Exact Analytical Formulation of Three-Dimensional Pennes Bioheat Model in Regional Hyperthermia with Modified Initial Condition

  • Jaideep Dutta
  • Balaram KunduEmail author
Original Contribution


The present research work is intended to develop an exact analytical solution of three-dimensional Pennes bioheat model subjected to regional hyperthermia. Existing literature survey indicates that a large number of research papers have been reported in one-dimensional and two-dimensional approaches, but researchers have rarely focused on three-dimensional modelling apart from a few. In practical sense, energy always propagates in multi-dimensional manner; hence, three-dimensional analysis would be considered as better approach in comparison with existing one and two-dimensional investigation particularly for living tissues. Living tissues are highly susceptible to temperature variation. As suggested by the treatment protocol of thermal therapy, regional hyperthermia occurs in large-sized organs (prostrate, breast, lung, etc.) which are deep seated inside the human body and in such cases three-dimensional analysis is essential. To solve three-dimensional Pennes bioheat model, a hybrid analytical scheme comprising of ‘shift of variables’ and ‘finite Fourier transform’ has been employed in the present research paper. The prime novelty of this research work is implementation of spatially dependent initial condition as highly non-homogeneous and non-uniform anatomical structures are observed in living tissues, whereas existing research work has been carried out on the constant temperature initial condition. The boundary conditions are imposed in the present modelling in relation with the realistic approach subjected to thermal therapies. The research output has been validated and justified with the published numerical research work, and it indicates the maximum temperature deviation of 0.215%.


Bioheat transfer Three dimensional Regional hyperthermia Hybrid analytical scheme 

List of Symbols


Biot number (dimensionless)


Specific heat of tissue (J Kg-1 °C-1)


Specific heat of blood (J Kg-1 °C-1)


Dimensionless constant


Non-dimensional therapeutic exposure time


Convective heat transfer coefficient (W m-2 °C-1)

i, j, k

Non-negative integers (0, 1, 2, 3 …)


Thermal conductivity of tissue (W m-1 °C-1)


Length of the physical domain (m)


Heat flux imposed on outside of the tissue (W m-2)


Non-dimensional heat flux, qL/k(To − Tb)

Qm, QS

Metabolic heat generation rate (W m-3) and spatial heat generation rate (W m-3), respectively


Dimensionless metabolic heat generation rate


Therapeutic exposure time (s)


Local temperature of skin tissue (°C)


Arterial temperature of skin tissue (°C)


Reference temperature of heat flux (°C)


Ambient temperature (°C)

x, y, z

Three spatial directions of physical domain

X, Y, Z

Non-dimensional spatial length

Greek Symbols


Thermal diffusivity (m2 s-1)


Blood perfusion term (dimensionless)


Blood perfusion rate (m3 s-1 m-3)


Local tissue temperature (dimensionless)


Steady-state temperature (dimensionless)


Constant (dimensionless)


Density of blood (kg m-3)



This paper is a revised and expanded version of an article entitled, “Analytical Model for Tri-Dimensional Fourier Bioheat Transfer Encountered in Regional Hyperthermia, Paper No., “166” of the conference “Recent Innovations and Developments in Mechanical Engineering” organized by Mechanical Engineering Department, NIT Meghalaya, Shillong.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© The Institution of Engineers (India) 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia
  2. 2.Department of Mechanical EngineeringMCKV Institute of EngineeringHowrahIndia

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