Boundary Element Method Solution of a Fractional Bioheat Equation for Memory-Driven Heat Transfer in Biological Tissues

Abstract

This work develops a Boundary Element Method (BEM) formulation for simulating bioheat transfer in perfused biological tissues using the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). The ABC operator incorporates a nonsingular Mittag–Leffler kernel to model thermal memory effects while preserving compatibility with standard boundary conditions. The formulation combines boundary discretization with cell-based domain integration to account for volumetric heat sources, and a recursive time-stepping scheme to efficiently evaluate the fractional term. The model is applied to a one-dimensional cylindrical tissue domain subjected to metabolic heating and external energy deposition. Simulations are performed for multiple fractional orders, and the results are compared with classical BEM $(a=1.0)$, Caputo-based fractional BEM, and in vitro experimental temperature data. The fractional order $a\approx 0.894$ yields the best agreement with experimental measurements, reducing the maximum temperature error to 1.2% while maintaining moderate computational cost. These results indicate that the proposed BEM–ABC framework effectively captures nonlocal and time-delayed heat conduction effects in biological tissues and provides an efficient alternative to conventional fractional models for thermal analysis in biomedical applications.

1. Introduction

Realistic heat conduction simulation in biological tissue is critical in biomedical treatments, i.e., hyperthermia therapy, cryosurgery, and thermal diagnostics [1,2,3,4]. The Fourier-type conduction is used in traditional models, e.g., Pennes’ bioheat equation, which does not support memory-dependent behavior simulation in perfused biological media [5,6,7,8]. Fractional calculus can be used for heat memory effects modeling, but traditional fractional derivatives (Riemann-Liouville, Caputo) have singular kernels and are computationally intensive [9,10,11,12].

The Atangana–Baleanu Caputo (ABC) derivative is a relatively recent development in fractional calculus, introduced to describe dynamical systems with non-singular and nonlocal memory effects [13,14,15]. Unlike the Riemann–Liouville and classical Caputo derivatives, which use a power-law kernel that is singular at the origin, the ABC derivative employs a Mittag–Leffler function-based kernel, resulting in a smooth and bounded representation of memory [16,17,18]. This formulation enables the modeling of fading memory effects observed in physical and biological processes without the complications introduced by kernel singularities. In the Caputo sense, the ABC derivative remains compatible with standard initial and boundary conditions, which supports its applicability in engineering and biomedical contexts [19,20]. In bioheat transfer, this approach allows the delayed thermal response in heterogeneous, perfused tissues to be represented in a physiologically consistent way [21,22]. Furthermore, its kernel properties can improve numerical convergence and stability when applied in computational techniques such as the Boundary Element Method [23].

Despite its theoretical advantages, solving the fractional bioheat equation (FBE) involves notable computational challenges. The presence of fractional time derivatives introduces history-dependent operators, which complicate both spatial and temporal discretization [24,25,26]. Many bioheat transfer problems also involve irregular geometries, non-uniform boundary conditions, and internal heat sources such as metabolic activity and externally applied energy. These factors can make conventional numerical methods, including the Finite Difference Method (FDM) [27] and Finite Element Method (FEM) [28], computationally demanding and less effective in accurately representing problems dominated by boundary effects.

The Boundary Element Method (BEM) offers an alternative by reformulating partial differential equations as boundary integral equations, reducing the problem’s dimensionality, and restricting discretization to the boundary [29,30]. This approach can lower computational cost, particularly for problems with large or unbounded domains, and provides high accuracy in enforcing boundary conditions [31,32]. BEM is also well-suited for cylindrical and layered geometries, which commonly arise in bioheat transfer applications such as simulating heat distribution in tumors, blood vessels, or multilayered tissue structures [33,34].

This work presents a numerical approach that combines the Atangana–Baleanu fractional derivative with the Boundary Element Method (BEM) to solve the fractional bioheat equation (FBE) in biological tissues. The formulation uses cell-based domain integration to incorporate volumetric heat sources and a time-stepping scheme to evaluate the memory-dependent fractional operator. The method is applied to a one-dimensional cylindrical tissue model that includes both metabolic heating and external energy deposition. Simulations are carried out for different fractional orders (α), examining their influence on temperature evolution and heat flux. The optimal agreement with simulated experimental data is obtained at $a\approx 0.894$. A sensitivity analysis is performed to quantify the effects of thermal conductivity, blood perfusion rate, and external heating on the temperature response. Results indicate that the method can reproduce physiologically consistent temperature distributions and capture the effects of time-lagbtainged heat conduction. The fractional heat flux trends further illustrate the role of nonlocal thermal effects in biological tissues. The primary aim of this work is to develop and validate a Boundary Element Method (BEM) formulation that incorporates the Atangana–Baleanu fractional derivative in the Caputo sense (ABC) for modeling memory-dependent bioheat transfer in biological tissues. The method integrates the nonsingular Mittag–Leffler kernel into a boundary-based numerical framework, employing cell-based domain integration for volumetric sources and a recursive time-stepping scheme for fractional time derivatives. The novelty lies in combining the physical suitability of the ABC kernel for bounded thermal memory with the computational efficiency and accuracy of BEM. A secondary aim is to apply the validated formulation to investigate how the fractional order αα influences temperature response and heat flux in a cylindrical tissue model, and to compare the results with synthetic and experimental data. This dual focus allows both the demonstration of a robust numerical tool and the extraction of physical insights into memory-driven bioheat transfer.

The paper is structured as follows: Section 1 provides context for the research by summarizing bioheat transfer models, pointing out deficiencies of conventional and fractional formulations, and discussing why the Atangana–Baleanu derivative has been used under a Boundary Element Method approach. Section 2 defines the fractional bioheat equation, declares the physical parameters, and re-states the problem to be solved numerically. Section 3 describes the BEM implementation, i.e., boundary discretization, cell-based domain integration for volumetric sources, and time-stepping strategies for the ABC operator, ensuring reproducibility of the technique. Section 4 presents simulation results in a consistent order: first examining the fractional order effect, then parameter sensitivity, fixed-time temperature variation, heat flux evolution, and finally comparisons with experiments and computational performance evaluation. Section 5 gives an overview of main achievements, ensures the superiority of the proposed method over existing approaches, and outlines avenues of future research. Other technical details and derivations are given in the Appendix A and Appendix B to support the main text without interrupting its flow.

2. Formulation of the Problem

The fractional bioheat transfer equation in one-dimensional form is:

$${\rho c }^{ABC}{D}_t^{\alpha }T\left(x,t\right)=k{\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\omega }_b{c}_b\left(T_a-T\left(x,t\right)\right)+Q_{met}+Q_{ext}$$

(1)

The classical Fourier-based cylindrical heat flux is:

$$Q\left(t\right)=-{\displaystyle \frac{2\pi LK\left(T_i-T_0\right)}{\mathit{ln}\left(\frac{R_0}{R_i}\right)}}$$

(2)

Replacing the standard derivative with a fractional Atangana-Baleanu operator gives:

$$Q\left(t\right)=-{\displaystyle \frac{2\pi LK}{\mathit{ln}\left(\frac{R_0}{R_i}\right)}}{}^{ABC}D_t^{\alpha }\left(T_i-T_0\right)$$

(3)

It is when the BEM is applied to solve the FBE that one finds domain integrals arising naturally due to internal heat sources such as metabolic heating, external energy deposition, and the perfusion-related sink term. Since classical BEM is boundary-only, some appropriate means is required to solve these volumetric contributions throughout the problem domain. In this paper, domain integrals are treated using an internal cell discretization scheme, wherein the BEM elements are compatible with the subdivision of the computational domain into subregions. This allows one to have direct numerical integration over the interior including source terms, such as terms containing the fractional Atangana–Baleanu derivative. For time-dependent and nonlocal terms, such as the fractional derivative operator applied to temperature, the scheme gives clear representation of temporal history by discretizing the time domain and employing convolution-type memory kernels. This allows for consistent and precise calculation of nonhomogeneous source terms, making the BEM framework compatible with simulation of complex thermal response under nonlocal fractional dynamics within biological tissues.

The choice of the Atangana–Baleanu fractional derivative in the Caputo sense (ABC) in this study is guided primarily by physical considerations of heat transfer in perfused biological tissues, rather than solely by numerical convenience. The ABC formulation employs a non-singular, nonlocal Mittag–Leffler kernel, which captures finite and smoothly decaying thermal memory effects consistent with measured tissue responses. In biological systems, conduction, and blood perfusion act together to dissipate earlier thermal states over a finite period, leading to a rounded initial temperature rise rather than the singular onset predicted by traditional Caputo or Riemann–Liouville derivatives. This behavior has been observed in experimental bioheat measurements and is compatible with physical models such as the Cattaneo–Vernotte and dual-phase lag theories. The ABC kernel’s bounded nature avoids nonphysical infinite memory while still accounting for spatial nonlocality, making it a more realistic representation of thermal relaxation in heterogeneous, vascularized media. Its computational advantages, such as enhanced numerical stability and convergence when implemented within the Boundary Element Method, are therefore complementary benefits, supporting but not dictating the choice of operator.

3. BEM Implementation

Let us rewrite the fractional PDE in a simplified 1D form:

$$\rho c{}^{ABC}D_t^{\alpha }T\left(x,t\right)=k{\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+S\left(x,t\right)$$

(4)

where

$$S\left(x,t\right)={\omega }_b{c}_b\left(T_a-T\right)+Q_{met}+Q_{ext}$$

(5)

The BEM implementation for the ABC fractional bioheat model is described in Appendix A.

To handle the fractional derivative, take the Laplace transform of both sides:

$$\rho c·B\left(\alpha \right)·{\displaystyle \frac{s^{\alpha }}{s^{\alpha }+\alpha }}\overline{T}\left(x,s\right)=k{\displaystyle \frac{{\partial }^2\overline{T}}{\partial x^2}}+\overline{S}\left(x,s\right)$$

(6)

Solve the homogeneous part:

$${\displaystyle \frac{d^2\overline{T}}{d{x}^2}}-{\lambda }^2\overline{T}=0$$

(7)

with

$${\lambda }^2={\displaystyle \frac{\rho c B\left(\alpha \right) s^{\alpha }}{k(s^{\alpha }+\alpha )}}$$

(8)

Solution (1D infinite medium Green’s function) can be expressed as:

$$G\left(x,\xi \right)={\displaystyle \frac{1}{2\lambda }}{e}^{-\lambda \left|x-\xi \right|}$$

(9)

The temperature at a point $x$ is given by:

$$c\left(x\right)\overline{T}\left(x\right)+{\int }_{\Gamma }\left[G\left(x,\xi \right){\displaystyle \frac{\partial \overline{T}}{\partial n}}\left(\xi \right)-\overline{T}\left(\xi \right){\displaystyle \frac{\partial G\left(x,\xi \right)}{\partial n}}\right]d\mathsf{\Gamma }={\int }_{\Omega }G\left(x,\xi \right){\displaystyle \frac{\overline{S}\left(\xi \right)}{k}}d\xi .$$

(10)

Discretize boundary into elements, approximate temperature $T$ and flux $q$ using interpolation functions, then solve the system

$$\left[H\right]\left\{T\right\}=\left[G\right]\left\{q\right\}+\left\{b\right\}$$

(11)

The improved Talbot method [35] was used because it provides a more reliable and efficient way to numerically invert Laplace transforms by enhancing the classical approach through modified contour paths and better stability management.

Figure 1 illustrates discretization of 1D cylindrical tissue domain into 8 boundary elements (E₁–E₈) separated by 9 nodes (N₁–N₉). Each node is a discrete point at which the temperature or flux is computed, and each element is the interface on which integral equations are computed. This arrangement is key to the application of the Boundary Element Method (BEM) in having the capability to enforce boundary conditions effectively and solve the fractional bioheat equation over the cylindrical domain.

4. Numerical Results and Discussion

The results presented in this section evaluate the proposed BEM–ABC formulation for its ability to accurately and efficiently model memory-dependent bioheat transfer in a one-dimensional cylindrical tissue domain. The evaluation focuses on (i) the effect of fractional order $\alpha$ on temperature and heat flux, (ii) sensitivity to key thermal parameters, and (iii) agreement with synthetic and experimental data, in line with the study’s stated objective.

The section begins with a description of the discretization scheme used in the Boundary Element Method (BEM), as illustrated in Figure 1. This is a sketch of the discretization of the computational domain into boundary elements and nodes, on which the solution of the fractional bioheat equation is founded. Building on this as a basis, the section proceeds to outline simulation results that examine the effects of fractional order, model accuracy, parameter sensitivity, and heat flux behavior in biological tissue.

For an investigation of the dynamics of the proposed fractional bioheat model, numerical computations are performed using the BEM in combination with the Atangana–Baleanu Caputo (ABC) fractional derivative. The influences of fractional order, thermal conductivity, perfusion rate, and external heating on temperature profile and heat flux are discussed in the following subsections

4.1. Simulation Parameters

The computations were carried out with a set of standard thermal and physical properties of biological tissue. Presented in

Table 1. Thermal and physiological parameters used in numerical simulations.

Variable	Description	Value
$\rho$	Tissue density	1000 kg/m3
$c$	Tissue specific heat	3600 J/kg·K
$k$	Thermal conductivity	0.5 W/m·K
${\omega }_b$	Blood perfusion rate	0.001 s−1
$c_b$	Blood specific heat	3770 J/kg·K
$T_a$	Arterial temperature	37 °C
$Q_{met}$	Metabolic heat	1000 W/m3
$Q_{ext}$	External heat	5000 W/m3
$R_i, R_0$	0.005 m, 0.01 m
$L$	Length of cylindrical shell	0.02 m
$\alpha$	Fractional order	0.9

, these parameters serve as the baseline for all subsequent computational experiments.

4.2. Effect of Fractional Order on Temperature

Figure 2 provides the temperature-time response of three fractional orders ($\alpha$ = 1.0, 0.9, and 0.8), highlighting the effect of memory on heat transfer. In the classical case ($\alpha$ = 1.0), the temperature increase is the fastest, which reflects instantaneous heat conduction with no memory. For the decrease in the fractional order to $\alpha$ = 0.9, the curve becomes increasingly sloping, which reflects moderate thermal memory. For $\alpha$ = 0.8, the growth is much slower, revealing strong nonlocal and memory-dependent behavior characteristic of biological tissues.

4.3. Sensitivity to Thermal Parameters

Figure 3 illustrates the sensitivity of tissue temperature to the primary thermal parameters as a function of time: thermal conductivity ($k$), blood perfusion rate (${\omega }_b$), and external heat source ($Q_{ext}$). In the first panel (a), decreased thermal conductivity results in greater retention of heat and higher temperatures, while higher conductivity allows for more effective dissipation of heat. The second panel (b) demonstrates that higher blood perfusion enhances cooling by more effective removal of heat and decreases the rate of temperature rise. In the third panel (c), the more intense external heating source produces a faster and larger temperature rise. These findings confirm the physical consistency of the model and its capacity to represent the effects of physiological and thermal parameters on tissue heating dynamics.

4.4. Fractional Order Impact at Fixed Time

Figure 4 presents the temperature at a fixed time point ($t$ = 100 s) as a function of the fractional order $\alpha$, ranging from 0.7 to 1.0. The plot clearly demonstrates that as α increases, the temperature rises more rapidly. Lower values of α correspond to stronger memory effects and delayed heat propagation, resulting in reduced temperatures at the same time. This trend confirms the inverse relationship between thermal memory and heat accumulation, emphasizing the ability of fractional-order models to capture time-dependent diffusion behaviors in biological tissues.

4.5. Heat Flux Dynamics

Figure 5 illustrates the fractional heat flux $Q\left(t\right)$ as a function of time for different $\alpha$ values (1.0, 0.9, and 0.8), demonstrating the effect of memory on thermal transport in biological tissue. The classical model ($\alpha$ = 1.0) exhibits the greatest and earliest heat flux, corresponding to instantaneous conduction. As $\alpha$ is decreased, the initial heat flux is considerably smaller and the rise in the flux is more gradual, illustrating the nonlocal and thermal memory effects. This kind of slowed response, which is characteristic of fractional-order models, shows the ability of the approach to simulate non-instantaneous, physiologically realistic heat transfer processes in non-homogeneous media.

4.6. Best-Fit Simulation and Experimental Comparison

Figure 6 shows the bioheat equation with memory (BEM) model response using the best-fit fractional order $\alpha$ = 0.894. The curve exhibits a moderate but realistic temperature rise over time, capturing thermal memory effects more accurately than classical models. This highlights the model’s effectiveness for simulating bioheat transfer in tissues during treatments like hyperthermia.

Figure 7 is a comparison of the temperature–time solutions for the three models: the standard bioheat model ($\alpha$ = 1.0), the reduced-order fractional model ($\alpha$ = 0.8), and the best-fit BEM-based fractional model ($\alpha$ = 0.894). The experimentally obtained data simulated (red dots) are very close to the BEM solution, confirming its capability to simulate actual thermal behavior in tissues. The classical model overshoots prematurely, and the lower-order model underestimates heating, illustrating the accuracy of the BEM technique to thermal memory effects modeling by fractional calculus. Simulated experimental data were designed based on the fractional bioheat model with α ≈ 0.894 to simulate realistic thermal responses observed in biological tissues, as commonly used for validation studies of bioheat transfer models.

4.7. Validation with Real Experimental Data

To validate the proposed BEM–ABC approach, simulated temperature profiles were compared with experimental data of Xu et al. [36] in Figure 8, where in vitro rat skin tissue was locally surface heated, and subdermal temperature was monitored as a function of time. The numerical solution was implemented under the identical thermal loading and geometric conditions but with tissue thermal properties as given: $k=0.5 \mathrm{W}/\mathrm{m}·\mathrm{K}, \rho =1050 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3, c=3600 \mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}, \mathrm{and} \mathrm{perfusion} \mathrm{rate} {\omega }_b=0.0015{ \mathrm{s}}^{-1}$. Temperature–time curve through the proposed BEM–ABC formulation had satisfactory agreement with experimental results [36] as shown in Figure 8, particularly at a fractional order $\alpha =0.894,$ matching the thermal lag and subdiffusive nature observed in biological tissue. Such comparison demonstrates the ability of the BEM–ABC model to mimic the observed thermal behavior in bio media and the suitability of fractional modeling using non-singular kernels for bioheat analysis.

4.8. Computational and Physical Advantages of the BEM–ABC Model over Classical Fractional Bioheat Formulations

To highlight the advantages of the proposed approach, we compare the BEM–ABC formulation with existing models based on classical fractional derivatives. The proposed BEM–ABC (Boundary Element Method with Atangana–Baleanu in Caputo sense) approach significantly enhances the simulation of thermal memory in biological tissue over existing fractional bioheat models with traditional Riemann–Liouville or Caputo derivatives. Traditional fractional models are based on power–law kernels with singularities at the origin, and this is plagued with mathematical as well as numerical instabilities, especially in the case of biological systems when smooth thermal responses are required. Alternatively, the ABC derivative employs a nonlocal, nonsingular Mittag–Leffler kernel with a bounded and smooth form of memory effects. This property enables the BEM–ABC model to reproduce the physiological impact of thermal lag more and more.

Unlike Riemann–Liouville and Caputo models, which struggle to impose initial and boundary conditions in a physical way, the ABC derivative is endowed naturally with classical boundary conditions, making it more compatible with actual biological measurements and interventions. Computationally, the ABC kernel also enforces higher convergence and lower computational cost, particularly when combined with boundary-based discretization such as BEM, which reduces problem dimensionality naturally and is better suited to boundary-dominant phenomena.

Empirically, the BEM–ABC model outperforms predecessors with temperature-time responses optimally fitting synthetic experimental data, most prominently at an optimum fractional order of $\mathsf{\alpha }\approx 0.894$. Over-estimation of thermal diffusion (i.e., memory-less) is seen with the conventional models ($\mathsf{\alpha }=1$), while under-estimation is observed for lower-order models ($\mathsf{\alpha }<0.8$). The ABC-based method is a balanced representation in capturing the retarded and yet smooth heat transfer characteristic of perfused, heterogenous tissues. This benefit is not only theoretical but is supported by its greater predictive accuracy, better stability, and physical consistency, and so the BEM–ABC model is an easy-to-use and trustworthy tool for bioheat transfer modeling in complex biomedical applications such as hyperthermia, cryotherapy, and thermal diagnosis.

4.9. Computational Efficiency and Accuracy Assessment of the BEM–ABC Formulation

For completeness, a brief comparison of computational efficiency was also carried out between the CBEM, FBEM–C, and FBEM–ABC models. While the ABC operator benefits from a nonsingular kernel that reduces numerical cost, the principal reason for its superior performance lies in its more accurate physical representation of bioheat transfer with bounded memory effects. Since detailed computational efficiency metrics are secondary to the physical modeling results, these are presented in Appendix B.

5. Conclusions

This study presented a Boundary Element Method (BEM) formulation for solving the fractional bioheat equation using the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). The proposed approach integrates a nonsingular Mittag–Leffler kernel with a boundary-only discretization strategy, enabling efficient handling of volumetric heat sources and memory-dependent operators through cell-based integration and a recursive time-stepping scheme. Validation against synthetic and experimental temperature data confirmed the method’s accuracy and numerical stability, with an optimal fractional order of approximately $a\approx 0.894$ for the studied tissue model. Beyond its numerical performance, the method was applied to explore the influence of fractional order on temperature evolution and heat flux trends, providing evidence that bounded-memory kernels can capture physiologically realistic thermal relaxation in perfused tissues. The results demonstrate that the ABC–BEM framework offers a versatile and computationally efficient tool for simulating bioheat transfer with finite memory effects, supporting potential applications in thermal therapy planning, noninvasive diagnostics, and real-time biomedical simulations. The comparative analysis between CBEM, FBEM–C, and FBEM–ABC demonstrates that while CBEM remains computationally fast, it fails to capture the memory-driven effects essential in bioheat transfer. FBEM–C improves accuracy but at the cost of high computational burden due to the singular kernel. The proposed FBEM–ABC strikes the best balance, achieving the lowest temperature error (1.2%) with moderate computational demand. This consistent three-model evaluation strengthens the reliability of the proposed method and confirms its suitability for biomedical applications requiring accurate yet efficient numerical predictions. Future work will extend the approach to multilayered heterogeneous tissues, incorporate perfusion–metabolism coupling, and implement three-dimensional and GPU-accelerated solvers for clinical and embedded applications.