A-to-Z Guide to Thermodynamics,
Heat & Mass Transfer, and Fluids Engineering
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## THE MODIFIED ARRHENIUS EQUATION FOR THE LIVING TISSUES

The main result of any computational analysis of thermal treatment of human tumors is the degree of damage to the tumor cells. According to Friebel et al. (2006), some important transformations in tumor cells take place even at temperatures in the range of 40–42°C, where the cells are not destroyed but some molecular-level damage appears. These processes can be approximately described using the Arrhenius-like kinetic equation. At the same time, the regeneration of all human tissues because of oxygen supplied by arterial blood should be taken into account. Obviously, the balance between thermal degradation and continuous regeneration also takes place in healthy tissues under ordinary quasi-steady thermal conditions. The formal use of the Arrhenius-like kinetic equation by ignoring the tissue regeneration would give us the monotonic degradation of a healthy living tissue. Fortunately, this is not the case because of the continuous supply of oxygen with arterial blood and the removal of the products of the vital activity of living cells with venous blood.

As a result, the traditional Arrhenius kinetic equation employed by many authors to estimate the thermal damage to tumors from hyperthermia treatment should be modified. Such a modification was suggested by Dombrovsky and Timchenko (2015) and reproduced in the review paper by Dombrovsky (2019):

 (1)

where i is the conventional number of tissue; ξi is the degree of thermal degradation; and Bi is the dimensionless coefficient. It was assumed that Bi = Bh for all of the healthy tissues, whereas B6 = 0 (for the totally destroyed regeneration of the tumor tissue). The last assumption is supported by experimental data on the very high sensitivity of red blood cells to overheating (Fasano et al., 2010). The value of Bh = 100 was chosen to minimize the thermal damage of healthy tissues at T < 41°C. Additional input data used in the calculations can be found in Dombrovsky et al. (2012) and Dombrovsky and Timchenko (2015). Note that more detailed kinetic models should be considered that take into account multi-stage conversions in tumor cells (Feng and Fuentes, 2011). However, the computational estimates based on the aforementioned simple kinetic model look physically correct. Figure 1 shows that thermal degradation of biological tissues is localized in the tumor (from green to red), whereas there is no damage to the surrounding healthy tissue (in the area colored blue).

Figure 1. Thermal damage (ξ) of a superficial human tissue after the first session of thermal treatment

#### REFERENCES

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, Berlin: Springer, vol. 4: 71–127.

Dombrovsky, L.A. and Timchenko, V.M. (2015) Laser Induced Hyperthermia of Superficial Tumors: Computational Models for Radiative Transfer, Combined Heat Transfer, and Degradation of Biological Tissues, Therm. Proc. Eng., 7(1): 24–36 (in Russian).

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.

Fasano, A., Hömberg, D., and Naumov, D. (2010) On a Mathematical Model for Laser-Induced Thermotherapy, Appl. Math. Model., 34(12): 3831–3840.

Feng, Y. and Fuentes, D. (2011) Model-Based Planning and Real-Time Predictive Control for Laser-Induced Thermal Therapy, Int. J. Hyperthermia, 27(8): 751–761.

Friebel, M., Roggan, A., Müller, G., and Meinke, M. (2006) Determination of Optical Properties of Human Blood in the Spectral Range from 250 to 1100 nm Using Monte Carlo Simulations with Hematocrit-Dependent Effective Scattering Phase Functions, J. Biomed. Opt., 11(3): 034021.

#### References

1. 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, Berlin: Springer, vol. 4: 71–127.
2. Dombrovsky, L.A. and Timchenko, V.M. (2015) Laser Induced Hyperthermia of Superficial Tumors: Computational Models for Radiative Transfer, Combined Heat Transfer, and Degradation of Biological Tissues, Therm. Proc. Eng., 7(1): 24–36 (in Russian).
3. 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.
4. Fasano, A., Hömberg, D., and Naumov, D. (2010) On a Mathematical Model for Laser-Induced Thermotherapy, Appl. Math. Model., 34(12): 3831–3840.
5. Feng, Y. and Fuentes, D. (2011) Model-Based Planning and Real-Time Predictive Control for Laser-Induced Thermal Therapy, Int. J. Hyperthermia, 27(8): 751–761.
6. Friebel, M., Roggan, A., Müller, G., and Meinke, M. (2006) Determination of Optical Properties of Human Blood in the Spectral Range from 250 to 1100 nm Using Monte Carlo Simulations with Hematocrit-Dependent Effective Scattering Phase Functions, J. Biomed. Opt., 11(3): 034021.