The proposed solution consists of implementing the physics of the MWA procedure, generating geometry, and meshing in a form suitable for FEM simulation. We show these methods in this section.
2.2. Geometry and Mesh Generation
Gmsh comprises two modules for geometry creation. The built-in module requires model creation from lower to higher dimensions (points < curves < surfaces < volumes). However, for complex structures, this type of modeling can quickly get out of control. The OpenCASCADE module supports work with solids and has the same capabilities as any commercially available CAD design package. Three-dimensional models can be directly composed of defined volumes, and the most important factor is complete Boolean support, which means that multiple solid models can be united and intersected, effectively creating new volumes. Geometry can be manipulated in any way: translated, rotated, copied, scaled, and so on. The OpenCASCADE module supports the import, export, and processing of the STEP and IGES file formats, which are commonly used 3D geometry formats, and the BREP file format, which is native to OpenCASCADE.
The Gmsh mesh module is extremely versatile and powerful. Gmsh can operate with most standard FEM shapes, such as lines, triangles, quadrangles, tetrahedrons, hexahedrons, prisms, and pyramids [
32,
33], as shown in
Figure 2. The standard Gmsh unstructured mesh algorithm comprises planar (2D) elements that are generally meshed with triangles (some structures can be recombined into quadrangles), whereas 3D elements are meshed with tetrahedrons or tetrahedrons and pyramids when quadrangles are present [
43].
Global control of the mesh size can be achieved by specifying the minimal and maximal cell sizes that can be specified at any point, curve, volume, etc. The mesh can be interpolated linearly at the geometric edge or with geometric progression. The mesh size can be adapted to the geometric curvature. The quality of each created element is determined, and appropriate adaptive and refinement algorithms can be applied. An equilateral triangle has a quality of 1, and a degenerated shape with a zero area has a quality of 0. Similarly, for a tetrahedron, the maximum quality is obtained when the tetrahedron is constructed from equilateral triangles, whereas zero quality occurs when the element is shattered and has zero volume.
The major advantage of unstructured meshing is that it can be applied to any shape. However, for good quality meshing, multiple refinement steps are often necessary, with a massive increase in the number of elements, computing time, and resource demands, and generally lower accuracy in FEM simulations in comparison with structured meshing. Hexahedrons with small aspect ratios are ideal mesh elements in FEM and should be used whenever possible. Some geometrically highly irregular structures, such as tumors, are difficult to represent using this algorithm, whereas probes with antennas can be represented using structured meshing. There are numerous advantages of structured meshing in FEM simulations. The number of elements in the same structure with structured meshing was several times smaller. With accurately defined structured meshing, mesh quality is superior, mesh generation is very quick because the best quality mesh is already defined, and there is no need for multiple refinement steps.
In MWA, the liver represents the computational domain where the process of MWA takes place, as seen in
Figure 3a. In MWA, we are interested in simulating the processes around the tumor, which is usually much smaller than the entire liver volume; hence, the computational domain does not have to consider the whole liver, and it can be a part of the liver around the tumor in the form of a regular geometrical shape. The most commonly used shape is a cylinder. However, for structured meshing, the hexahedron domain is a better choice because of the possibility of using mostly identical hexahedrons as finite elements across the entire domain. Two models were developed in this study. One had an unstructured tetrahedral mesh and a cylindrical computational domain with a radius of 30 mm and height of 90 mm, and the other had a mixed meshing and hexahedral computational domain with dimensions of 60 mm × 60 mm × 90 mm.
Figure 3b shows the geometry of the probe used for MWA [
10,
13]. The antenna was a coaxial slot antenna, where the inner conductor passed through the center of the geometry and was connected to the outer conductor at the end of the antenna. Radiation occurred through 10 slots in the outer conductor. A matching network exists in front of the radiating slots. The entire antenna is placed inside the catheter. There are four different material regions in the probe structure: the metal conductor, the dielectric in the antenna between conductors, air, and the dielectric material of the catheter (
Table 3). The basic geometry of the antenna was a long cylindrical volume. From electromagnetic analysis through the FEM, the value of the electric field is necessary for heat transfer estimation. From electromagnetic theory, we can assume that there is no field propagation inside the conductor volume, and that the faces of the conductor volume behave as field boundaries. Field propagation occurs only through dielectric materials. Based on this, there is no need for volume meshing of the conductor part; therefore, the basic geometry for the meshing of the antenna consists of multiple concentric hollow cylinders.
An arbitrary hollow cylinder is shown in
Figure 4. Using the OpenCASCADE module, the geometry can be created with a few commands, and the cylinder can be defined by specifying its position in space, radius, and height. All other geometric components of lower orders, such as surfaces, curves, and nodes, were automatically created. All elements of the same order have unique tags in Gmsh that are necessary for any geometrical manipulation.
The construction of the unstructured mesh was simple. It can be seen that the mesh from the curvature is 20, which means that the circular cross-section is represented by 20 segments, which is sufficient for an accurate circle geometry representation, and the maximal size of the cells is specified at 2 mm. The number of tetrahedrons in the volume per quality factor is listed in
Table 4. After the two mesh optimization steps, the mesh became suitable for the simulation. In the second iteration, all elements with a quality lower than 0.3 are eliminated, and the average mesh quality is now 0.76, which is very good. The mesh contained 3969 tetrahedrons and 2704 triangles.
Structured meshing cannot be applied directly to arbitrary geometries. The basic condition in Gmsh is that the volume must have five or six faces. Neighboring volumes must follow a defined grid pattern. Geometric volumes must be sliced or adjusted to conform to the structured meshing. In this case, the geometry has four faces. The simplest way to conform this geometry to structured meshing is to slice the geometry through the symmetry plane to form two half-ring volumes, as shown in
Figure 5a. Instead of a full cylinder, two symmetrical half-cylinders were created, each with six faces, as displayed in
Figure 5b.
The number of segments per curve has to be specified for each plane. The desired meshing parameters were the same as before, with 20 segments for circular representation and a maximal cell size of 2 mm. There are no automatic meshing algorithms for structured meshing; however, the process of specifying the number of segments per curve can be easily automated with simple scripting. There are various options for dividing the physical length of the residual curves by 2 mm; therefore, the number of segments can be obtained. The curves that define the half circumference should be divided into 10 segments. We obtained a meshing composed of 250 quadrangles and 100 identical hexahedrons. The base of the hexahedron had a very small deviation from the rectangle, and the structure was elongated in a single direction. The mesh quality of this element was close to perfect.
The difference in the number of finite elements for these two types of meshing is large. In unstructured meshing, there are 11 times more planar elements and 40 times more volume elements. All volume elements in the structured elements are identical, which is highly desirable, and the mesh quality is superior, resulting in a shorter simulation time, smaller resource usage, and higher accuracy. In this simple example, it is clear that there is a significant benefit to using structured meshing.
In Gmsh, it is possible to combine both structured and unstructured meshing with a transitional layer of pyramids between hexahedrons and tetrahedrons. In this model, the tumor represents an irregular unpredictable geometry, which is meshed with unstructured meshing, and the catheter volume layer is between the antenna and the tumor and can have a layer of pyramids.
Figure 5c shows an example of a hollow cylinder comprising an internal ring meshed with structured meshing (hexahedrons) and an outer ring meshed with unstructured meshing. The middle ring contains pyramids, which are transitional layers between two types of meshing.
The geometry of the probe can be simply constructed as a group of solid objects directly in Gmsh based on the antenna dimensions. The geometry is clearly defined and does not change during the analysis. Tumor geometry can be obtained from medical scans in the STL format, which represents an object in the form of a shell made from multiple triangles [
17,
32]. STL geometry represents the surface meshing of an object with triangles. The STL file does not represent solid geometry; therefore, it cannot be used with OpenCASCADE. The built-in module can work with the STL format and create a surface and volume geometry based on it, after which meshing can be performed. The geometry of STL files typically has a very high resolution and can consist of a very large number of triangles. A tumor is a live structure with blood circulation and other metabolic processes, and its geometry can undergo slow and fast changes over time. For the purpose of accurate simulation of MWA, extremely high resolution is not required as long as the tumor volume boundaries are accurately represented.
In Gmsh, it is possible to use STL remeshing, where the surface geometry can be reconstructed in a much smaller number of triangles with higher mesh quality. In
Figure 6a, an STL file of the live liver tumor with the dimensions of 21.9 mm × 22.7 mm × 25.4 mm is shown. The original model contained 3000 triangles. Remeshing can be performed, depending on the required resolution. In
Figure 6b, one version of remeshing is shown, with 1122 triangles. With all the details, this model provides a good representation of the tumor for adequate analysis. In
Figure 6c, the more aggressive remeshing model has 336 triangles, and some details are lost. The number of smaller triangles is decreased; hence, the resolution is lower, but the basic geometry is still present.
The built-in Gmsh module can create the geometry and mesh for the STL file, but it does not support Boolean operations. The geometry of the probe with the antenna is inserted into the tumor, and the tumor is inside the liver tissue, so new geometries must be created based on the interaction of the probe, tumor, and liver tissue. Therefore, it was necessary to use the OpenCASCADE module. For this, STL must be converted to a format that is suitable for the OpenCASCADE module, such as the BREP file format. In Gmsh, there is no direct way for STL conversion, but it can be achieved through external scripting, for example, with the Python API. The BREP file of the tumor can then be imported and used to create a solid volume that can be used in the OpenCASCADE module as any other solid geometry.
A very important operation for simulation preparation is grouping finite element volumes and surfaces based on whether they belong to a particular material or have the same boundary conditions. Each material listed in
Table 3 belonged to a unique physical zone. In addition, there are tumor and liver volumes; hence, there are five physical volume zones. There is no electric field in the conductor; hence, the volume meshing of the conductors is not required.
There are three specific surface boundary regions: absorbing, port, and Perfect Electric Conductor (PEC) boundaries. The largest is the absorbing boundary, which consists of two parts: the surface at the exterior of the computational domain (liver) (
Figure 3a) and a small ring surface at the exterior of the catheter, which represents the catheter’s absorbing boundary. The absorbing boundary has a very important purpose in the FEM simulation. The electric field intensity decreases with distance from the source; for infinite or very large domain spaces (relative to wavelength), the entire electric field is negligible, and there is no reflection from domain surface boundaries. The purpose of the absorbing boundary is to absorb the electric field, which essentially enables the small computational domain to behave in a manner similar to that of a much larger domain (the entire liver).
The port boundary was defined as the planar ring surface on the exterior of the antenna dielectric (
Figure 3a). This is where the electric-field source is defined. The conductor surfaces played a major role in the simulation. This essentially represents a reflective boundary that controls the direction of propagation of the electric field. The PEC boundary is a region consisting of surface zones at the interface between the dielectric and the conductors. Three surface zones define the PEC interface: the antenna dielectric conductor, air conductor, and catheter conductor.
The frequency of interest here is 2.45 GHz, and the approximate value of the dielectric constant for the liver tissue is 44, which indicates that the finite element size in the given domain should be . The dielectric constant is a critical parameter for electromagnetic simulations. A higher value of the dielectric constant requires a higher meshing density.
Figure 7a shows a cylindrical domain cross-section with the probe (dark), tumor (yellow), and liver (blue). The total volume of the domain was 254.47 cm
3, and the total number of tetrahedrons was 640,604. In
Figure 7b, we show a view of the location where the probe with the antenna exits the domain. The conductors are not meshed. The inside parts of the geometry can be seen: air (green), which is further inside, and the catheter (purple), which is at the end of the antenna. The port boundary surface (orange) and the absorbing boundary of the catheter (red) are shown. The blue surface around the catheter represents the absorbing boundary of the domain on the exterior of the liver.
As shown in
Figure 8, the model has a hexahedron domain meshed with a combination of structured and unstructured meshing. The total volume of this domain was 324 cm
3, which was 27% higher than that of the cylindrical domain. The total number of hexahedrons was 49,552 (green), the total number of tetrahedrons was 99,063 (red), and the total number of pyramids was 2709 (blue). Even with larger volumes, the total number of volume elements is much smaller; the number of tetrahedrons is 6.46 times smaller, and the total number of volume elements is 4.38 times smaller. The hexahedrons used in the domain are mostly identical cubes, which are perfect elements for FEM analysis. For higher mesh control in the geometry, we can create dummy objects whose purpose is to facilitate a specific mesh pattern. Around the tumor, a dummy hexahedron volume is created; the volume was meshed with tetrahedrons (red) in the same manner as the tumor, which enabled a smooth transition to structured meshing (
Figure 8a). A similar dummy object was placed around the catheter, as shown in
Figure 8b,c. The catheter was meshed with a small layer of pyramids intertwined with surrounding tetrahedrons. The boundary regions are the same as those shown in
Figure 7b.