Open access peer-reviewed chapter

Boundary Element Modeling and Simulation Algorithm for Fractional Bio-Thermomechanical Problems of Anisotropic Soft Tissues

Written By

Mohamed Abdelsabour Fahmy

Submitted: 09 October 2020 Reviewed: 27 January 2021 Published: 05 April 2021

DOI: 10.5772/intechopen.96268

From the Edited Volume

Recent Developments in the Solution of Nonlinear Differential Equations

Edited by Bruno Carpentieri

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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.

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2. Formulation of the problem

Consider an anisotropic soft tissue in the Cartesian coordinate system Ox1x2x3 as shown in Figure 1. It occupies the region Ω=x1x2x3:0<x1<α¯0<x2<β¯0<x3<γ¯ with boundary Γ that is subdivided into two non-intersective parts ΓD and ΓN.

Figure 1.

Geometry of the current problem.

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

TσT+F=ρu¨+ϕρfu¨fu¨E1
ζ̇+Tq=0E2
σ=CailgtrApIBθ,E3
=12uT+uTTE4
ζ=Atr+ϕ2RPE5

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

q=Kp+ρfu¨+ρa+ϕρfϕu¨fu¨E6

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

KTrτ+WbCbTbT+Qmet+Qext+τ¯α!WbCbDταT+DταQmet+DταQext=ρCτ¯α!Dτα+1T+Tτ,0<α1E7

where σ, , Cajlg, ρ=ρs1ϕ+ϕρf, ρs, ρf, u, uf, FF and qq 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, B are stress-temperature coefficients, tr denotes the trace, A=ϕ1+Q/R is Biot’s coefficient, Q and R are the solid–fluid coupling parameters, p is the fluid pressure in the vasculature, ζ is the fluid volume variation measured in unit reference volume, ϕ=VfV is the porosity, Vf is the fluid volume, V=Vf+Vs is the bulk volume, Vs is the solid volume, τ is the time, K is the permeability, ρa=Cϕρf where C=0.66 at low frequency [38], K is the soft tissue thermal conductivity, Wb is the blood perfusion rate, Cb is the blood specific heat, Tb is the arterial blood temperature, T is the soft tissue temperature, τ¯ is the thermal relaxation time ρ is the soft tissue density, C is the soft tissue specific heat, Qmet is the metabolic heat generation and Qext is the external heat generation.

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

B̂x˜ûgx˜=0forx˜Ωûgx=ĝDforxΓDt̂gx=ĝNforxΓNE8

where

B̂x˜=Bx˜e+s2ρβρfIαβx˜Bx˜sαβx˜TβsρfΔx˜+sϕ2R0E9
t̂gx=Txeαnx0nxTβsρfnxTx0ûxp̂xθx,β=ϕ2sKρfϕ2+sKρa+ϕρfE10
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3. Boundary element implementation for bioheat transfer field

Through this chapter, we supposed that Qmet and Tb are constants and θrτ=TrτTr0. Thus, Eq. (7) can be written as

ρCτ¯α!Dτα+1θ+ρCθτ+τ¯α!WbCbDταθ+WbCbθ=K2θx2+q,0<α1E11

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

Ciθi+ΓqθdΓΓθqdΓ=j=1N+LαjCjθ̂ij+Γqθ̂jdΓΓθq̂jdΓE12

in which

Ci=γ2π,q=θn,q=θn,θ=ln1rE13

where n is the outward unit normal vector to boundary Γ, r is the distance between source point i and considered point j, N is the number of boundary nodes and L is the number of internal nodes.

where

α=R1f˜=R1a˜2θτ2+b˜θτ+c˜θ+d˜E14

The discretization process for Eq. (12) leads to

Ciθi+k=1NΓkqθdΓk=1NΓkθqdΓ=j=1N+LαjCiθ̂ij+k=1NΓkZikθ̂kjdΓk=1NΓkGikq̂kjdΓE15

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

Ciθi+k=1NZikθkk=1NGikqk=j=1N+LαjCiθ̂ij+k=1NZikθ̂kjk=1NGikq̂kjE16

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

Gq=ZΘ̂GQ̂R1a˜2θτ2+b˜θτ+c˜θ+d˜E17

which also can be written

Xa˜2θτ2+b˜θτ+c˜θ+d˜+=GqE18

where

X=ZΘ̂GQ̂R1

The boundary and initial conditions

θxyτ=0E19
θxy0τ=ϑxy0=0E20
θxy0=1C0.02x,y0.020otherx,yE21

The time discretization has been performed as follows

q=1θqqm+θqqm+1E22
θ=1θuθm+θuθm+1E23
θτ=1Δτθm+1θmE24
2θτ2=1Δτ2θm+1+θm12θmE25

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

Xa˜Δτ2+Xb˜Δτ+Xc˜θu+θuZθm+1θqGqm+1+Xd˜=2Xa˜Δτ2+Xb˜ΔτXc˜1θuZ1θuθmXa˜Δτ2θm1+1θqGqmE26

Thus, with the temperature θ determined, the remaining task is to solve the problem (8).

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4. Boundary element implementation for the poro-elastic fields

The representation formula of (8) that describes the unknown field ûg can be written as

ûgx˜=V̂t̂gΩx˜K̂ûgΩx˜forx˜ΩE27

where

V̂t̂gΩx˜=ΓÛTyx˜t̂gydsyE28
K̂ûgΩx˜=ΓT̂yÛTyx˜ûgydsyE29

For anisotropic case, the Laplace domain fundamental solution Ûr and the corresponding traction T̂v can be expressed as [40].

Ûr=ÛsrÛfr0P̂sTrP̂fr0,T̂y=Tyesany0βnyTβsρfnyT0withryxE30

where the solid displacement fundamental solution Ûsr may be expressed as

Ûsr=14πrρβρfR1k42k22k12k22ek1rR2k42k12k12k22ek2r+Ik32R3ek3rE31

with

Rj=3yryTrIr2+kj3yryTrIr+kj2yryTrE32

which can be expressed as [36].

Ûsr=14πμrλ+2μλ+μyryTr+Iλ+3μ+Or0E33

The fundamental solution of solid displacement Ûsr can be dismantled into singular Ûssr and regular Ûrsr parts as

Ûsr=Ûssr+Ûrsrwithryx=1μIΔyλ+μλ+2μyyTΔyx̂r1μk12+k22Δyk12k22Ik12+k22k42k12k22k32yyTx̂rE34

in which

x̂r=14πrek1rk22k12k32k12+ek2rk22k12k22k32+ek3rk12k32k22k32=1k12k22k12k32k32k22+Or2E35

The remaining parts of Ûr as in (30) can be described as [36].

Ûfr=ρfαβyr4πrβλ+2μk12k22k1+1rek1rk2+1rek2r=Or0E36
P̂sr=Ûfrs=Or0E37
P̂fr=sρf4πrβk12k22k12k42ek1rk22k42ek2r=sρf4πrβ+Or0E38

On the basis of limiting process x˜ΩxΓ on (28) we get

limx˜ΩxΓV̂t̂gΩx˜=V̂x̂gxΓÛTyxt̂gydsyE39

According to limiting process x˜ΩxΓ on (28) we obtain [42].

limx˜ΩxΓK̂ûgΩx˜=Ix+Cxûgx+K̂ûgxE40

where

Cx=limε0yΩ:yx=εT̂yÛTyxdsyE41

and

K̂ûgx=limε0yxεT̂yÛTyxûgydsyE42

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

Cxûgx=V̂t̂gxK̂ûgxE43

By applying the inverse Laplace transform, we obtain

Cxugxt=VtgxtKugxtE44

where is the time convolution.

According to [40], the fundamental solution is

T̂yÛT=T̂yesanyβnyTβsρ0fnyTyÛsÛfP̂sTP̂fT=T̂sT̂fQ̂sTQ̂fE45

On the basis of Stokes theorem, we obtain

Γy×anydsy=Γ,avdγy=ϕ,avdγy=0E46

which can be expressed as

Γny×yadsy=0E47

On the basis of [40], we get

ΓMyadsy=0E48

in which My=yyTTyyT.

By applying (48) to a formula a=vu we obtain [43].

ΓMyvudsy=Γ,vMyudsyE49
ΓMyvTudsy=ΓvTMyudsyE50

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

T̂sT=TyeÛsings+ÛregsT+P̂snyT=TyeÛsingsT+Or0E51

On the basis of [40], we obtain

T̂sT=λ+2μnyyTÛsingsμny×y×Ûsings+2μMyûsings+Or0E52

which may be expressed using (34) as

T̂sT=MyΔy2X̂+InTyΔy2X̂+2μMyÛsingsT+or0E53

By applying (29) (53), we obtain

k̂ûΩsx˜=ΓMyΔy2X̂û+InTyΔy2X̂û+2μMyÛsingsTû+or0ûdsyE54

Based on [42], we have

K̂ûΩsx˜=ΓΔy2X̂Myû+InTyΔy2X̂û+2μÛssMyû+or0ûdsyE55

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

nTyΔy2x̂r=nTyr4πr2+Or0E56

in which

Csx=Ixcxwithcx=ϕx4πE57

According to [40], we can write

limΩx˜xΓK̂ûΩsx˜=Ix1+cxûx+K̂ûsxE58

By augmenting Ûss to Ûs, we obtain

k̂ûΩsx˜=ΓΔy2x̂Myû+InTyΔy2x̂û+2μÛsMyû+Or0ûdsyE59

According to [41], we get

fgt=0tftτgτfort0TE60

where

fgtnk=0nωnkΔtf̂gtkE61

On the basis of Lubich [44, 45], the integration weights ωn are calculated using Cauchy’s integral formula as

ωnkΔtf̂12πiz=Rf̂γzΔtzn+1dzE62

Polar coordinate transformation z=Re is used with the trapezoidal rule to approximate the integral (62) as

ωnΔtf̂R1L+1=oLf̂sζnwithζ=e2πiL+1ands=γRζΔtE63

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

fgtnk=0NRnkN+1=0Nf̂sζnkgtkRnN+1=0Nf̂sĝsζnE64

with

ĝs=k=0NRkgtkζk.E65

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

Cxugxt=vtgxtkugxtE66

which may be written as

Cxûgxs=v̂t̂gxsk̂ûgxs,=0..NE67

Let the boundary Γ is discretized into Ne boundary elements τe as follows

ΓΓh=e=1NeτeE68

Now, we assume that we have

ShkΓN,hspanφiαki=1i,α1E69
ShkΓD,hspanψiβkj=1j,β0E70

where

ûgkxûhgkx=i=1iûh,igkφiαkxShkΓN,hE71
t̂gkxt̂hgkx=j=1jt̂h,jgkψjβkxShkΓD,hE72

where k=1,2,3,4 are the poro-elastic degrees of freedom, φiαk are i continuous polynomial shape functions and ψiβk are j piecewise discontinuous polynomial shape functions.

Thus, from (67), we can write the following N+1 algebraic systems of equations

V̂DDK̂DNV̂NDC+K̂NNt̂D,hgûN,hg=V̂DNC+K̂DDV̂NNK̂NDĝN,hgĝD,hg=0NE73
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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

Cablg=C11C12C13000C12C11C13000C13C13C33000000C44000000C44000000C66E74

in which

C11=E2v02EE01+v2Ev02+E0v1,C12=E2v02+EE0v1+v2Ev02+E0v1C13=EE0v2Ev02+E0v1,C33=E02v12Ev02+E0v1C44=μ0,C66=12C11C12E75

where

v=0.196,v0=0.163,μ0=20.98GPa,E=68.34GPa,E0=51.35GPaE76

and therefore

k1=108.39GPa,k2=21.70GPaE77

where E and E0 are the respectively, v and v0 are Poisson’s ratio in the isotropy plane and in the fiber direction respectively, and μ0 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 [48].

v=0.95,v0=0.49,μ0=20.98GPa,E=22kPa,E0=447kPaE78

and therefore

k1=1243kPa,k2=442kPaE79

and other constants considered in the calculations are as follows.

ρs=1600kg/m3,ρF=1113kg/m3, p=25MPap=25MPa,ϕ=0.15 and Q/R=0.65.(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 2.

Boundary element model of the current problem.

Figure 3 shows the variation of the temperature T along xaxis 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 3.

Variation of the temperature T along xaxis.

Figure 4 illustrates the variation of the displacement u1 along xaxis 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 u1.

Figure 4.

Variation of the displacement u1 along xaxis.

Figure 5 shows the variation of the displacement u2 along xaxis 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 u2.

Figure 5.

Variation of the displacement u2 along xaxis.

Figure 6 shows the variation of the fluid pressure p along xaxis 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 p.

Figure 6.

Variation of the fluid pressure p along xaxis.

Figure 7 shows the variation of the bio-thermal stress σ11 along xaxis 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 σ11.

Figure 7.

Variation of the bio-thermal stress σ11 along xaxis.

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 σ11 along xaxis 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 [49] and FEM results of Torvi and Dale [50] 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.

Figure 8.

Variation of the bio-thermal stress σ11 along xaxis for BEM, FDM and FEM.

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6. Conclusion

  1. 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.

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

  3. 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.

  4. 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.

  5. 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.

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

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

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

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. 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.

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Nomenclature

A=ϕ1+Q/RBiot’s coefficient
Bx˜elinear elastostatics operator
Γconsidered boundary
ΓDDirichlet boundary
ΓNNeumann boundary
Cspecific heat of soft tissue
Cshape factor
Cbspecific heat of the blood
Cpjklspecific heat of the blood
Fbulk body forces
ĝDDirichlet datum
ĝNNeumann datum
Kdynamic permeability
Kthermal conductivity of soft tissue
miterative parameter
ppore pressure
P0τheating power
Q,Rsolid–fluid coupling parameters
Qmetmetabolic heat source
Qextexternal heat source
S0scattering coefficient
Tsoft tissue temperature
T⌣barterial blood temperature
Txetraction derivative
usolid displacement
uffluid displacement
V=Vf+Vsbulk volume
Vffluid volume
Vssolid volume
Wbblood perfusion rate
Bstress-temperature coefficients
linear strain tensor
ζfluid volume variation
ρ=ρs1−ϕ+ϕρfbulk density
ρs=Cϕρfmass density of soft tissue
ρfblood density
σtotal stress tensor
τtime
τqphase lag for heat flux
τTphase lag for temperature gradient
φiαkcontinuous polynomial shape functions
ϕ=VfVporosity
ψjβkdiscontinuous polynomial shape functions
Ωconsidered region

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Written By

Mohamed Abdelsabour Fahmy

Submitted: 09 October 2020 Reviewed: 27 January 2021 Published: 05 April 2021