Voltage and Time Required for Irreversible Thermal Damage of Tumor Tissues during Electrochemotherapy under Thomson Effect

The essential target of the tumor's treatment is how to destroy its tissues. This work is dealing with the thermal damage of the tumor tissue due to the thermoelectrical effect based on the Thomson effect. The governing equation of tumor tissue in concentric spherical space based on the thermal lagging effect is constructed and solved when the surface of the tumor tissue is subjected to a specific electric voltage. Different voltages and resistances effects have been studied and discussed for three different types of tumor tissues. The thermal damage quantity has been calculated with varying values of voltages and times. The voltage has significant effects on the temperature and the amount of irreversible thermal damage to the tumor. Electrotherapy is a successful treatment. This work introduces a different model to doctors who work in clinical cancer to do experiments about using electricity to damage the cancer cells.


Introduction
Electrochemotherapy with a low-level direct electric current (DEC) is a therapeutic approach that consists of applying a direct current electric across the tumor tissues, and it has been proved to be a sure, effective, and relatively cheap treatment of tumors [1]. Currently, electrochemotherapy procedure has been performed in several cancer clinics based on standardized clinical protocol [2]. The electrical properties of living tissues have been studied by many researchers to evaluate the effect of electromagnetic fields [3]. Nuccitelli discussed the use of pulsed electric fields in cancer therapy based on pulse length, millisecond domain, microsecond domain, and nanosecond domain [4]. Gabriel et al. introduced one of the most essential and complete data on the electrical properties of tissues in the full range of 2 1 Hz z 0 0GH  [5]. Many authors have confirmed that the electrical properties of all types of tumor tissue (muscle, fat, breast, etc.) vary after applying electroporation pulses [5][6][7][8][9][10][11]. Tasi et al. have determined the in-vivo dielectric properties, resistivity, and relative permittivity of living epidermis and dermis of human skin soaked with a physiological saline solution for one minute between 1 kHz and 1 MHz [12].
We have three common types of tumors. The first type is called a lipoma tumor, which is a fatty tissue tumor. The second type is the liposarcoma tumor that arises in deep and fat cells of the soft tissue. The third type is called myxoid liposarcoma tumor, which is characterized by round to oval cells. The resistance values of the above three types of tumors have been calculated. The 4 www.videleaf.com myxoid liposarcoma tumors show resistance values (50 Ω -100 Ω). The liposarcoma tumors show resistance values (250 Ω -970 Ω), while lipoma tumors show resistance (800 Ω -1800 Ω) [13].
The dual-phase-lag (DPL) model is a heat conduction model that describes the temperature on a macroscopic scale with the microstructural effect by taking into account the phase-lag-times of temperature gradient and heat flux [14,15]. It has been applied to studying various problems of heat transfer. Many authors investigated the effects of the phase lag times of heat flux and temperature gradient on the thermal wave transfer inside the tissues [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Mathematical modeling is essential for scientific studies, and industrial and is widely used in treatment planning [30]. Nuccitelli solved an application of Pulsed Electric Fields to cancer therapy, where he found that nanosecond pulsed fields to be effective in treating skin lesions but have not yet been approved for cancer therapy [4]. Calzado et al. constructed Simulations of the electrostatic field, temperature, and tissue damage generated by multiple electrodes for electrochemical treatment [31]. Soba et al. have integrated the analysis of the potential, electric field, temperature, P.H., and tissue damage, which has been generated by different electrode arrays in a tumor under electrochemical treatment [32]. Aguilera et al. studied the electric current density distribution in planar solid tumor and its surrounding healthy tissue generated by an electrode elliptic array used in electrotherapy [33]. Luo et al. studied the tumor treating fields for high-grade gliomas [34].
Any electrical conductor has two different thermoelectric effects, the Peltier and Thomson effects, which are responsible for the thermal dissipation. Those effects are going in a conductor material when the electrical current passes through it. Within the Thomson effect, the absorption of heat occurs when the electric current goes through a circuit composed of a single material that has a gradient of temperature along its length. The Thomson effect is considered as a heat source/sink, commonly, which is added to the Joule-heating. Some attempted has to consider the Thomson effect, but only for a specific case [35]. Chen et al. studied a model in which the Thomson effect and determined a 5 www.videleaf.com threshold criterion for neglecting the Thomson effect based on material properties [36].
No data for the value of the Thomson effect coefficient (Seebeck coefficient) has been found for the skin tissue or to the tumor as an electrical conductor material. So, the values of that coefficient have been assumed to study the effect of this phenomenon on tumors.
Goodarzi et al. develop the lattice Boltzmann method to simulate the slip velocity and temperature domain of buoyancy forces of FMWCNT nanoparticles in water through a microflow imposed to the specified heat flux and constructed the numerical simulation of natural convection heat transfer of nanofluid with nanoparticles in a cavity with different aspect ratios [37][38][39].
The DPL bioheat conduction model could significantly predict the different temperatures and thermal damage in any tissues from the hyperbolic equation of thermal wave and Fourier's heat conduction models. Moreover, the DPL bioheat conduction equations can be reduced to the Fourier heat conduction equations only if both phase lag times of the temperature gradient and the heat flux is zero [40].
This work is a theoretical investigation for the thermoelectrical effect on three different types of tumor tissues that have different known values of resistance. This study does not discuss the impact of the phase-lag parameters, which already has been done in many publications. The aim is to know the values of the suitable electric voltage and time required to do enough irreversible thermal damage for the three different types of tumors.

Materials and Methods
A small volume of tumor is considered as a solid sphere with the radius R [41]. Cancer occupies the region 0 rR  , and the temperature distributes over it with a function of the distance r from the center of the sphere and time t (see Figure 1). 6 www.videleaf.com We consider three different types of tumors with the same histological features; myxoid liposarcoma, liposarcoma, and lipoma. The heat conduction equation takes the form [15,21,28,29,41]: The energy conservation formula of bioheat transfer is given by [15,21,28,29,41,42]: The term which gives 2 11 Almost all the tumor types take a spherical shape with different sizes, so we assumed the body of our application is a spherical body.
The differential equation of bio-heat transfer in the spherical coordinate system is obtained from equations (2) and (4) as: Consider the following functions [41]: Thus, we have Hence, we have from equations (5)-(7) that: Consider that the tumor is working as a conductor with electrical resistance   e R  , and the surface of the tumor rR  is subjected to a particular heating source that comes from the thermal effect due to connection with electric voltage V(V) and current I(A). Then, the heat flux is given by: where is the Seebeck coefficient.
The Seebeck coefficient (also known as thermoelectric power, thermopower, and thermoelectric sensitivity) of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material.
Applying Laplace transform for equation (8) and (9) defined as: 9 www.videleaf.com Then, we have The following initial conditions have been used within applying the Laplace transform: where   The general solution of equation (15) where 12 , kk are the roots of the characteristic equation: Apply the following boundary conditions: is the applied external heat flux due to the electric voltage V and negative value because it goes towards the origin in a negative direction.
Using Laplace transform defined in (11), we get: Thus, we have the following system of linear equations: 12 2 which is the final solution in the Laplace transform domain.

The Thermal Damage
Henriques and Moritz used the Arrhenius formula to calculate the thermal damage [43]. They proposed that the thermal damage of the tissue could be considered as a chemical rate process, and calculated by using the first-order Arrhenius rate equation. The measure of thermal damage  was introduced, and its rate   T  was supposed to satisfy [43][44][45]: where A is a frequency factor,  is the universal gas constant, and a E is the activation energy.
Equation (23) Hence, we obtain the thermal damage in the form: To simulate the thermal response within a small spherical tumor of radius R = 0.003 m, we will use the tumor tissue with material properties, as shown in Table 1 [13,22,41].  [13]. In the figures based on the distance of the tumor, the horizontal axis has been taken in reverse order to scale the temperature and the damage from the surface of the tumor rR  up to the center 0 r  .

Discussion
There are no significant histological differences between the electroporated tumor volume and the remaining regions of the tumor mass. On the other hand, and interestingly, different tumor types show different electrical conduction. So, the resistance values have been considered according to tumor histotype [13].   Figure 10 shows Considering the Thomson effect leads to a smaller quantity of irreversible thermal damage to the tumors more than the case that does not include this effect.

Conclusions
In this work, governing partial differential equation of tumor tissue in concentric spherical space based on thermal lagging effect and Thomson effect is solved in the Laplace transform domain. The surface of the tumor tissue is subjected to an electric voltage. The results represent the effects of the different values of voltages, Thomson effect, and times on myxoid liposarcoma tumor, liposarcoma tumor, and lipoma tumor. We focused our attention on the difference value of electrical resistance of the three types of tumors which have been used.
The time and the applied voltages on the surface of the tumors have significant effects on the absolute temperature and the quantity of the irreversible thermal damage to the three types of tumors used.
The Thomson effect has a significant impact on the absolute temperature and the quantity of the irreversible thermal damage of the three types of tumors used. www.videleaf.com Applying electrical potential within the electrochemotherapy for some seconds maybe is enough to get the irreversible damage of the myxoid liposarcoma tumor, liposarcoma tumor, and lipoma tumor, which makes electrochemotherapy is a successful treatment.
This research offers a new and potentially effective method in the treatment of cancer that has the characteristic sin of the specific place and its known electrical qualities within the patient's body.
Therefore, we direct the clinical trial operators to try these results on a sample of patients, and we emphasize that these trials are likely to be very successful if the accuracy and study are done.