Figure 1: CFD in Biomedical Applications: Structured multi-block grid for a brain aneurysm
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Numerics in Bio-Medical Field
Modern medicine, or medicine as we know it, started to emerge after the Industrial Revolution in the 18th century. At this time, there was rapid growth in economic activity in Western Europe and the Americas. During the 19th century, economic and industrial growth continued to develop, and people made many scientific discoveries and inventions.
Scientists made rapid progress in identifying and preventing illnesses and in understanding how bacteria and viruses work. However, they still had a long way to go regarding the treatment and cures for many diseases.
Recent advancements in biomedical computing are changing the face of biology and medicine in research as well as clinical practice. The benefits of CFD in biomedical applications are immense and multifaceted. They provide crucial help to acquire a better understanding of human physiology, enable exciting new biomedical scientific discoveries and help in coming up with new clinical treatments.
For simulation purposes, various types of numerical techniques are used in the biomedical field. Approaches like the Finite element method (FEM) are used for structural analysis. For example, stress analysis of human skull during head impact or understanding stress distribution in hip implants, etc. Techniques like Computational fluid dynamics (CFD) help to gain insight into fluid motion in and around the body. Analysis of blood flow in arteries, simulation of airflow in the respiratory passages, etc, are some examples of CFD in biomedical applications.
Thermal analysis is another interesting technique that helps in understanding the heat transfer mechanism between various parts of the body and the external environment. Thermal analysis of cooling of a human heart during cardiac surgery is one such application.
Another important simulation approach is the multi-body dynamics, which helps in understanding the biomechanics of human body movements. This is particularly very handy in understanding the body movements of a physically disabled or injured person.
The latest tool in usage in biomedical computing is the optimization technique. Optimization algorithms are now used for improving the designs of medical devices implanted in the patient’s bodies. Optimization of patient-specific stents, hip-implants are some common examples.
Steps in Biomedical Computing
Unlike the steps adopted in other engineering fields such as aerospace, marine, automotive, the numerical steps in the bio-medical field is slightly different. Though the steps of meshing, solver, and post-processing stand, the geometry construction is different.
In the biomedical field, the geometry is not constructed in CAD but is reconstructed from scanned images. This step of extracting geometric surfaces from medical images is called ‘Segmentation’. The outcome of this step is a series of differently oriented circular sections lined up along the centerlines. Figure 2 shows the segmentation for a carotid artery. Organ or body-specific images are obtained from any one of the imaging techniques like 3D Ultrasound, biplane angiography, CT, MRI, and rotational angiography. For a coronary artery, for example, biplane angiography is used. The images of the patient-specific vascular domain are taken and segmented into groups of circular sections.
Subsequently, the circular sections are used and the boundary surface is created using Bezier splines.
Meshing Problems for Medical Images
Lately, the automation of segmenting the medical images, constructing the boundary surface using splines has been made possible. However, mesh generation has been a bottleneck in efficiently conducting bio-medical simulations.
One of the challenges is the qualitative representation of the meshed surfaces confirming the domain boundaries. For instance, when meshing internal body parts like the heart, liver, lungs, bones, etc, accurate capturing of these surfaces is very essential for a reliable simulation. Further, if any medical device like a pacemaker is been simulated, proper modeling of the equipment is also necessary.
Secondly, the geometric shapes of the mesh elements, especially the parameters like angle, spacing, orientation between the neighbouring elements are critical. Though non-intuitive, cell shapes have a large say in the outcome of the simulation. For example, the dihedral angles of each tetrahedron have a direct effect on the numerical conditioning of matrices used in the FEM solver. As the angles tend towards 0 or 180 degrees, solutions degenerate due to lack of computational precision. This results in additional computational effort. A meshing process that can meet both of these quality concerns is a major challenge as improvement in one of them often comes at the expense of the other.
Hex Meshing – The Most Preferred Meshing Approach
Bio-medical geometries are quite complex often having intricate connectivity, multiple bifurcations, loops and vessels, and other organic parts with high tortuosity. As a consequence, more often than not unstructured meshing is adopted for biomedical simulations, while the structured approach is limited to simpler geometries. Though quick and easy to generate, unstructured approaches fail to generate high-quality results. Experts in the field, prefer to use structured meshes as they feel, the effort to build the multi-blocks is worth the high-quality end results they provide. Although there is no fully automated Hexa-meshing code, structured meshes are still preferred over tet meshes for the following reasons:
1. Tet meshes typically need 4-10 times more cells than a hex mesh to achieve the same level of accuracy. So, unstructured meshes demand higher computational time and memory. Also, it has been observed that mesh refinement does not necessarily reduce mesh-dependent error in calculating parameters like wall shear stress. On the other hand, structured meshes produce stable results with fewer elements to reach mesh independence.
2. In certain FEM applications like high deformation structural analysis with linear elements, tet cells are found to be mathematically ‘stiffer’ due to reduced degrees of freedom associated with such geometrical shapes. This problem is known as ‘tet-locking’. Tets and wedges are good to generate acceptable displacement data but fail to predict stress accurately. This is because, these elements are known to be ‘stiff shapes’ – i.e., shapes with poor bending and near incompressibility behavior.
3. Structured meshes have the distinct advantage of placing cells aligned in the predominant flow direction. In structured meshes, cell edges i.e grid lines are oriented to the flow direction, while tet meshes cannot generate such flow-oriented cells. This favorable alignment between mesh edges and flow in structured meshes reduces the computational errors due to numerical diffusion. On the other hand, in unstructured meshes, the randomly oriented cells introduce numerical diffusion in the flow solution, especially in predominantly unidirectional flow systems.
4. For FEM analysis of arterial structures, hex meshes bring in some unique advantages. With the built-in ability to generate stretched anisotropic cells without losing the quality, structured meshes help in efficient layering of the arterial wall, without the need for increasing the circumferential and longitudinal resolution. Also, structured meshes bring in the possibility of using the orientation of the elements as a surrogate of the local material orientations to specify material anisotropy.
5. Finally, the arrangement of near-orthogonal hexahedral cell layers adjacent to the geometric surfaces makes them ideal for analysis involving fluid-structure interaction and for capturing the gradients near the wall surface.
The above points highlighting the superiority of the structured meshes are not one-off study conclusions. Following two independent research studies involving the grid effect on flow solution in bio-medical applications, brings out the positives of using structured meshes in a more lucid way.
Coronary Artery Flow- A Case Study
In a research study done at the Institute of Biomedical Technology, Belgium [2], the influence of mesh type in predicting flow parameters for arterial blood flow has been investigated and the results reveal interesting insight.
A series of structured and unstructured hybrid meshes comprising tets and prisms are generated for an arterial geometry to understand the wall shear stress (WSS) distribution using CFD. For similar flow solver inputs and boundary conditions, the structured mesh needed a lower cell count to obtain a grid-insensitive wall shear stress value within an acceptable tolerance of 5 percent when compared to the unstructured mesh. This translates to performing a more accurate, faster, and less memory-demanding computation with structured meshes.
The mean WSS computed using the Hex grid family shows a low mesh dependency, implying that very coarse meshes are good enough to predict an average hemodynamic condition. On the other hand, the mean WSS calculated using the unstructured hybrid grid family shows a high mesh-dependency. Also, the unstructured solution tends to oscillate without indicating a clear converging trend.
The final result shows that structured grids converge better than unstructured grids. For the same solution accuracy, structured grids take 6 times fewer cells ( fine Hex ~ 0.36 million, fine Tet-Prism ~ 2 million) and 14 times less CPU time (Hex ~ 3 min, Tet-Prism ~ 43 min). Further, in unstructured meshes with additional refinement, the mesh-dependent error did not decrease below 5 percent. The predicted WSS values were oscillatory in nature, whereas in the structured family, WSS values decreased progressively towards a stable solution.
On closer observation, it revealed that the coarsest Hex mesh starts with a WSS profile which is qualitatively similar to that obtained from the finest mesh with a difference in peak value by 24 percent. On the other hand, the Tet-Prism series starts with a WSS profile, which is qualitatively and quantitatively wrong. In fact, the peak value was observed to be in the wrong position if the number of cells is less than one million.
Case study 2: CFD Simulation of Cerebral Arterial Trees
Another similar research work [1] with cerebral arterial trees brings out other unique advantages of structured meshes over unstructured meshes.
Post analysis of the computations reveals that, for the same time discretization, more than 80 percent of the unstructured cells exhibited a higher CFL number (>1), while only fewer structured cells needed higher CFL value. Structured grids were found to have superior grid quality. On average, cell skewness and orthogonality were 22 percent and 37 percent respectively higher than that for unstructured meshes. Figure 7 shows the CFL contour and cell quality histogram comparisons between the two grid types.
In this case study also the unstructured meshes needed more cells to attain a grid-independent solution. Unstructured meshes needed nearly 13.7 times more cells to reach a mesh-independent solution than structured meshes. This resulted in 27 times longer CPU time for unstructured grids relative to their structured counterpart.
Trailing Thoughts
Though the argument on how the shape of the elements and alignment affects the solution accuracy is still debated, case studies like the ones discussed show that hexahedral meshes still continue to provide the highest quality solution. With matured structured meshing techniques the traditional classification of “complexity of the geometry” for structured meshes has become debatable too. This leaves the engineers and researchers to decide on their trade-off, whether it is manual labor vs machine time and is it to understanding the flow with some gross numbers or are they seeking stable and reliable high-quality results.
Further Readings
References
1. “Large-scale subject-specific cerebral arterial tree modeling using automated parametric mesh generation for blood flow simulation”, Mahsa Ghaffari, et al, Computers in Biology and Medicine, 91 (2017) 353-365.
2. “Novel Mesh Generation Method for Accurate Image-Based Computational Modelling of Blood Vessels”, Gianluca De Santis, Biofluid, Tissue and Solid Mechanics for Medical Applications, Institute Biomedical Technology, Ghent University, Belgium.
3. “Detailed Comparison Of Numerical Flow Predictions In Cerebral Aneurysms Using Different CFD Software”, Philipp Berg, et al, Conference on Modelling Fluid Flow (CMFF’12), The 15th International Conference on Fluid Flow Technologies Budapest, Hungary, September 4-7, 2012.
4. “Airflow simulation inside a model of the human nasal cavity in a virtual reality based rhinological operation planning system”, Th. Van Reimersdahl, et al, International Congress Series, 1230 (2001), 87–92.
5. “Numerical Solution of Ocular Fluid Dynamics in a Rabbit Eye: Parametric Effects”, Satish Kumar, et al, Annals of Biomedical Engineering, Vol. 34, No. 3, March 2006, pp. 530–544.
6. https://www.sci.utah.edu/cibc.html
Hello,
Your work is very impressive! I have a very similar geometry with a 3D aneurysm. Is there any tutorial about meshing this geometry? May you provide me the mesh for an academic use?
Thank you very much