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.
- boundary element method
- modeling and simulation algorithm
- bio-heat transfer
- fractional bio-thermomechanical problems
- anisotropic soft tissues
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 . 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  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  introduced analytical solutions for the transient Fourier and non-Fourier bio-heat transfer equations. Li et al.  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 as shown in Figure 1. It occupies the region with boundary that is subdivided into two non-intersective parts and .
where the fluid was modeled by the following Darcy’s law .
The fractional order equation which describes the TWMBT can be expressed as .
where , , , , , , , , and are total stress tensor, linear strain tensor, constant elastic moduli, bulk density, solid density, fluid density, solid displacement, fluid displacement, bulk body forces and specific flux of the fluid, respectively, are stress-temperature coefficients, denotes the trace, is Biot’s coefficient, and are the solid–fluid coupling parameters, is the fluid pressure in the vasculature, is the fluid volume variation measured in unit reference volume, is the porosity, is the fluid volume, is the bulk volume, is the solid volume, is the time, is the permeability, where at low frequency , is the soft tissue thermal conductivity, is the blood perfusion rate, is the blood specific heat, is the arterial blood temperature, is the soft tissue temperature, is the thermal relaxation time is the soft tissue density, is the soft tissue specific heat, is the metabolic heat generation and is the external heat generation.
3. Boundary element implementation for bioheat transfer field
Through this chapter, we supposed that and are constants and . Thus, Eq. (7) can be written as
According to finite difference scheme of Caputo  and using the fundamental solution of difference equation resulting from fractional bio-heat Eq. (11) , we can write the following dual reciprocity boundary integral equation
where is the outward unit normal vector to boundary , is the distance between source point and considered point , is the number of boundary nodes and is the number of internal nodes.
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
The boundary and initial conditions
The time discretization has been performed as follows
Thus, with the temperature determined, the remaining task is to solve the problem (8).
4. Boundary element implementation for the poro-elastic fields
The representation formula of (8) that describes the unknown field can be written as
For anisotropic case, the Laplace domain fundamental solution and the corresponding traction can be expressed as .
where the solid displacement fundamental solution may be expressed as
which can be expressed as .
The fundamental solution of solid displacement can be dismantled into singular and regular parts as
The remaining parts of as in (30) can be described as .
On the basis of limiting process on (28) we get
According to limiting process on (28) we obtain .
By using (39)-(42), we can write
By applying the inverse Laplace transform, we obtain
where is the time convolution.
According to , the fundamental solution is
On the basis of Stokes theorem, we obtain
which can be expressed as
On the basis of , we get
in which .
By applying (48) to a formula we obtain .
Making use of (34) and (45), we can express as
On the basis of , we obtain
which may be expressed using (34) as
By applying (29) (53), we obtain
Based on , we have
In the in right-side of (55), we can write second term as follows
According to , we can write
By augmenting to , we obtain
According to , we get
Polar coordinate transformation is used with the trapezoidal rule to approximate the integral (62) as
According to the procedure , the convolution operator (44) can be expressed as
which may be written as
Let the boundary is discretized into boundary elements as follows
Now, we assume that we have
where are the poro-elastic degrees of freedom, are continuous polynomial shape functions and are piecewise discontinuous polynomial shape functions.
Thus, from (67), we can write the following algebraic systems of equations
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)  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 :
The elasticity tensor
where and are the respectively, and are Poisson’s ratio in the isotropy plane and in the fiber direction respectively, and is the shear moduli in any direction within a plane perpendicular to isotropy plane.
Since for strongly anisotropic soft tissue both bulk moduli are positive, we used the following physical parameters for anisotropic soft tissue .
and other constants considered in the calculations are as follows.
, , , and .(80)
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 along for different values of fractional order parameter. It can be seen from this figure that the fractional order parameter has a significant influence on the temperature.
Figure 4 illustrates the variation of the displacement along for different values of fractional order parameter. It can be seen from this figure that the fractional order parameter has a significant influence on the displacement .
Figure 5 shows the variation of the displacement along for different values of fractional order parameter. It can be seen from this figure that the fractional order parameter has a significant influence on the displacement .
Figure 6 shows the variation of the fluid pressure along for different values of fractional order parameter. It can be seen from this figure that the fractional order parameter has a significant influence on the fluid pressure .
Figure 7 shows the variation of the bio-thermal stress along for different values of fractional order parameter. It can be seen from this figure that the fractional order parameter has an important influence on 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 along for BEM, FDM and FEM, where the boundary of the special case problem has been discretized into 21, 42 and 84 boundary elements (bes). The validity, accuracy and efficiency of our proposed technique have been confirmed by a graphical comparison of the three different boundary elements (21, 42 and 84) with those obtained using the FDM results of Shen and Zhang  and FEM results of Torvi and Dale  for the special case under consideration, the increase of BEM boundary elements leads to improve the accuracy and efficiency of the BEM, also, it can be noted that the BEM findings are in excellent agreement with the FDM and FEM results, we refer the interested reader to recent work [51, 52, 53, 54, 55] for understanding the BEM methodology.
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.
linear elastostatics operator
specific heat of soft tissue
specific heat of the blood
specific heat of the blood
bulk body forces
thermal conductivity of soft tissue
solid–fluid coupling parameters
metabolic heat source
external heat source
soft tissue temperature
arterial blood temperature
blood perfusion rate
linear strain tensor
fluid volume variation
mass density of soft tissue
total stress tensor
phase lag for heat flux
phase lag for temperature gradient
continuous polynomial shape functions
discontinuous polynomial shape functions