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ArticlesAdaptive Tetrahedral Mesh Generation for Non-uniform Soft Tissue Simulation
• Jion Kim1, Koojoo Kwon2, and Byeong-Seok Shin1,*

Human-centric Computing and Information Sciences volume 11, Article number: 29 (2021)
https://doi.org/10.22967/HCIS.2021.11.029

Abstract

An accurate representation of the object’s movement is important in carrying out a realistic simulation. In the case of human soft tissues, depicting their deformation realistically is not easy because the local changes on the surface of human organs have a substantial influence on their internal structure. Moreover, manipulating an organ is even more difficult when it is composed of multiple materials that are different from one another. A method of representing an organ’s interior with uniform-sized tetrahedrons is commonly used. However, this method is problematic because the processing time and the memory space need to be expansive. Therefore, we propose a method for simulating human organs that divides the interior into non-uniform tetrahedrons with different sizes depending on the density. When an organ contains tissues with multiple densities, the tetrahedron may deform differently when an external force is applied. Through this experiment, we show that it is possible to represent realistic movements inside an organ model in which low-density soft tissues and rigid tumors are mixed. We designate a few tetrahedrons to rigid tumors with little deformation and more tetrahedrons to soft tissues with a wide range of deformation. The results underscore a more accurate soft tissue simulation for the same number of tetrahedrons.

Keywords

Tetrahedron Subdivision, Level of Detail, Non-rigid Body Model Visualization, Mesh Simplification

Instruction

A surgical simulation is a tool used to plan and evaluate scenarios before carrying out a surgery [1]. It helps ensure patient safety and enables doctors to adapt to the medical environment. Several techniques that are crucial for the surgical environment are currently under development [2]. In particular, research on the surgical simulation of soft tissues in the human body is actively underway [3]. Since low-density objects are softer compared to rigid ones, multiple changes occur internally when the body is subjected to external force, no matter how minimal. In addition, the soft tissues that make up human organs have various density value distributions [4]. Thus, even if the same force is applied, the direction and intensity of deformation may differ greatly depending on the soft tissue. Therefore, an accurate density-based representation is necessary to show these deformations during surgical simulations.
Soft tissues can be modeled with formulas based on experimental data. However, it is difficult to carry out simulations in real time using this technique because the applied formula is very complex [6]. Surface models are mainly used to represent human organs. A typical method for creating a surface model is to reconstruct the geometry from computed tomography (CT) images using the Marching cubes algorithm [6, 7]. However, it cannot describe the internal soft tissue deformation, as only the surface of the model is shown. Therefore, to model the interior of an organ, it must be filled with geometric information, such as tetrahedrons [8]. When the interior of a soft tissue is composed of tetrahedrons, the deformation of the organ model can be expressed by changing the vertex position of the tetrahedrons during the simulation.
Methods for dividing the interior of an object into tetrahedrons of equal size have been studied [9]. However, when an object’s structure is considerably complex and it is subdivided into tetrahedrons of the same size, its internal area must be divided into a large number of tiny tetrahedrons. This requires a tremendous volume of memory to store those tetrahedrons; manipulating them also takes a very long time. When a part with high density (such as a bone structure or a tumor) and a part with low density (soft tissue) are situated together in an object [10], the low-density material deforms more even if the same external force is applied. Therefore, when the object is non-uniformly subdivided according to the density of the material, the number of unnecessary tetrahedrons can be reduced and the external and internal movements can be efficiently modeled according to the density distribution of the object. Notably, soft tissue manifests many fine movements due to interactions with the surgical tool during the surgical process. Yet, existing simulation methods have only focused on visualizing the process of cutting a part of soft tissue using surgical devices [11]. To attain realistic simulation results, the movements inside the soft tissue must also be considered.
To divide an object model into tetrahedrons of different sizes, we first divide the interior of the geometric object into tetrahedrons of equal size as an initial step. Then, we recursively subdivide each tetrahedron so that it is inversely proportional to the calculated density value. After subdivision, there are a few tetrahedrons in the high-density area and many tetrahedrons in the low-density area. As a result, the number of divisions can be increased only at the necessary area which has lower density values. Thus, we can represent a non-uniform object more accurately while using the same number of tetrahedrons. This strategy provides a method for pre-calculating the distribution function of the density value for all vertices to control the total number of subdivisions. When external force is applied to the surface of a soft tissue, the vertices of the tetrahedron move toward the direction of the force, and the shape of the tetrahedrons in some regions changes. The position of the tetrahedral vertex with higher density value changes slightly, while the position of the vertex with lower density value changes substantially.
The contribution of this paper is a non-uniform subdivision method of tetrahedrons based on the density values of an object’s interior. Previous methods divide the object model uniformly with the same resolution regardless of the density, so unnecessary tetrahedrons may be generated inside the model. This entails large memory usage and lengthy processing time. By contrast, the proposed method in this paper can subdivide soft tissues with adequate resolution in accordance to the density. This makes it possible to perform realistic deformation while using the same number of tetrahedrons.
Section 2 introduces related work on density value distribution, soft tissue simulation, and tetrahedron subdivision. Section 3 describes the process of recursively dividing an object into tetrahedrons of different sizes according to the density of the soft tissue. It also explains the process used in deforming the tetrahedrons. Section 4 shows the experimental results of the modeling and simulation of tetrahedrons. We conclude our work in Section 5.

Related Work

The density value distribution of soft tissue has been used in various medical fields, such as disease detection, classification, and segmentation. Chawla et al. [12] proposed a method for detecting stroke symptoms using CT images of the brain. This method finds the abnormality by comparing the density distribution histograms of the left and right brains. Meanwhile, Nelson et al. [13] segmented and classified skin, fat, and glandular tissues in the breast using the difference in the density distribution of breast tissues. This method makes use of a threshold that is calculated based on the histogram peak maximum value.
The finite element method (FEM) and mass-spring model (MSM) are often used to visualize the deformation of an object when simulating the soft tissues of the human body. FEM is developed to solve complex differential equations defined based on the constraints in the process of deforming a specific region [14]. The complex equations are computed by simplifying them into multiple polynomial equations. MSM divides the surface of the mesh into m × n virtual points with assigned mass values [15]. Virtual points can be connected in the form of springs to reflect forces, such as the tension and elastic force acting between the points. FEM allows an accurate modeling of differential equation-based deformations, but it is computationally time consuming. MSM has the advantages of simpler structure, easier implementation, and faster calculation time than FEM [16].
A tetrahedral model is a data structure that is mainly used to model the interior of a three-dimensional object. The quality of the tetrahedron is an indicator of how the tetrahedron closely approximates the shape of the actual model [17]. Several factors determine the quality of a tetrahedral model. The dihedral angle is typically used. Many studies have improved the quality of tetrahedra by adjusting the range of the dihedral angle [18]. For example, Labelle and Shewchuk [9] applied a tetrahedral model to a simulation that is generated at a faster division speed. However, these methods divide the tetrahedron uniformly and, in this case, many unnecessary divisions might occur, which would cause the performance to decline. To solve this problem, Park and Choi [19] categorized the model into a surface area and an internal area. They proposed a method for creating a model by assigning different resolutions to the two abovementioned regions. However, this method does not consider the material property inside the model. While handling an object that is made up of soft tissues, the deformable range changes depending on the density. Therefore, the deformation cannot be expressed realistically by only categorizing it into the surface and internal regions. A density-based non-uniform subdivision is required.
Various research on simulations using the tetrahedral model have been undertaken. The tetrahedron model is used to visualize the process by which surgical instruments cut a portion of soft tissue [11]. In some cases, a tetrahedral model is used to accurately determine the process of changes in the internal structure due to deformation [20]. However, these previous studies used a uniform tetrahedron, assuming that the interior of the object is composed of the same density. When a tumor is located in a soft tissue, the density value is different. That is, the range of deformation is different even if the same external force is applied.

Soft Tissue Deformation Using Density-based Level-of-Detail Models

The entire process for our density-based level-of-detail (LOD) method is shown in Fig. 1. The geometric model is reconstructed from CT data via surface reconstruction. The tetrahedral model is generated from the geometric model through the tetrahedral subdivision process. In previous tetrahedral subdivision methods, the interior of the geometric model is uniformly divided several times without considering the physical characteristics. Therefore, all of the divided tetrahedrons are of the same size. The method we propose is undertaken as follows. First, the inside of the geometric model is divided into uniform-sized tetrahedrons at the uniform subdivision stage. At this stage, the uniform-sized tetrahedral model is created. Afterward, those tetrahedrons are recursively subdivided based on the density value distribution by region in the non-uniform subdivision stage. Density value distribution is calculated based on the CT data. As a result, the non-uniform-sized tetrahedral model is created, with each tetrahedron having a different size. After the tetrahedral subdivision, the generated tetrahedral model is deformed in the simulation stage.

Fig. 1. The entire process of tetrahedral non-uniform subdivision and simulation.
The light gray boxes mean the methodology, and the dark gray boxes are the data manipulated.

Density-based Tetrahedron Subdivision
Geometric models are generated from consecutive CT images using surface reconstruction algorithms, such as marching cubes [6]. The density value is stored in one voxel v(s, t, w) of the volume data. Therefore, the density of a vertex p(x, y, z) of the mesh reconstructed from the volume data can be accurately calculated through the trilinear interpolation of eight voxel values adjacent to p. Since the human body is composed of elastic soft tissues, the density indicates the degree of hardness of a particular tissue or organ. A rigid object hardly deforms even when a strong external force is applied; only its position changes. Meanwhile, soft tissue is severely deformed when an external force is applied. Thus, non-uniform objects need to be subdivided into tetrahedrons of different sizes according to their density to achieve a realistic simulation.
The proposed tetrahedral subdivision process is as follows. First, the inside of the geometric model is divided into nTinit tetrahedrons of the same size by applying Labelle’s method [9]. The density Ti of each tetrahedron is calculated as the average of the density Vi,j of the four vertices that constitute it.

$T_i=\frac{∑_{j=0}^4 V_{i,j}}{4}$(1)

Next, the number of recursive division l applied to each evenly divided tetrahedron is determined to be inversely proportional to the density value, where N is the user-defined maximum number of subdivisions (0 < l < N). When the density value is large (rigid area), those tetrahedrons are divided into fewer ones. When the density value is small (soft area), they are divided into more tetrahedrons. We define an array C that stores the number of tetrahedrons with a density of k. Since Ti is a floating-point number, we quantize it to an integer value with 0 ≤ k ≤ $d_max$.

$C(k)=C(k)+1 if ⌊T_i ⌋=k (0 ≤ k ≤ d_{max})$(2)

Using C($k$), the array CD($k$) that stores the cumulative number of tetrahedra up to a specific density k can be defined as shown in Equation (3). CD($k$) is the cumulative distribution function of C($k$) with a maximum of $nT_{init}$.

$CD(k)=∑_{j=0}^k C(j) (0 ≤k≤d_{max})$(3)

To measure the number of recursive subdivisions to be inversely proportional to the density, CD($k$) is divided into N sections on the y-axis with the same interval as the cumulative distribution as shown in Fig. 2.

$[0,⌊\frac{nT_{init}}{N}⌋],[⌊\frac{nT_{init}}{N}⌋+1,2×⌊\frac{nT_{init}}{N}⌋],…,[(N-1)×⌊\frac{nT_{init}}{N}⌋+1,N×⌊\frac{nT_{init}}{N}⌋]$(4)

Let the density value corresponding to the boundary of each section be the density threshold dth(n). This can be calculated from the inverse of the cumulative distribution function CD(k). CDs are stored in a sorted array, so they can easily be found through a binary search.

$d_{th}(n) = CD^{-1} (n×⌊\frac{nT_{init}}{N}⌋) (0≤n≤N)$(5)

Consequently, the l of a tetrahedron is calculated through a comparison between the density thresholds. The density of the tetrahedron is shown in Equation (6).

$l_i=N-m if d_th (m)≤T_i(6) Each tetrahedron is additionally subdivided by l times, in accordance to the method proposed by Schaefer et al. [21] (Fig. 3). Fig. 2. The process of calculating the density threshold dth(n) in CD. Fig. 3. Pseudocode for the density-based tetrahedron subdivision. Simulation of the Proposed Tetrahedral Model When external force is applied to the soft tissue, the tetrahedron’s vertices move and its shape is deformed. To achieve a realistic deformation, the adjacent vertices in a particular range must also move along with the vertices that first moved. The target vertex is a point where force is applied. Using the mass-spring model, it is assumed that there is a massless spring between the target vertex and its adjacent vertices. When the target vertex moves, its adjacent vertices that are connected to the spring move together. The displacement of the adjacent vertices depends on the density and the distance between the target vertices. The process of moving a specific vertex from va to vb can be expressed using Equation (7). The movement of the vertices is represented as the product of the direction vector n of the target vertex and the displacement h. Since the vertices are connected, the direction vector n depends on the relationship between the vertices. For simplification, we assume that the direction vector n applied to the adjacent vertices is the same.$v_b=v_a+hn$(7) The greater the distance between the target vertex and its adjacent vertices, the weaker the transmitted force, so the displacement becomes smaller. Let d =$|v_{ta} – v_a|$be the distance between the target vertex and its adjacencies. The movement attenuation factor$f_{att}$(d) can be calculated as shown in Equation (8), where R is the maximum range of deformations. Vertices within a radius of R relative to$v_{ta}$can only be moved. The closer the adjacent vertices are to the target vertex, the more it will move.$f_{att}(d)=(\frac{R-d}{R}) (0≤d<R)$(8) The density$ρ$is the mass per unit volume. If all the vertices occupy the same volume, the density can be considered as the mass. When the same force is applied, the mass and the acceleration of the vertex are inversely proportional based on Newton’s second law. However, since the density and mass are the same, the density and acceleration can also be inversely proportional.$F=ma=ρa$(9) The displacement$h$is defined in Equation (10) when it is assumed that the vertices move by equivalent velocity. When the initial velocity$v_0$is 0, the movement distance of vertex$h$is proportional to acceleration$a$.$ h=v_0 t+\frac{1}{2} at^2$(10) As a result, when the same force is applied for a constant time period, the moving distance is inversely proportional to the density, as shown in Equation (11). The external force F applied to the target vertex can be defined as the displacement ht =$|v_{tb} – v_{ta}|$of the target vertex moved by the user.$h=\frac{Ft^2}{2ρ}=\frac{h_t t^2}{2ρ}$(11) Applying these two properties, the h of a particular vertex can be computed as follows (Fig. 4):$h=f_{att} (d)\frac{h_t t^2}{2ρ}\$(12)

Fig. 4. Pseudocode for computing vertex movement.

Experimental Results

The system used in the experiment is an Intel Core i7-8700 CPU 3.20 GHz with 64 GB RAM. The graphics accelerator is NVIDIA GeForce RTX 2080 Ti and the operating system is Microsoft Windows 10. For the soft body simulation, we created a brain tetrahedral model and a breast tetrahedral model. The brain tetrahedral model was created using the CT image of Anderson Winkler’s Brain for Blender [22] and the entire brain surface model. The total number of vertices in the surface model is 289,561. Here, simplification using multi-edge mesh collapse was performed [23], and the number of vertices was reduced to 72,004. The resolution of CT volume data is 224×320×190. Meanwhile, the breast tetrahedral model was created using the CT image and the breast surface model. The total number of vertices in the surface model is 1,577. The resolution of the CT volume data is 512×512×124.
We compared the conventional uniformly divided tetrahedral model with our non-uniformly divided tetrahedral model. Figs. 5 and 6 show the internal structures of the brain and breast models while the two approaches are applied. The numbers of tetrahedrons in the uniformly divided model and in the non-uniformly divided model are 581,944 and 582,882 for the brain tetrahedral model and 696,076 and 697,218 for the breast tetrahedral model, respectively. The pixel value indicating the density of the tumor inserted in the CT image was set to a maximum value of 255 to reveal the change in the internal structure of the tetrahedral model clearly. For simplification, the location of the tumor is the center of the model. The shape is set by the sphere, and the radius is 30 pixels for the brain model and 60 pixels for the breast model, respectively. We can see that the previous method divides the entire model into tetrahedrons of the same size on both the surface and the interior. However, the proposed method divides it into tetrahedrons of different sizes according to the density of the materials. The total number of tetrahedrons is nearly the same. In general, malignant tumors have a very high density compared with normal brain tissue. As shown in Figs. 5 and 6, the cancerous region splits into a few large tetrahedrons, and its surrounding soft tissue is divided into many tiny tetrahedrons with inverse proportion to the density of each area.
Table 1 shows a comparison of the amount of memory required and the processing time of the two methods while using nearly the same number of tetrahedrons. Experiments were performed twice with different numbers of vertices for each brain and breast model. The non-uniform subdivision method recursively subdivides the uniformly divided tetrahedral model. Therefore, this method generates a much larger number of vertices than the uniform subdivision method. Given the foregoing, we nearly match the vertices of the tetrahedral model for the two subdivision methods by using different initial division sizes. The tetrahedron model generation time of the proposed method is slightly shorter while using less memory space than the existing uniform subdivision method. This implies that the proposed method performs surgical simulation more realistically since it generates more tetrahedrons in areas where there are many deformations.
When an external force is applied to a specific location on the brain surface, the vertices of the inner tetrahedral model move in a specific direction. Most parts of the human body comprise elastic material with high resilience, so when the force is removed, the tetrahedral vertices return to their original positions. By simulating this process, more realistic soft tissue deformation can be expressed.

Fig. 5. Comparison of uniformly and non-uniformly divided brain models.
The left column is the CT image of a malignant tumor. We artificially increased the density of some parts of the left brain (marked as a white circle) to evaluate the results generated by the two methods. The middle columns show the cross-section of the tetrahedral models, where the upper image is for the uniformly divided model and the lower image is for the non-uniformly divided model. The right column is a magnification of the rectangular area of the images.

Fig. 6. Comparison of uniformly and non-uniformly divided breast models.

Table 1. Performance comparison of uniform and non-uniform subdivision method for the brain and breast models
Case 1 Case 2
Uniform subdivision Non-uniform subdivision Uniform subdivision Non-uniform subdivision
Brain Number of tetrahedrons 1,544,691 1,553,356 792,563 795,212
Ratio of vertices to tetrahedrons 21.453 21.573 11.007 11.044
Generation time (s) 5.346 2.717 2.957 1.445
memory usage (MB) 503.5 410.5 292.8 262.4
Breast Number of tetrahedrons 871,392 872,169 463,327 460,117
Ratio of vertices to tetrahedrons 552.563 553.056 293.803 291.767
Generation time (s) 2.056 1.425 1.206 0.843
Memory usage (MB) 373.1 251 181.2 133
Fig. 7. Internal shape change of the uniform subdivision model for brain.
The first row shows the surface and the inside of the original model, the second row presents the surface and the inside of the slightly deformed model, and the bottom row illustrates the images when the model is considerably deformed. The first column shows an image of the surface model, the second column demonstrates a wireframe of tetrahedron mesh, and the third column is an overlapped image.

Figs. 7–10 show the shape of the brain and breast after pushing the uniform and non-uniform models in a specific direction. Equation (8) is applied to the uniform subdivision model because density is not considered. Meanwhile, Equation (12) is applied to calculate the displacement in the non-uniform subdivision model. The non-uniform subdivision model is rendered in different colors according to the number of divisions, that is, the density of the tetrahedron. Depending on the number of divisions, red color (no subdivision), green color (one subdivision), and blue color (two subdivisions) are used. A red-colored area indicates the presence of an object with a very high density, like a tumor. Although external force is applied, almost no deformation occurs; only its position changes. On the other hand, we can see that the displacement of the individual tetrahedron is large and the shape of the tetrahedron changes substantially as external force is applied to the areas marked in green or blue.
Fig. 8. Internal shape change of the non-uniform subdivision model for brain.
The model representation of each row and column is same as Fig. 7. The red color indicates a high-density area comprising a small number of large tetrahedrons (cancerous region). The green and blue regions are the soft tissues surrounding the malignant tumor where shape changes occur.

Fig. 9. Internal shape change of the uniform subdivision model for breast.

Fig. 10. Internal shape change of the non-uniform subdivision model for breast.

Conclusion

The shape change that occurs in the internal structure of a soft tissue with uneven density varies depending on its position when the external force is applied. In this study, we proposed a technique for dividing the tetrahedron of a non-uniform soft tissue and a concomitant simulation method. Unlike existing algorithms that uniformly divide the tetrahedron using only the geometric information of the object, with our method, the tetrahedron is recursively and non-uniformly divided using density. With this technique, the hard area is divided less and the soft area is divided more, so when external force is applied, the hard part is deformed at a lesser degree and the soft part is deformed more substantially, thereby achieving a more realistic deformation.

Acknowledgements

Not applicable.

Author’s Contributions

Conceptualization, KK, BS. Funding acquisition, BS. Investigation and methodology, JK, KK, BS. Project administration, BS. Resources, KK. Supervision, KK, BS. Writing of the original draft, JK. Writing of the review and editing, KK, BS. Software, JK. Validation, JK. Formal Analysis, JK, KK, BS. Data Curation, JK, KK. Visualization, JK.

Funding

This work was supported by a grant from the National Research Foundation of Korea (NRF) funded by the Korea government (No. NRF-2019R1A2C1090713).

Competing Interests

The authors declare that they have no competing interests.

Author Biography

Name : Ji-on Kim
Affiliation : Department of Computer Engineering, Inha University, Incheon, Korea
Biography : He received his master’s degree in computer engineering from Inha University, Incheon, Korea. He is currently Ph.D. student under the supervision of Prof. Byeong-seok Shin. His current research interests include computer graphics, simulation, VR, AR, global illumination.

Name : Koojoo Kwon
Affiliation : Department of Smart Information Technology, Baewha Women’s University, Seoul, Korea
Biography : He received Ph.D. degrees from Inha University, Incheon, Korea. He is now an assistant professor at the Department of Smart Information Technology, Baewha Women’s University, Seoul, Korea. His current research interests include real-time rendering, medical imaging, VR and AR.

Name : Byeong-seok Shin
Affiliation : Department of Computer Engineering, Inha University, Incheon, Korea
Biography : He received Ph.D. degrees from Seoul University, Seoul, Korea. He is now a professor at the Department of Computer Engineering, Inha University, Incheon, Korea. His current research interests include real-time rendering, medical imaging, HCI, fluid simulation and next generation computing.

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Jion Kim1, Koojoo Kwon2, and Byeong-Seok Shin1,*, Adaptive Tetrahedral Mesh Generation for Non-uniform Soft Tissue Simulation, Article number: 11:29 (2021) Cite this article 3 Accesses

• Recived25 May 2021
• Accepted2 July 2021
• Published15 July 2021

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