**4.2 Challenging the mathematical modeling of MDD**

#### *4.2.1 The magnetostatic problem*

Considering the geometry of **Figure 4**, the analysis domain is limited to the scaffold and the fracture gap, neglecting the bone tumor and assuming that only healthy bone is present, in a way similar to [13]. The MagS and the gap have a radius *Biomedical Applications of Biomaterials Functionalized with Magnetic Nanoparticles DOI: http://dx.doi.org/10.5772/intechopen.89199*

of 5 mm. An external uniform and static magnetic flux density field of strength B0 is supposed to be applied along the z-axis of the system. The magnetic composite nanomaterial will magnetize in a nonlinear way according to the following relationship [13]:

$$\overline{M} = M\_\prime \phi \left( \coth \left( \zeta \right) - \frac{1}{\zeta} \right) \tag{16}$$

where all symbols have the previous definition. As presented in **Table 1**, the magnetization response of the scaffolds varies from a minimum of 0.4 emu�g�<sup>1</sup> to a maximum of 25 emu�g�1. Considering this nonlinear material property, the problem is the determination of the spatial distribution of the magnetic field, i.e., the solution of the following magnetostatic problem employing the scalar magnetic potential *ψ <sup>m</sup>* [13]:

$$\begin{aligned} \nabla \times H &= \mathbf{0} \\ \overline{H} &= -\nabla \boldsymbol{\mu}\_m \end{aligned} \tag{17}$$

Due to the presence of the magnetic material, the magnetic field flux lines concentrate in the prosthetic implant, implying that the norm of the gradient of magnetic density field between the MagS and the diamagnetic tissues is relevant [6]. In the literature, it is reported that if the magnetic density field gradients are higher than 1.3 Tm�1, then the magnetic force exerted on a population of surrounding MNPs would be sufficient to overcome their weight force and set them in motion toward the scaffold [13, 28]. This is a very simplified view of the problem. Indeed, several relevant physical and biological factors took part to the transport phenomena of MNP attraction to the MagS in the presence of a static magnetic field. As defined by Grief and Richardson, the magnetic force vector *Fm* on an ensemble of MNPs in saturation regime can be evaluated as follows [12]:

$$\overline{F}\_m = \frac{M\_{s2} V\_{m2}}{\mathfrak{G} k\_B T} \nabla \left| \overline{B} \right|^2 \tag{18}$$

where M*<sup>s</sup>*<sup>2</sup> and V*<sup>m</sup>*<sup>2</sup> are the saturation magnetization, in Am�1, and the volume of the spherical magnetic nanoparticles, in nm�3, to be attracted, respectively. The nanoparticles conjugated with growth factors or drugs are hence set in motion with a velocity v*<sup>m</sup>* equal to [12, 13]:

$$
\overline{v}\_m = \frac{\overline{F}\_m}{6\pi\eta r\_{m2}}\tag{19}
$$

where r*<sup>m</sup>*<sup>2</sup> is the radius of the magnetic carriers. The term *η* is the viscosity of the medium in which the nanoparticles move, in Pa�s. This medium is often assumed to be water (*ηw*=1�10�3Pa�s); however, actually the MNPs that move from the capillaries of bone tissues into the fracture gap are dragged in a solution of water, proteins (e.g., collagen, fibrin, and plasmin), and other macromolecules. Therefore, the extracellular matrix (ECM) can be assumed to be the medium in which the MNPs move, implying that *<sup>η</sup>w*=1�103Pa�s [13].

After having solved Eq. (17) and calculated Eqs. (18) and (19), the spatiotemporal distribution of the concentration of MNPs (C*mnp*, mol�m3) functionalized with the drug can be obtained computing the following diffusion-convection equation [13]:

the temperature rise is noticeable, and the external field should be modulated (or

*(a) 2D temperature distribution after 60 min of treatment using a RF magnetic field of 30 mT and working at 293 kHz. A OST with r*t*=0.5 mm is considered. (b) Average temperature in the region with residual FIB cells. (c) Average temperature in the region with residual OST cells. (MHA = magnetic hydroxyapatite).*

OST, give the high value of blood perfusion (see **Table 3**); not all MagS are able to treat successfully the residual cancer cells. This is the most challenging tumor. Indeed, the Fe-doped PCL scaffold fails to reach the lethal HT temperature for an OST of 0.5 mm, even increasing the amplitude to 40 mT or the frequency to 409 kHz. The magnetic hydroxyapatite scaffold is more effective in treating the residual osteosarcoma cells, as can be observed in **Figure 5c**. These results demonstrate that the in HT treatment of residual bone tumor cells is feasible, and, with the knowledge of the physico-chemical properties of the nanomaterial, the treatment

Magnetic scaffolds were conceived as a multifunctional platform for tissue engineering applications (see **Figure 1**) [1–5]. As presented in the Introduction, they are a platform for magnetically targeted drug delivery of growth factors to control and enhance tissue healing, such as in the case of bone tissue [1, 11]. The bio-nanotechnology research developed magnetic carriers of biomolecules such as VEGF or TGF-*β* [11, 27]; however, the problem of maximizing and controlling their delivery to the site of injury is still addressed in the literature [1–4, 13]. This section will cover the use of MagS, implanted in a damaged bone, as an in situ magnet, i.e., as attraction site for external MNPs carrying GFs. The influence on the cellular response is assessed employing a multiphysics model [13]. The prediction of the magnetic force required to attract the MNPs, the velocity in the extracellular matrix (ECM) medium, and the final spatial distribution is fundamental to foresee a treatment planning procedure, while evaluating the influence parameters in the drug

Considering the geometry of **Figure 4**, the analysis domain is limited to the scaffold and the fracture gap, neglecting the bone tumor and assuming that only healthy bone is present, in a way similar to [13]. The MagS and the gap have a radius

. In the case of

turned off) to keep the temperature closest to the target value of 42<sup>∘</sup>

*Smart Nanosystems for Biomedicine, Optoelectronics and Catalysis*

can be planned against different type of tumors.

delivery and the cellular migration process.

*4.2.1 The magnetostatic problem*

**12**

**4.2 Challenging the mathematical modeling of MDD**

**4.1 Magnetic drug delivery**

**Figure 5.**

**4. Magnetic scaffolds and regenerative medicine**

$$\frac{\partial \mathcal{C}\_{mnp}}{\partial t} = \nabla \cdot \left[ D\_{mnp} \nabla \mathcal{C}\_{mnp} \right] - \overline{\nu}\_m \nabla \mathcal{C}\_{mnp} \tag{20}$$

D*mnp* is the diffusion coefficient of MNPs in the medium, assumed to be equal to 10�9ms�2. The analytical mass balance is subject to the outflow condition at the scaffold surface, while a constant initial concentration of MNPs (C*m*, 0) is assumed at the host bone interface. In the fracture gap, it is assumed that C*mnp*ð Þ¼ *r*, *z*, *t* ¼ 0 0 as initial condition.

The magnetic field distribution (Eq. (17)) is derived by solving numerically the magnetostatic problem for the geometry depicted in **Figure 4** using the *Magnetic Fields No Currents* package from the *AC/DC module* of COMSOL Multiphysics. Then *vm* is inserted in the *Transport of Dilute Species* interface to solve Eq. (20).
