. 2017 Mar 23;16:36. doi: 10.1186/s12938-017-0327-x

Table 3.

Equations used in describing bioheat transfer phenomena based on basic Pennes model [133]

Eq. no.	Parameter	Equation expression	Nomenclature
	Pennes model (1948)
12	Bioheat transfer equation	$ρ c \frac{\partial T}{\partial t} = - k [\frac{\partial^{2} T}{\partial r^{2}} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{1}{r^{2}} \frac{\partial T}{\partial \emptyset} + \frac{\partial^{2} T}{\partial Z^{2}}] + q_{m} + q_{b}$	$ρ$ density of medium (kg m⁻³) $c$ medium specific heat (J kg⁻¹ K⁻¹)
13	Heat transfer from blood to tissue, $q_{b}$ (J m⁻³ s⁻¹)	$q_{b} = {(ρ c)}_{b} ω (T_{a} - T)$	$T$ tissue temperature (K) $k$ tissue specific thermal conductivity (W m⁻¹ K⁻¹)
14	Temperature dependent blood flow, $ω$ (s⁻¹)	$ω = ω_{0} (1 + γ T)$	$q_{m}$ heat production rate of tissue (J m⁻³ s⁻¹) $q_{b}$ heat transfer rate from blood to tissue (J m⁻³ s⁻¹) $r, \emptyset, Z$ cylindrical coordinate $ω$ blood perfusion (s⁻¹) $ρ_{b}$ blood density (kg m⁻³) $c_{b}$ blood specific heat (J kg⁻¹ K⁻¹) $T_{a}$ arterial blood temperature (K) $ω_{0}$ baseline of volumetric flow rate of blood (s⁻¹) $γ$ time dependent blood flow coefficient (K⁻¹)
	Chen and Holmes model (1980)
15	Thermal equilibrium length, $l_{e}$ (m)	$l_{e} = \frac{A {(ρ c)}_{b} \bar{V}}{U \cdot P}$	$A$ flow area (m²) $\bar{V}$ local blood velocity (m s⁻¹) $U$ overall heat transfer coefficient (W m⁻² K⁻¹) $P$ circumference (m) $\bar{u}$ net volume flux (m s⁻¹) $k_{p}$ heat transfer coefficient of contributing vessel(W m⁻² K⁻¹) $*$ properties of large blood vessel
16	Bioheat transfer equation	$ρ c \frac{\partial T}{\partial t} = \nabla \cdot k \nabla T + {(ρ c)}_{b} ω^{} (T_{a}^{} - T) - {(ρ c)}_{b} \bar{u} \nabla T + \nabla \cdot k_{p} \nabla T + q_{m}$
	Weinbaum, Jiji and Lemons model (1984)
17	Countercurrent arteries	${(ρ c)}_{b} π r_{b}^{2} \bar{V} \cdot \frac{d T_{a}}{d s} = - q_{a}$	$r_{b}$ blood vessel radius (m) $q_{a}$ heat loss rate at artery wall (J m⁻¹ s⁻¹) $q_{v}$ heat gain rate at vein wall (J m⁻¹ s⁻¹) $g$ blood bleed off rate (m s⁻¹) $T_{v}$ venous blood temperature (K) $n$ vessel number density (m⁻²) $s$ direction along a blood vessel
18	Countercurrent veins	${(ρ c)}_{b} π r_{b}^{2} \bar{V} \cdot \frac{d T_{v}}{d s} = - q_{v}$
19	Bioheat transfer equation	$ρ c \frac{\partial T}{\partial t} = \nabla \cdot k \nabla T + 2 n π r_{b} {(ρ c)}_{b} g (T_{a} - T_{v}) - n π r_{b}^{2} {(ρ c)}_{b} \bar{V} \cdot \frac{d (T_{a} - T_{v})}{d s} + q_{m}$