Cattaneo-Vernotte bioheat transfer equation. Stability conditions of numerical algorithm based on the explicit scheme of the finite difference method

Mochnacki, B.; Tuzikiewicz, Wioletta

doi:10.17512/jamcm.2016.4.15

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…At each time step t j+1 = (j + 1)∆t for j = 0, (N − 1), using the discretized Robin boundary conditions given by (26), the difference equations given by (27) and (28) can be reformulated as a two-step implicit FDM procedure of the form:…”

Section: Numerical Solution Of Direct Problemmentioning

confidence: 99%

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Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

Alosaimi

Lesnic

Niesen

2020

HFF

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Purpose This study aims to at numerically retrieve five constant dimensional thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements. Design/methodology/approach The thermal-wave model of bio-heat transfer is used as an appropriate model because of its realism in situations in which the heat flux is extremely high or low and imposed over a short duration of time. For the numerical discretization, an unconditionally stable finite difference scheme used as a direct solver is developed. The sensitivity coefficients of the dimensionless boundary temperature measurements with respect to five constant dimensionless parameters appearing in a non-dimensionalised version of the governing hyperbolic model are computed. The retrieval of those dimensionless parameters, from both exact and noisy measurements, is successfully achieved by using a minimization procedure based on the MATLAB optimization toolbox routine lsqnonlin. The values of the five-dimensional parameters are recovered by inverting a nonlinear system of algebraic equations connecting those parameters to the dimensionless parameters whose values have already been recovered. Findings Accurate and stable numerical solutions for the unknown thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements are obtained using the proposed numerical procedure. Research limitations/implications The current investigation is limited to the retrieval of constant physical properties, but future work will investigate the reconstruction of the space-dependent blood perfusion coefficient. Practical implications As noise inherently present in practical measurements is inverted, the paper is of practical significance and models a real-world situation. Social implications The findings of the present paper are of considerable significance and interest to practitioners in the biomedical engineering and medical physics sectors. Originality/value In comparison to Alkhwaji et al. (2012), the novelty and contribution of this work are as follows: considering the more general and realistic thermal-wave model of bio-heat transfer, accounting for a relaxation time; allowing for the tissue to have a finite size; and reconstructing five thermally significant dimensional parameters.

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Section: Numerical Solution Of Direct Problemmentioning

confidence: 99%

“…Elimination of the heat flux q from the non-Fourier’s law (5) and the heat balance equation (2) yields the thermal-wave model of bio-heat transfer (or sometimes called the Cattaneo–Vernotte equation) given by (Liu et al , 1995; Mochnacki and Tuzikiewicz, 2016): …”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

Alosaimi

Lesnic

Niesen

2020

HFF

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“…As one knows, a stability condition should be formulated for these types of algorithms. The considerations concerning the formulation of the condition discussed are presented in [2] and [3]. The implicit scheme of the FDM was also the topic of our research work, here the papers [4,5] should be mentioned.…”

Section: Microscale Heat Transfer 1d and 2d Problemsmentioning

confidence: 99%

Application of numerical methods for solving the non-fourier equations. A review of our own and collaborators’ works

Majchrzak¹,

Mochnacki²

2018

J. Appl. Math. Comput. Mech.

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Abstract. Thermal processes occuring in the solid bodies are, as a rule, described by the well-known Fourier equation (or the system of these equations) supplemented by the appropriate boundary and initial conditions. Such a mathematical model is sufficiently exact to describe the heat transfer processes in the macro scale for the typical materials. It turned out that the energy equation based on the Fourier law has the limitations and it should not be used in the case of the microscale heat transfer and also in the case of materials with a special inner structure (e.g. biological tissue). The better approximation of the real thermal processes assure the modifications of the energy equation, in particular the models in which the so-called lag times are introduced. The article presented is devoted to the numerical aspects of solving these types of equations (in the scope of the microscale heat transfer). The results published by the other authors can be found in the references posted in the works cited below. MSC 2010: 35L10, 65M06Keywords: heat transfer, non-Fourier heat diffusion models, numerical methods, computational mechanics Governing equationsIn the well-known Fourier law, the relationship between the heat flux and the temperature gradient for the heat conduction process is of the formwhere q is a heat flux vector, λ is a thermal conductivity, T, X, t denote the temperature, spatial co-ordinates and time.

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“…One can refer here to the paper [12] in which the FDM equations for the Cattaneo-Vernotte model are presented in details. Generally speaking, the 'start point' for the successive mathematical transformations is the equation…”

Section: The Explicit Fdm Approximation Of the Pennes Equation Is Takmentioning

confidence: 99%

Comparison of bio-heat transfer numerical models based on the Pennes and Cattaneo-Vernotte equations

Ciesielski¹,

Duda²,

Mochnacki³

2016

J. Appl. Math. Comput. Mech.

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Abstract. The homogeneous soft tissue domain subjected to an external heat source is considered. Thermal processes in this domain are described using the well known Pennes equation and next the Cattaneo-Vernotte one. Within recent years the prevailing view is that the Cattaneo-Vernotte equation better describes the thermal processes proceeding in the biological tissue (it results from the specific internal tissue structure). Appearing in this equation the delay time of heat flux with respect to the temperature gradient (τ q ) is of the order of several seconds and the different values of τ q are taken into account. At the stage of numerical modeling the finite difference method is used. In the final part of the paper, the examples of computations are shown.

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Cattaneo-Vernotte bioheat transfer equation. Stability conditions of numerical algorithm based on the explicit scheme of the finite difference method

Cited by 4 publications

References 11 publications

Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

Application of numerical methods for solving the non-fourier equations. A review of our own and collaborators’ works

Comparison of bio-heat transfer numerical models based on the Pennes and Cattaneo-Vernotte equations

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