Extended Generalized Riccati Equation Mapping for Thermal Traveling-Wave Distribution in Biological Tissues through a Bio-Heat Transfer Model with Linear / Quadratic Temperature-Dependent Blood Perfusion *

Analytical thermal traveling-wave distribution in biological tissues through a bio-heat transfer (BHT) model with linear/quadratic temperature-dependent blood perfusion is discussed in this paper. Using the extended generalized Riccati equation mapping method, we find analytical traveling wave solutions of the considered BHT equation. All the travelling wave solutions obtained have been used to explicitly investigate the effect of linear and quadratic coefficients of temperature dependence on temperature distribution in tissues. We found that the parameter of the nonlinear superposition formula for Riccati can be used to control the temperature of living tissues. Our results prove that the extended generalized Riccati equation mapping method is a powerful tool for investigating thermal traveling-wave distribution in biological tissues.


Introduction
Heat transfer in biological systems is relevant in many diagnostic and therapeutic applications involving either decrease or increase of temperature and often requires precise monitoring of the spatial distribution of thermal histories that are produced during a treatment protocol [1][2][3][4][5][6][7][8][9].Since the pioneering work by Henriques and Moritz [10] and by Pennes [11] on heat transfer in biological system, the problem of heat transfer in biological systems has received renewed attention and has been the focus of considerable research [12][13][14][15][16].The investigation of such heat transfer problems requires the evaluation of spatiotemporal distributions of temperature and has been traditionally addressed using the Pennes model which is originally designed for predicting temperature fields in the human forearm [10,11].Pennes model of the heat transfer in biological systems accounts for the ability of tissue to remove heat by both passive conduction (diffusion) and perfusion of tissue by blood.Here, perfusion is defined as the nonvectorial volumetric blood flow per tissue volume in a region that contains sufficient capillaries that an average flow description is considered reasonable.
After performing a series of experiments which measured temperatures of human forearms, Pennes derived a thermal energy conservation equation called the "bioheat transfer" (BHT) equation or the traditional BHT equation.The generalized one-dimensional Pennes BHT equation can be written in the following form [17][18][19][20][21]   where   is the local tissue temperature (temperature elevation); and t x denote the time and the distance from the skin surface to the body core, respectively; is the arterial blood temperature (ambient temperature of perfusing blood);  is the heat generation in the body, that is, the heat generated by the normal processes in the body, which is usually very small compared to the external power deposition term p Q [22].When the metabolic heat generation rate ( metabolic m Q Q where Q is the total heat) is positive, body temperature will rise and when m is negative, it will go down [23]; we remember that the total heat transported across a flat surface of area  is related to the temperature gradient across the surface and the thermal conductivity of T  the material:    in the right hand side (r.h.s.) of Equation (1.1) models the perfusion heat loss and can be the dominant form of energy removal when considering heating processes.Vascular tissues generally experience increased perfusion as temperature increases.The above term is always considered in cases of tissues with a high degree of perfusion, such as liver.Regarding cardiac ablation, the perfusion heat loss is incorporated in some models, but is generally ignored since its effect is negligible [24].In general, b  is assumed to be uniform throughout the tissue.However, its value may increase with heating time because of vasodilatation and capillary recruitment.
Spatially distributed heating occurs in skin exposed to penetrating, dissipative radiation such as microwave, ultrasound, and laser light [25][26][27].These heating methods often involve an exponentially decaying power transmission accompanied by reflection at the interface of regions with different electrical properties.For a uniform plane wave incident normally upon the skin surface, with a layer of air included to model the reflection at the skin/air interface, the average absorbed power density p Q is given by: where and E  are the total electric field in the tissue and the electric conductivity, respectively.
Prediction of temperature in tissue models generally assumes a constant-rate (temperature independent) blood perfusion within each tissue and has long been carried out by both analytical and numerical methods based on Equation (1.1).In this case, the temperature rise can be easily predicted by traditional analytical methods based on Equation (1.1) and given boundary conditions, which can either be solved analytically for simple geometries [28,29] or by finite element method or finite-difference method for more realistic, complicated tissue geometry [30,31].
Several experiments have shown that the response of vasculature in tissues to heat stress is strongly temperature-dependent [32].Obviously, models which include temperature-dependent increases in perfusion are more difficult to solve.The case of a linear temperature-dependence has been described combining analytical simulations [9,33,34].Lang et al. [33] described an optimization process specially designed for regional hyperthermia of deep-seated tumors in order to achieve desired steadystate temperature distributions.To predict the temperature, they applied a nonlinear three-dimensional heat transfer model based on temperature-dependent blood perfusion.Using linearly implicit methods in time and adaptive multilevel finite elements in space, they were able to integrate efficiently the instationary nonlinear heat Equation with high accuracy.Temperature distributions for two individual patients calculated on coarse and fine spatial grids were compared.Most recently, Partridge et al. [35] presented an inverse analysis procedure based on a coupled numerical formulation first to identify the coefficients of linear and quadratic variations of blood perfusion, and then to investigate temperature distribution at the skin surface for tumor perfusion.Their procedure has the advantage of not requiring the calculation of derivatives or sensitivities, or an initial estimate of these values.
We assume in this work a linear and a quadratic temperature-dependent blood similar to the expression identified respectively by Gowrishankar et al. [36] and Partridge et al. [35]: 2), we aim in the present work to analytically investigate thermal traveling-wave distribution in biological tissues.We consider only the case of constant spatial heating, that is, the case when In our knowledge, there has not yet being reported a scientific work that treats thermal traveling-wave distribution in biological tissues by means of Riccati Equation  are two real functions of  .The Riccati Equation, though a nonlin- ear Equation, is characterized by the fact of possessing a superposition formula, as it is the case for all linear equations.Moreover, Riccati equation can be linearized; indeed, we can use the Cole-Hopf transformation [37,38] to reduce it to a linear Schrödinger spectral problem.Riccati Equation (1.3) appears in many fields of applied mathematics, in many instances when we can find exact solutions of nonlinear partial differential equations.In mathematical literatures, Riccati Equation (1.3) is known as the only first order nonlinear ordinary differential equation which has no movable singularity, i.e., which possesses the Painlevé property [39,40].As it was shown by Lie et al. [41], Riccati Equation (1.3) admits a (nonlinear) superposition formula will also be a solution of Equation (1.3), with being a constant parameter.The nonlinear superposition Formula (1.4) allows us to build a denumerable set of solutions of Riccati Equation (1.3), as soon as three of its solutions are found.
The aim of the present work is to apply Riccati Equation (1.3) to analytically investigate thermal traveling-wave distribution in biological tissues through the BHT model (1.1) with the temperature-dependent blood perfusion (1.2).The rest of the paper is organized as follows.In Section 2, we explore the solutions of Riccati Equation (1.3) in the special case of constant coefficients In Section 3, we apply the generalized Riccati equation mapping method [42] to find solutions to Equation (1.1) with perfusion (1.2).Here, we distinguish the initial processus (temperature at each distance x from the skin surface to the body core at time ), instantaneous temperature distribution of biological tissues (temperature at each distance x from the skin surface to the body core at a given time ), and the steady-state temperature distribution (the temperature of each point t x of the biological tissue is constant with respect with time).The results are discussed in Section 4 and summarized in Section 5.

Solutions of Riccati
respectively, where 1 and 2 C are two arbitrary real constants.Expression (1.8) gives the following general solution of Riccati Equation (1.5) while expression (1.9) gives the following general solution of Riccati Equation (1.5) We first apply the generalized Riccati equation mapping method, and then a simple coupling method to find analytical solutions of Equation (1.14).

Generalized Riccati Equation Mapping Method for Temperature Distribution in Biological Tissues
The generalized Riccati equation mapping method assumes that Equation (1.12) admits solutions that can be expressed as a polynomial of degree in m  : G G  Really speaking, solution (1.10) contains only one free parameter, namely, 0

Thermal Traveling-Wave Distribution in Living Tissues
where 0 1 , , , m A A A  m are real constants to be determined and is solution of the Riccati Equation (1.5).The first step in this method consists of determining .To determine , we consider the homogenous balance between the highest order derivative and the highest order nonlinear terms.After finding , one substitutes Equation (1.15) into Equation (1.14) and use the fact that is a solution of Equation (1.5).Then, one collects all terms with the same order of 1 n G and .If the coefficients of and vanish separately, one will obtain a system of algebraic equations in , a  that physical solutions can be found with the help of Mathematica (here, physical solution means solutions of the algebraic system that define physical solutions of Equation (1.1) with perfusion (1.2)).Finally, one uses the solutions of Riccati Equation (1.5) found in the previous section, and goes back in original variables x and t through ansatz (1.13) to obtain travelling wave solution of Equation (1.12).
With the help of Riccati equation, we investigate in this section the thermal traveling-wave distribution in biological tissues through the BHT Equation (1.1) with the temperature-dependent blood perfusion (1.2).In other words, we present in this section the application of the Riccati equation to find analytical traveling-wave solutions of equation with constant p Q .We start by transforming Equation (1.12) into a nonlinear ordinary differential equation by introducing the variable given by z where and 0 a   are two arbitrary constants ( is the wave velocity).Inserting ansatz (1.13) into Equation (1.12) yields   Let us now turn to the concrete case of Equation (1.5).We start by inserting Equation (1.15) into Equation (1.14).Using the fact that is a solution of Equation (1.5) and balancing the highest order of derivative term G and nonlinear term, we get that for a quadratic blood perfusion, and for a linear blood perfusion.So that Equations (1.15) and (1.5) give for quadratic and linear blood perfusion, respectively.By combining solution (1.10) of Riccati equation and expression (1.16a) and going back to original coordinates x , , and one easily sees that The steady-state analytical solution to the bio-heat Equation (1.1) with blood perfusion (1.2.) differs with a spatially distributed uniform sink given in Ref. [43].

Case of a quadratic temperature-dependent blood perfusion
Let us start with the case of a quadratic blood perfusion, that is, when 0 0   .Then Equation (1.14) admits solutions of form (1.16a).Inserting Equation (1.16a) into Equation (1.14) and imposing to the coefficients of and to vanish separately, we obtain, after the compatibility requirement, the algebraic system (see (1.17

 
; ; ; It should be noted that condition 0   corresponds to either the initial temperature distribution or the steady-state temperature distribution.Indeed, the initial temperature distribution and the steady-state temperature distribution in biological tissues is obtained by integrating the ordinary differential equations Q any real constant.We notice that in the case of absence of heating sources, the initial temperature coincides with the steady-state temperature fields.

Case of a linear temperature-dependent blood perfusion
Here, we consider the case of a linear blood perfusion, that is, when 0 0   and 0 0 . As it is shown above, Equation (1.14) admits solutions of form (1.17).Inserting Equation (1.17) into Equation (1.14) and imposing to the coefficients of and to vanish separately, we obtain, after the compatibility requirement, the algebraic system (see (1.20) (1.17)   means the generalized Riccati equation mapping method in the case of linear blood perfusion can be applied just for the initial temperature distribution and the steady-state temperature distribution.

Coupling Method for Thermal Traveling-Wave Distribution in Biological Tissues
In this section, we split Equation (1.14) into two ordinary differential equations which lead to a Riccati equation that will help us to find analytical solutions.
It is obvious that will satisfy Equation (1.14) as soon as it verifies the following two equations One easily verifies that any solution   u z of Equation (1.21a) will satisfy equation and 0 C is an arbitrary real constant.

Results and Discussion
In this section, we use the analytical solutions found in the previous section to discuss thermal traveling-wave distribution in biological tissues.
In the following calculations, the typical tissue properties are applied as given in [20]: for the healthy tissue and 3 4200 J m s m Q   for the tumor.In our simulations, the skin surface and the body core are assumed to be defined at and 0 x  x L  , respectively.As it is shown by several investigators [44,45], the interior tissue temperature usually tends to a constant within a short distance such as 2 -3 cm.Therefore we will choose   2,3 cm L  in our simulations.Following Partridge et al. [35], we will use for the temperature-dependent blood perfusion (1.2), the coefficients where for tumour 1 takes values up to 7, 2 takes values up to 1.9, and 3 takes values up to 0.07.In the case of linear blood perfusion, i.e., when we will take 1 and 2 so that 0 p p  and 0  coincide with the coefficients of linear blood perfusion used in Ref. [36].Because the variation of the constant spatial heating r can only affect the temperature magnitude at a given depth Q , x we will, without loss of generality, consider in all our simulations that r so that the initial temperature field for the basal state of biological bodies will coincide with the steady-state temperature fields.
We start our analysis with the case of linear temperature-dependent blood perfusion.A and .These two parameters can be defined in a concrete problem of heat transfer, for example, when the boundary conditions are given.For plots of Figure 1  give the same temperature at surface of the skin and at the body core.The top right and bottom left and right plots show that one may use the superposition parameter to manage (increase or de-crease) temperature of the biological tissues.
Let us discuss the case of a quadratic temperaturedependent blood perfusion.We will focus our attention on the case 0   and 0 p  , that is, the case of thermal traveling-wave distribution.We will investigate the effect of linear and quadratic coefficients of temperature dependence on temperature distribution, and then, show how one may used the parameter of the nonlinear superposition formula of Riccati equation to control the temperature in time and space.
Figure 2 shows the effect of the quadratic coefficient of temperature dependence on temperature distribution.For this figure, we have used the perfusion coefficients , , 3, 0.5, 0.01 , 3, 0.5, 0.02 , p p p     3, 0.5, 0.03 .The 3 cm long tissue has being used.For these values of 1 2 and 3 physical solutions of system (1.17) are given in  (a) A and s appear as free parameters and can be particularized with the help of additional condition of the bio-heat transfer problem such as boundary conditions.For all plots of Figure 2, we have used 1 0.1 A  and .25 s   The left and right plots of Figure 2 respectively show the temporal and the spatial distribution of temperature at different distance x from the skin surface and at different time for different values of 3 (i.e., for different values of the quadratic coefficient of temperature dependence of blood perfusion).As expected, increased perfusion causes a decline in local temperature.As it is seen from the left plots, the temperature of the tissue at each distance x from the skin surface decreases and approaches the steady-state temperature.It is also seen that with smallest 3 the tissue temperature will quickly reach a steady state.As seen in right plots, tissue temperature decreases as one goes from the skin surface to body core.To investigate the effect of linear coefficient of temperature dependence of the blood perfusion on temperature distribution, we fix and and vary as follows: Figure 3 shows the temporal (left plots) and the spatial (right plots) temperature distribution for a 3 cm long tissue.The plots of this figure have been generated with the help of Riccati solution (1.10).As solutions of system (1.17), we have used data from the first row of (a), (b), and (c) in x  , increased perfusion causes a decline in the temperature of skin surface.Plots (b) and (c) show that inside the tissue (when ), for a short period of heating, decreased perfusion causes a decline in local temperature.As one tends to the steady-state, higher perfusion gives lower local temperature.From plots (d) and (f) which respectively show the initial temperature distribution and the steady-state temperature fields, it is seen that increased perfusion causes an increase in the initial and steady-state local temperature of tissue.Let us now validate solutions (1.24) and (1.25).This solution presents some restriction on blood perfusion (1.2).Indeed,   tending the time to plus infinity.After the initial time t 0 t  and before the steady-state time St T (time after which the temperature at depth x is constant with respect to time ), the obtained analytical solutions are used for temperature prediction at any depth t x at any time .Our analytical solutions allowed us to investigate the effect of both linear and quadratic coefficients of temperature dependence on temperature distribution.We mainly showed that the increase in perfusion via the increased in the linear coefficient causes a decline in local temperature.On the other hand, it has been found that decreased perfusion via the quadratic coefficient of temperature-dependence causes decline in interior local temperature (that is, away from the skin surface).The application of the nonlinear superposition formula of the Riccati equation is useful for temperature control in the tissue.The analytical solutions found in this paper can be used to predicate the evolution of the detailed temperature within the tissues during thermal therapy.(f).From plot (a) which shows the temporal distribution of temperature close to skin surface, it is seen that near the skin surface, increased perfusion causes a decline in local temperature.Away from the skin surface, the temperature curves intersect before the time of the steady-state temperature (see plots (b) and (c)).This means that away from the skin surface and before the steady-state time, increased perfusion does not necessary cause a decline in local temperature.This last fact can also be observed in plots (d) and (e).As it is seen from left plots, the higher steady-state temperature is reached at the body core.Plot (f) shows that higher perfusion gives lower steady-state temperature at any depth x of the tissue.
Analytical traveling-wave solutions of the (1 + 1) generalized Pennes Equation obtained using the extended generalized Riccati equation mapping method for the hyperbolic (rational in exponential functions), trigonometric and rational function types are presented in this paper.This method is characterized by the fact that the construction of three particular solutions of the considered equation leads to an infinity of solutions via the nonlinear superposition formula of Riccati equation.With the use of the obtained solutions, we have analytically investigated the thermal traveling-wave distribution in biological tissue through a bio-heat transfer model with linear and quadratic temperature-dependent blood perfusion.For linear temperature-dependent blood perfusion, the method can give only initial temperature distribution which is useful for numerical simulations, and the steady-state temperature distribution which is useful to set the initial and boundary conditions of bio-heat transfer problem [46,47].In the case of quadratic temperature-dependent blood perfusion, the initial condition may be obtained by setting the propagating velocity  to zero, while the steady-state temperature of the tissue can be obtained by


 is specific heat of blood; b  is the density of blood;   ml s ml b  is the blood perfusion;  the tissue thermal conductivity; 1.1) describes the influence of blood flow on the temperature distribution in the tissue in terms of volumetrically distributed heat sinks or sources.The second term

0 
are respectively the linear and quadratic coefficients of temperature dependence.The unit employed for coefficient 0 ,  and 0  is such that b b   represents mass flow rate of blood per unit volume of tissue.Based on the Pennes BHT Equation (1.1) with the blood perfusion (1.


satisfy condition(1.23),then any solution of Equation (1.21a) will be solution of Equation(1.14).Henceforth, we will focus ourselves on Equation (1.12a).When 0 particular and the general solutions of Equation (1.21a) the following solutions of Equation (1.1) with perfusion Equation(1.2) Figure shows the initial/steady-state temperature distribution obtained in the case of linear temperature-dependent blood perfusion with the same baseline perfusion 0 0.1   , 0.15, 0.2 and the linear perfusion coefficient 0 0.1   , respectively.The physical solutions of system (1.20) used for this figure are showed in

Figure 1 .
Figure 1.(Color online) Initial/steady-state temperature-dependent perfusion distributions for different linear coefficient of temperature dependence in the case of linear temperature-dependent blood perfusion.The perfusion level was dependent on local temperature with the same baseline perfusion α 0 = 0.1 and a linear coefficient of temperature dependence shown in inset.Top left: Effect of linear coefficient of temperature dependence on temperature distribution; here, we have used (C 1 , C 2 ) = (0.01, 2) (see Equations (1.4)).Top and bottom right, bottom left: Spatial temperature distribution for different and different (C 1 , C 2 ) appearing in Equations (1.4) and (1.10), respectively; the same values of (C 1 , C 2 ) and , shown in the bottom right plots have been used.The dashed and dotted curves show the temperature distribution obtained with the help of the nonlinear superposition (1.4). k  k

C 2 C 0. 2 Figure 4
Dashed and dotted curves ofFigure 4 show the temperature distribution obtained with the help of nonlinear superposition formula (1.4) of Riccati equation.Other three curves of this figure show the temperature distribution obtained with the use of solution (1.10) after particularizing parameters 1 and 2 C ; we have used the same parameter and three different C 1 = 17, 20, 23.To generate the solutions of the superposition, we have used 1 7, p  2 1, p  and 3 and the following solution of system (1.17): α = 0.5291502622129182, υ = 0.00012855305358758568, A 0 = 100.49771736484482,r = −634.2109797960809,A 1 and s = 4. Plots (a) and (b) show respectively the temperature of skin surface and body core as function of time .Plots (c) and (d) depict the temperature curves as function of skin depth show that one can easily readjust (increase or decrease) the temperature of the biological tissues, just by varying parameter of the nonlinear superposition formula (1.4) of Riccati equation.

Figure 3 .
Figure 3. (Color online) Temporal and spatial temperature-dependent perfusion distributions: Effect of the linear coefficient of temperature dependence on temperature distribution.The left and right plots respectively show the temporal and spatial temperature distribution at different distance x from skin surface and at different time t.Plots (a), (b), and (c) show the temperature distribution at skin surface, at distance x = 0.015 m, and at the body core, respectively.Plots (d), (e), and (f) show the initial temperature distribution, the temperature distribution at time t = 100 s, and the steady-state temperature distribution, respectively.For all these plots, we have used solution (1.4)) with (C 1 , C 2 ) = (−17, 0.2).Here, we have used p 1 = 7, p 3 = 0.07, and different p 2 shown in plot (a) (see Equation (1.26)).We have used a = 0.5291502622129182, υ = 0.00012855305358758568, A 0 = 100.49771736484482,and r = −634.2109797960809for p 2 = 1; a = 0.5291502622129182, υ = 0.00013226855259313876, A 0 = 102.1353600675383,and r = −650.6257480827784for p 2 = 1.2; and a = 0.5291502622129182, υ = 0.00013594278617606382, A 0 = 103.73333731695307,and r = −666.6402113289303for p 2 = 1.4.Other parameters used for the realization of these plots are given in the text.

Figure 4 .Figure 5 .
Figure 4. (Color online) Temperature management via the parameter of nonlinear superposition formula of Riccati equation.(a), (b): Temperature of skin surface and temperature of the body core as a function of time.(c), (d): Temperature distribution as a function of skin depth x for two different time points t = 0 s and t = 90 s.Dashed and dotted curves show the temperature distribution obtained with the help of the nonlinear superposition formula (1.4) with = 1.1 (dashed curve) and = 0.3 (dotted curve).Solution (1.10) of Riccati equation has been used with to generate the three solutions of the superposition; for these three solutions, we used in Equation (1.10) C 2 = 0.2 and C 1 shown in plot (a).The values of other parameters and constants are given in the text.

tFigure 5 2 ..
investigate the effect of the linear and quadratic coefficients of temperature dependence on temperature distribution.In fact, 3 appears in Equation (1.23) as a function of .We will work with 1 and different values of 2 and, consequently, different value of 3 shows the thermal traveling-wave distribution obtained with the use of solution (1.24) with different Inserting these values of 2 and 1 in Equation (1.23), we obtain p 3 = 0.0367006, 0.0421037, 0.0475073, respectively.Thus, 3 increases as increases.For all the plots of this figure, we have used and The left plots show the temperature distribution as a function of time t at different depths (a), (b), and 0 The right plots show the spatial temperature distribution for different time t = 0 s (d), t = 500 s (e) , and

.
As one can see from

Table 1 ,
the physical solutions contain two parameters, 2

Table 1 . Solutions of system (1.20) obtained for the same baseline perfusion α 0 and different linear coefficient β 0 .
, we r

Table 2 .
For these physical

Table 3 .
This table shows two free parameters, 1

Table 3
C 