Non-Fourier Heat
Conduction in Soft Tissues: Experiments, Models, Waves

Martin Ostoja-Starzewski

Department
of Mechanical Science & Engineering,

Institute
for Condensed Matter Theory, and Beckman Institute

University
of Illinois at Urbana-Champaign

https://martinos.mechanical.illinois.edu/

**ABSTRACT**

Electrosurgery of soft
tissue organs involves application of high voltage at
high frequency to the interface between the surgical probe and the tissue
boundary. In order to accomplish correct
interaction of the electrosurgical probe with the tissue (i.e., coagulation of
blood at the probe/tissue interface) as opposed to charring, it is imperative
to understand the heat conduction phenomena due to moving heat sources. Recent research [1] has established
experimental evidence for the damped-hyperbolic character of transient heat
conduction – described by a Maxwell-Cattaneo (not a
Fourier) model – in porcine muscle tissue and blood. In fact, a fractional derivative of order α = 0.5
offers the most appropriate model for heat conduction in blood while an integer
model is sufficient to describe heat conduction in muscle. Since the thermal signal speeds are on the
order of a few millimeters per second, by way of
analogies, one needs to consider subsonic (M < 1) and even supersonic (M > 1) *second sound phenomena* during
electrosurgery. In 2d (resp., 3d)
problems, this translates into a possible formation of Mach wedges (resp., Mach
cones). The subsonic case has recently
been examined in [2]. Two types of
sensitivity studies of supersonic 2d problems have been performed so far: (i) non-rectilinear paths of a heat
source in two-phase inhomogeneous media and (ii) rectilinear paths of heat
source in random fractal media. The
latter case is motivated by the widely reported fractal character of many biological
tissues, while its solution relies on a methodology outlined in [3]. Both types of studies are richly illustrated
by 2d computer simulations of transient heat fields.

[1] A.
Madhukar, Y. Park, W. Kim, H.J., R. Berlin, L.P. Chamorro, J. Bentsman, and M. Ostoja-Starzewski,
“Heat
conduction in porcine muscle and blood: Experiments and time-fractional
telegraph equation model,” *J.
Roy. Soc. Interface* **16**,
20190726, 2019.

[2] Y.
Povstenko and M. Ostoja-Starzewski,
“Doppler effect described by the solutions of
the Cattaneo telegraph equation,”
*Acta Mech.* **232**, 725-740, 2021.

[3] X. Zhang and M. Ostoja-Starzewski, “Impact
force and moment problems on random mass density fields with fractal and Hurst
effects,” in special issue
"Advanced materials modelling via fractional calculus: challenges and perspectives,"
*Phil. Trans. Roy. Soc. A* **378**(2172), 20190591, 2020.