Identification of tumor region parameters using evolutionary algorithm and multiple reciprocity boundary element method

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Abstract

In the paper the inverse problems consisting in the simultaneous estimation of unknown thermophysical and/or geometrical parameters (thermal conductivity, perfusion coefficient, metabolic heat source, location, size) of the tumor region are solved. The additional information concerning the knowledge of local skin surface temperature at the selected set of points is assumed to be known. The problem of thermal processes proceeding in the domain considered is described by the system of the Pennes equations and boundary conditions given on the outer and contact surfaces. On the stage of numerical solution the evolutionary algorithm coupled with the multiple reciprocity boundary element method has been applied.

Introduction

The body surface temperature is controlled by the blood perfusion, local metabolism and the heat exchange between skin and environment. The apparition of tumor region can lead to the increase of a local blood perfusion and a capacity of metabolic heat source (Liu and Xu, 2000). In this case the temperature of skin surface can also increase (Miyakawa and Bolomey, 1996). So, the abnormal temperature at the skin surface can be used in order to predict the location, size and thermal parameters of tumor region.

The non-invasive diagnosis using skin surface temperature measurements requires solution of inverse bioheat transfer problem. In the paper three different inverse problems are presented. First problem concerns the identification of thermal parameters of tumor (thermal conductivity, perfusion coefficient and capacity of metabolic heat source), second one concerns the estimation of location and size of tumor region. The third inverse problem consists in the simultaneous identification of thermal and geometrical parameters of tumor.

Section snippets

Governing equations

From the mathematical point of view the heat transfer processes in the domain of biological tissue with a tumor are described by the Pennes equations (Huang et al., 1994; Majchrzak, 2001; Majchrzak and Mochnacki, 1998, Majchrzak and Mochnacki, 1999, Majchrzak and Mochnacki, 2002). If we consider the steady-state problem then we are looking for temperatures TeC2(Ωe), e=1, 2 that satisfyxΩe:λe2Te(x)+ke[TB-Te(x)]+Qme=0,where e=1, 2 identifies the sub-domains of healthy tissue and tumor (Fig. 1,

Multiple reciprocity boundary element method (MRBEM)

From the mathematical point of view the Pennes equations (1) are the Poisson equations in which the source functions are temperature dependent. So, in the case of standard boundary element method application the boundary and also the interior of the domain considered should be discretized (Banerjee, 1994; Brebbia and Dominguez, 1992; Majchrzak, 2001). If the multiple reciprocity BEM (Nowak, 1992, Nowak, 1993; Nowak and Neves, 1994) is used then Eq. (1) can be solved without internal grid

Inverse problems

Three different inverse problems have been discussed. The first problem concerns the identification of thermophysical parameters λ2, k2, Qm2 of tumor region (Table 1, variant 1). In this case we assume that other quantities appearing in the formulation of the direct problem are known. The second inverse problem is connected with the estimation of geometrical parameters of the tumor region, this means the position of tumor center ((x1S,x2S) for 2-D problem and (x1S,x2S,x3S) for 3-D problem) and

Evolutionary algorithm

The evolutionary algorithm (EA) operates on population of chromosomes. Each chromosome contains the genes (Arabas, 2001; Michalewicz, 1996; Rutkowska et al., 1997; Schaefer, 2002). In the inverse problem considered the genes contain information about the thermophysical and geometrical parameters of tumor region (Burczynski et al., 2004; Burczynski and Majchrzak, 2002). Each chromosome is evaluated with use of the fitness functionS=i=1M(Ti-Tdi)2,where Tdi, Ti are the measured and calculated

Gradient method

The inverse problems formulated in this paper have been also solved using the gradient methods. For example, we assume that the thermal parameters λ2, k2, Qm2 of tumor region are unknown (c.f. Table 1, variant 1). These parameters can be estimated through the minimization of the least squares criterion (23).

Function Ti=T(xi) is expanded in a Taylor seriesTi=Tis+Tiλ2|λ2=λ2s(λ2s+1-λ2s)+Tik2|k2=k2s×(k2s+1-k2s)+TiQm2|Qm2=Qm2s(Qm2s+1-Qm2s),where TiS are calculated temperatures under the

Results of computations

At first the direct problem both for 2-D and 3-D domains has been solved. The following thermophysical parameters have been assumed: λ1=0.5 [W/(mK)], k1=1998.1 [W/(m3K)], Qm1=420 [W/m3], λ2=0.75 [W/(mK)], k2=7992.4 [W/(m3K)], Qm2=4200 [W/m3], blood temperature TB=37 °C. On the skin surface Γ1 the boundary condition (4) has been accepted (α=10 [W/(m2K)], Ta=25 °C), on the arbitrary assumed internal boundary Γ2 the temperature Tb=37 °C has been assumed.

In the case of 2-D problem the domain of dimensions

Conclusion

The algorithms of tumor region parameters identification using the information about skin surface temperature have been presented. This type of inverse problems have been solved using evolutionary method coupled with the multiple reciprocity BEM. It should be pointed out that this approach can be used to solve an inverse heat transfer problems without internal grid generation. As has been shown above, always real values of searched parameters have been found but the algorithm is time consuming

Acknowledgement

This paper is a part of research Project No. 3 T11F 018 26 sponsored by KBN.

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