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## HEAT TRANSFER PROBLEM FOR THE LASER-INDUCED HYPERTHERMIA OF SUPERFICIAL TUMORS

The energy transport in a biological system is usually expressed by the so-called bioheat equation. The bioheat equation developed by Pennes (1948) is one of the earliest models for energy transport in tissues. It was assumed that the arterial blood temperature (Tb) is uniform throughout the tissue, while the venous blood temperature is equal to the local tissue temperature (Tt). The resulting transient energy equation is as follows:

 (1)

where (ρc)t and kt are the volumetric heat capacity and thermal conductivity of human tissue, respectively; the second term on the right-hand side is responsible for the heat transfer due to the arterial blood perfusion rate (vb); and Wm is the metabolic heat generation within the tissue. A more detailed model for heat transfer in human tissues should be based on two coupled energy equations for the tissue and arterial blood with the spatial and time variations of the arterial blood temperature (Khaled and Vafai, 2003; Nakayama and Kuwahara, 2008). The following coupled energy equations were suggested by Dombrovsky et al. (2012):

 (2a)
 (2b)

where εa is the volume fraction of arterial blood, and αλ,b is the spectral absorption coefficient of arterial blood. The volumetric heat generation due to absorption of laser radiation is taken into account in both energy equations. The absorbed laser power (Wλ) is determined as a sum of the absorbed collimated and diffuse radiation. The diffuse radiation field is calculated using the P1 approximation for the two-dimensional radiative transfer problem (Dombrovsky and Baillis, 2010; Dombrovsky et al., 2012). Of course, the radiation is not uniformly absorbed in a composite medium characterized by total absorption coefficient αλ; therefore, the corresponding terms in Eqs. (2a) and (2b) are different. It is important that the αλ,b/αλ ratio is greater in the case when there are no highly absorbing plasmonic nanoparticles in ambient tissue. In this case, the radiative heat generation in arterial blood may be considerably greater than that in the tissue. This effect is known, and it is used in selective thermal heating and damage of blood by the action of pulsed laser radiation at wavelength λ = 0.532 or 0.585 μm characterized by extremely high spectral values of αλ,b (Pfefer et al., 2000; Jia et al., 2006). It is clear that the same effect of relatively strong absorption of laser radiation also takes place for venous blood. The latter may be a physical basis for the more detailed three-temperature heat transfer model with a third energy equation for venous blood. The term Wch in Eq. (2a) takes into account the heat of endothermic chemical conversions in human tissues and venous blood during strong hyperthermia. According to Dombrovsky et al. (2012), one can neglect Wch in soft thermal treatment.

The energy equations for arterial blood and other tissues [(2a) and (2b)] are based on a presentation of real complex tissue as a two-temperature continuous porous medium. It is assumed that there are no large blood vessels in the computational region. In relation to arterial blood velocity, this can be determined using an additional Darcy-type equation for filtration of the blood. In many cases, one can neglect a variation of the arterial blood flow (Dombrovsky et al., 2012; Dombrovsky, 2019). Instead, the field of this physical quantity [u(r)] is usually assumed to be known. The fields of εa(r) (the volume fraction of arterial blood) and hb,t (r) (the volumetric heat transfer coefficient between arterial blood and ambient tissue) are also considered as input parameters of the problem. These parameters depend on the fine structure of the peripheral vascular system.

The initial and boundary conditions for Eq. (2a) are as follows:

 (3)
 (4a)
 (4b)
 (4c)

where boundary condition (4a) is the natural convective heat transfer with ambient air, and condition (4b) refers to water cooling of the body surface (using a transparent water jacket). The first two conditions in Eq. (4c) are the symmetry and adiabatic conditions, whereas the last one approximately describes the heat transfer from the internal part of the body, which has a constant temperature Te,2. The initial steady-state temperature profile T (z,r) is a solution to the simple boundary-value problem (Dombrovsky et al., 2012). With regard to the boundary conditions for Eq. (2b) (for arterial blood), these are slightly different at the body surface and at the boundary with the rest of the massive body: Tb = Tt at the body surface and Tb = Tbe  = Te,2 at z = H. It was shown by Dombrovsky et al. (2012) that the effect of the convective terms in Eqs. (2a) and (2b) is insignificant and can be neglected in the calculations for superficial human tissues.

#### REFERENCES

Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, New York: Begell House.

Dombrovsky, L.A., Timchenko, V., and Jackson, M. (2012) Indirect Heating Strategy of Laser Induced Hyperthermia: An Advanced Thermal Model, Int. J. Heat Mass Transf., 55(17–18): 4688–4700.

Dombrovsky, L.A. (2019) Scattering of Radiation and Simple Approaches to Radiative Transfer in Thermal Engineering and Bio-Medical Applications, In A. Kokhanovsky, Ed., Springer Series in Light Scattering, chapter 2, Berlin: Springer, vol. 4: 71–127.

Jia, W., Aguilar, G., Verkruysse, W., Franco, W., and Nelson, J.S. (2006) Improvement of Port Wine Stain Laser Therapy by Skin Preheating Prior to Cryogen Spray Cooling: A Numerical Simulation, Lasers Surg. Med., 38(2): 155–162.

Khaled, A.-R.A. and Vafai, K. (2003) The Role of Porous Media in Modeling Flow and Heat Transfer in Biological Tissues, Int. J. Heat Mass Transf., 46(26): 4989–5003.

Nakayama, A. and Kuwahara, F. (2008) A General Bioheat Transfer Model Based on the Theory of Porous Media, Int. J. Heat Mass Transf., 51(11–12): 3190–3199.

Pennes, H.H. (1948) Analysis of Tissue and Arterial Blood Temperature in the Resting Human Forearm, J. Appl. Physiol., 1(2): 93–122.

Pfefer, T.J., Choi, B., Vargas, G., McNally, K.M., and Welsh, A.J. (2000) Pulsed Laser-Induced Thermal Damage in Whole Blood, ASME J. Biomech. Eng., 122(2): 196–202.

#### References

1. Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, New York: Begell House.
2. Dombrovsky, L.A., Timchenko, V., and Jackson, M. (2012) Indirect Heating Strategy of Laser Induced Hyperthermia: An Advanced Thermal Model, Int. J. Heat Mass Transf., 55(17–18): 4688–4700.
3. Jia, W., Aguilar, G., Verkruysse, W., Franco, W., and Nelson, J.S. (2006) Improvement of Port Wine Stain Laser Therapy by Skin Preheating Prior to Cryogen Spray Cooling: A Numerical Simulation, Lasers Surg. Med., 38(2): 155–162.
4. Khaled, A.-R.A. and Vafai, K. (2003) The Role of Porous Media in Modeling Flow and Heat Transfer in Biological Tissues, Int. J. Heat Mass Transf., 46(26): 4989–5003.
5. Nakayama, A. and Kuwahara, F. (2008) A General Bioheat Transfer Model Based on the Theory of Porous Media, Int. J. Heat Mass Transf., 51(11–12): 3190–3199.
6. Pennes, H.H. (1948) Analysis of Tissue and Arterial Blood Temperature in the Resting Human Forearm, J. Appl. Physiol., 1(2): 93–122.
7. Pfefer, T.J., Choi, B., Vargas, G., McNally, K.M., and Welsh, A.J. (2000) Pulsed Laser-Induced Thermal Damage in Whole Blood, ASME J. Biomech. Eng., 122(2): 196–202.