## Abstract

The main purpose of this chapter is to propose a novel boundary element modeling and simulation algorithm for solving fractional bio-thermomechanical problems in anisotropic soft tissues. The governing equations are studied on the basis of the thermal wave model of bio-heat transfer (TWMBT) and Biot’s theory. These governing equations are solved using the boundary element method (BEM), which is a flexible and effective approach since it deals with more complex shapes of soft tissues and does not need the internal domain to be discretized, also, it has low RAM and CPU usage. The transpose-free quasi-minimal residual (TFQMR) solver are implemented with a dual-threshold incomplete LU factorization technique (ILUT) preconditioner to solve the linear systems arising from BEM. Numerical findings are depicted graphically to illustrate the influence of fractional order parameter on the problem variables and confirm the validity, efficiency and accuracy of the proposed BEM technique.

### Keywords

- boundary element method
- modeling and simulation algorithm
- bio-heat transfer
- fractional bio-thermomechanical problems
- anisotropic soft tissues

## 1. Introduction

Human body is a complex thermal system, Arsene d’Arsonval and Claude Bernard have shown that the temperature difference between arterial blood and venous blood is due to oxygenation of blood [1]. A large number of research papers in bio-heat transfer over the past few decades have focused on an understanding of the impact of blood flow on the temperature distribution within living tissues. Pennes [2] was the first attempt to explain the temperature distribution in human tissue with blood flow effect. The improvement of numerical models for determination of temperature distribution in living tissues has been a topic of interest for numerous researchers. Askarizadeh and Ahmadikia [3] introduced analytical solutions for the transient Fourier and non-Fourier bio-heat transfer equations. Li et al. [4] studied the bio-thermomechanical interactions within the human skin in the context of generalized thermoelasticity.

Analytical solutions for the current problem [5, 6] are very difficult to obtain, so numerical methods have become the main way for solving these problems [7, 8, 9, 10]. The boundary element method (BEM) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] is one of the numerical methods used to solve the current general problem [22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Generally, Laplace-domain fundamental solutions are easier to obtain than time-domain fundamental solutions for engineering and scientific problems [32, 33]. consequently, the BEM is very helpful for problems that did not have time-domain fundamental solutions, because it requires the Laplace-domain fundamental solutions of the problem’s governing equations. So, BEM expands the range of engineering problems that can be solved with the classical time-domain BEM.

The main aim of this chapter is to propose a new boundary element fractional model for describing the bio-thermomechanical properties of anisotropic soft tissues. The dual reciprocity boundary element method has been used to solve the TWMBT for obtaining the temperature distribution, and then the BEM has been used to obtain the displacement and stress at each time step. The linear systems from BEM were solved by the TFQMR solver with the ILUT preconditioner which reduces the number of iterations and the total CPU time.

A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of bio-thermomechanical problems in anisotropic soft tissues Section 2 describes the BEM modeling of the bio-thermomechanical interactions and introduces the partial differential equations that govern its related problems. Section 3 outlines the dual reciprocity boundary element method (DRBEM) for temperature field. Section 4 discusses the convolution quadrature boundary element method (CQBEM) for poro-elastic field. Section 5 presents the new numerical results that describe the bio-thermomechanical problems in anisotropic soft tissues.

## 2. Formulation of the problem

Consider an anisotropic soft tissue in the Cartesian coordinate system

The governing equations which model the fractional bio-thermomechanical problems in anisotropic soft tissues can be written as follows [34, 35].

where the fluid was modeled by the following Darcy’s law [36].

The fractional order equation which describes the TWMBT can be expressed as [37].

where

According to Bonnet [39], our problem can be expressed as a matrix system as [40].

where

## 3. Boundary element implementation for bioheat transfer field

Through this chapter, we supposed that

According to finite difference scheme of Caputo [22] and using the fundamental solution of difference equation resulting from fractional bio-heat Eq. (11) [41], we can write the following dual reciprocity boundary integral equation

in which

where

where

The discretization process for Eq. (12) leads to

After interpolation and integration processes over boundary elements, Eq. (15) can be expressed as

The matrix form of Eq. (16) can be written using (14) as

which also can be written

where

The boundary and initial conditions

The time discretization has been performed as follows

Substituting from Eqs. (22)–(25) into (20), we obtain

Thus, with the temperature

## 4. Boundary element implementation for the poro-elastic fields

The representation formula of (8) that describes the unknown field

where

For anisotropic case, the Laplace domain fundamental solution

where the solid displacement fundamental solution

with

which can be expressed as [36].

The fundamental solution of solid displacement

in which

The remaining parts of

On the basis of limiting process

According to limiting process

where

and

By using (39)-(42), we can write

By applying the inverse Laplace transform, we obtain

where

According to [40], the fundamental solution is

On the basis of Stokes theorem, we obtain

which can be expressed as

On the basis of [40], we get

in which

By applying (48) to a formula

Making use of (34) and (45), we can express

On the basis of [40], we obtain

which may be expressed using (34) as

By applying (29) (53), we obtain

Based on [42], we have

In the in right-side of (55), we can write second term as follows

in which

According to [40], we can write

By augmenting

According to [41], we get

where

On the basis of Lubich [44, 45], the integration weights

Polar coordinate transformation

Substitution of Eq. (63) into Eq. (61), we get

with

According to the procedure [43], the convolution operator (44) can be expressed as

which may be written as

Let the boundary

Now, we assume that we have

where

where

Thus, from (67), we can write the following

## 5. Numerical results and discussion

In the current study, a Krylov subspace iterative method is used for solving the resulting linear systems. In order to reduce the number of iterations, a dual threshold incomplete LU factorization technique (ILUT) which is one of the well-known preconditioning techniques is implemented as a robust preconditioner for TFQMR (Transpose-free quasi minimal residual) [46] to accelerate the convergence of the solver TFQMR.

To illustrate the numerical calculations computed by the proposed technique, the physical parameters for transversely isotropic soft tissue are given as follows [47]:

The elasticity tensor

in which

where

and therefore

where

Since for strongly anisotropic soft tissue both bulk moduli are positive, we used the following physical parameters for anisotropic soft tissue [48].

and therefore

and other constants considered in the calculations are as follows.

The domain boundary of the current problem has been discretized into 21 boundary elements and 42 internal points as depicted in Figure 2. The computation was done using Matlab R2018a on a MacBook Pro with 2.9GHz quad-core Intel Core i7 processor and 16GB RAM.

Figure 3 shows the variation of the temperature

Figure 4 illustrates the variation of the displacement

Figure 5 shows the variation of the displacement

Figure 6 shows the variation of the fluid pressure

Figure 7 shows the variation of the bio-thermal stress

Since there are no findings available for the problem under consideration. Therefore, some literatures may be regarded as special cases from our general problem. In the special case under consideration, the results of the bio-thermal stress caused by coupling between the temperature and displacement fields are plotted in Figure 8 to illustrate the variation of the bio-thermal stress

## 6. Conclusion

A novel boundary element model based on the TWMBT and Biot’s theory was established for describing the bio-thermomechanical interactions in anisotropic soft tissues.

The bio-heat transfer equation has been solved using the dual reciprocity boundary element method (DRBEM) to obtain the temperature distribution.

The mechanical equation has been solved using the convolution quadrature boundary element method (CQBEM) to obtain the displacement and fluid pressure for different temperature distributions at each time step.

Due to the advantages of DRBEM and CQBEM such as dealing with more complex shapes of soft tissues and not needing the discretization of the internal domain, also, they have low RAM and CPU usage. Therefore, they are a versatile and powerful methods for modeling of fractional bio-thermomechanical problems in anisotropic soft tissues.

The linear systems resulting from BEM have been solved by TFQMR solver with the ILUT preconditioner which reduces the number of iterations and the total CPU time.

Numerical findings are presented graphically to show the effect of fractional order parameter on the problem variables temperature, displacements and fluid pressure.

Numerical findings confirm the validity, efficiency and accuracy of the proposed BEM technique.

The proposed technique can be applied to a wide variety of fractional bio-thermomechanical problems in anisotropic soft tissues.

For open boundary problems of soft tissues, such as the considered problem, the BEM users need only to deal with real geometry boundaries. But for these problems, FDM and FEM use artificial boundaries, which are far away from the real soft tissues. Also, these artificial boundaries are also becoming a big challenge for FDM users and FEM users. So, BEM becomes the best method for the considered problem.

The presence of fractional order parameter in the current study plays a significant role in all the physical quantities during modeling and simulation in medicine and healthcare.

From the research that has been performed, it is possible to conclude that the proposed BEM is an easier, effective, predictable, and stable technique in the treatment of the bio-thermomechanical soft tissue models.

It can be concluded from this chapter that Biot’s equations for the dynamic response of poroelastic media can be combined with the bio-heat transfer models to describe the fractional bio-thermomechanical interactions of anisotropic soft tissues.

Current numerical results for our complex and general problem may provide interesting information for researchers and scientists in bioengineering, heat transfer, mechanics, neurophysiology, biology and clinicians.

## Nomenclature

A=ϕ1+Q/R | Biot’s coefficient |

Bx˜e | linear elastostatics operator |

Γ | considered boundary |

ΓD | Dirichlet boundary |

ΓN | Neumann boundary |

C | specific heat of soft tissue |

C | shape factor |

Cb | specific heat of the blood |

Cpjkl | specific heat of the blood |

F | bulk body forces |

ĝD | Dirichlet datum |

ĝN | Neumann datum |

K | dynamic permeability |

K | thermal conductivity of soft tissue |

m | iterative parameter |

p | pore pressure |

P0τ | heating power |

Q,R | solid–fluid coupling parameters |

Qmet | metabolic heat source |

Qext | external heat source |

S0 | scattering coefficient |

T | soft tissue temperature |

T⌣b | arterial blood temperature |

Txe | traction derivative |

u | solid displacement |

uf | fluid displacement |

V=Vf+Vs | bulk volume |

Vf | fluid volume |

Vs | solid volume |

Wb | blood perfusion rate |

B | stress-temperature coefficients |

∈ | linear strain tensor |

ζ | fluid volume variation |

ρ=ρs1−ϕ+ϕρf | bulk density |

ρs=Cϕρf | mass density of soft tissue |

ρf | blood density |

σ | total stress tensor |

τ | time |

τq | phase lag for heat flux |

τT | phase lag for temperature gradient |

φiαk | continuous polynomial shape functions |

ϕ=VfV | porosity |

ψjβk | discontinuous polynomial shape functions |

Ω | considered region |