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Modeling Heat Transfer in Tumors: A Review of Thermal Therapies

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Abstract

It is quite challenging to describe heat transfer phenomena in living systems because of the involved phenomena complexity. Indeed, thermal conduction and convection in tissues, blood perfusion, heat generation due to metabolism, complex vascular structure, changing of tissue properties depending on various conditions, are some of the features that make hard to obtain an accurate knowledge of heat transfer in living systems for all the clinical situations. This theme has a key role to predict accurately the temperature distribution in tissues, especially during biomedical applications, such as hyperthermia treatment of cancer, in which tumoral cells have to be destroyed and at the same time the surrounding healthy tissue has to be preserved. Moreover, the lack of experimentation in this field, due to ethical reasons, makes bioheat models even more significant. The first simple bioheat model was developed in 1948 by Pennes (J Appl Physiol 1:93–122, 1948) but it has some shortcomings that make the equation not so accurate. For this reason, over the years it has been modified and more complex models have been developed. The purpose of this review is to give a clear overview of how the bioheat models have been modified when applied in various hyperthermia treatments of cancer.

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Abbreviations

a :

Specific surface area between blood and tissue (m−1)

a c :

Antenna constant (m−1)

A :

Maximum power density (W m−3)

A tm :

Area of the tumor (m2)

c v :

Nanoparticle concentration in the blood flow (kg L−1)

C :

Specific heat (J kg−1 K)

E :

Modulus of the electric field (V m−1)

f :

Frequency (Hz)

\(f_{\text{T}} \left( {c_{\text{v}} } \right)\) :

Source term caused by alternating magnetic field (W m−3)

G :

Incident radiation (W m−2)

g, F :

Functions depending on the radiative heat transfer (W m−3)

h :

Heat transfer coefficient (W m−2 K−1)

H :

Magnetic field (T)

I :

Scattered diffuse intensity (W m−2)

I ac :

Local acoustic intensity (W m−2)

j 0 :

Current density (A m−2)

k :

Thermal conductivity (W m−1 K−1)

L :

Specific thermal effects of chemical conversions (m2 s−2)

L LFp :

Lymphatic permeability (m Pa−1 s−1)

n :

Refractive index (–)

n inj :

Total number of injection points (–)

n np :

Number of nanoparticles (–)

P :

Transmitted antenna power (W)

q :

Heat flux (W m−2)

Q :

Power density (W m−3)

R mil :

Radius of magnetic loop (m)

r :

Radial coordinate (m)

r 0 :

Electrode radius (m)

r dist :

Tissue/outer surface relative distance (m)

\(\bar{r}\) :

Distance covered by the heat generated by nanoparticles (m)

r inj :

Radial distance of the injection (m)

r np :

Mean radius of nanoparticles (m)

S :

Antenna constant (m−1)

SAR :

Specific absorption rate (W kg−1)

t :

Time (s)

T :

Temperature (K)

u :

Velocity (m s−1)

u, w :

Velocity components (m s−1)

u q(t):

Step function (–)

W :

Tissue water density (kg m−3)

∞ :

Far away from heating focus

a:

Arterial

b:

Blood

cr:

Critical

ch:

Channel conversion

dis:

Dispersion

e:

Effective

E:

Energy to vaporize water

Ext:

External

Fat:

Referred to fat

Laser:

Referred to laser energy

max:

Maximum

met:

Metabolism

muscle:

Referred to muscle

np:

Nanoparticles

p:

Probe

r:

Radiative

ref:

Reference

t:

Tissue

tm:

Tumor

v:

Venous

α :

Absorption coefficient (Np Hz−1 m−1)

α diff :

Effective thermal diffusivity (m2 s−1)

δ Λ :

Parameter that refers to the microvascular network (–)

γ :

Water latent heat constant (J kg−1)

Φ:

Phase function (–)

Γ:

Coordinates index (–)

Γf :

Euler gamma function (–)

ε :

Porosity (–)

θ :

Nanoparticles concentration (–)

χ ″ :

Imaginary part of susceptibility of the magnetic nanoparticles (–)

ρ :

Density (kg m−3)

P :

Arithmetic average of each segment contained into the tumor (m)

Ψ :

Density of nanoparticles on the vascular walls (L m−2)

σ s :

Stefan–Boltzmann constant (W m−2 K−4)

σ :

Electric conductivity (S m−1)

τ :

Relaxation time (s)

τ q :

Phase-lag of the heat flux (s)

τ T :

Phase-lag of temperature gradient (s)

μ cr :

Critical cosine of an angle (–)

μ 0 :

Dielectric vacuum permeability (H m−1)

ω :

Blood perfusion (s−1)

ω tr :

Nanoshell transport albedo (–)

ω b0 :

Constant blood perfusion (s−1)

Ω :

Solid angle (–)

Ω(t):

tissue injury degree (–)

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Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale, Università di Napoli Federico II, Piazzale Tecchio 80, 80125, Naples, Italy

    Assunta Andreozzi & Marcello Iasiello

  2. Dipartimento di Medicina e Scienze della Salute “Vincenzo Tiberio”, Università del Molise, Via Francesco De Sanctis 1, 86100, Campobasso, Italy

    Luca Brunese, Claudio Tucci & Giuseppe Peter Vanoli

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  1. Assunta Andreozzi
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  2. Luca Brunese
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Correspondence to Claudio Tucci.

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Associate Editor Estefanía Peña oversaw the review of this article.

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Andreozzi, A., Brunese, L., Iasiello, M. et al. Modeling Heat Transfer in Tumors: A Review of Thermal Therapies. Ann Biomed Eng 47, 676–693 (2019). https://doi.org/10.1007/s10439-018-02177-x

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