Table 3 Equations used in describing bioheat transfer phenomena based on basic Pennes model [133]

Pennes model (1948)

12

Bioheat transfer equation

\(\rho c\frac{\partial T}{\partial t} = - k\left[ {\frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial T}{\partial r} + \frac{1}{{r^{2} }}\frac{\partial T}{\partial \emptyset } + \frac{{\partial^{2} T}}{{\partial Z^{2} }}} \right] + q_{m} + q_{b}\)

\(\rho\) 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)

\(q_{b} = (\rho c)_{b} \omega (T_{a} - T)\)

\(T\) tissue temperature (K)

\(k\) tissue specific thermal conductivity (W m⁻¹ K⁻¹)

14

Temperature dependent blood flow, \(\omega\) (s⁻¹)

\(\omega = \omega_{0} (1 + \gamma T)\)

\(q_{m}\) 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

\(\omega\) blood perfusion (s⁻¹)

\(\rho_{b}\) blood density (kg m⁻³)

\(c_{b}\) blood specific heat (J kg⁻¹ K⁻¹)

\(T_{a}\) arterial blood temperature (K)

\(\omega_{0}\) baseline of volumetric flow rate of blood (s⁻¹)

\(\gamma\) time dependent blood flow coefficient (K⁻¹)

Chen and Holmes model (1980)

15

Thermal equilibrium length, \(l_{e}\) (m)

\(l_{e} = \frac{{A(\rho c)_{b} \bar{V}}}{U \cdot P}\)

\(A\) 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

\(\rho c\frac{\partial T}{\partial t} = \nabla \cdot k\nabla T + (\rho c)_{b} \omega^{*} \left( {T_{a}^{*} - T} \right) - \left( {\rho c} \right)_{b} \bar{u}\nabla T + \nabla \cdot k_{p} \nabla T + q_{m}\)

Weinbaum, Jiji and Lemons model (1984)

17

Countercurrent arteries

\(\left( {\rho c} \right)_{b} \pi r_{b}^{2} \bar{V} \cdot \frac{{dT_{a} }}{ds} = - q_{a}\)

\(r_{b}\) 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

\(\left( {\rho c} \right)_{b} \pi r_{b}^{2} \bar{V} \cdot \frac{{dT_{v} }}{ds} = - q_{v}\)

19

Bioheat transfer equation

\(\rho c\frac{\partial T}{\partial t} = \nabla \cdot k\nabla T + 2n\pi r_{b} \left( {\rho c} \right)_{b} g\left( {T_{a} - T_{v} } \right) - n\pi r_{b}^{2} \left( {\rho c} \right)_{b} \bar{V} \cdot \frac{{d\left( {T_{a} - T_{v} } \right)}}{ds} + q_{m}\)