Heat Transfer in Vivo: Phenomena & Models

Abstract

Physical phenomena encountered in heat transfer in living tissues and mathematical models used to simulate them are discussed. Effects of high or low temperature on the biological systems at the different levels – cell, tissue, organism – as well a role of the blood circulation and of the structure of the vascular network on heat transfer are considered. The classic Pennes bioheat equation, a number of non-Fourier heat transfer models (including single-phase-lag and dual-phase-lag models), porous media models, models based on the fractional differential equations, and discrete vascular models are analyzed. A few selected exact solutions are presented.

When your feet are cold, cover your head – Inuit saying

Nomenclature

C :

Concentration (m⁻³)

c :

Specific heat (Jkg⁻¹m⁻³)

c _b :

Blood specific heat (Jkg⁻¹ m⁻³)

E _a :

Activation energy (J)

G _m , _n :

Multi-regional Green function

Gz:

The Graetz number

K :

The permeability of the porous medium (m²)

L _e :

Thermal equilibration length (m)

l _i :

The direction cosines of the vessels

Nu:

The Nusselt number

Pe:

The Pecle number

Q :

Heat source (Wm⁻³)

q :

Heat flux (Wm⁻²)

q _c :

Heat source due to blood convection (Wm⁻³)

q _p :

Heat source due to blood perfusion (Wm⁻³)

${\dot{q}}_{\mathrm{met}}$ :

Metabolic heat rate (Wm⁻³)

Re:

The Reynolds number

T :

Temperature (K)

T _a :

Temperature of the arterial blood (K)

T _w :

Temperature of the vessel’s wall (K)

T _∞ :

Ambient temperature (K)

t :

Time (s)

V :

Volume (m³)

$\overline{u}$ :

Mean blood velocity (ms⁻¹)

1.1 Greek Symbols

α :

Order of fractional derivative

Γ :

Euler Gamma function

δ (s):

Dirac delta function

γ :

The angle between the direction of the blood vessels and the tissue temperature gradient

ε :

Porosity

κ :

Thermal diffusivity (m² s⁻¹)

λ :

Thermal conductivity (Wm⁻¹K⁻¹)

λ _b :

Blood thermal conductivity (Wm⁻¹K⁻¹)

λ _p :

“Perfusional” thermal conductivity (Wm⁻¹K⁻¹)

λ _eff :

Effective thermal conductivity (Wm⁻¹K⁻¹)

μ :

Dynamic viscosity (kgm⁻¹ s⁻¹)

ρ :

Density (kgm⁻³),

σ ^ij :

Surface tension on the interface between phases i and j

τ :

Relaxation time (s)

τ _q :

Phase lag for the heat flux vector (s)

τ _T :

Phase lag temperature gradient (s)

Ω :

Irreversible thermal ‘damage”

ω _b :

The blood perfusion rate (kgs⁻¹m⁻³)

1.2 Mixed Symbols

Δt :

Time interval (s)

δV :

Elementary volume (m³)