Fractal Pennes and Cattaneo–Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth

Abstract

In this study, the Pennes and Cattaneo–Vernotte bioheat transfer equations in the presence of fractal spatial dimensions are derived based on the product-like fractal geometry. This approach was introduced recently, by Li and Ostoja-Starzewski, in order to explore dynamical properties of anisotropic media. The theory is characterized by a modified gradient operator which depends on two parameters: R which represents the radius of the tumour and R 0 which represents the radius of the spherical living tissue. Both the steady and unsteady states for each fractal bioheat equation were obtained and their implications on living cells in the presence of growth of a large tumour were analysed. Assuming a specific heating/cooling by a constant heat flux equivalent to the metabolic heat generation in the tissue, it was observed that the solutions of the fractal bioheat equations are robustly affected by fractal dimensions, the radius of the tumour growth and the dimensions of the living cell tissue. The ranges of both the fractal dimensions and temperature were obtained, analysed and compared with recent studies. This study confirms the importance of fractals in medicine.

Micropolar mechanics of product fractal media

“…The Carpinteri column [23] is a simple example of an anisotropic product-like fractal porous structure with a microstructure modelled by the Sierpi ński carpet in the cross-section and a Cantor set along the longitudinal axis. Natural examples of product-like fractals are found in [10][11][12][13] with applications in molecular physics, superconductivity, tumour growth and fluid flows. Also, the geometry and mechanics of planetary rings, like those of Saturn, can be handled by fractal products in polar coordinate systems [24].…”

“…Applications range over solid mechanics of fractal media (elasticity, beams, elastic-brittle fracture, viscoelasticity, among others) as well as fluid mechanics, molecular dynamics and quantum mechanics [e.g. 4,[10][11][12][13][14][15][16][17].…”

Mathematical modeling of the phenomena that occur in a biological tissue containing photosensitizer

“…which is introduced into the equation describing the bioheat transfer in the tissue domain [6,7,9]. In the current work, a description based on the Pennes bioheat transfer equation with adequate boundary initial conditions is used [11][12][13][14][15] 2 0 : ,…”

Emergence of lump-like solitonic waves in Heimburg–Jackson biomembranes and nerves fractal model

“…In fact, this new concept was introduced recently by Li and Ostoja-Starzewski in order to describe dynamics in anisotropic and continuum media [42][43][44] and was motivated by Tarasov fractal calculus arguments [45,46]. It is considered a successful approach, which has proved to have several successful implications in sciences and engineering at different scales [47][48][49][50][51][52][53][54][55][56][57][58]. In the Li and Ostoja-Starzewski approach (LOSA henceforth), the dynamic equations of motion hold mathematical forms involving integer-order integrals, whereas their local forms are expressed through partial differential equations with integerorder derivatives except that they contain coefficients involving fractal dimensions.…”