1 Introduction

Brain temperature, as an independent therapeutic target variable, has received increasingly intense clinical attention. The impact of the brain temperature is prevalent in a number of common human diseases. This, generally means that efforts to understand the fundamental aspects of the brain temperature regulation are critical for the development of safe, effective, and pragmatic clinical treatments for patients with brain tumors [1]. However, it seems that there is no accurate numerical simulation exist for the temperature distribution in the human body and the equations must be simplified by making assumptions. These assumptions must preserve the parameters and the main properties of the tissue.

As far as the heat transfer models in human body metabolism, some researchers have proposed different equations to study the heat transfer in vital tissues, including the Pennes bio-heat transfer equation [2], the Chen–Holmes model [3], the three-temperature model for peripheral tissue [4], and the Weinbaum–Jiji equation [5], which comes with specific applications and limitations. One of the limitations of the Pennes equation is the assumption of non-directional blood perfusion in the exchange of the energy between the veins and tissues. In addition, the local structure of capillaries was ignored; therefore, the energy exchange with arteries (larger blood vessels) was not considered. Generally, This equation (Pennes bio-heat transfer equation) is used in thermal simulation of the body and measuring the blood perfusion [6]. Unlike Pennes equation, the Chen–Holmes equation considers a balance between the blood and tissue in arteries, as well as the directional blood perfusion in the capillary geometry. Despite resolving the Pennes equation limitations to some extent, this model is still more difficult to analyze and is often used for capillary networks and blood perfusion [3]. The three-temperature model, considers the temperature in arteries, veins, and tissues. This equation also assumes directional blood flow for the thermal analysis and is mostly performed for peripheral tissues [7]. Similarly, to the previous one, solving equations by this method is also riddled with complexities. The Weinbaum–Jiji equation simplifies the three-temperature model, but still, it involves several variables and is relatively complex [5]. As discussed earlier, for the thermal simulation of body tissues, the Pennes equation is the most straightforward and practical option among the four cases.

Detection of cancerous tumors is one of the main objectives in the thermal simulation of vital tissues. On the other hand, the metabolism of cancerous cells is higher than the normal cells. Thus, as a general fact, the temperature of cancerous tissues is higher than that of the healthy tissues [8]. The thermal analysis serves a critical role in the thermal simulation and geometrical modelling of a tissue. Bousselham et al. [9] studied on the thermal analysis of the brain tumor for different sizes. They showed that the magnetic resonance parameters are highly temperature-dependent. They relied on Pennes bio-heat transfer equation to solve the temperature distribution in the brain tumor and modelled the brain in three layers of cerebrospinal fluid (CSF), gray matter (GM), and skull, as concentric spheres, with the spherical tumor was located at the center of this head model. Their results showed that in comparison with the normal tissue, the maximum temperature rise at the center of the tumor is 1.5 °C. Tang et al. [10] implemented the Penne’s bio-heat transfer equation for hyperthermia purposes considering the blood perfusion rate term as a function of ambient temperature, and then solved it separately for both normal and tumor tissues. They also assumed a spherical shape for normal and tumor tissues. Bousselham et al. [11] presented a 3D model of the brain, representing normal tissues by a cuboid containing a spherical embedded tumor and used a genetic algorithm with the inverse solution technique to estimate the presence and dimensions of the tumor. Li et al. [12] conducted an analytical solution of Penne’s bio-heat transfer equation to obtain the temperature distribution using a Laplace transform for hyperthermia treatment applications. They investigated the temperature distribution separately in the scalp and the brain matter. Their results showed that controlling the blood perfusion rate and the temperature inside the brain are useful for hyperthermia treatment purposes. They also simplified the geometry and assumed the tissue to be spherical. Majchrzak and Turchan [13] took into account the artery effects on the thermal simulation of tissues and solved the Pennes bio-heat transfer equation assuming a cubic geometry for both normal and tumor tissues. Kenna and Lakhssassi [14] used the Pennes bio-heat transfer equation for body tissues considering a linear temperature-dependent term for blood perfusion. Hossain and Mohammadi [15] applied an analytical solution to the Pennes bio-heat transfer equation for a cancerous tissue with spherical geometry. In a study by Kohani et al. [16] on the hyperthermia treatment of the cancerous tissues, the Pennes bio-heat transfer equation was employed for a spherical geometry. Bousselham, et al. [17] investigated the thermal analysis of brain tumor in an MRI image segmentation using finite difference method for discretization of Pennes bio-heat transfer equation. Considering various values for the blood perfusion and body metabolism, Shirkavand and Nazif [18] studies the temperature control of the human forearm under hyperthermia conditions. The effect of changing two aforementioned parameters (blood perfusion and body metabolism) on the temperature of a tumor and normal tissues was investigated in this study.

From the above literature review, it can be concluded that the brain tissue is highly sensitive to the temperature fluctuations. Meanwhile, it seems that the scientific and clinical aspects of the brain in terms of spatial and temporal distribution of the temperature are not well-established yet. In the current research work, a realistic geometric model of the human brain, as well as the tumor is presented. Using a numerical simulation based on the finite element scheme, one may be to be able to conduct a variety of analyses at any points of the brain including the effects of the temperature variations.

In this paper, the brain and tumor's actual geometry are both applied. Bio-heat transfer equation with proper boundary conditions is solved numerically using FEM. The temperature profiles of a simple model of the brain and the real model are presented. Furthermore, the necessary temperature for hyperthermia by changing the volume of tumor metabolism and blood perfusion rate are also analyzed, and the temperature controlling process is presented.

2 Mathematical modeling

The human head consists of many regions with different layers. The schematic, location and the thermal properties of each layer of the head can be found in the literature [19]. Generally, brain matter, cerebrospinal fluid, skull, and scalp are the four main layers of the head. The brain matter contains both gray and white matters. In present work, three major layers of the head (brain matter, cerebrospinal fluid, skull) are modeled. These layers can be derived from the output of the geometric model, namely, the Menelik surface head mesh model. This file is available in an online database [20]. The geometry of the human head layers is demonstrated in Fig. 1. Here, the Pennes bio-heat transfer equation was solved under steady state conditions for all of the head layers considering appropriate thermo-physical properties and boundary conditions (Table 1). The general form of the Pennes bio-heat transfer equation can be expressed as follows [2]:

$$\rho c\frac{\partial T}{{\partial t}} = \nabla \cdot \left( {k\nabla T} \right) - \rho_{b} c_{b} w_{b} \left( {T {-}T_{b} } \right) + Q_{met} ,$$
(1)

where ρ is the density of tissue, c is the specific heat of tissue, k is the thermal conductivity, wb is the blood perfusion rate, ρb is the density of blood, cb is the specific heat of blood, Tb is the temperature of blood, and Qmet is the metabolic heat generation. The physical properties of blood are ρb = 1052 kg/m3 and cb = 3800 J/(kg °C), and the temperature is assumed to be equal to 37 °C [21].

Fig. 1
figure 1

Schematics of the three-dimensional (3D) geometry of the human head layers and surface meshing

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Table 1 Physical properties of the head layers and the tumor
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From Eq. (1), it can be noted that obtaining an exact analytical solution for complex shapes like the human tissue is almost impossible [22]. Hence, different numerical methods (finite element, finite volume, finite difference) are often utilized to solve differential equations with respect to the problem context. In present research, based on the Pennes bio-heat transfer equation, the temperature distribution of the so-called “Menelik head mesh model” is analyzed, numerically, via finite element scheme. As far as the structural analysis of the human body, it is very crucial to conduct any mechanical experiment. In recent years, many researchers has conducted the finite element method via numerical packages as powerful and effective tools to simulate the mechanical experiments related to the human body [23]. Developing the mesh formation, here, the surface elements are in triangular shape while the volumetric elements are of tetrahedron type.

2.1 Boundary conditions

In order to solve the Pennes equation, appropriate boundary conditions must be imposed to the problem domains. Temperature and the heat flux continuity conditions are set for the internal boundaries, as follows [24]:

$$T_{1} = T_{2}$$
(2)
$$k_{1} \nabla {\text{T}} = k_{2} \nabla {\text{T}}$$
(3)

while the convective boundary condition is assumed at the outer surface of the skull as [25]:

$$- K \nabla T = h_{a} \left( {T - T_{e} } \right)$$
(4)

where ha = 8 Wm−1 K−1, and the ambient temperature, Te, is equal to 25 °C [25].

3 Results and discussion

3.1 Validation

In order to simulate the heat transfer inside the brain tissue, the head layers (CSF, GM, and skull) are considered as the concentric spheres. The spherical tumors with three different sizes were placed at the center of the head model. As far as the verification of the present model, the obtained results of the temperature distribution at the centerline of the tumors are compared to those of the Bousselham et al. [9] work for three different diameters, namely, 10, 15, and 20 mm (Fig. 2). As the figure shows, very good agreement exists between the results using the present FEM model and those reported by Bousselham et al.; however, they obtained the temperature distribution using finite difference method whereas in the current study, the finite element method was conducted for the solution of the Pennes bio-heat transfer equation. Noting that the thermo-physical properties of the tissues are also presented in Table 1.

Fig. 2
figure 2

Comparison between the present results of temperature distribution at the centerline of a tumor with those reported by Bousselham et al. [9] for tumors with three different diameters, namely, 10, 15, and 20 mm

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3.2 Tumor temperature distribution: effects of diameter

Figure 3 indicates the effects of the tumor size on the temperature distribution for three different tumor diameters, namely, 10, 15, and 20 mm. Due to this figure, it can be seen that the center of the cancerous tumor has higher temperature than its surrounding normal tissue (about 3 °C). Moreover, the maximum temperature at the center of the tumor is increased with increasing the tumor diameter. For the sake of a better visual understanding, results for the tumor temperature fields are also represented in Fig. 4 with magnification.

Fig. 3
figure 3

Effect of tumor diameter on temperature distribution for three different diameters; a 20 mm, b 15 mm, c 10 mm

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Fig. 4
figure 4

Magnified visuals of tumor temperature distribution for three different diameters; a 20 mm, b 15 mm, c 10 mm

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In present study, in order to simplify the problem solving process and reducing the computational effort, some assumptions have been made. From those which is very important here and have been taken under consideration, is the convective boundary condition between the skull and the air. Basically, as a matter of fact, this condition should be applied at the scalp/air interface. However, a simple simulation of the layers of the head, addressed in the validation section, may allow one to model the scalp layer by using the thermo-physical properties mentioned by Ma et al. [12]. Based on this ground, the results of the temperature distribution are compared with those obtained from the simulation without considering the scalp (Fig. 5). It can be observed from the figure that, due to a very small thickness of the scalp (3–11 mm) [26] in relative comparison with the larger size of the head, the temperature values at the center of the tumor for both of the models are extremely close (absolute difference is lower than 0.025%) (Fig. 5). Therefore, the effect of the scalp on the heat transfer process can be ignored. Here, the scalp is modeled considering the maximum thickness equal to 11 mm).

Fig. 5
figure 5

Temperature distribution along a tumor with diameter of 20 mm and the layers of the head (GM-CSF-skull); a with scalp (of thickness of 11 mm), b without scalp

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3.3 Temperature distribution in tumor and healthy tissues (without hyperthermia)

Having a geometrically-accurate representation of the vital body tissues, and using the right boundary conditions are established as two major basis in obtaining the temperature distribution in the brain and tumor tissues via the numerical simulation of the bio-heat transfer analysis. In the present study, the thermal characteristics of a simplistic model of the head consists of three spheres representing the skull, the cerebral spinal fluid (CSF), and the gray matter (GM), as well as a spherical tumor at the center of the model, are examined utilizing the finite element method. For this purpose, a realistic thermal simulation was carried out for the human head layers considering the modelled tumor with the volumetric size of 5990 mm3. Figure 6 shows the exact location of the maximum temperature in the tumor for different cross sections. As one may be expected, the maximum values of temperature occurs at nearly center of each cross section while drops to its minimum at the vicinity of the tumor surface. It can also be noted that the maximum tumor temperature is marked on Sect. 2 (Fig. 6). As far as the temperature profile across the tumor central plane, Fig. 7 demonstrates more clear representation of the temperature profile in the line passing through the center of tumor.

Fig. 6
figure 6

Temperature distribution in a 5990 mm3 tumor for the three sections

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Fig. 7
figure 7

Temperature profile at the plane passing through the tumor center

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It should be noted that in current study, the tumor is located inside WM, while the other three layers (GM, CSF, and skull) are considered to enable a realistic representation of the human head. The absolute values of temperature, i.e., minima and minima, throughout the tumor and its surrounding tissues are tabulated in Table 2.

Table 2 Temperature maxima and minima in the tumor and head layers
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Basically, regulating the brain temperature within the permissible limits is a highly sensitive process while can affects the brain tumors. So, having a proper temperature range at each layer of the head can be really effective for development of the safe clinical treatments. The characteristics of the temperature fields for the tumor and the White Matter (WM) are numerically investigated which are presented in Figs. 8, 9, 10, respectively. Figure 8 shows the temperature distribution in the WM and the tumor. The minimum WM temperature was found to be equal to 36.18 °C (Table 2). Figure 9 illustrates the temperature distribution over the tumor surface. Accordingly, the maximum temperature of the WM is found to be equal to 38.36 °C, which is marked in Fig. 10, at the plane passing through a point with maximum temperature.

Fig. 8
figure 8

Temperature distribution in tumor (5990 mm3) and the WM (white matter)

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Fig. 9
figure 9

Temperature distribution over the tumor surface

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Fig. 10
figure 10

Temperature distribution over WM (white matter) and the tumor on the cross-section passing through the surface with maximum tumor temperature

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Figure 11a shows the temperature distribution over the brain matter (WM and GM) and the tumor. As it is evident, the maximum temperature of the head corresponds to the tumor whereas the minimum temperature was obtained as 34.83 °C. Figure 11b indicates the geometry and the cross-sectional plane passing through the maximum temperature point of the GM. This plane and its respective hot-spot are also presented in Fig. 11c. The maximum value for the temperature is calculated as 37.7 °C.

Fig. 11
figure 11

a Temperature distribution in WM (white matter)–GM (gray matter) layers and the tumor; b the cross section passing through the maximum temperature point of GM; c temperature distribution and the hot-spot on the cross section passing through the maximum temperature point of GM

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Figure 12a shows the temperature distribution over the layers of WM, GM, CSF, and the tumor. It can be observed from the figure that the minimum CSF temperature is equal to 33.02 °C. For the sake of a better illustration of the temperature distribution, Fig. 12b depicts the temperature field on the 2D-plane passing through the center of the tumor. The three-dimensional (3D) meshing frame-work for the CSF and the mentioned cross-sectional plane are also presented in Fig. 12c. Figure 12d exhibits the temperature distribution in the CSF tissue.

Fig. 12
figure 12

a Temperature distribution over WM (white matter), GM (gray matter), CSF (cerebro spinal fluid), and tumor; b temperature distribution on the cross section passing through the center of tumor; c cross section passing through the maximum temperature in CSF; d temperature distribution in the CSF

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Figure 13 shows the thermal analysis of a brain tumor in an MRI image belongs to the study of Bousselham et al. [17]. The size of the tumor is about 27,400 mm3. They reported the peak temperature at the center of the tumor as 40.03 °C. Due to Table 2, the maximum temperature difference between the present results and the work of Bousselham et al., at the center of the tumor is about 1.39 °C

Fig. 13
figure 13

Thermal analysis of brain tumor in MRI image segmentation [17]

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Figure 14a shows the temperature distribution over the entire head comprised of WM, GM, CSF, skull, and the tumor. The minimum temperature is found to be about 28.18 °C which occurs at the outer surface of the skull. A cross-sectional plane runs through all layers of the head layers, as well as the tumor which graphically illustrates the temperature distribution at the 2D plane (Fig. 14b, c).

Fig. 14
figure 14

a Temperature distribution over the entire layers of head; b, c Temperature distribution on the cross-sectional plane passing through the head layers and the tumor

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Figure 15a indicates the temperature field at the cross-sectional plane passing through all of the head layers. Figure 15b shows the temperature profile in the head model for two different cases: without the tumor, and in the presence of the tumor. Evidently, it can be observed from the profile that the minima at the start and at the end of the line correspond to the skull, while the peak is located at the center of the tumor (green line). Meanwhile, for the case of a brain without the tumor, the peak temperature is located on the brain matter (blue line).

Fig. 15
figure 15

a A line through the tumor center and all of the head layers; b Temperature distribution in the head model with tumor and without tumor

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3.4 Tumor temperature distribution: effects of tumor blood perfusion and metabolism (with hyperthermia)

One of the prominent approaches to eliminate the cancerous cells is thermal therapy. In this method, a temperature range of 40–46 °C is required for the elimination of the tumor [10]. Temperature control is a critical part of the thermal simulation of human body tissues for eliminating cancerous tumors. In fact, normal tissue is sensitive to the temperature rising. Therefore, it is essential that the temperature is raised up to 41–45 °C in the cancerous tissue, while it must not exceed 41 °C in the healthy tissue [27]. In this section, the effects of two thermo-physical parameters, namely, the blood perfusion rate, and the tumor metabolism, on the temperature of the tumor are investigated. Different values for the two mentioned parameters are considered which presented in Table 3. According to the results of the study conducted by Bousselham et al. [17] on an 11,600 mm3 tumor, with wb = 0.0016 (ml/(ml.s)), the maximum temperature of the tumor obtained as 39.36 °C; however, for wb = 0.001 (ml/(ml.s)) there is a rise of 0.17 °C in temperature at the center of the tumor. If, in present study, the same values for blood perfusion rate be considered for the tumor (the size of the tumor in the present work is half of the volume of the tumor mentioned in the work of Bousselham et al.), the maximum temperature will be increased from 38.35 to 38.52 °C. It should be noted that the difference between the maximum temperatures in the two studies is about 1 °C. Based on the initial results, if wb = 0.0005 (ml/(ml.s)) be considered for the tumor, the temperature at the center of the tumor will be 38.64 °C.

Table 3 Different values for blood perfusion rate (wb), and metabolic heat generation (Qmet)
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Figure 16 shows the effect of three different blood perfusion rates on the temperature profile along the line passing through the center of the tumor. The maximum temperatures at the center of tumors show no remarkable differences for the three illustrated plots implying that decreasing the blood perfusion rate does not increase the temperature of the tumor, significantly. Hence, it does not provide the desired condition of hyperthermia.

Fig. 16
figure 16

The tumor temperature distribution along the line passing through the center of tumor, considering different values of wb

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Figure 17 indicates the tumor temperature along a line passing through the center of the tumor for increasing values of the tumor metabolism, Qmet, while the blood perfusion rate is constant at wb = 0.001 (ml/(ml.s)). As the figure shows, the ideal temperature for tumor elimination can be obtained when the tumor metabolism is increased five times (equal to 125 × 103 W/m3) than the normal rate (Qmet = 25,000 W/m3) (Fig. 18). As it is shown in Fig. 18, the maximum temperature in the healthy tissue obtained as 41.37 °C. In fact, the obtained numerical results in Fig. 18 shows that when the amount of the heat produces inside the tumor reaches 125 × 103 (W/m3), the temperature control creates to provide hyperthermia condition without causing damage through rising temperature in the healthy tissue.

Fig. 17
figure 17

The tumor temperature distribution on the path passing through the center of tumor for different values of Qmet

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Fig. 18
figure 18

The temperature distribution in the tumor and the surrounding healthy tissue (white matter) on the cross section passing the maximum temperature of WM for Qmet = 125 × 103 (W/m3) and wb = 0.001 (ml/(ml.s))

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Figure 19 illustrates the temperature variations along a line passing through all the layers of the head for two different values of tumor metabolism, namely, Qmet = 25 × 103 (W/m3), and Qmet = 125 × 103 (W/m3). Due to this figure, when the tumor metabolism increases five times, the temperature distribution does not varies substantially in all of the head layers but only in few regions including a small portion of white matter (WM). This means that the rest of the layers (Gray Matter, Cerebrospinal Fluid and Skull) are not affected. In other words, the major temperature difference between the two plots occurs in the tumor and in a small region of he WM adjacent to the tumor.

Fig. 19
figure 19

Comparing the temperature distribution across all the layers of the head, when tumor metabolism is increased 5 times and blood perfusion rate remains constant at wb = 0.001 (ml/(ml.s))

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Generally, there are different techniques for thermal therapy. Magnetic fluid hyperthermia (MFH) is a common modern method for the elimination of the cancerous tumor. In this method, by injecting a magnetic fluid into the tumor and then applying an AC magnetic field to the area, the Magnetic Nanoparticles (MNPs) act as a heat source inside the tumor, and consequently, the temperature enhances inside the tumor. Figure 20 indicates that Qmet equal to 125 × 103 W/m3 provides the great effects on the tumor temperature in hyperthermia implementation, if one employs the method of MFH in order to increase the temperature in the tumor considering 125 × 103 W/m3 heat produced by nanoparticles. The temperature variations along the line passing through the center of the tumor can be seen in Fig. 20, considering the the heat of metabolism and the MFH technique.

Fig. 20
figure 20

Temperature distribution of the tumor for constant blood perfusion rate of wb = 0.001 (ml/ (ml.s)), under conditions of hyperthermia for two different techniques: heat produced by nanoparticles QMNP = 125 × 103 W/m3 (green line); heat produced by increasing the tumor metabolism Qmet = 125 × 103 W/m3 (blue line), along the cross section passing through the center of the tumor

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Figure 21 shows that without changing the tumor metabolism (Qmet = 25,000 W/m3), the produced heat of 125 × 103 W/m3 by nanoparticles inside the tumor may cause that some parts of the normal tissue reach to a temperature equal to 42.52 °C. It seems that this amount of temperature may damage the healthy tissue of the brain. In fact, using the MFH method, the total volumetric heat source in tumor becomes equals to 150 × 103 W/m3 (the sum of tumor metabolism and the heat produced by nanoparticles). Therefore, in order to supply the heat required for tumor ablation and prevent the damage to the normal tissue, the total heat source of 125 × 103 W/m3 provide the desired conditions of hyperthermia. Hence, the heat produced by the nanoparticles for a tumor with the size of 5990mm3 is calculated and found to be equal to 105 W/m3.

Fig. 21
figure 21

The temperature distribution in the tumor and the surrounding healthy tissue (white matter) on the cross section passing the maximum temperature of WM for QMNP = 125 × 103 (W/m3) without modifying the tumor metabolism while blood perfusion is constant at wb = 0.001 (ml/(ml.s))

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4 Conclusion

As a matter of fact, it is difficult to present an analytical solution for complex shapes like human tissue. Therefore, different numerical methods (finite element, finite volume and finite difference) may be used for such complicated cases. In present study, the major aim was to present a realistic geometry of the main layers of the head (Skull, CSF, Gray Matter and White Matter) to achieve a thermal simulation of the tumor using the finite element scheme. The Pennes bio-heat transfer equation was utilized to conduct a thermal analysis of the layers of the head and the tumor. The obtained numerical results related to the temperature of the tumor and the layers of the head were presented in different plots and figures. Subsequently, the conditions of thermal therapy and controlling the temperature were examined. Finally, the elimination of the tumor were investigated via changes in two parameters, i.e., perfusion and metabolism rate in the tumor. A summary of the results is presented as below:

  1. 1.

    The change observed in the slope of the temperature plot for the different layers of the head is a result of their different thermo-physical properties. The horizontal line in the whole head temperature distribution plot, showing 37.3 °C, is related to the white matter and gray matter layers. This indicates the fact that the temperature equals 37.3 °C on average, in most parts of the gray matter and white matter. Furthermore, the maximum temperature on the plot belongs to the tumor due to the significant difference in the rate of metabolism in this area compared to the rest of the head.

  2. 2.

    Regulating brain temperature is very effective in brain tumors, so having temperature range in each layer of the head, can be effective for development of safe clinical treatments. Table 2 showed the temperature minima and maxima in each layer of head. In fact, by thermally simulating the layers of the head, the temperature of any point of the layers (WM-GM-CSF-SKULL) can be available.

  3. 3.

    If only one decreases the blood perfusion rate, the tumor temperature does not increase, significantly. Therefore, it does not provide the desired condition of hyperthermia.

  4. 4.

    Changing the values of the tumor metabolism affects its temperature significantly; consequently, in present study, the necessary temperature for tumor elimination was achieved in the whole of the tumor by increasing the metabolism to 5 times of its original value. Although, the shaped of tumor was asymmetrically, this amount of heat allowed for temperature control, all over the tumor, to prevent damage to the surrounding normal tissue.

  5. 5.

    If magnetic fluid hyperthermia method being used, the necessary heat is 4 times the tumor metabolism value, which is equivalent to 105 W/m3. This value is sufficient for hyperthermia conditions, and damage to the surrounding healthy tissue is prevented.

  6. 6.

    The asymmetric shape of the tumor is one of the main reasons why temperature control is essential to the process of hyperthermia. Therefore, it is crucial to pay attention to how fluid is injected to the tumor, in magnetic fluid hyperthermia (MFH).

  7. 7.

    The value of metabolism parameter in a cancerous tissue is a good criterion for estimating the amount of heat required for tumor destruction in MFH method. Changing this parameter in a thermal simulation allows us to estimate the heat required for the elimination of the tumor, before injecting nanoparticles to the cancerous tissue and creating the appropriate magnetic field.

In this research, many assumptions were made to simplify the problem solving process. These simplifications can be neglected in other studies in order to achieve the most optimal and accurate estimation of hyperthermia treatment. In future works, a bio-heat transfer equation more accurate than the Pennes bio-heat transfer equation can be used.