brain is based on model generation from medical images, and we term it anatomical model generation. I demonstrate this in an example of the generation of the mathematical model for the mouse as well as the synthesis of vascular trees in humans. The guiding principle of generating mathematical models shown in **Figure 5** is the convergence of medical images into mathematical model representations. The schematic outlines this process. For instance, in this case study of the mouse brain, two-photon microscopy was used to acquire the anatomy of the brain cells and blood vessels in a large section of the primary somatosensory cortex of the mouse. My lab then used image segmentation to create vectorized image data to create an inventory of all the blood vessels and the cells capturing their precise location, diameter, and connectivity. This vectorized image data were encoded into a network graph using adjacency matrices to store connections

**29**

**Figure 6.**

*representation of vectorized data using graphs (fight).*

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

of nodes with arcs and property vectors for Cartesian dimensions, diameters, and sizes. Based on this image-derived domain representation, we have generated an anatomically concise network topology of the primary somatosensory cortex of the mouse. **Figure 4** shows examples of four different sample sections, which are digital representations of the same cortical region of the brain of four different animals. Once we have generated the topological representation, in this case the anatomy of the cerebral cortex, we can automatically generate the equations from this network. The network representation of the mouse brain enables the computer to perform the

task of synthesizing transport equations using a set of rules automatically.

oxygen extraction at an unprecedented level of detail as is shown in **Figure 8** for four samples of the primary somatosensory cortex. The diagram shows with

*Information flow for anatomical model generation from medical image. Two photon image acquisition of the mouse brain in an open cranial window (left), vectorization of multiscale data (middle) and network* 

**Figure 6** depicts the phenomenological description of the model generation methodology. Biphasic blood flow equations for mass conservation as well as a simplified momentum equation can be generated automatically for each node of the vascular network. Blood flow in the microcirculation does not behave like a single fluid, but at the microlevel, blood plasma and red blood cells act as a bi-phasic suspension. It is also very well known that red blood cells in the microcirculation do not travel uniformly between the branches of the microcirculation but essentially tend to concentrate in vessels with higher flow and larger diameter. Therefore, we need a biphasic representation of blood flow which was implemented with a very simple drift flux model of biphasic blood flow [16] depicted in **Figure 6**. This kinematic model is equivalent to a mixture with species of different volatilities; thus, red blood cells tend to divide into the thinner or thicker branch of a bifurcation as a function of their relative kinematic affinity (volatility) described by an empirical drift flux parameter, m. This simple hematocrit split rule allows us to predict the uneven distribution of red blood cells; the descriptive equations of the biphasic drift flux models are again automatically synthesized. Additional modeling details instantiate equations for oxygen transport to brain tissues, expressed by molar flux balances for red blood cells and oxygen unbinding from hemoglobin into plasma according to dissociation kinetics. The entire set of these complex model equations fit on a single piece of paper as shown in **Figure 7**. The anatomical modeling approach enables the automatic generation of system equations from information encoded in the network topology. Once these network equations are automatically generated, the numerical solution of these equations on the computer yields predictions of microcirculatory blood flow patterns and

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

#### *Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*

of nodes with arcs and property vectors for Cartesian dimensions, diameters, and sizes. Based on this image-derived domain representation, we have generated an anatomically concise network topology of the primary somatosensory cortex of the mouse. **Figure 4** shows examples of four different sample sections, which are digital representations of the same cortical region of the brain of four different animals. Once we have generated the topological representation, in this case the anatomy of the cerebral cortex, we can automatically generate the equations from this network. The network representation of the mouse brain enables the computer to perform the task of synthesizing transport equations using a set of rules automatically.

**Figure 6** depicts the phenomenological description of the model generation methodology. Biphasic blood flow equations for mass conservation as well as a simplified momentum equation can be generated automatically for each node of the vascular network. Blood flow in the microcirculation does not behave like a single fluid, but at the microlevel, blood plasma and red blood cells act as a bi-phasic suspension. It is also very well known that red blood cells in the microcirculation do not travel uniformly between the branches of the microcirculation but essentially tend to concentrate in vessels with higher flow and larger diameter. Therefore, we need a biphasic representation of blood flow which was implemented with a very simple drift flux model of biphasic blood flow [16] depicted in **Figure 6**. This kinematic model is equivalent to a mixture with species of different volatilities; thus, red blood cells tend to divide into the thinner or thicker branch of a bifurcation as a function of their relative kinematic affinity (volatility) described by an empirical drift flux parameter, m. This simple hematocrit split rule allows us to predict the uneven distribution of red blood cells; the descriptive equations of the biphasic drift flux models are again automatically synthesized. Additional modeling details instantiate equations for oxygen transport to brain tissues, expressed by molar flux balances for red blood cells and oxygen unbinding from hemoglobin into plasma according to dissociation kinetics.

The entire set of these complex model equations fit on a single piece of paper as shown in **Figure 7**. The anatomical modeling approach enables the automatic generation of system equations from information encoded in the network topology. Once these network equations are automatically generated, the numerical solution of these equations on the computer yields predictions of microcirculatory blood flow patterns and oxygen extraction at an unprecedented level of detail as is shown in **Figure 8** for four samples of the primary somatosensory cortex. The diagram shows with

#### **Figure 6.**

*Information flow for anatomical model generation from medical image. Two photon image acquisition of the mouse brain in an open cranial window (left), vectorization of multiscale data (middle) and network representation of vectorized data using graphs (fight).*

*Technology, Science and Culture - A Global Vision*

*batch operations and stream table for a batch recipe.*

BDK system was a model generation framework that used natural language input, synthesized equations, and created a virtual representation of materials and eventually solved these equations to predict the physiochemical state changes resulting from material transformations, phase separations, and reactions. The idea of using formal algorithms to support mathematical model generations has received attention in the community and continues to be an interesting research task in system science.

*List of operational tasks in the BDK batch for natural language input of chemical recipes (right). Flowsheet of* 

**4. The practicality of computer-aided mathematical model generation**

I will now turn to the question of whether model generation is practical. Numerous examples of my work in brain research provide evidence that formal generation of mathematical models has an attractive place in knowledge discovery. I will highlight oxygen exchange and blood flow in the aging brain to show how mathematical models can serve as an instrument of knowledge discovery. The mathematical modeling paradigm we propose for the brain is based on model generation from medical images, and we term it anatomical model generation. I demonstrate this in an example of the generation of the mathematical model for the mouse as well as the synthesis of vascular trees in humans. The guiding principle of generating mathematical models shown in **Figure 5** is the convergence of medical images into mathematical model representations. The schematic outlines this process. For instance, in this case study of the mouse brain, two-photon microscopy was used to acquire the anatomy of the brain cells and blood vessels in a large section of the primary somatosensory cortex of the mouse. My lab then used image segmentation to create vectorized image data to create an inventory of all the blood vessels and the cells capturing their precise location, diameter, and connectivity. This vectorized image data were encoded into a network graph using adjacency matrices to store connections

**28**

**Figure 5.**

$$0 = \nabla \cdot D\_{\mu \wr \rhd\_1} \cdot \nabla C\_{\mu \wr \rhd\_1 \smile \mu\_{\rho \wr}} \cdot \omega\_{\rho \wr} \cdot \nabla C\_{\mu \wr \rhd\_1} + \dot{R}\_{\rho \smile \nu^p} - S\_b \frac{U}{\varkappa \sigma\_b} \left(C\_{\mu \wr \rhd\_1} - C\_{B \boxtimes \heartsuit \rhd\_1}\right)$$

**31**

**Figure 9.**

*(capillaries).*

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

in the anatomical vessel hierarchy. Our research shows that the hematocrit distribution, the hydrostatic pressure, and red blood cell saturation all experience large variability as a function of the different pathways red blood cells can take to traverse the network. These findings are nonintuitive and have been revealed with the help of the anatomical mathematical model. Key findings include that hemodynamic states in the microcirculation are not uniform, but that the tissue is evenly oxygenated, and that pressure drop occurs mainly in the capillary evenly [17]. None to these findings were previously predicted by prior models that did not offer the fine-grained level of anatomical detail

**5. Growing circulatory trees to explore cortical oxygen transport in the** 

I attempt to address how oxygen is supplied to the human brain. In humans, it is not possible to access microcirculation data through open cranial windows as was shown for rodents; rather, a noninvasive approach was needed. My lab successfully deployed a model generation methodology to overcome this limitation. In humans, we used a modified constructive constrained optimization algorithm [18] originally developed by Wolfgang Schreiner for the synthesis of coronal arterial networks [19]. Schreiner randomly added segments to a main coronal arterial tree and determined the optimal segment location in the tree hierarchy, its coordinates, and segment diameters by minimizing the vascular tree volume subjected to flow conditions. Remarkably, when sequentially repeating the process of random segment addition followed by deterministic optimization, a tree emerges whose topology resembles natural vasculature. This discovery suggests that in nature, vascular trees grow in a manner that perfuses capillaries evenly, while at the same time, the segment diameters as well as locations of bifurcations are chosen so that the total required blood filling volume is at a minimum. We have modified this original algorithm to generate vascular structures for very complicated organs such as the brain. Our modified algorithm is versatile and is capable of delineating vascular structures in quite complicated domains. The example in **Figure 9** shows the initials of my laboratory (lppd-laboratory for product and process design) literally painted in blood. Each letter constitutes a physiological vascular tree that discharges the exact same amount of flow through its terminal nodes (capillaries). We have successfully used vascular synthesis to generate cerebrovascular models for rodents as well as for humans that are virtually indistinguishable from real vascular structures. Specifically, we made a computer-generated anatomical model of human microcirculation. **Figure 10** depicts a comparison between the synthetic vasculature structure and a real sample. Using an artificially generated human cortical structure, we were able to predict oxygen exchange in humans at a length scale that has not been acquired

*Demonstration of vascular growth in a complex domain. The example shows a synthetic blood vessel networks delineated the initials of the laboratory for product and process design-lppd. Each letter constitutes a physiological vascular tree that discharges the exact same amount of flow through its terminal nodes* 

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

shown in this research.

**human brain**

#### **Figure 7.**

*Overview of the equation generation mechanism for cortical blood flow and oxygen exchange mechanism.*

#### **Figure 8.**

*Predictions of hemodynamic states in microcirculation of mouse, blood pressure, hematocrit, and red blood cell (RBC) oxygen saturation.*

microscopic detail the distribution of red blood cells, blood oxygen saturation, the uneven distribution of hematocrit, and the patterns of blood pressure for any capillary and its surrounding tissue in the mouse cortex. The computer-generated mathematical model allows analysis at an unprecedented scale, down to the detail of individual cells or capillaries. The model predicted that the blood pressure is not uniform, but there are large deviations of hemodynamic states along different paths traversing the microcirculatory network. Previously, it was believed that in the capillary bed there are representative average conditions of pressures or oxygen saturation as a function of level *Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*

in the anatomical vessel hierarchy. Our research shows that the hematocrit distribution, the hydrostatic pressure, and red blood cell saturation all experience large variability as a function of the different pathways red blood cells can take to traverse the network. These findings are nonintuitive and have been revealed with the help of the anatomical mathematical model. Key findings include that hemodynamic states in the microcirculation are not uniform, but that the tissue is evenly oxygenated, and that pressure drop occurs mainly in the capillary evenly [17]. None to these findings were previously predicted by prior models that did not offer the fine-grained level of anatomical detail shown in this research.
