Heat Transfer in Biological Tissues - Selected Topics of Computational and Experimental Fluid Mechanics - page 300

interventions. Given the importance of the issue, several authors have studied various
methods for describing the energy transfer process in tissues. However, the main
contribution was made in 1948 by Harry H. Pennes, who published a study on the
temperature distribution in the human body Pennes ( 1948 ). Thismodel uses one of the
most successful equations in continuum physics, the traditional Fourier's law of heat
conduction to describe the energy flux. One of the main shortcomings of Fourier's
law is that it leads to a parabolic equation for the temperature field. This means that
any initial disturbance is felt instantly throughout the entire medium. This behavior is
said to contradict the principle of causality. To correct this unrealistic feature, which
is known as the “paradox of the heat conduction”, various modifications of Fourier's
lawhave been proposed over the years Jordan et al. ( 2008 ). Of these, the best known is
the Maxwell-Cattaneo model Joseph and Preziosi ( 1989 , 1990 ), Chandrasekharaiah
( 1998 ), Ostoja-Starzewski ( 2007 ).
This study uses a modified expression of the Pennes model and focuses on the
phenomena that occur when there is a perturbation in the environment that modifies
the surface temperature of the tissue. This happens when the tissue is exposed to a
stream of ambient air and to temperatures with stochastic values Fiala et al. ( 1999 ),
Deng and Liu ( 2002 , 2004 ). The analysis focuses on the response of the tissue when it
is subjected to a temperature gradient due to the variation of the thermal environment.

2 Model Description

Energy transfer in biological systems was firstly described through the biological
energy equation developed by Pennes in 1948. His work involved theoretical and
experimental investigation of the temperature distribution in the forearm of a group
of people Pennes ( 1948 ). The derived mathematical model consists on an energy
balance in the tissue that incorporates the effects of metabolism and blood perfusion.
The biological heat transfer equation in expanded form (2D) is given by Eq. ( 1 ).

k ∂

2 T 2 T

ˁ m C p m ∂

T
∂

+ ∂

w b ˁ b C p b (

t =

+ T a − T ) + Q m + Q r (1)

x 2 y 2 ∂ ∂

where x and y are spatial variables, t is the temporal variable, T the temperature of
the tissue, k thermal conductivity, C pm the specific heat of tissue,

ˁ m the tissue's

density, Q m the energy that generates within the tissue as a result of the metabolic

activity, ˁ b is the blood density, C p b the specific heat of blood, W b the volumetric

flow of blood perfusion, Q r is an external heat source and T a the temperature of the

blood in the arteries.
The definition of the system of study is based on a portion of epithelial tissue.
To ensure that the internal temperature remains constant and that the temperature
distributions in the tissue are symmetrical, the depth of the section should not be
greater than 3 cm. Given the maximum temperature that a normal tissue can stand
before degrading, the highest temperature allowed will be 42 ⓦ C. It will be considered