Cattaneo-Vernotte bioheat transfer equation. Stability conditions of numerical algorithm based on the explicit scheme of the finite difference method
Journal of Applied Mathematics and Computational Mechanics
CATTANEO-VERNOTTE BIOHEAT TRANSFER EQUATION. STABILITY CONDITIONS OF NUMERICAL ALGORITHM BASED ON THE EXPLICIT SCHEME OF THE FINITE DIFFERENCE METHOD
Bohdan Mochnacki 1, Wioletta Tuzikiewicz 2
of Occupational Safety Management in Katowice
2 Institute of Mathematics, Czestochowa University of Technology
Received: 06 October 2016; accepted: 10 November 2016
Abstract. The Cattaneo-Vernotte (CVE) equation is considered. This equation belongs to the group of hyperbolic PDE. Supplementing this equation by two additional terms corresponding to perfusion and metabolic heat sources one can apply the CVE as a mathematical model describing the heat transfer processes proceeding in domain of the soft tissue. Such an approach is recently often preferred substituting the classical Pennes model. At the stage of numerical computations the different numerical methods of the PDE solving can be used. In this paper the problems of stability conditions for the explicit scheme of the finite difference method (FDM) are discussed. The appropriate condition limiting the admissible time step have been found using the von Neumann analysis.
Keywords: bioheat transfer, Cattaneo-Vernotte equation, finite difference method, stability conditions of FDM explicit scheme
Bioheat transfer processes proceeding in the domain of soft tissue are, as a rule, described using the well known Pennes equation [1-4] being a certain modification of the Fourier parabolic equation. As is well known the mathematical form of this equation, results from the assumption of the infinite velocity of thermal wave propagation. In the case of materials with a specific internal structure (e.g. biological tissue), the Fourier equation should be modified. To take into account the delay effect of the local and temporary heat flux with respect to the temperature gradient, the so-called relaxation time τq is introduced, and then the Cattaneo-Vernotte equation should be considered [5, 6]. According to literature, the relaxation time for the processed meat is the order of seconds (2¸5 s) [6, 7]. As mentioned, the bioheat transfer equation (the tissue model) contains two additional terms determining the perfusion and metabolic heat sources. The first is proportional to the local differences between blood and tissue temperatures. The second term (in this paper) is treated as a constant value. It should be pointed out, that the mathematical form of perfusion heat source results from the assumption that the tissue is supplied by the large number of blood capillaries uniformly distributed in the area under consideration. To take into account the presence of thermally significant vessels of considerable size the so-called vessels models are considered but these problems will not be discussed here.
The primary goal of this paper is to establish the stability conditions for the Cattaneo-Vernotte bioheat transfer equation (in the case when at the stage of numerical modeling, the explicit scheme of the finite difference method is used). The FDM equations are constructed in the version proposed in , while at the same time the 2D problem for domains oriented in the Cartesian co-ordinate system is considered.
2. The FDM equations
We consider the following energy equation
where c is the volumetric specific heat of tissue, λ is the thermal conductivity, Q is the capacity of internal heat sources, τq is the relaxation time, T is the temperature, x, y, t denote the geometrical co-ordinates and time.
where GB [m3 blood/m3 tissue/s] is the perfusion coefficient, cB is the volumetric specific heat of blood, TB is the arterial blood temperature, Qmet is the metabolic heat source (treated here as a constant value).
So, in the case of bio-heat problems the CVE is of the form
The equation (2) is supplemented by the appropriate boundary and initial conditions. It should be pointed out that the form of typical boundary conditions in the case of CVE is somewhat different than the classic ones. For example, the Neumann condition takes a form
One can see, that for the constant value of the boundary heat flux the condition (4) takes a well known form. The initial conditions determine the initial tissue temperature and initial cooling (heating) rate.
The numerical solution of the problem discussed can be obtained using the explicit scheme of the FDM. Let us consider the 2D differential mesh being the Cartesian product of the geometrical mesh (Fig. 1) and temporal one . Both the geometric h, k and time ∆t mesh steps are assumed to be the constant values.
Fig. 1. Rectangular mesh
Below, the FDM equation for the internal nodes will be presented. To simplify the mathematical notation the local numbering of nodes is introduced, in particular the nodes (i, j), (i, j +1), (i, j–1), (i+1, j), (i–1, j) are denoted as 0, 1, 2, 3 and 4.
The FDM approximation of the Cattaneo-Vernotte equation can be taken in the following form
In the case of rectangular differental mesh
while are the mesh shape functions and the thermal resistances between the neighboring nodes.
The equation (5) can be transformed as follows
Let us denote
3. Stability condition
The problem of numerical schemes stability is closely associated with a numerical error. The FDM scheme is stable when the errors made at one time step of the calculation do not cause the errors to increase as the computations are continued . If, on the contrary, the errors grow with time the numerical scheme is said to be unstable. The stability of numerical schemes can be investigated by performing von Neumann stability analysis. According to this theory, the approximation error carried by at every node of space (i, j) = (e) and time s is assumed to have a wave form with the wave numbers denoted by w1, w2 and the amplitude by δ:
As time progresses, to assure convergence, the amplitude of an approximation error must be less than unity, i.e. [8-10].
Let us introduce the formula (14) into the FDM equation (13)
One can see, that for the rectagular mesh and the constant value of thermal conduc- tivity
Additionally the source term can be neglected, because it has no effect on the FDM equation stability. So
Dividing by δs-2 one obtains
or using the Euler formulas
one obtains the equation
According to  the absolute values of the roots of equation (20) will be less than 1 when
So, the first inequality takes a form
From the last inequality one obtains
This inequality is unconditional and does not limit the time step.
Let us consider the second inequality, this means:
The left hand side of (25) can be trasformed in the following way
From the view point of FDM equation stability the most ‘safe’ variant of the last inequality corresponds to and then
For the constant value of thermal conductivity (see (6)) one obtains
but the transition from (27) to (28) is very tedious.
The final form of CVE stability condition is the following
In the case of non-linear tasks the stability condition can be also found. Then each FDM star for transition must be considered individually and the critical time step corresponds to the lowest value, of course.
This work is supported by the project No. 2015/19/B/ST8/01101 sponsored by National Science Centre (Poland).
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