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We consider the one-dimensional bio-heat transfer equation with quadratic temperature-dependent blood perfusion, which governs the temperature distribution inside biological tissues. Using an extended mapping method with symbolic computation, we obtain the exact analytical thermal traveling wave solution, which describes the non-uniform temperature distribution inside the bodies. The found exact solution is used to investigate the temperature distribution in the tissues. It is found that the surrounding medium with higher temperature does not necessarily imply that the tissue will quickly (after a short duration of heating process) reach the desired temperature. It is also found that increased perfusion causes a decline in local temperature.

Using the Pennes bio-heat transfer (BHT) equation [

Here,

to the usual boundary conditions 1) temperature prescribed,

whole or a part of the boundary of domain

surrounding medium; or radiation,

is the radiative interchange factor between the surface and the exterior ambient temperature

The parameters considered in Equation (1.1) are usually assumed to be constant except for the blood perfusion, which varies with temperature

The analytical study of Equation (1.1) as a nonlinear evolution equation is of great interest. As in the study of nonlinear physical phenomena, the investigations of the travelling wave solution of Equation (1.1) play an im- portant role in the analytical study of the nonuniform thermal distribution in biological tissues. The importance of obtaining the analytical solutions, if available, of Equation facilitates 1) the investigation of temperature distribution inside the biological bodies, 2) the verification of numerical solvers, and aids in the stability analysis of solutions. In the present work, we aim to find analytical thermal traveling wave solution of one-dimensional (1D) BHT equation

with a quadratic temperature-dependent blood perfusion [

where

In this section, we aim to apply the extended mapping method to find analytical solutions of Equation (1.1) with quadratic temperature-dependent blood perfusion (1.3). For the traveling wave solutions of Equations (1.2), (1.3), we introduce the ansatz

where

Then, we seek for the solutions of Equation (1.5) in the form [

where

determined by balancing the second order derivative and the cubic terms in Equation (1.5), and

solution (satisfying condition

where

Equation (1.7) is a perfect square so that (1.7) can be solved in the derivative:

equations

ourselves to only one of these equation (the case of the second equation can be done similarly). Without loss of generality, we consider the equation with sign “

Because we are interesting in the solutions

where

We now turn to the search of different parameters appearing in Equations (1.6), (1.7), and (1.8). Inserting

Inserting expression (1.10) for

for

It should be noted that

is a third degree polynomial with respect to

and condition

solution (1.13) must be particularized from condition either

From what have being saying above, it is clear that solution (1.15) contains three parameters,

where

In the present section, we use the analytical solution (1.15) with sign “

The maximal value of

as

Using tissues’ properties given in table 1, the various temperature profiles have been studied.

. tissue parameters

Tissue | Thermal conductivity | Density | Specific heat | Metabolic level | Blood perfusion |
---|---|---|---|---|---|

Fat | 0.210 | 900 | 3500 | 33800 | |

Kidney | 0.577 | 1000 | 3500 | 33800 | |

Bladder | 0.600 | 1000 | 3500 | 33800 | |

Tumor | 0.642 | 1000 | 3500 | 33800 |

(Color online) Temperature-dependent perfusion distributions at the metabolic level at for four types of biological tissues fat (solid line), kidney (dashed line), bladder (dotted line), and tumor (dash-dotted line) with tissue properties shown in table 1. All the four figures have been obtained with the same blood perfusion parameters, , , and. Top: Temperature of the tissues as a function of depth integrated over time (a) and (b). Bottom: Temperature of the skin surface (c) and body core (d) as a function of time. Other parameters are given in the text

of other tissues near the body core.

In what follows, we concentrate ourselves to the temperature distribution in tumor tissue.

(Color online) Effect of the surrounding medium on the temperature response for tumor tissue for, , , and with. Solid lines: Temperature distribution associated with the surrounding medium temperature and the solution parameters and; Dashed lines: Temperature distribution associated with the surrounding medium temperature and the solution parameters and; Dotted lines: Temperature distribution associated with the surrounding medium temperature and the solution parameters and; dash-dotted lines: Temperature distribution associated with the surrounding medium temperature and the solution parameters and. Top: Profile of the temperature distribution at given time as a function depth, (a) initial temperature distribution, (b) temperature profiles at, and (c) temperature profile at time. Bottom plots: Temperature profiles at given depths as a function of time (d) at, (e) at, and (f) at. Other parameters are given in the text

(Color online) Temperature-dependent perfusion distributions for tumor tissue with properties given in table 1. The temperature of the surrounding medium is maintained constant at. The perfusion level was dependent on local temperature with three values of the linear coefficient of temperature dependence (solid line), (dashed line), and (dotted line). The basal perfusion rate was, while the quadratic coefficient of temperature dependence was. All the plots are obtained with the traveling wave parameters and for, for, and for. The solution parameters are defined by system (1.12) with and for, and for, and and for. (a): Temporal distribution of temperature close to skin surface for different values of; (b) Temporal distribution of temperature close to core body for different values of. Other parameters are given in the text

(Color online) Temperature-dependent perfusion distributions for tumor tissue with properties given in table 1. The temperature of the surrounding medium is maintained constant at. The perfusion level was dependent on local temperature with three values of the quadratic coefficient of temperature dependence (solid line), (dashed line), and (dotted line). The basal perfusion rate was, while the linear coefficient of temperature dependence was. All the plots are obtained with the traveling wave parameters and for, for, and for. The solution parameters are defined by system (1.12) with and for and for, and and for. (a): Spatial distribution of temperature at the initial time for different values of; (b) Spatial distribution of temperature at the initial time for different values of; (c) Spatial distribution of temperature at the initial time for different values of

perfusion on the nonlinear temperature distribution in tumor tissue with properties given in table 1. The layer of air farthest from the skin was set at

Using the extended mapping method with symbolic computation, we found exact analytical solution of the BHT equation with temperature-dependent blood perfusion, that describes the nonuniform temperature distribution in biological tissues. Using this solution, we have explicitly investigated temperature distribution in living tissues. The effects of the surrounding medium and the effects of the temperature-dependent blood perfusion on tem- perature distribution are also addressed. The exact solutions found in this work can be used to predicate the evolution of the detailed temperature within the tissues during thermal therapy.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grants Nos. 7033009.