A Semi-Analytical Heterogeneous Model for Thermal Analysis of Cancerous Breasts

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A Semi-Analytical Heterogeneous Model for Thermal Analysis of Cancerous Breasts

A. Ramírez-Torres, Reinaldo Rodríguez-Ramos, Aura Conci, Federico J.

Sabina, C. García-Reimbert, Luigi Preziosi, J Merodio, Frédéric Lebon

To cite this version:

A. Ramírez-Torres, Reinaldo Rodríguez-Ramos, Aura Conci, Federico J. Sabina, C. García-Reimbert, et al.. A Semi-Analytical Heterogeneous Model for Thermal Analysis of Cancerous Breasts. Eddie Y.

K. Ng; Mahnaz Etehadtavakol. Application of Infrared to Biomedical Sciences, Springer, pp.175-190, 2017, Series in BioEngineering, �10.1007/978-981-10-3147-2_11�. �hal-01697070�

A Semi-Analytical Heterogeneous Model for Thermal Analysis of Cancerous

Breasts

A. Ramírez-Torres, R. Rodríguez-Ramos, A. Conci, F.J. Sabina, C. García-Reimbert, L. Preziosi, J. Merodio and F. Lebon

Abstract In the present work coupled stationary bioheat transfer equations are considered. The cancerous breast is characterized by two areas of dissimilar thermal properties: the glandular and tumor tissues. The tumorous region is modeled as a two-phase composite where parallel periodic isotropic circularfibers are embedded in the glandular isotropic matrix. The periodic cell is assumed square. The local problem on the periodic cell and the homogenized equation are stated and solved.

The temperature distribution of the cancerous breast is found through a numerical computation. A mathematical and computational model is integrated by FreeFem++.

Keywords TemperatureCancerous breastAsymptotic homogenization^Finite

element method

A. Ramírez-Torres (&)F.J. SabinaC. García-Reimbert

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, CP 20-126, 01000 CDMX, Mexico

e-mail: ariel.ramirez@iimas.unam.mx F.J. Sabina

e-mail: fjs@mym.iimas.unam.mx C. García-Reimbert

e-mail: cgr@mym.iimas.unam.mx R. Rodríguez-Ramos

Facultad de Matemática y Computación, Universidad de La Habana, CP 10400 La Habana, Cuba

e-mail: reinaldo@matcom.uh.cu A. Conci

Instituto de Computação, Universidade Federal Fluminense, CEP:24210-346 Rio de Janeiro, Brazil

e-mail: aconci@ic.uff.br L. Preziosi

Dipartimento di Matematica, Politecnico di Torino, CP 10129 Turin, Italy e-mail: luigi.preziosi@polito.it

1 Introduction

Actually, clinical examination, ultrasound, mammography, thermography, among others, are employed to identify and treat breast cancer [1, 2]. In particular, mammography is considered the standard procedure for detecting breast cancer.

Yet, it presents difficulties for finding tumors in dense breasts. Thermography technique has arisen as a prospective method with the aim of increase the efficacy of the early discovery of breast cancer [3, 4]. Then, mathematical and numerical models have been proposed for studying thermal distribution on healthy and cancerous breasts, with the aim of using thermography as a complementary tool.

For instance, [5] modeled a three-dimensional tumorous breast and sensitivity parameters are analyzed. Moreover, [3] were able to set a method to approximate thermal properties, where the physical process was ruled by a bioheat transfer equation. A three-dimensional breast, taking into account thermal and elastic properties, was modeled and the influence of both properties on the surface tem- perature was considered by [6]. In the aforementioned works, the numerical sim- ulation was performed via FEM.

In the present study, a semi-analytical method is proposed for studying the breast thermal properties for different parameter data. Then, mathematical and computa- tional modeling are integrated for solving two coupled stationary bioheat transfer equations. To separate micro and macro variables of the heterogeneous problem, the two-scale asymptotic expansion is used [7,8]. In fact, multiscales methods have been successfully applied to various physical systems. For example, a formal two-scale asymptotic expansion for studying the macroscopic behavior of a porous and linear elastic solid was used in [9]. On the other hand, the homogeneous problem associated with the healthy breast tissues (without tumor) and the homogenized problem resulting by the application of the two-scale homogenization method to the heterogeneous tumor tissue, are solved usingFreeFem++. Finally, numerical results are shown and discussed.

J. Merodio

Departamento de Mecánica de los Medios Continuos y T. Estructuras, E.T.S. de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, CP 28040 Madrid, Spain

e-mail: jose.merodio@upm.es F. Lebon

2 Mathematical Model

The aim of the present work is tofind the stationary temperaturefieldsuandu^ethat are described by the following bioheat transfer equations [10].

_@^@_x

i K_ij^g_@^@_x^u

j

þq_bc_bx^g_bu¼q^g_mþq_bc_bx^g_bu_a inX1; K_ij^g_@^@_x^u_jni¼h uð ueÞ on@Xⁿ1;

u¼uc on@X^d1: 8>

>>

<

>>

>:

ð1Þ

_@^@_x

i K^e_ij^@_@^u_x^e

j

þq_bcbx^e_bu^e¼q^e_mþq_bcbx^e_bua inX2; K_ij^e@_@^u_x^e

jni¼ K_ij^g@_@^u_x^g

jni on@X2;

u^e¼u_g on@X2

8>

>>

<

>>

>:

ð2Þ

whereK_ij^g¼k_gdij denotes glandular tissue thermal conductivity,q_b is blood mass density,cbblood specific heat capacity,uais the arterial blood temperature,ucthe temperature at the boundary between breast and chest,ue is the surrounding tem- perature and h represents the combined effective heat transfer coefficient due to convection, radiation and evaporation of 13:5 W=m²K [11]. Besides, the rapidly oscillating coefficients K^e_ij;x^e_b andq^e_mare defined as follows

K_ij^eðxÞ ¼ kgdij;x2X^eg

ktdij;x2X^et

; x^e_bðxÞ ¼ x^g_b;x2X^eg

x^t_b;x2X^et

and q^e_mðxÞ ¼ q^g_m;x2X^eg

q^t_m;x2X^et

:

Note that in the case of the healthy breast model only (P₁) has to be solved.

For the sake of simplicity, we will work in a two-dimensional section where the breast geometry is represented by a hemispherical shape with a diameterLas done in [11]. The healthy breast will be represented by a homogeneous tissue (glandular tissue) and associated with the open, bounded, and connected domain X1 with Lipschitz boundary@X1¼@Xⁿ1[@X^d1, where@Xⁿ1\@X^d1¼ ;. On the other hand, the cancerous tissue will be characterized by two regions of dissimilar thermal properties: the tumoral area (X^et—fibers) and the glandular area (X^eg—matrix). In this sense, the cancerous region will consist of a periodic microstructure associated with the open, bounded, and connected domain X2¼X^e_g[X^e_t[@X^e_g with Lipschitz boundary@X2¼@X^eg and with X^eg\X^et ¼ ;. Then, the cancerous breast is repre- sented byX¼X2[X1 (Fig. 1). Let e[0 be the size of the microstructure and y¼x=ethe fast scale coordinate. The reference periodic cell will be denoted byY, which contains one inclusion occupying the domainY_twith Lipschitz boundary@Y_t such thatY ¼Yg[Yt[@Yt, withYtYandYg\Yt¼ ;. It is also assumed thatX^eg

is connected and that the inclusions do not intersect the boundary@X^eg. In previous works, soft tissues assume to present this type of arrangement. In fact, Penta et al. [9]

used the same periodic geometry to depict a porous tissue microstructure. Boundary conditions for (1) are heat transfer by convection between the surface of the tissue and the external environment on@Xⁿ1 and a prescribed temperature on@X^d1. In the case of (2) we assume heat and temperature continuity on@X2. Moreover, continuity conditions for temperature and heatflow are imposed onC^e(boundary between the glandular tissueX^eg and the tumor inclusionsX^et), i.e.,

u^e

½

½ ¼0 onC^e and ½½K^erxu^en ¼0 onC^e: ð3Þ

3 Two-Scale Homogenization

Here, the two-scale homogenization technique is applied tofind the homogenized equation and corresponding effective coefficients. An overview of how this method is applied and its main assumptions can be found in [12]. Specifically, afterfind- ing the solutionu of problem (1), an asymptotic expansion ofu^e [problem (2)] is sought as a function ofefore!0, namely

u^eðxÞ ¼u⁰ðxÞ þev_pðyÞ@u⁰ðxÞ

@x_p þe²u²ðx;yÞ þ. . .; ð4Þ where the functions v_pðyÞ;u²ðx;yÞ, are Y-periodic in y. In particular, the vector functionvðyÞsatisfies the unit cell problem

_@^@_yiK_ijðyÞ^@v_@^p_y^ðyÞ_j þK_ipðyÞ

¼0 inYnC;

½ v

½ ¼0 onC;

KijðyÞ^@v_@^p_y^ðyÞ

j þKipðyÞ

ni

h i

h i

¼0 onC 8>

><

>>

: ð5Þ

andu⁰ðxÞthe homogenized problem solution Fig. 1 Decomposition of the

macroscopic domain (left) and the corresponding unit periodic cell (right)

_@^@_x

i K^ij@u⁰ðxÞ

@xj

þ q_bcbx^g_b^jY_jYj^gjþq_bcbx^t_b^jY_jYj^tj

u⁰ðxÞ ¼ q^g_mþq_bcbx^g_bua

_jYgj

jYjþ þ q^t_mþq_bcbx^t_bua

_jYtj

jYj inX2;

u⁰ðxÞ ¼ug on@X2; 8>

><

>>

: ;

ð6Þ where j j represents volume fraction. The effective constant coefficients K^_ip are given by

K^ip¼ Kij

@vp

@yj

þKip

ð7Þ

wherep¼1;2 andhidenotes volume average.

3.1 Analytical Solution of the Unit Cell Problem

In particular, the theory of analytical functions by Muskhelishvili [13] is applied to solve the cell problem (5). In this sense, the solutions of the local problems are written as

v^ðgÞ₁ ¼Re a¹₀zþX^1o

k¼1

a¹_kf^ðk1ÞðzÞ ðk1Þ!

( )

and v^ðtÞ₁ ¼Re X^1o

l¼1

c¹_lz^l

( )

; ð8Þ

v^ð₂^gÞ¼Im a²₀zþX^1o

k¼1

a²_kf^ðk1ÞðzÞ ðk1Þ!

( )

and v^ð₂^tÞ¼Im X^1o

l¼1

c²_lz^l

( )

; ð9Þ

whereðcÞwithc¼g;t denotes the constituent, the superscriptospecifies that the sum is carried out over odd indices, the coefficientsa^p₀;a^p_kandc^p_lðp¼1;2Þare real and f is the zeta quasi periodic Weierstrass function. Now, using Laurent’s expansion offand the quasi-periodicity property offand its derivatives

v^ðgÞ₁ ¼Re X^1o

l¼1

a¹_lz^lA¹_lz^l

( )

and v^ðgÞ₂ ¼Im X^1o

l¼1

a²_lz^lA²_lz^l

( )

; ð10Þ

where forp¼1;2

A^p_l ¼X^1o

k¼1

ka^p_kg_kl with g_kl ¼ ð1Þ^pþ1p; kþl¼2

ðkþl1Þ!

k!l! Skþl; kþl[2 (

and Sk are called the reticulate sums and are defined as Sk¼ P

w2L 1

w^kðk3;koddÞwithL representing the lattice excluding the numberw¼ 0 andw¼mw1þnw2wherem;n2Zand w1;w2 are the periods. In particular, in the present workw1 ¼1 andw2 ¼i, due to we are in presence of square unit cells.

Substitution of (8)–(10) in boundary conditions of problem (5) and taking into account that onC;z¼Re^ih whereR is the circumference radius give that coeffi- cientsa^p_k can be found through solution of the following infinite linear system (for finding the effective properties it is truncated into an appropriate orderk¼N)

n¹Iþ ð1Þ^pþ1W^p

A^^p¼V^p; ð11Þ

whereA^^p¼ ð^a^p₁;^a^p₂;. . .Þ^T;^a^p_k ¼a^p_k ffiffiffi pk

=R^k;V^p¼ ðð1Þ^pþ1R;0;. . .Þ^T,

n¹¼kgþkt

k_gk_t and W^p¼ ð1Þ^pþ1pR²; kþl¼2 P

1o k¼1

ffiffiffik p ffiffi

pl

g_klR^kþl; kþl[2 8<

: :

Now, from Eq. (7)

K^ip¼ Kij

@vp

@yj

þKip

:

Using the form ofK_ij, Green’s theorem, the double periodicity of v_p and for- mulas (8)–(10), then

K^_pp¼ k_g12pa¹₁

; ifp¼1 kt1þ2pa²₁

; ifp¼2

: ð12Þ

In fact, ifkg¼kt. Then,K^ ¼K^11¼K^22.

4 Numerical Solution and Analysis of Results

This section is devoted tofind the temperature distribution of problems (1) and (6) where we define asg¼q_bcbx^g_b;g^eðxÞ ¼q_bcbx^e_bðxÞ,f ¼q^g_mþq_bcbx^g_bua;f^eðxÞ ¼ q^e_mðxÞ þq_bc_bx^e_bðxÞua. With this aim, we follow the following procedure.

4.1 Solution of (1)

Forfinding solutionuof (1), we useFreeFem++. First, the problem must be written in its weak formulation. In this sense, let H_d¹ðX1Þ ¼ fu2H¹ðX1Þs:t:cðuÞ ¼ 0 on@X^d₁g. Using the trace theorem foru_c2H¹⁼²ðX1Þ, there exists a continuous linear operatorR0 :H¹²ð@X1Þ !H¹ðX1Þ such that cðR0ucÞ ¼uc. Now, we define

~

u¼uR0uc2H_d¹ðX1Þ. Then, on@X^d1

cð~uÞ ¼cðuÞ cðR0ucÞ ¼ucuc¼0:

In this way, the equivalent variational formulation of problem (1) is Find~u2H_d¹ðX1Þsuch that

að~u;vÞ ¼LðvÞ; 8v2H¹_dðX1Þ (

; ð13Þ

where

að~u;vÞ ¼ Z

X1

K^grx~u rxvdxþ Z

X1

g~uvdxþ Z

@Xⁿ1

h~uvdS;

LðvÞ ¼ Z

X1

fvdx Z

X1

gðR0ucÞvdx Z

X1

K^grxðR0ucÞ rxvdx

þ Z

@Xⁿ1

hðueR0ucÞvdS:

In particular, the weak solution existence and uniqueness of problem (13) can be proved by standard methods using the Lax–Milgram theorem. In this sense, the following must be proved:

(i) The bilinear formað~u;vÞis continuous

In this sense, observe thatK^g2Mða;b;X1Þand by Cauchy–Schwartz jað~u;vÞj bkrx~uk_L2ðX1Þkrxvk_L2ðX1Þþgk~uk_L2ðX1Þkvk_L2ðX1Þ

þhk~uk_L2ð@Xⁿ1Þkvk_L2ð@Xⁿ1Þ:

Furthermore, by the Poincaré–Friedrichs II theorem k~uk_L2ðX1ÞC~1krx~uk_L2ðX1Þ¼C~1k~uk_H1

dðX1Þ: On the other hand, by the trace theorem

k~uk_L2ð@Xⁿ1Þ¼ kcð~uÞk_L2ð@Xⁿ1Þ¼ Z

@Xⁿ1

jcð~uÞj²þ Z

@X^d1

jcð~uÞj² 2

64

3 75

1 2

¼ k~uk_L2ð@X1ÞC2k~uk_H1ðX1Þ: Moreover,

k~uk_H1ðX1Þ¼ k~uk²_L2ðX1Þþ krx~uk²_L2ðX1Þ

h i¹₂

C~1krx~uk²_L2ðX1Þþ krx~uk²_L2ðX1Þ

h i¹₂

¼ ð1þC~₁Þ¹²krx~uk_L2ðX1Þ¼C~₃k~uk_H1

dðX1Þ; ~u2H_d¹ðX1Þ:

Therefore,

k~uk_L2ð@Xⁿ1ÞC₂k~uk_H1ðX1ÞC~₄k~uk_H1 dðX1Þ;

whereC~₄¼C₂C~₃¼C₂ð1þC~₁Þ¹². Finally, for~u2H_d¹ðX1Þ jað~u;vÞj bk~uk_H¹

dðX1Þkvk_H¹

dðX1Þþg~C1C1k~uk_H¹

dðX1Þkvk_H¹

dðX1Þþh~C4C4k~uk_H¹

dðX1Þkvk_H¹

dðX1Þ

C5k~uk_H¹

dðX1Þkvk_H¹

dðX1Þ; withC5¼bþgC~1C1þhC~4C4.

(ii) The bilinear formað~u;vÞisH¹_d-elliptic Letu2H¹_dðX1Þ,

aðu;uÞ ¼ Z

X1

K^gðrxuÞ²dxþ Z

X1

gu²dxþ Z

@Xⁿ1

hu²dS

C6

Z

X1

ðrxuÞ²dxþ Z

X1

u²dxþ Z

@Xⁿ1

u²dS 0

B@

1 CA

¼C₆ krxuk²_L2ðX1Þþ kuk²_L2ðX1Þþ kuk²_L2ð@Xⁿ1Þ

C6krxuk²_L2ðX1Þ¼C6kuk²_H1

dðX1Þ; withC6 ¼minða;g;hÞ:

(iii) The linear formLðvÞis continuous inH_d¹ðX1Þ Letv2H_d¹ðX1Þ,

jLðvÞj ¼ Z

X1

fvdxZ

X1

gðR0ucÞvdxZ

X1

K^grxðR0ucÞ rxvdxþ Z

@Xⁿ1

hðueR0ucÞvdS

Z

X1

j jdxfv þZ

X1

gðR0ucÞv j jdxþ Z

X1

K^grxðR0ucÞ rxv

j jdxþ Z

@Xⁿ1

hðueR0ucÞv

j jdS:

But, by Cauchy–Schwarz inequality, the Poincaré–Friedrichs II theorem and the fact thatR₀u_c2H¹ðX1ÞandrxðR0u_cÞ 2ðL²ðX1ÞÞⁿ

jLðvÞj C10kvk_H1 dðX1Þ;

where C₁₀¼C₁kfk_L2ðX1ÞþgC₁kR0u_ck_L2ðX1ÞþbkrxðR0u_cÞk_L2ðX1ÞþðhC₇juej þ hC₈kuck_H1=2ð@Xⁿ1ÞÞC3:

Thus, (i)–(iii) proves the existence and uniqueness of solution ~u⁰ by using the Lax–Milgram theorem.

Now, it must be shown that the maphf^ei 2L²ðX1Þ !u2H_d¹ðX1Þis continuous in order to prove the regularity of the weak solution. In fact, from theH_d¹-ellipticity of the bilinear form

jaðu;uÞj C₆kuk²_H1 dðX1Þ

and the continuity of the linear operator inH_d¹ðX1Þ jLðuÞj C10kuk²_H1

dðX1Þ:

Then,

C6kuk²_H1

dðX1Þ jaðu;uÞj ¼ jLðuÞj C10kuk²_H1 dðX1Þ;

i.e.

kuk_H¹

dðX1ÞC10

C6

¼C11kfk_L2ðX1Þ;

with

C₁₀¼ 1 C6

C₁þgC1kR0uck_L2ðX1Þ

kfk_L2ðX1Þ

þbkrxðR0ucÞk_L2ðX1Þ

kfk_L2ðX1Þ

þ hC₇juej þhC₈kuck_H1=2ð@Xⁿ1Þ

C₃ kfk_L2ðX1Þ

0

@

1 A:

Now, the contribution of R0uc may be difficult in some cases. However, FreeFem++ replaces the Dirichlet condition by a Robin condition of the form

rxunþu=e¼uc=eon@X^d1 and solves the problem with a very small value ofe.

In particular, we approximate the involved functions by piecewise linear continuous finite elements.

4.2 Solution of (6)

The last step in the homogenization procedure is to solve the homogenized problem (6). Here we prove thatu⁰is its solution and that the problem is well posed. In this sense, letH₀¹ðX2Þ ¼ fu2H¹ðX2Þs:t:cðuÞ ¼0 on@X2g. Using the trace theorem for u_g2H¹⁼²ðX2Þ there exists a continuous linear operator R₀:H¹²ð@X2Þ ! H¹ðX2Þsuch thatcR₀u_g

¼u_g. Now, we define~u⁰¼u⁰R₀u_g2H₀¹ðX2Þ. Then, on@X2

cð~u⁰Þ ¼cðu⁰Þ cðR0ugÞ ¼ugug¼0:

In this way, the equivalent variational formulation of problem (6) is Find~u⁰2H₀¹ðX2Þsuch that

að~u⁰;vÞ ¼LðvÞ; 8v2H₀¹ðX2Þ

; ð14Þ

where

að~u⁰;vÞ ¼ Z

X2

Kr^ x~u⁰ rxvdxþ Z

X2

hg^ei~u⁰vdx;

LðvÞ ¼ Z

X2

hf^eivdx Z

X2

Kr^ xðR0ugÞ rxvdx Z

X2

hg^eiðR0ugÞvdx:

In particular, K^ 2Mða;b;X2Þ see Cionarescu and Donato [14], and hf^ei 2 L²ðX2Þandhg^ei[0. The existence and uniqueness of solution~u⁰ can be proved through the Lax–Milgram theorem. Then, we must show that:

(i) The bilinear formað~u⁰;vÞis continuous

In this sense, observe that using the fact thatK^ 2Mða;b;X2Þand by Cauchy– Schwartz inequality

jað~u⁰;vÞj bkrx~u⁰k_L2ðX2Þkrxvk_L2ðX2Þþ hg^eik~u⁰k_L2ðX2Þkvk_L2ðX2Þ:

Now, from remark 3.37 p. 32 by Cionarescu and Donato [14], for

~ ; 2 ðXÞ,