Study of the one dimensional and transient bioheat transfer equation: multi-layer solution development and applications

D. B. Rodrigues, P. J. S. Pereira, P. Limão-Vieira, P. R. Stauffer, P. F. Maccarini

Research output: Contribution to journalArticlepeer-review

37 Citations (Scopus)


In this work we derive an analytical solution given by Bessel series to the transient and one-dimensional (1D) bioheat transfer equation in a multi-layer region with spatially dependent heat sources. Each region represents an independent biological tissue characterized by temperature-invariant physiological parameters and a linearly temperature dependent metabolic heat generation. Moreover, 1D Cartesian, cylindrical or spherical coordinates are used to define the geometry and temperature boundary conditions of first, second and third kinds are assumed at the inner and outer surfaces. We present two examples of clinical applications for the developed solution. In the first one, we investigate two different heat source terms to simulate the heating in a tumor and its surrounding tissue, induced during a magnetic fluid hyperthermia technique used for cancer treatment. To obtain an accurate analytical solution, we determine the error associated with the truncated Bessel series that defines the transient solution. In the second application, we explore the potential of this model to study the effect of different environmental conditions in a multi-layered human head model (brain, bone and scalp). The convective heat transfer effect of a large blood vessel located inside the brain is also investigated. The results are further compared with a numerical solution obtained by the Finite Element Method and computed with COMSOL Multiphysics v4.1

Original languageEnglish
Pages (from-to)153-162
Number of pages10
JournalInternational Journal Of Heat And Mass Transfer
Issue number1
Publication statusPublished - 2 Apr 2013


  • Bessel functions
  • Bioheat transfer equation
  • Eigenfunctions and eigenvalues
  • Finite Element Method
  • Hyperthermia applications
  • Method of separation of variables


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