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A computational study of non-Fourier temperature distribution in HIFU ablation of 3D liver tumor

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Abstract

So far, many researches have been done on temperature distribution due to thermal therapy. In these studies, the temperature field is obtained using the Fourier heat transfer equation. In biological tissues, because of the inhomogeneity nature of materials, temperature distribution has time delay and temperature oscillations. In this case, using the Fourier heat transfer creates an error, and for higher accuracy, using non-Fourier models for obtaining temperature distribution is necessary. In this study, the three-dimensional non-Fourier temperature distribution in liver tumor under HIFU radiation in the presence of significant blood vessel are studied. Temperature results for different distances of the ultrasonic transducer axis relative to the vessel are obtained and compared. The Helmholtz equation is used to simulate the acoustic pressure field, and the Fourier heat transfer model and non-Fourier TWMBT and DPL models are used to simulate the temperature distributions. The results show that by moving the wave axis away from the center of the vessel until the coordinates of the axis are smaller than the radius of the vessel, the temperature distribution decreases, and then as the transducer axis moves away from the vessel wall, the resulting temperature distribution values increase. The results also show the high accuracy of the non-Fourier heat transfer models, especially the DPL model compared to the Fourier heat transfer model in simulating the temperature distribution in biological tissues.

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Abbreviations

C :

Specific heat \({\text{J}} \left( {{\text{kg}}.{\text{K}}} \right)^{ - 1}\)

\(C_{0}\) :

Speed of sound \(\left( {{\text{ms}}^{ - 1} } \right)\)

\(C_{\rm C}\) :

Complex speed of sound \(\left( {{\text{ms}}^{ - 1} } \right)\)

\(f_{0}\) :

Initial frequency \(\left( {{\text{MHz}}} \right)\)

I :

Sound intensity \(\left( {{\text{W}}/{\text{m}}^{2} } \right)\)

K :

Thermal conductivity \({\text{W}}\left( {{\text{m}}.{\text{K}}} \right)^{ - 1}\)

m :

Order of PML

p :

Acoustic pressure \(\left( {{\text{MPa}}} \right)\)

u :

Blood velocity(m/s)

\(p_{0}\) :

Focal acoustic pressure \(\left( {{\text{MPa}}} \right)\)

q :

Heat flux \(\left( {{\text{Wm}}^{ - 2} } \right)\)

\(Q_{{{\text{ext}}}}\) :

External heat source \(\left( {{\text{Wm}}^{ - 3} } \right)\)

\(Q_{{{\text{met}}}}\) :

Metabolic heat source \(\left( {{\text{Wm}}^{ - 3} } \right)\)

\(R_{0}\) :

Reflection coefficient of PML

t :

Time \(\left( {\text{s}} \right)\)

\(t_{0}\) :

Initial time \(\left( {\text{s}} \right)\)

T :

Temperature \(\left( {^\circ {\text{C}}} \right)\)

X :

Coordinate in X direction

Y :

Coordinate in Y direction

Z :

Coordinate in Z direction

\(Z_{\rm p}\) :

PML thickness \(\left( {\text{m}} \right)\)

\(\alpha\) :

Absorption coefficient \(\left( {{\text{Np}}\left( {{\text{MHz}}.{\text{m}}} \right)^{ - 1} } \right)\)

\(\eta\) :

Attenuation power

\(\lambda\) :

Wavelength \(\left( {\text{m}} \right)\)

\(\rho\) :

Density \(\left( {{\text{kg m}}^{ - 3} } \right)\)

\(\tau_{\rm q}\) :

Heat flux thermal relaxation time \(\left( {\text{s}} \right)\)

\(\tau_{{\text{t}}}\) :

Temperature gradient thermal relaxation time \(\left( {\text{s}} \right)\)

\(\omega\) :

Angular frequency \(\left( {{\text{rad}} {\text{s}}^{ - 1} } \right)\)

\(\omega_{{\text{b}}}\) :

Blood perfusion rate \(\left( {{\text{s}}^{ - 1} } \right)\)

b :

Blood

t :

Tissue

HIFU:

High-intensity focused ultrasound

TWMBT:

Thermal wave model of bio-heat transfer

DPL:

Dual phase lag

PML:

Perfect matched layer

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Authors and Affiliations

  1. Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Ali Karimi Roknabadi

  2. Department of Mechanical Engineering, Arak University of Technology, Arak, 38181-46763, Iran

    Somayeh Davoodabadi Farhani

  3. Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran

    Mohammad Karimi Roknabadi

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  1. Ali Karimi Roknabadi
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Correspondence to Somayeh Davoodabadi Farhani.

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Roknabadi, A.K., Farhani, S.D. & Roknabadi, M.K. A computational study of non-Fourier temperature distribution in HIFU ablation of 3D liver tumor. J Therm Anal Calorim 147, 12933–12946 (2022). https://doi.org/10.1007/s10973-022-11469-3

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