Applied Mathematics and Computation 259 (2015) 21–31
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Implementation of Smoothed-Particle Hydrodynamics for non-linear Pennes’ bioheat transfer equation J. Ghazanfarian a,⇑, R. Saghatchi a, D.V. Patil b,1 a b
Faculty of Engineering, Department of Mechanical Engineering, University of Zanjan, University Blvd., 45371-38791 Zanjan, Iran 42 George Nuttall Close, Cambridge, Cambridgeshire CB4 1YE, UK
a r t i c l e
i n f o
Keywords: Bioheat transfer Pennes’ equation SPH Dual-phase lag
a b s t r a c t Non-linear Pennes’ bioheat equation including the thermal wave model of bioheat transfer as well as the dual-phase-lag model has been discretized using a mesh-free Smoothed-Particle Hydrodynamics (SPH) procedure. The time evolution of the temperature distribution within a living tissue and effect of non-linearity in the PDEs on the solution have been investigated. The thermal conductivities for Fourier and non-Fourier based bioheat models are varied. The results obtained using the SPH method were compared with the previously published data for benchmark problems. It is shown that these SPH results show an excellent agreement in comparison with the previous results obtained using other numerical methods which proves suitability of SPH scheme towards simulations for bioheat equation. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction The determination of temperature distribution in biological living tissues is important in many medical therapies and physiological studies. These studies play a vital role in situations such as cryosurgery [1], frostbite [2], hyperthermia [3], skin burns [4], body thermal regulation [5], responses to the environmental conditions [6] as well as the tissue thermal stress cycles [7]. This motivates to establish an appropriate bioheat transfer model for the living tissues e.g. human body to obtain insights on thermal effects. So far, various bioheat transfer models have been introduced by some researchers [8–10]. One of the prominent and earliest bioheat models is the celebrated Pennes’ bioheat equation [10]. In this article, we have proposed a novel numerical approach of using a particle based method namely, Smoothed-Particle Hydrodynamics (SPH) for the solution of the Pennes’ bioheat equation. Due to the simplicity of this model, it has been used for the simulations involving various diverse biological applications. Here, we have reviewed few important earlier contributions. Lv and Liu [11] presented an approach to measure the local blood perfusion based on an analytical solution of the Pennes’ bioheat equation. Mital and Tafreshi [12] used this model to determine the optimal thermal damage in the magnetic nanoparticle hyperthermia cancer treatment. Ezzat et al. [13] introduced a mathematical model for Pennes’ bioheat equation. Their solution is based on the fractional calculus to study the temperature transients in the skin exposed to an instantaneous surface heating. On the other hand, a traditional model based on the Fourier’s law for heat conduction predicts an infinite rate of heat propagation known as the heat transfer paradox. However, for the modeling of problems such as heat transport in the living
⇑ Corresponding author. 1
E-mail addresses:
[email protected] (J. Ghazanfarian),
[email protected],
[email protected] (D.V. Patil). Present address: School of Engineering, The University of Edinburgh, King’s Building Campus, Mayfield Road, Edinburgh EH9 3JL, Scotland, UK.
http://dx.doi.org/10.1016/j.amc.2015.02.036 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
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tissues specially in the presence of a heat source (laser heating, or nanoscale heat transport [14]), the rate of heat propagation is finite. For the bioheat transfer applications, involving non-equilibrium heat transport a lagging behavior exists which leads to the finite time gap for heat exchange between participating tissue and the blood [15]. Therefore, a classical model of the Fourier-based Pennes’ bioheat equation does not precisely predict the temperature distribution within the tissue. Accordingly, in order to overcome the paradox of an infinite sound speed, numerous modifications have been proposed and employed in the past. We briefly review these models here. Cattaneo [16] and Vernotte [17] independently suggested a modified heat flux constitution law by considering a finite speed of propagation of the thermal disturbance. This model is referred as the CV model abbreviated using the first letters of the author’s names. In this model, a phase-lag is implemented for the heat flux vector. The first application of the CV model in the bioheat equation is known as the thermal wave model of bioheat transfer (TWMBT) and has been presented in [18,19]. Next, we review other known approximation of the non-Fourier heat conduction known as the dual-phase-lag (DPL) model. This model is proposed by Tzou [20,21] with considering the effect of a finite relaxation time by using the phase-lags of heat flux and temperature. A mathematical modeling of the skin bioheat transfer has been presented by Xu et al. [22] with three different versions of the DPL model. The literature is full of other articles which discuss the simulations of bioheat transfer by employing the DPL model as the constitutive law [23–27]. Zhao et al. [28] developed a two-level finite difference scheme for one-dimensional Pennes’ bioheat equation with a spacedependent thermal conductivity. Further, few researchers have focused on the temperature-dependency of the thermal conductivity in the bioheat transport (e.g. Refs. [29,30]). Mitra and Balaji [31] numerically simulated the heat conduction in a cancerous breast via COMSOL, a finite-element based commercial solver. Next, Dehghan and Sabouri [32] have developed a spectral-element method for the numerical solution of Pennes’ bioheat transfer equation for one- and two-dimensional geometries, They obtained the temperature distribution within the skin layers, healthy tissue, and the tumor-containing tissue. Other numerical approaches are also applied by Dehghan and group to investigate heat conduction problems [33,34]. The reminder of this article is arranged as follows. We introduce the mathematical formulation for the Pennes’ bioheat equation including the Fourier’s law, the CV model, and the DPL model in Section 2. The boundary conditions for the case studies are introduced and the Pennes’ bioheat transfer equation has been discretized using the SPH formulation in Section 3. The Pennes’ equation is solved including the Fourier- and non-Fourier-based models as case studies. For the first category, various parameters including a constant, space- and temperature-dependent thermal conductivities have been investigated in Section 4. Finally, in Section 4, the temperature distributions within the tissue are presented and the capability of the SPH method for an accurate solution of the phase-lagging bioheat equation is presented. The major conclusions of this work are given in Section 5. 2. Pennes’ bioheat transfer model 2.1. Fourier-based Pennes’ bioheat transfer equation In the middle of the 20th century, Pennes [10] proposed a bioheat transfer equation based on the experimental analysis of a human forearm. The equation is a modified form of the transient heat conduction equation which includes the effects of blood flow and metabolic heat generation rates. It is one of the simplest blood perfusion models and has been successfully applied for the analysis of biological and medical systems [9]. In the work by Pennes, the properties relating the thermal storage terms were considered for the participating tissue which relates the blood properties to the blood perfusion term. The blood perfusion term has been modeled to be proportional to the difference between the arterial temperature and the temperature at a given location. Pennes [10] assumed that the arterial blood temperature (T B ) is uniform throughout the tissue while he considered the vein temperature to be equal to the tissue temperature (T) at the same point [35]. The one-dimensional form of the Pennes’ equation may be written as follows [25],
qc
@T @q ¼ þ qb cb wb ðT B T Þ þ qm þ qs ; @t @x
ð1Þ
where x; t; q; c; qb ; cb ; wb ; k; q; qm ; qs are one-dimensional spatial coordinate, the physical time, the tissue density, the specific heat for the participating tissue, the blood density, the blood specific heat, the blood volumetric perfusion rate, the thermal conductivity of the tissue, the x-component of heat flux vector, the heat generation within the tissue, and the external heat source, respectively. In the Fourier-based form with no external heat source [8] this equation reads
qc
@T @ @T þ qb cb wb ðT B T Þ þ qm : ¼ k @t @x @x
ð2Þ
Next, defining the elevated temperature as h ¼ T T B in Eq. (2) and later dividing the both sides by the volumetric heat capacity (qc) we get [28],
@h 1 @ @h q cb wb qm b : k ¼ h @t qc @x @x qc qb cb wb Now, by substituting we have h ¼ h q qcmw , a ¼ qb cqbcwb , and b ¼ q1c in Eq. (3) yields [28], b b
b
ð3Þ
J. Ghazanfarian et al. / Applied Mathematics and Computation 259 (2015) 21–31
@h @ @h ah: ¼b k @t @x @x
23
ð4Þ
The equation above is the final form of the one-dimensional Pennes’ bioheat transfer equation which is subjected to the following initial and boundary conditions [28],
hð0; tÞ ¼ h0 ;
hðx ¼ 0; 0Þ ¼ 0;
@h ¼ 0; @xx¼l
ð5Þ
where, l is the length of the participating tissue and h0 is the initial temperature. It is usual that an increase in temperature at the skin surface is kept constant in the case of skin burns. On the other hand, for the case of biological body, the heat flux approaches zero deep in the tissue (x ¼ l). 2.2. Dual phase-lag (non-Fourier) based Pennes’ bioheat transfer equation Tzou [20] proposed a non-Fourier approximation for heat conduction wherein the heat flux vector at certain location at time (t þ sq ) corresponds to the temperature-gradient at the same location at time (t þ st ),
~ qðt þ sq ;~ rÞ ¼ krTðt þ st ;~ rÞ;
ð6Þ
where sq and st are the heat flux and temperature-gradient phase-lags, respectively. Here, these phase-lags are positive and form intrinsic properties of the material (participating tissue). However, the values of the two phase-lags e.g. the thermal conductivity and the thermal diffusivity must be experimentally determined. On the contrary, the appropriate values of the two phase-lags for the living tissues especially the skin are still not available [19,23]. However, we have used a data set proposed by few researchers [23,36]. Next, using the Taylor series expansion with respect to time, Eq. (6) in 1-D becomes,
! @q @T @2T q þ sq ffi k þ st : @t @x @t@x
ð7Þ
The above equation may now be combined with the Pennes’ bioheat transfer equation eliminating the heat flux vector. Further, using a simple algebra, the Pennes’ bioheat equation under the effect of the DPL model is derived in [25] and is given here as follows with the assumption of h ¼ T T B ,
sq
@ 2 h qb cb wb @h @ 2 h @ 3 h q cb wb qm qs þ 1 þ s ¼ a þ a s b h þ þ ; q t qc @t @x2 @t@x2 qc qc qc @t 2
ð8Þ
where a ¼ k=qc is the thermal diffusivity of the participating tissue. Further, defining ðh ¼ h qm =qb cb wb Þ, ða ¼ qb cb wb =qcÞ, and b ¼ 1=qc, the equation above is simplified as,
sq
@h @2h @2h @3h þ 1 þ asq ¼ a 2 þ ast ah þ bqs : 2 @t @x @t@x2 @t
ð9Þ
It is noted that this equation is reduced to the bioheat CV model with st ¼ 0 and is solved with identical initial and boundary conditions as given in Eq. (5). 3. Numerical solution procedure The smoothed particle hydrodynamics (SPH) is a mesh-free, fully-Lagrangian, particle-based scheme which has been applied for simulating multidimensional complex geometries, deformable materials and moving boundaries. The researchers usually assume a regular simple simulation geometry, which is not always be the case for the biological tissues. In order to deal with such geometric complexity, the domain should be mapped requiring complex and time-consuming mapping kernels. One of the alternative is to use a mesh-free method such as the SPH scheme. The SPH method is a mesh-free Lagrangian particle-based method and is proved to be applicable for the simulation of multidimensional complex geometries, deformable materials or moving boundaries [37], non-Fourier nanoscale applications [38], and solving the Poisson’s problem [39]. Thus, important advantage of the SPH method over classical methods such as finite-difference (FD), finite volume (FV) or finite element (FE) is the fact that the SPH method does not apply any restrictions over the mesh movement or it does not need any curved boundary capturing strategy. In the SPH technique, the particles are free to move in any direction or could be placed within any arbitrarily geometry. Also, the algorithm is simpler relative to the FD scheme. Hence, the SPH method could potentially be applied to simulate the heat conduction in irregular biological geometries [40]. However, the issue of finding a proper interpolation kernel function for various applications, could be a disadvantageous of the SPH method. The governing equations in SPH are numerically solved by replacing the local substance elements by a set of particles [41]. The conservation laws and properties of each particle are determined from the continuum equations of mechanics and by interpolating from other neighboring particles. The interpolation kernel function may be constructed using the
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analytical functions, and the spatial derivatives of the interpolated quantities can be found using the ordinary calculus [42]. Next, we first review the SPH formulation and then give the details on the SPH discretization for the Fourier- and DPL-based Pennes’ bioheat transfer models. As a final sub-section under numerical solution procedure, details on the boundary conditions and independence test cases are given. 3.1. The SPH formulation The fundamental principle of the SPH method lies in the utilization of the integral interpolants for particles. Any typical rÞ may be rewritten in the integral form using the Dirac delta function as follows [42], function such as Að~
Að~ rÞ ¼
Z
r Þdð~ r Þd~ r ; Að~ r ~
ð10Þ
r’ is the position vector. This integral may be approximated by replacing the Dirac delta function by a kernel, W in where, ‘~ order to assign an integral interpolant to Að~ rÞ as,
Að~ rÞ ¼
Z
r ÞWð~ r ; hÞd~ r ; r ~ Að~
ð11Þ
where Wð~ r; hÞ is the kernel function and h is the smoothing length which is a controlling parameter for the area around particle i within which the contribution of remaining particles are taken into consideration. Finally, the particle approximation of a typical function A at a typical particle identified as i, is given below.
rÞ ¼ Að~
X
mj
j
Aj
qj
W ij :
ð12Þ
Here, the summation is carried out over all SPH particles within the supporting area of particle i; m is the mass of particles, and W ij ¼ Wð~ r; hÞ is the kernel function. The kernel function plays a critical role in the SPH method and should satisfy the basic requirements such as positivity, compact support, and normalization. The kernel function depends on the smoothing length of the particle, (h), and the nondimensional distance between two particles (q ¼ r=h), where r being the distance between particles i and j which is given in [41]. In this article, the super Gaussian kernel function [43] has been used which is defined as 2
WðR; hÞ ¼ ad ð3=2 R2 ÞeR ;
ð13Þ
pffiffiffiffi where, ad is 1= p in a one-dimensional coordinate. The approximation for the spatial derivative of Að~ rÞ is obtained simply by substituting Að~ rÞ with r Að~ rÞ in Eq. (12). Using the divergence theorem we have
r Að~ rÞ ¼
X j
mj
Að~ rj Þ
qj
rWð~ r~ r j ; hÞ:
ð14Þ
This equation implies that instead of evaluating the derivative of a function itself, a derivative of the kernel function should be extracted. Further, Eq. (4) contains a second-order spatial derivative term which could be discretized by using a similar procedure. However, it is to be noted that the resulting expression is too sensitive to the particle disorder. Cleary and Monaghan [44] have introduced an alternative formulation for the second-order derivative evaluation as, N @2T X mj 1 @W ij ¼ ðT i T j Þ : 2 ~ @x q j r r ij j ij j @j~ j j¼1
ð15Þ
On the other hand, for a variable thermal conductivity, the following equation is recommended [44],
X N @ @T 4mj ki kj 1 @W ij ¼ k ðT i T j Þ : ~ @x @x q k þ k j r r ij j i j ij j @j~ j j¼1
ð16Þ
3.2. SPH discretizations for Fourier- and DPL-based models In the above discussion, some important concepts of the SPH method were introduced. In this section, the details are given on the SPH discretizations of the Pennes’ bioheat transfer equations following the Fourier’s law as well as the DPLbased model. Referring to Fig. 1, the one-dimensional computational domain is discretized into N particles distributed equally with a distance of Dx. Hence, we denote the position of the particle, i as xðiÞ and the distance between particles i and j is computed rij ¼ xðjÞ xðiÞ . Now, Eq. (4) is discretized using Eq. (16) as follows, as ~
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Fig. 1. The distribution of SPH particles within the numerical domain.
htþ1 ¼ hti þ i
( N X
b
j¼1
1 @W 4mj ki kj t ij hi htj qj ki þ kj j~ r ij j @j~ rij j
!
) ahti Dt
ð17Þ
and forms SPH discretization for the Fourier-based Pennes’ bioheat equation. For the case of Pennes’ bioheat transfer equation under the framework of the DPL model, Eq. (9) is discretized using a second-order [44] SPH formulation. For the sake of simplicity, the following parameters are defined.
J¼
@h ; @t
/¼
@2h ; @x2
b¼
@3h @/ ¼ : @t@x2 @t
ð18Þ
Next, using these parameters, we rewrite Eq. (9) as,
sq
@J þ ð1 þ asq ÞJ ¼ a/ þ ast b ah þ bqs : @t
ð19Þ
Further, the newly introduced terms (a and b) on the right-hand-side of the above equation are discretized as follows,
/ðiÞ ¼
N X mj j¼1
bðiÞ ¼
N X mj j¼1
ðhi hj Þ
qj
qj
ðJ i J j Þ
1 @W ij ; j~ r ij j @j~ r ij j
ð20Þ
1 @W ij : j~ r ij j @j~ r ij j
ð21Þ
Here, the term JðiÞ is computed as,
JðiÞ ¼ JðiÞ þ
Dt
sq
a/ðiÞ þ ast bðiÞ ah þ bqs ð1 þ asq ÞJðiÞ :
ð22Þ
Finally, the elevated temperature of the participating tissue at location i and at new time-step (t þ 1) may be depicted as
hðiÞ
tþ1
t
¼ hðiÞ þ JðiÞDt:
ð23Þ
3.3. Boundary conditions In the above sub-section, the SPH discretization for the Pennes’s bioheat transfer equation has been discussed. Next, we impose boundary conditions as per Eq. (5) for the governing equations and with variable heat conductivity. The boundary condition implementation procedure given in [44] is followed in the present work. Referring to Fig. 1, particle nodes 1 and N are the boundary nodes and the temperature at these locations are assigned to the given value (h0 ) for Dirichlet-type boundaries. It is noted that for a constant-temperature boundary condition in the SPH method, Eqs. (17) and (19) do not contribute to the boundary nodes. In order to impose the adiabatic boundary condition, two SPH particles are needed e.g. the particles at N and N 1 are used to impose the Neumann boundary condition. Similar to the constant-temperature condition, Eqs. (17) and (19) are not used to evaluate the temperature of particle N. Alternatively, at the end of each time-step, the temperatures at particle nodes N and N 1 are equalized. 3.4. Stability of SPH method Cleary and Monaghan [44] have suggested the following expression to evaluate the time-step size for the SPH scheme,
Dt ¼
lqcp h2 k
;
ð24Þ
where 0 < l < 0:15 is a stability factor and h is the smoothing length. Here, considering the stability of our algorithm, we set Dt ¼ 0:005 s. The stability factor can be calculated from a try and error process as an optimal value. Below this critical value, the exact stable results were produced and the convergence of the solution is guaranteed. A similar stability analysis for the
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SPH study of conduction problem can be found in [44,45]. They performed a wide range of test simulations to obtain a limit of stability. Further details about the stability analysis of the SPH method for conduction problems are presented in [46]. 3.5. Particle-number independence test In order to show the independence of solution to the particle-number, series of simulation runs with the Fourier- and non-Fourier based (st ¼ 0; sq ¼ 20 s, respectively) bioheat transfer models have been carried out for a total time of t ¼ 100 s with three different particle numbers. The relative errors evaluated from the case studies and the analytical solution of Liu et al. [19] are listed in Table 1 for temperature values at two locations within the tissue, x ¼ 0:00208 and 0.01. Further, listed are the corresponding CPU times for each case i.e. Fourier-based and the thermal wave model bioheat equation. It is well worth to notice that the relative error has been computed using the following relation:
h h eRE ¼ max Analytical Numerical ; hNumerical
ð25Þ
where ‘eRE ’ and ‘max’ stand for the ‘relative error’ and ‘maximum value’, respectively. These simulations were carried out on a compute system equipped with 2.53 GHz, Intel Core i5 processor. The comparison in Table 1 clearly indicates that the selection of 60 and 200 particles for Fourier- and non-Fourier-based bioheat models, respectively, generates particle-number independent solutions and requires a relatively low computational cost. 4. Numerical results In this section, the numerical results for various case studies are given. These results are compared with the existing literature and exhibit the ability of the formulated SPH method to accurately simulate thermal bioheat transfer equation. The details of all case studies including the Fourier- and non-Fourier based models are described in Table 2. 4.1. Fourier-based Pennes’ bioheat transfer model Here, Eq. (17) is solved for three cases, which involve a constant, space-, and temperature-dependent thermal conductivities. The physical model consists of a one-dimensional skin structure (tissue) with the thickness of 0.01208 m [19]. The details of all physical properties used in our simulations are listed in Table 3. The imposed temperature-rise on the left hand side boundary of the tissue is also presented as the last column of this table. 4.1.1. Case 1: constant thermal conductivity The Pennes’ bioheat transfer equation is solved using a constant thermal conductivity (k = 0.2 W/m K) [19]. It is noted that, in this case Eq. (15) is employed wherein the thermal conductivity is not involved in the summation. The time variation of the temperature (rise) at two locations in the tissue, namely, x = 0.00208 m and x = 0.01 m, have been shown in Fig. 2. The results were compared with the analytical solutions given by Liu et al. [19]. An excellent agreement is seen from the figure for the SPH data. Further, from Fig. 2, we analyze that in approximately 30s the heat flux penetrate into the skin tissue rising the temperature by 2 °C and reach the location of x = 0.01 m. The time period is referred as the heat diffusion time of the Fourier’s law and different than the relaxation time introduced in the phase-lagging concept, which is microscopic in nature. 4.1.2. Case 2: space-dependent thermal conductivity It is known that the skin tissue is composed of different sub-parts with spatially varying specific thermal conductivities. As a most commonly used model [19,23], three different but constant thermal conductivities for each layer are assumed in a three-layered skin model. However, a more realistic model of the human tissue [28] with the thermal conductivity variation as a continuous function of the skin tissue depth x, has been used here. This space-dependency of thermal conductivity is considered as follows [28],
kðxÞ ¼ 0:7ð1 þ 3xÞ;
ð26Þ
where, k is now a linear function of the depth (distance), x. For this case, Eq. (16) is employed to obtain the temperature distribution and in order to accommodate the thermal conductivity variation inside the summation. The results of our simulations are shown in Fig. 3 and compared with the results by Zhao et al. [28]. Here, the temperature (rise) is illustrated as a function of time and location. Zhao et al. [28] used a traditional discretization technique based on finite differences as the numerical method to obtain their results. Thus, in Fig. 3, the newly formulated SPH in this article for the bioheat transfer equation show acceptable accuracy in comparison with the solutions obtained from traditional numerical method. Further, the value of conductivity at x = 0.00208 m is computed using Eq. (26), as 0.704368 W/m K which implies that the thermal conductivity is greater by approximately 3.5 times to that of a constant conductivity considered in case 1. This illustrates that the conductive power of the tissue is strengthened and the temperature is increased from 8.6 °C to 9.1 °C at time 150 s.
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Table 1 Relative errors for particle-number independence test performed at t ¼ 100 s and x ¼ 0:01; 0:00208 m for Fourier-based bioheat equation and four different particle numbers. No. of particles
R.E., x = 0.01 m
R.E., x = 0.00208 m
CPU time (s)
Fourier’s law 1 2
50 60
6.799 451 861 0.001 606 677
0.231 235 425 0.0013 952 258
3.837 5.226
TWMBT model 1 2 3
50 100 200
0.080 202 544 0.024 594 130 0.001 077 272
26.348 36.675 68.983
Table 2 The description of details for all simulation cases considered in the paper. Fourier-based models Case 1 Case 2 Case 3
Constant thermal conductivity [19] Space-dependent thermal conductivity [28] Temperature-dependent thermal conductivity [29]
k ¼ 0:2 W/m K kðxÞ ¼ 0:7ð1 þ 3xÞ kðTÞ ¼ :4882 þ :001265T
Non-Fourier-based models Case 1 Case 2
TWMBT model DPL model
Table 3 Physical properties of the biological tissue [19]. c; cb (J/kg °C)
wb qb (kg/s)
q (kg/m3)
L (m)
h0 (°C)
4200
0.5
1000
0.01208
12
1.8 8 1.6
SPH results Results of Liu et al.
1.4 1.2
θ (o C)
θ (o C)
6
4
1 0.8 0.6
2
0
0.4
SPH results Results of Liu et al.
0.2 0
0
50
time (s)
100
150
0
50
time (s)
100
150
Fig. 2. Time variation of temperature rise within the tissue for the case of constant thermal conductivity; solid lines: the SPH results, symbols: results of Liu et al. [19]; left column at x ¼ 0:00208 m, right column at x ¼ 0:01 m.
4.1.3. Case 3: temperature-dependent thermal conductivity In this case, the temperature (solution) dependent thermal conductivity is considered for the skin tissue. It is noted that the non-linearity of the partial differential equation in this case, add complexity to the numerical solution of the Pennes’ bioheat transfer equation. However, the obtained results show that the presently formulated SPH scheme is powerful enough to overcome such non-linearity in the governing equations. As mentioned in Section 1, few researchers have focused on the dependence of the local tissue temperature on the thermal conductivity in the bioheat transfer [29,30]. These researchers have experimentally evaluated the thermal conductivity of biological tissue at various temperatures. Valvano et al. [29] suggested a linearly varying temperature dependent thermal conductivity for the biological tissue as follows,
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J. Ghazanfarian et al. / Applied Mathematics and Computation 259 (2015) 21–31 12 8
SPH results Results of Zhao et al.
10
8
θ (o C)
θ (o C)
6
4
4
SPH results Results of Zhao et al.
2
6
2
0 0
50
100
Time (s)
150
0
0.002
0.004
0.006
x (m)
0.008
0.01
0.012
Fig. 3. Left column: time variation of temperature-rise at x ¼ 0:00208 m, right column: spatial variation of temperature rise at t ¼ 150 s, with spacedependent thermal conductivity; solid lines: the SPH results, symbols: results of Zhao et al. [28].
kðTÞ ¼ 0:4882 þ 0:001265T;
ð27Þ
where, T is skin temperature in °C at a given instant and location, and next, recalling the relation h ¼ T T B , Eq. (27) is written as,
kðh Þ ¼ 0:4882 þ 0:001265ðh þ T B Þ;
ð28Þ
where, T B ¼ 36:8 °C is selected which is close to the mean arterial blood temperature measured by Pennes [10]. From the numerical point of view, since the thermal conductivity is a function of temperature, the updated magnitude of conductivity tþ1
(kðh Þ ) should be evaluated at the beginning of each time-step before substituting it in Eq. (16). The obtained temperaturerise profiles as a function of time at two locations in the tissue corresponding to x = 0.00208 m and x = 0.01 m are shown in Fig. 4. With the time evolution, the temperature within the tissue and the conductivity of the material would increase and hence from Fig. 4, the temperature at time t = 150 s, at locations x = 0.00208 m and x = 0.01 m, are 10.1 °C and 5.4 °C, respectively. This in comparison to the respective counterparts from case 1, temperatures were computed as 8.6 °C and 1.4 °C, which signify very high heat flux with temperature dependent conductivity. The dashed line in Fig. 4 indicates that the diffusion time has been reduced from 30 s to 15 s relative to the case 1, which is with a constant conductivity. The spatial–temporal variation of the temperature rise profiles within the skin tissue for the above three mentioned cases are plotted in Fig. 5. It is very obvious from the figure and above discussion that the condition on the thermal conductivity critically influences the temperature variation.
12
10
θ (o C)
8
6
4 x=0.00208 m x=0.01 m
2
0 0
50
100
150
200
250
300
time (s) Fig. 4. Time variation of temperature-rise within the tissue with temperature-dependent thermal conductivity; solid line at x ¼ 0:00208 m, dashed line at x ¼ 0:01 m.
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J. Ghazanfarian et al. / Applied Mathematics and Computation 259 (2015) 21–31 12
10
t=500s
θ (o C)
8
6 k=f(T) k=f(x) k=cte.
t=50s
4
2
t=5s 0 0
0.002
0.004
0.006
0.008
x (m)
0.01
0.012
Fig. 5. Spatial–temporal variation of temperature rise distribution within the tissue with three different cases considered for thermal conductivity.
1.6 1.4
8
SPH results Results of Liu et al.
1.2 1
θ (o C)
θ (o C)
6
4
0.8 0.6 0.4
2 SPH results Results of Liu et al. 0
0.2 0 -0.2
0
50
100
Time (s)
150
0
50
100
Time (s)
150
Fig. 6. Time variation of temperature-rise within the tissue using the TWMBT model (st ¼ 0 s and sq ¼ 20 s); solid lines: the SPH results, symbols: results of Liu et al. [19]; left column at x ¼ 0:00208 m, right column at x ¼ 0:01 m.
4.2. Non-Fourier based Pennes’ bioheat transfer model As discussed in the introduction section, the classical model of the Fourier’s law overestimates the heat conduction in the biological tissues and a modified model based on the phase-lagging concept (e.g. the dual-phase-lag model) is required [20]. Here, Eq. (23) is solved as a realistic bioheat model for two cases i.e. when st ¼ 0 (the TWMBT model) and st – 0 (the DPL model). Again, we consider the physical domain as a one-dimensional skin tissue structure with the thickness of 0.01208 m [19]. The boundary conditions, initial condition, and used physical properties except the thermal conductivity are selected from Table 3. As proposed by Liu et al. [19], the thermal conductivity for simulations in this sub-section is fixed e.g. at k = 0.2 W/m K. 4.2.1. Case 1: TWMBT model After introducing the thermal wave model of bioheat transfer in Section 2.2 and the SPH discretization of the TWMBT model in Section 3.2, here, the numerical solution of Eq. (23) (with st ¼ 0) is presented. As per the guidelines from Liu et al. [19], for TWMBT case, the heat flux phase-lag equals to 20 s (i.e. sq ¼ 20 s). The comparison of the SPH results with the analytical solution of Liu et al. [19] is shown in Fig. 6. This figure demonstrates the time evolution of tissue temperature at two locations within the tissue (x ¼ 0:00208 m and x ¼ 0:01 m). The SPH results show a good accuracy when compared with the existing data except some fluctuations around the sharp wave front. The fluctuation appeared around the sharp temperature-gradient in Fig. 6, indicates the existence of mixed dissipation-dispersion truncation error in the discretized form of Eq. (23). It can be best seen from Fig. 6 that the diffusion time at x ¼ 0:00208 m and x ¼ 0:01 m are approximately 25 s and 130 s, respectively, which illustrates the wave-like features of the TWMBT heat transfer.
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Fourier-based TWMBT model (τq= 20 s) DPL model ( τq= 16 s, τt= 0.043 s) DPL model ( τq= 14 s, τt= 0.056 s)
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θ (o C)
θ (o C)
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Fourier-based TWMBT model (τq= 20 s) DPL model ( τq= 16 s, τt= 0.043 s) DPL model ( τq= 14 s, τt= 0.056 s)
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0 0
50
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12 Fourier-based TWMBT model (τq= 20 s) DPL model ( τq= 16 s, τt= 0.043 s) DPL model ( τq= 14 s, τt= 0.056 s)
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θ (o C)
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t=5 s 6
t=50 s
t=500 s
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0 0
0.002
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Fig. 7. Comparison of temporal variation of the temperature-rise within the tissue at (a) x ¼ 0:00208 m, (b) x ¼ 0:01 m, and (c) all along the tissue; obtained from the DPL model using sq ¼ 16 s; st ¼ 0:043 s and sq ¼ 14 s; st ¼ 0:056 s, the TWMBT model with st ¼ 0 s; sq ¼ 20 s, and the Fourier-based TWMBT model with both phase-lags equal to zero.
4.2.2. Case 2: the DPL bioheat model As discussed in Section 2.2, the DPL model of bioheat equation is the more realistic form of the TWMBT bioheat transfer equation. Here, Eq. (23) is solved with st – 0 which makes the numerical algorithm more complex than the TWMBT. Unfortunately, no previously published data concerning the thermal relaxation time especially for the skin tissue has been reported yet [19,23]. However, in our simulations the values presented by Xu et al. [23] have been utilized which were evaluated by fitting the experimental data of Mitra et al. [47] for the bologna meat without the blood perfusion terms. Xu et al. [23] have proposed two sets involving (sq ¼ 16 s, st ¼ 0:043 s) and (sq ¼ 14 s, st ¼ 0:056 s) and these are considered in the present simulations. Fig. 7 compares the results of the SPH method for the solution of the DPL based bioheat transfer equation with the data obtained from the solution of the TWMBT model and the Fourier-based Pennes’ bioheat equation. As expected, Fig. 7(c) shows that going through the steady-state, the results of all thermal models lie on each other. So that at t ¼ 500 s, the computed temperature distribution is approximately the same for all models. At t ¼ 50 s, the Fourier’s model predicts that relatively the entire tissue is affected by the excess temperature on the left boundary. However, the CV (TWMBT) and the DPL models predict more localized temperature rise distribution. 5. Conclusions The non-linear, Fourier- and non-Fourier based Pennes’ bioheat transfer equations in 1D system are discretized and subsequently solved using the mesh-free smoothed particle hydrodynamics (SPH) with appropriate set of boundary conditions. The thermal conductivity of the living biological tissue has been assumed to be either a constant, a function of the spatial coordinate, or a function of the temperature-field. In the present article, for the case of a non-Fourier based bioheat transfer equation, two models namely, the TWMBT and DPL model have been studied. A number of benchmark cases have been considered and their solutions using the SPH method show a good accuracy in comparison with the results obtained using existing approaches.
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