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

### Bohdan Mochnacki

,### Wioletta Tuzikiewicz

Journal of Applied Mathematics and Computational Mechanics |
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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}

^{1 }University
of Occupational Safety Management in Katowice

Katowice, Poland

^{2} Institute of Mathematics, Czestochowa
University of Technology

Częstochowa, Poland

bmochnacki@wszop.edu.pl, wioletta.tuzikiewicz@im.pcz.pl

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*

1. Introduction

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 τ* _{q}* 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 [8], 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

(1) |

where *c* is the volumetric specific heat
of tissue, λ is the thermal conductivity, *Q* is the capacity of
internal heat sources, τ* _{q}* is the relaxation time,

*T*is the temperature,

*x*,

*y*,

*t*denote the geometrical co-ordinates and time.

Additionally

(2) |

where *G _{B}* [m

^{3}blood/m

^{3}tissue/s] is the perfusion coefficient,

*c*is the volumetric specific heat of blood,

_{B}*T*is the arterial blood temperature,

_{B}*Q*is the metabolic heat source (treated here as a constant value).

_{met}So, in the case of bio-heat problems the CVE is of the form

(3) |

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

(4) |

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

(5) |

In the case of rectangular differental mesh

(6) |

while are the mesh shape functions and the thermal resistances between the neighboring nodes.

The equation (5) can be transformed as follows

(7) |

and finally

(8) |

Let us denote

(9) |

(10) |

(11) |

(12) |

and then

(13) |

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 [9]. 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 *w*_{1}*,* *w*_{2} and the amplitude by δ:

(14) |

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)

(15) |

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

(16) |

Dividing by δ^{s-2}^{ }one obtains

(17) |

or using the Euler formulas

(18) |

Denoting

(19) |

one obtains the equation

(20) |

According to [9] the absolute values of the roots of equation (20) will be less than 1 when

(21) |

So, the first inequality takes a form

(22) |

From the last inequality one obtains

(23) |

or

(24) |

This inequality is unconditional and does not limit the time step.

Let us consider the second inequality, this means:

(25) |

The left hand side of (25) can be trasformed in the following way

(26) |

From the view point of FDM equation stability the most ‘safe’ variant of the last inequality corresponds to and then

(27) |

For the constant value of thermal conductivity (see (6)) one obtains

(28) |

but the transition from (27) to (28) is very tedious.

The final form of CVE stability condition is the following

(29) |

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.

Acknowledgement

*This work is supported by the project No.** 2015/19/B/ST8/01101** sponsored by
National Science Centre (Poland).*

References

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[5] Cattaneo M.C., A form of heat conduction equation which eliminates the paradox of instantaneous propagation, C.R. Acad. Sci. I Math. 1958, 247, 431-433.

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[8] Mochnacki B., Suchy J.S., Numerical Methods in Computations of Foundry Processes, PFTA, Cracow 1995.

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