# Skull Thickness Calculation Using Thermal Analysis and Finite Elements
## 2 Materials and Methods
### Thermal Analysis of 3D Solid Models  
Heat transfer is thermal energy that is in transition due to differences in spatial temperature. The most efficient method of heat transfer is conduction. This heat transfer method occurs whenever there is any temperature change within an object. In this case, energy is transferred from a high-temperature region to a low-temperature region. The current equation of thermal conduction in continuous environments can be obtained on the basis of the principle of conservation of thermal energy. The application of the heat transfer relationship for systems with isotropic materials is as follows [34]:  
\[\rho C\frac{\partial T}{\partial t}=div\bigg{(}\overset{\rightarrow}{k} gradT\bigg{)}+Q \tag{1}\]  
where \(\rho\) represents density, \(C\) is the specific heat, \(T\) is the environmental temperature, \(t\) is the time, \(\overset{\rightarrow}{k}\) is the thermal conduction coefficient, and \(Q\) is the internal heat produced per volume. The amount of heat transferred by convection is determined by [PERSON]'s law of cooling, as follows:  
\[q_{c}=-h_{c}(T_{a}-T_{sb}) \tag{2}\]  
where \(h_{c}\) is the convection coefficient, \(T_{sb}\) is the temperature of the surface boundaries, and \(T_{a}\) is the environmental temperature. The convection coefficient is a function of properties such as fluid velocity \((V_{f})\) and surface roughness. Simplified formulas are available for calculating this coefficient [35]. \[h_{c}=3.8V_{f}+5.7 \tag{3}\]  
In this section, thermal analysis of the modeled solid object is examined. Thus, we aim to carry out an experimental application in order to verify the application of the model for skull images. ANSYS Workbench-2020 software was used for analysis. The ANSYS Workbench interface consists of a toolbox, project diagram, toolbar, and menu bar. Two main types of thermal analysis are commonly used. These are steady-state thermal analysis and transient thermal analysis. Steady-state analysis is performed at a constant temperature, while transient thermal analysis is performed at varying temperatures. Steady-state thermal analysis is important for determining temperatures, heat fluxes, etc. In the steady-state method, a constant known heat flow is assumed to stream through an object [36]. Steady-state thermal analysis is used in this study. This type of analysis calculates the effect of a constant thermal load on a solid 3D object. To this end, thermal analysis is used to determine the temperature, thermal slope, heat flow rates, and heat flow in the object [37]. One of the important issues in creating 3D thermal models is to create appropriate boundary conditions. The steps for realizing the steady-state thermal analysis of a 3D structure are as follows: (1) the designed 3D model is taken; (2) the model is meshed; (3) the outside  
Figure 2: Images of meshed solid models with different numbers of nodes and elements: (**a**) mesh model-1, (**b**) mesh model-2, and (**c**) mesh model-3. temperature is given a value other than 0 \({}^{\circ}\)C; and (4) 0 \({}^{\circ}\)C is applied to the entire inner surface as a constant heat source. Since the modeled solid object contains experimental applications for the bone structure of the human skull, it was designed with different thicknesses. Temperature differences were analyzed by applying different amounts of heat to the interior and exterior surfaces. The temperature value of the inner surface was accepted as 0 \({}^{\circ}\)C. This process, which is applied to solid models, can also be applied to living human skulls (see Section 4). However, no direct thermal application is made to the skull of the living person. First, CT images taken from the person are converted into a model for thermal analysis. The internal and external surface temperature values of the skull that is turned into the model are unimportant, because at this stage, the focus is on the internal-external surface temperature differences. In this case, choosing 0 \({}^{\circ}\)C or a different value for the inner surface makes no difference.