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Three phase bio-heat transfer model in three-dimensional space for multiprobe cryosurgery

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Abstract

This study addresses a three phase bio-heat transfer model in three-dimensional space consisting of tumor and normal tissues frozen with multiple cryoprobe. Mathematically, it is a three-dimensional, three region moving boundary problem with different types of boundary conditions. To solve this problem, we have developed a modified Legendre wavelet Galerkin method. We have calculated the operational matrix of integration of three dimensional Legendre wavelets and used in our problem. We have obtained the solution by using the idea of generalized inverse. In a particular case, when surface subjected to boundary condition of I kind, the results obtained are compared with exact solution and are in good agreement. This model is used to find the temperature in biological tissue when three, two or one cryoprobe are operated. Our results show that it is better to use three cryoprobes in place of two or one. In our problem, we have been obtained temperature distribution and moving layer thickness when three cryoprobes are placed. A variation is observed from the graphs of temperature distribution and moving layer thickness with respect to different boundary conditions. We see from the graphs of our results that temperature profile decreases and moving layer thickness increases when the time increases.

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Abbreviations

x :

Space coordinate in x-direction

y :

Space coordinate in y-direction

z :

Space coordinate in z-direction

\(\rho \) :

Density (\(\mathrm{kg m^{-3}}\))

c :

Specific heat (\(\mathrm{J kg^{-1} K^{-1}}\))

k :

Thermal conductivity of the tissue (\(\mathrm{W m^{-1} K^{-1}}\))

a :

Thermal diffusivity (\(\mathrm{m^2 s^{-1}}\))

\(\rho _\mathrm{b}\) :

Density of blood (\(\mathrm{kg m^{-3}}\))

\(c_\mathrm{b}\) :

Specific heat of blood (\(\mathrm{J kg^{-1} K^{-1}}\))

\(w_\mathrm{b}\) :

Blood perfusion rate (\(\mathrm{m^3 s^{-1} kg^{-1}}\))

T :

Temperature (°C)

t :

Time (s)

L :

Latent heat (\(\mathrm{kJ kg^{-1}}\))

\(T_\mathrm{b}\) :

Arterial temperature (°C)

\(T_0\) :

Initial temperature (°C)

\(T_\mathrm{c}\) :

Cryoprobe temperature (°C)

\(T_\mathrm{l}\) :

Liquidus temperature (°C)

\(T_\mathrm{s}\) :

Solidus temperature (°C)

l :

Length of the tissue (\(\mathrm{m}\))

\(s_\mathrm{i}\) :

Distance from origin

q :

Heat flux (\(\mathrm{W m^{-2}}\))

\(Q_\mathrm{m}\) :

Metabolic heat generation (\(\mathrm{W m^{-3}}\))

\(\tau _\mathrm{q}\) :

Phase lag in heat flux (s)

\(\tau _\mathrm{T}\) :

Phase lag in temperature gradient (s)

\(T_\mathrm{w}\) :

Surface temperature

\(T_\mathrm{r}\) :

Surrounding temperature

h :

Heat transfer coefficient (\(\mathrm{W m^{-2} K^{-1}}\))

X :

Dimensionless space coordinate

Y :

Dimensionless space coordinate

Z :

Dimensionless space coordinate

Fo :

Fourier number or dimensionless time

\(Fo_\mathrm{q}\) :

Dimensionless phase lag due to heat flux

\(Fo_\mathrm{T}\) :

Dimensionless phase lag due to temperature gradient

\(\lambda _\mathrm{i}\) :

Dimensionless distance

\(\theta \) :

Dimensionless temperature

\(\theta _\mathrm{b}\) :

Dimensionless blood temperature

\(\theta _\mathrm{u}\) :

Dimensionless unfrozen temperature

\(\theta _\mathrm{m}\) :

Dimensionless mushy temperature

\(\theta _\mathrm{f}\) :

Dimensionless frozen temperature

\(\theta _\mathrm{w}\) :

Dimensionless surface temperature

\(\theta _\mathrm{r}\) :

Dimensionless surrounding temperature

\(\theta _\mathrm{s}\) :

Dimensionless solidus temperature

\(P_\mathrm{f}\) :

Dimensionless blood perfusion coefficient

\(P_\mathrm{m}\) :

Dimensionless metabolic heat source coefficient

Ste :

Stefan number

\(K_\mathrm{i}\) :

Kirchhoff number

\(B_\mathrm{i}\) :

Biot number

u, 1:

Indication for unfrozen

m, 2:

Indication for mushy

f, 3:

Indication for frozen

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Acknowledgements

The first author is thankful to University Grant Commission New Delhi, India, grant no. 19/06/2016(i)EU-V for supporting this research. The authors are sincerely grateful to DST-CIMS, BHU, Varanasi, India, for allocating necessary amenities. The author is also thankful to the reviewer and editor as the paper quality increases with their useful comments.

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Authors and Affiliations

  1. DST-CIMS, Institute of Science, Banaras Hindu University, Varanasi, India

    Mukesh Kumar

  2. Department of Mathematics, I.I.T. BHU, Varanasi, India

    K. N. Rai

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Kumar, M., Rai, K.N. Three phase bio-heat transfer model in three-dimensional space for multiprobe cryosurgery. J Therm Anal Calorim 147, 14491–14507 (2022). https://doi.org/10.1007/s10973-022-11566-3

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