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Heat Transfer In Vivo: Phenomena and Models

  • Alexander I. Zhmakin
Reference work entry

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.

Nomenclature

C

Concentration (m−3)

c

Specific heat (Jkg−1m−3)

cb

Blood specific heat (Jkg−1 m−3)

Ea

Activation energy (J)

Gm,n

Multi-regional Green function

Gz

The Graetz number

K

The permeability of the porous medium (m2)

Le

Thermal equilibration length (m)

li

The direction cosines of the vessels

Nu

The Nusselt number

Pe

The Pecle number

Q

Heat source (Wm−3)

q

Heat flux (Wm−2)

qc

Heat source due to blood convection (Wm−3)

qp

Heat source due to blood perfusion (Wm−3)

\( {\dot{q}}_{\mathrm{met}} \)

Metabolic heat rate (Wm−3)

Re

The Reynolds number

T

Temperature (K)

Ta

Temperature of the arterial blood (K)

Tw

Temperature of the vessel’s wall (K)

T

Ambient temperature (K)

t

Time (s)

V

Volume (m3)

\( \overline{u} \)

Mean blood velocity (ms−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)

λb

Blood thermal conductivity (Wm−1K−1)

λp

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

λeff

Effective thermal conductivity (Wm−1K−1)

μ

Dynamic viscosity (kgm−1 s−1)

ρ

Density (kgm−3),

σij

Surface tension on the interface between phases i and j

τ

Relaxation time (s)

τq

Phase lag for the heat flux vector (s)

τT

Phase lag temperature gradient (s)

Ω

Irreversible thermal ‘damage”

ωb

The blood perfusion rate (kgs−1m−3)

Mixed Symbols

Δt

Time interval (s)

δV

Elementary volume (m3)

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.SoftImpact Ltd.St. PetersburgRussia
  2. 2.Ioffe InstituteSt. PetersburgRussia

Section editors and affiliations

  • Ram Devireddy
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringLouisiana State UniversityBaton RougeUSA

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