We have used a nonlinear one-dimensional heat transfer model based on temperature-dependent blood perfusion to predict temperature distribution in dermis and subcutaneous tissues subjected to point heating sources. By using Jacobi elliptic functions, we have first found the analytic solution corresponding to the steady-state temperature distribution in the tissue. With the obtained analytic steady-state temperature, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are numerically analyzed. Our results show that the derived analytic steady-state temperature is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.
The purpose of this work is to use Jacobian elliptic functions to construct a nonlinear heat transfer model in dermis and subcutaneous tissues. To predict the temperature in these two biological living tissues, we use a Pennes type of bio-heat transfer equation with a temperaturedependent blood perfusion term.
Since the pioneer work of Pennes [1] in which a linear mathematical model was proposed for describing the thermal interaction between human tissues and perfused blood, taking the effects of the metabolism into account, alternative nonlinear models for describing the heat exchange between tissues and blood have been developed [2-7]. The Pennes model assumes a constant rate blood perfusion within each type of tissues. However, it has been shown by several experiments and numerical simulations that physiological responses such as blood perfusion and metabolism in living tissues are temperaturedependent (see for example [5,8]). A more accurate description of the heat transfer in living biological tissues will then be obtained only by including, if possible, a temperature-dependent blood perfusion and a variable metabolic heat generation terms in Pennes equation.
By including a temperature-dependent blood perfusion term in Pennes equation, the temperature distribution in the living biological tissue at hand will be governed by a nonlinear time-dependent partial differential equation. In such biological tissues, the temperature will not be uniformly distributed in space and time. Moreover, it will be very difficult and even impossible to find analytical solutions of the governing equation; in such situations, only numerical solutions are attempted, and we talk of numerical temperature distribution in living biological tissues.
In the present work we use a Pennes type model of bio-heat transfer equation to numerically investigate temperature distribution in dermis and subcutaneous tissues; here, we include in Pennes equation a temperature-dependent blood perfusion term and maintain a constant metabolic heat generation term. Our investigation is based on the following one-dimensional (1D) modified Pennes bio-heat transfer model [8]:
where, , k are the density, specific heat and thermal conductivity of the tissue, respectively, is the blood specific heat, is the blood density, T is the local tissue temperature, is the arterial blood temperature which is treated as constant, is time, is the metabolic heat production per volume, and is the heat deposited per volume due to spatially distributed heating (it is the external spatial heating), is the temperature-dependent blood perfusion, and denotes the distance from the skin surface to the body core. In this work, we take the temperature-dependent blood perfusion to be of the form [8]
where is the baseline perfusion and is the linear coefficient of temperature dependence. Because the interior tissue temperature usually tends to a constant a short distance from the skin surface, such as 0.02 - 0.03 m (see for example [9,10]), we consider in Equation (1) that, with m.
The aim of this paper is to investigate via Equation (1) with blood perfusion (2) the temperature distribution in dermis and subcutaneous tissues in the hypothermia case when both skin surface and spatial heating are used. We restrict ourselves to positive temperatures. This paper is carried out in a case of hypothermia state where the tissue temperature is below 35˚C. In this work, we show how to increase the tissue temperature by applying point heating sources at different depths of the tissue and by controlling the equation parameters. The technique we used here is first to find the tissue temperature prior to heating, using Jacobi elliptic functions; this allows us to obtain the steady-state temperature (temperature of the tissue at time t = 0 s whose values are in the range of the temperature of a hypothermia of deep degree (below 20˚C). We then use the obtained steady-state temperature to numerically investigate the temperature distribution when the tissue is subjected to spatial point heating. The rest of the work is organized as follows: in Section 2 we use Jacobi elliptic functions to investigate the initial temperature distribution for the basal state of dermis and subcutaneous tissues. The numerical nonuniform (in time and space) temperature distribution is studied in Section 3, and our work is summarized in Section 4.
2. Investigation of the Initial Temperature Distribution for the Basal State of Dermis and Subcutaneous Tissues via Jacobi Elliptic Functions
Up to now, no author has applied Jacobi elliptic functions to analytically solve the steady-state problem of bioheat transfer equation with temperature-dependent blood perfusion. In most of the existing analytical studies, the solutions to the bioheat transfer problem for a steadystate are for temperature-independent blood perfusion, which may not be practical for real bio-thermal situations. Therefore it is still desirable to obtain possible way to analytically solve the most widely accepted Pennes’ equation in the bioheat field.
In this section, we investigate, by the means of Jacobi elliptic functions, the initial temperature field for the basal state of dermis and subcutaneous tissues. If we denote by the steady-state temperature field prior to heating, the surrounding air temperature, and the apparent heat convection coefficient between the skin surface and the surrounding air under physiologically basal state (is also considered as an overall contribution from natural convection and radiation), then will be the solution of the boundary value problem
Because a biological body tends to keep its core temperature to remain stable, the body core temperature is treated as a constant given by, whence the following additional boundary condition
Because we work in a hypothermia case, the core temperature will be chosen below 35˚C. If we multiply Equation (3) by and integrate the result, we obtain the first integral
where cst in a constant of integration. Using the boundary condition (4), we find that
where. The following notation,
reduces Equation (6) to
Particular solutions of Equation (9) can be found in terms of Jacobi elliptic functions [11]:
where is the Jacobi elliptic sine function with modulus and A, , , and m are four constants verifying the nonlinear algebraic system
Because of our restriction on the positivity of for all, we must have A > 0; moreover, if B < 0, the additional condition is required. Solving the first three equations of (11), we obtain, under the condition,
Inserting A in the fourth equation of system (11) yields
Because
for all m, we must have
Thus, the boundary value problems (3)-(5) will have solutions of the form (10) if and only if and The last condition means that the constant of integration, cst, in Equation (6) must satisfy the condition
Comparing the right-hand sides of Equations (7) and (13) we obtain that the apparent heat convection coefficient must satisfy the equation
It follows from Equation (10) that This last equality and condition (5) give, which together with the expression for A in Equation (12) give the equation
for determining m. A necessary condition for Equation (15) to have a solution is that should be non-positive.
We summarize the obtained result as follows: if the initial temperature at skin surface satisfies Equation (14) and if then the boundary value problems (3)-(5) admits particular solutions of the form (10) with, B, and given by Equation (12) and is a solution of Equation (15), where, , and are defined by Equation (8).
Solutions as (10) are expected to be very useful in a variety of bio-thermal practices: 1) for extreme situations where perfusion will change significantly with the external heating; 2) if the average perfusion in a specific temperature range was known (here, the analytical solution will provide intuitive temperature prediction); 3) for those bioheat transfers under small heating (in this case, a good accuracy from the analytical prediction can be expected).
Figure 1 shows the temperature distribution at the initial time t = 0 of dermis (left plot) and subcutaneous (right plot) tissues for different values of the linear coefficient γ of temperature dependence of blood perfusion. Here we have used for both dermis and subcutaneous tissues the following typical tissue parameters [12]: Ta = 37˚C, Ts = 15˚C, cb = 3500 J/kg˚C, Qm = 43800 w/m3, ω0 = 19 × 10–5 ml/100 g·min, ρb = 1060 kg/m3. As thermal conductivity of tissue, we used k = 0.45 w/m·˚C and k = 0.19 w/m·˚C for dermis and subcutaneous tissues, respectively. To investigate the effect of the linear coefficient of temperature dependenceγon the initial temperature distribution, we used three values of γ [13]: γ = 0.122, γ = 0.122597, and γ = 0.1226.
1) Dermis tissue: With the above equation parameters, we compute the elliptic moduli as m = 0.383359, m = 0.028886, and m = 0.0201884. Solution (10) then gives T0(0) = 12.7348˚C, T0(0) = 12.7˚C, and T0(0) = 12.6736˚C, respectively. Inserting these values of in Equation (14) yields h0 = 35.142 w/m˚C, h0 = 35.0382 w/m2, and h0 = 34.7618 w/m2·˚C, respectively.
2) Subcutaneous tissue: For the subcutaneous tissue, we found m = 0.383359, m = 0.028886, m = 0.0201884 T0(0) = 12.7797˚C, T0(0) = 2.7043˚C, and T0(0) = 12.6927˚C; h0 = 44.0957 w/m2·˚C, h0 = 43.513 w/m2·˚C, and h0 = 43.328 w/m2·˚C.
The temperature curves of Figure 1 show that increased perfusion causes a decline in the local temperature, while the local temperature decreases with the elliptic modulus. Also, the temperature distribution becomes more oscillatory when the blood perfusion increases. Because dermis tissue and subcutaneous tissue differ only in thermal conductivity, we can also conclude from the plots of Figure 1 that an increased thermal conductivity causes a decline in local temperature.
3. Numerical Simulations3.1. Numerical Techniques
Equation (1) with temperature-dependent blood perfusion (2) is a time-dependent nonlinear partial differential equation, and its numerical solutions can be obtained using a Crank-Nicolson scheme. In this section, we will stress on the solution of Equation (1) using a secondorder central difference scheme in space and a CrankNicholson type of scheme in time. The boundary conditions associated to Equation (1) is a segment that starts from the skin surface (x = 0) and ends at the body core (x = L) and is described as follows: a given temperature boundary condition is applied at the body core, i.e., T = Tc at x = L; a convective boundary condition is used at the skin surface, at x = 0, which is the normal case to which the skin surface is subjected. Because Equation (1) is time-dependent (first order differential equation in time), we need an initial condition. We will consider that prior to heating (i.e., at t = 0), the temperature at any depth x is, where is the steady-state temperature field found in the previous section (see Equation (10)); therefore at t = 0. Mathematically, we will numerically solve the following initial boundary value-problem
The Newton’s heating/cooling law
means that, at any
time, the skin surface has exactly the same temperature as the heating/cooling medium:, where denotes the time-dependent temperature of the cooling medium. This condition is typical at the skin surface for thermal comfort analysis; it is also used for cancer hyperthermia.
Denote by and h the time step discretization and the space mesh, respectively (here h is selected such that is a positive integer) and let, , and,. Let be the scalar numerical value of the temperature at depth xj and time, i.e., , and the vector value be an matrix (column vector), where. Then, by applying a second-order central difference scheme in space and a Crank-Nicholson type scheme in time for solving problem (16), i.e., for finding all the, we find that the vector is the solution of an algebraic linear system of form
where and are two tridiagonal matrices and F an matrix (column vector); moreover is diagonally dominant. Hence, system (17) admits a unique solution
3.2. Numerical Experiments and Discussion
For the numerical simulation, we use the following parameters in Equation (1): Ta = 37˚C, Ts = 15˚C, Tc = 12.7˚C, cb = 3500 J/kg·˚C, Qm = 43800 w/m3, ω0 = 19 × 10–5 ml/100 g·min, ρb = 1060 kg/m3, ρ = 1200 kg/m3, c = 3300 J/kg·˚C. As thermal conductivity of the tissue, we used k = 0.45˚C and k = 0.19 w/m·˚C for dermis and subcutaneous tissue, respectively. The linear coefficient of temperature dependence γ is chosen among the following values: γ = 0.122, γ = 0.122597, and γ = 0.1226. will be chosen in a way that Equation (10) will give a nontrivial solution of problems (3)-(5) (we point out that we shall use this solution as initial temperature distribution in the tissue). Although any spatial heating style like can be dealt with by the present numerical simulation, in this work we use only point heating to investigate the temperature response of the tissue [12,13]. Practical examples of point heating can be obtained in clinical treatments where heat is deposited by inserting a conducting heating probe in the deep tumor site or delivering thermal dose to it. Here, we use a point heating source of the form [12]
where is the strength of the point heating source (it is the time-dependent heating power on the skin surface), is the Dirac delta function, and the location of the point heating; is a constant, the constant oscillation amplitude of sinusoidal heating, the heating frequency, and the scattering coefficient. The reason for choosing an oscillating heating, especially a sinusoidal heating, is that sinusoidal surface heating can be generated by an instrument with repeated irradiation from regulated laser and used to estimate the blood perfusion (see [14,15]). It follows from Equation (18) that
and this, of course, means that all the heat coming from the heating source is concentrated at the point We will indicate for each example the values used for the heating parameters, , , , and. In all the following examples, we have applied a point heating source at three different depths between the skin surface and the body core, exclusively; in fact, the skin surface is maintained at the same temperature as the cooling medium, and the body core temperature is maintained at (core temperature). Without loss of generality, we only work with dermis tissue, and all the obtained results may be transferred to subcutaneous tissue.
Figure 2 depicts the spatial and temporal temperature distributions of dermis tissue in the absence of spatial heating for γ = 0.1226. Due to the surface heating by the flowing of medium, the tissue temperature increases and is above the initial (steady-state) temperature. Because of the oscillatory aspect of the steady-state temperature field (see the plot of Figure 1 with γ = 0.1226), temporal temperature curves at the beginning of the process tend to be oscillatory (see the left plots on the top Figure 2). It is also important to point out that, at the beginning of the process (during about the first 110 s of the heating process), the temperature of the tissue at any depth is below the temperature at the skin surface (see left bottom plots). With time passing, the temperature first increases when one goes from skin surface to body core, reaches a highest value at some depth between x = 0.01285 m and x = 0.01585 m (see the bottom middle plots), and then decreases when one approaches the body core (see the right top plots). As one can see from the right plots (top and bottom), each point of the tissue reaches it stationary state after some duration of the process (near 3000 s). Moreover the highest temperature of each point during the process remains in the hypothermia range (see the top and bottom plots). Because of the continuous heating, the temperature of each point of the tissue increases rapidly in the early heating time and then slowly approaches its stationary value; this is easily seen from the bottom plots.
Figure 3 depicts spatial and temporal temperature distributions in dermis tissue in the presence of spatial heating when a point-heating source is applied. In this example, we used γ = 0.1226, p0 = 500,000/3 w/m2, p1 = 0 w/m2 and applied a point-heating source at three different points, namely, at x0 = 0.008571 m, x0 = 0.01585 m, and x = 0.02357 m. We first point out that all effects observed in the absence of spatial heating occur when a spatial heating source is applied; particularly, the temperature at each depth of the tissue increases gradually until it reaches steady-state. As the top plots show, the positions of higher temperatures stay at the site of the point sources. This is very beneficial, not only for hypothermia therapy, but also for hyperthermia therapy; in fact, one can selectively apply a point-heating source to heat the deep regional tumor in case of hyperthermia therapy. A comparison of Figures 2 and 3 show that the point-heating source may affect other points of the tissue only after a long heating time. A suitable localization of the heating source may prevent the destruction of the tissue cell that are outside the tumor region (in the case
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, 
, k are the density, specific heat and thermal conductivity of the tissue, respectively, 
is the blood specific heat, 
is the blood density, T is the local tissue temperature, 
is the arterial blood temperature which is treated as constant, 
is time, 
is the metabolic heat production per volume, and 
is the heat deposited per volume due to spatially distributed heating (it is the external spatial heating), 
is the temperature-dependent blood perfusion, and 
denotes the distance from the skin surface to the body core. In this work, we take the temperature-dependent blood perfusion to be of the form [
is the baseline perfusion and 
is the linear coefficient of temperature dependence. Because the interior tissue temperature usually tends to a constant a short distance from the skin surface, such as 0.02 - 0.03 m (see for example [9,10]), we consider in Equation (1) that
, with 
m.
the steady-state temperature field prior to heating, 
the surrounding air temperature, and 
the apparent heat convection coefficient between the skin surface and the surrounding air under physiologically basal state (
is also considered as an overall contribution from natural convection and radiation), then 
will be the solution of the boundary value problem
is treated as a constant given by
, whence the following additional boundary condition
will be chosen below 35˚C. If we multiply Equation (3) by 
and integrate the result, we obtain the first integral
. The following notation,
is the Jacobi elliptic sine function with modulus 
and A, 
, 
, and m are four constants verifying the nonlinear algebraic system
for all
, we must have A > 0; moreover, if B < 0, the additional condition 
is required. Solving the first three equations of (11), we obtain, under the condition
,
and 
The last condition means that the constant of integration, cst, in Equation (6) must satisfy the condition
must satisfy the equation
This last equality and condition (5) give
, which together with the expression for A in Equation (12) give the equation
should be non-positive.
then the boundary value problems (3)-(5) admits particular solutions of the form (10) with
, B, and 
given by Equation (12) and 
is a solution of Equation (15), where
, 
, and 
are defined by Equation (8).
in Equation (14) yields h0 = 35.142 w/m˚C, h0 = 35.0382 w/m2, and h0 = 34.7618 w/m2·˚C, respectively.
at x = 0, which is the normal case to which the skin surface is subjected. Because Equation (1) is time-dependent (first order differential equation in time), we need an initial condition. We will consider that prior to heating (i.e., at t = 0), the temperature at any depth x is
, where 
is the steady-state temperature field found in the previous section (see Equation (10)); therefore 
at t = 0. Mathematically, we will numerically solve the following initial boundary value-problem
means that, at any
, where 
denotes the time-dependent temperature of the cooling medium. This condition is typical at the skin surface for thermal comfort analysis; it is also used for cancer hyperthermia.
and h the time step discretization and the space mesh, respectively (here h is selected such that 
is a positive integer) and let
, 
, and
,
. Let 
be the scalar numerical value of the temperature at depth xj and time
, i.e., 
, and the vector value be 
an 
matrix (column vector), where
. Then, by applying a second-order central difference scheme in space and a Crank-Nicholson type scheme in time for solving problem (16), i.e., for finding all the
, we find that the vector 
is the solution of an algebraic linear system of form
and 
are two 
tridiagonal matrices and F an 
matrix (column vector); moreover 
is diagonally dominant. Hence, system (17) admits a unique solution
will be chosen in a way that Equation (10) will give a nontrivial solution 
of problems (3)-(5) (we point out that we shall use this solution as initial temperature distribution in the tissue). Although any spatial heating style like 
can be dealt with by the present numerical simulation, in this work we use only point heating to investigate the temperature response of the tissue [12,13]. Practical examples of point heating can be obtained in clinical treatments where heat is deposited by inserting a conducting heating probe in the deep tumor site or delivering thermal dose to it. Here, we use a point heating source of the form [
is the strength of the point heating source (it is the time-dependent heating power on the skin surface), 
is the Dirac delta function, and 
the location of the point heating; 
is a constant, 
the constant oscillation amplitude of sinusoidal heating, 
the heating frequency, and 
the scattering coefficient. The reason for choosing an oscillating heating, especially a sinusoidal heating, is that sinusoidal surface heating can be generated by an instrument with repeated irradiation from regulated laser and used to estimate the blood perfusion (see [14,15]). It follows from Equation (18) that
We will indicate for each example the values used for the heating parameters
, 
, 
, 
, and
. In all the following examples, we have applied a point heating source at three different depths between the skin surface and the body core, exclusively; in fact, the skin surface is maintained at the same temperature as the cooling medium, and the body core temperature is maintained at 
(core temperature). Without loss of generality, we only work with dermis tissue, and all the obtained results may be transferred to subcutaneous tissue.