User Case Study: Applied mathematician uses Maple to simulate the biomechanics of human bone microstructure

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Bone is a complex tissue, which is highly optimized for biological and mechanical functions. Researchers are currently examining how bone architecture changes with age and how it alters when various pathologies are present or absent. In particular, the prediction of bone fracture remains elusive.

This detail of human bone (enlarged 6 times) shows the classification in compact and cancellous types.

Dr. Maria-Grazia Ascenzi of the Department of Orthopaedic Surgery at the University of California, Los Angeles (UCLA), performs research in her Bone Micro-Mechanics laboratory to better understand bone micro-structures. She conducts experimental biomechanics on bone micro-specimens and uses the powerful computational abilities of Maple™ to model the percentages and orientations of the various elementary components (such as collagen and carbonated hydroxyapatite, or “apatite”), and the details of fundamental micro-level units of compact bone. Compact bone tissue (Figure 1) is the dense type of bone that constitutes the long bone shafts and the outer layer that surrounds the otherwise more porous bone tissue, also known as cancellous bone (Figure 1). The compactness allows the bone to sustain high stresses. Dr. Ascenzi recently used Maple to investigate (i) the biomechanical behavior of the (secondary) osteon, the predominant micro-structural unit of adult human compact bone; and (ii) the role of orientation of very small antennae-like organelles in the bone’s growth plate of animal models.

Dr. Ascenzi used Maple in several major studies directed to human osteons. Because the bone’s components vary in terms of orientation and density within the millimeter scale, she focused on very small sized specimens to fix the tissue’s main parameters. This provided specimen groups with fairly homogeneous material properties.  The corresponding models, created at the micro scale, were more realistic than previous models. The experimentally tested mechanical behavior was well-explained by the parameters that geometrically and structurally described the tested specimens. In particular, the results obtained from Maple validated the characteristics of the bone specimens from which they were derived.

Mathematical modeling of human secondary osteons

To fully understand and predict how osteons behave as part of compact bone, a highly precise and accurate model of an osteon is helpful. Using observation techniques, such as circularly polarized light and high resolution micro-X-ray, thousands of measurements were collected from osteons and compiled in a database. A program was written in Maple to allow user to draw from the database to build models with specific data, random data, or a combination of both. Based on such specifications (Figures 2a, b, and c), highly detailed mathematical models of the osteons were created in Maple. With the ability to randomly generate data using the linear congruential generator in Maple, over a trillion unique models can be prepared. Further, Maple associates the osteon model with the material properties that correspond to specific biomechanical behavior observed in the laboratory. Specifically, the material properties strictly followed recent experimental findings.

An osteon is represented by a thick cylindrical shell containing pseudo-ellipsoids that represent the osteocyte lacunae.

The detail and accuracy of these models allow for some novel use of the data. The models generated allow the results gained from the highly detailed investigation on selected bone donors, to be applied to patients from whom only the information collected by conventional medical imaging is available. Using the generated models, it is now possible to simulate mechanical testing without performing costly experiments on actual bone. This is a great advantage in the study of fracture propagation. It holds promise for improved medical interventions, such as those pertaining to osteoporosis and prostheses. As the work of Dr. Ascenzi’s group continues to assemble random models to form the model of macroscopic bone specimens, researchers can observe how a fracture might propagate through bone and determine ways to avoid bone fracture.

Understanding bone strength through secondary osteons subjected to torsion

Human bone is a light and strong material, which provides structural support for the entire human body. Bone gets its strength, hardness and flexibility from a complex combination of calcification and proteins, the majority of which is composed of collagen fibrils. Dr. Ascenzi looked at indications for conjectures on the nature of the connections between collagen and apatite. She gathered experimental data on the mechanical properties of osteons that were subjected to torsion. Dr. Ascenzi could fit large quantities of data to cubic polynomial functions using regression techniques in Maple. This allowed her to study the stiffness and the variation of stiffness in terms of the collagen orientation, and the distribution of the modeled osteons.

The conjectures formulated on the effects of torsion on osteons will now guide researchers as they continue their study of bone nanostructure. This could lead to a better understanding of bone strength and provide the groundwork to apply this knowledge in creating new materials for prosthetics.

The primary cilia are antenna-like organelles of many cell types.

Mathematically analyzing bone growth

It has been hypothesized that as bone grows, small antennae-like organelles in the bone’s growth plate, called primary cilia, sense the environment around them and use this information to help the cells to function properly within a specific organized arrangement. Dr. Ascenzi gathered data from images of growth plates of animal models obtained through scanning multi-photon microscopy (Figure 3a) done by collaborators at Cornell University.  She created a database from which Maple can numerically reconstruct the length and orientation of these cilia in the three-dimensional (3-D) environment. To perform this analysis, a series of algorithms were written in Maple to reconstruct each cell with cilium (Figure 3b and 3c) from the data and then measure the ciliary length and orientation on the reconstruction with respect to any relevant direction.

Dr. Ascenzi employed Maple to create an accurate 3-D model suitable for further study. Maple was programmed to model the 2-D images of cells as ellipses and compute all geometric characteristics such as eccentricity. Additional algorithms modeled each cilium as a segment on each 2-D image. Each stack of ellipses and stack of segments relative to a ciliated cell was then reconstructed by the optimal approximation, with an ellipsoid modeling the cell in 3-D and a segment modeling the cilium in 3-D. Maple then computed the cilium orientation relative to the cell orientation.

The program obtained is a useful technique to investigate patterns of length and orientation of the primary cilium in conjunction with genetically manipulated murine models. Dr. Ascenzi’s group and collaborators are examining the relationship between form and function of cilium and cell.

Theoretical mathematics with Maple provides flexibility for biomechanics

The pioneering research on bone micromechanics began in the 1940s. The limited availability of computational power has constrained the bioengineering and mathematical aspects of the research; however, now that supercomputers have increased capabilities and programs such as Maple have developed highly advanced features, Dr. Ascenzi’s work can move much faster. “With Maple, I feel I can do anything a mathematician would do with a pencil and paper, only much, much faster,” said Dr. Ascenzi. “Maple is very flexible, allowing me to vary parameters and variables and test new models quickly.” Combine the power of Maple with the biomechanical studies of Dr. Ascenzi, and it becomes clear that full understanding of the complex organic systems in our body may be closer than we think.

About Dr. Maria-Grazia Ascenzi and the Bone Micro-Mechanics laboratory

Dr. Ascenzi is an Associate Researcher in the Department of Orthopedic Surgery at UCLA. She is trained as a theoretical mathematician. In the 1990s, she began to apply her grounding in theoretical mathematics to bone research. She began her work collaborating with biomechanics researchers, providing the mathematical models for their studies. Dr. Ascenzi now performs her own micro-biomechanical experiments in the Bone Micro-Mechanics laboratory at UCLA. In this laboratory, very small specimens are tested using highly specialized equipment. By testing small samples, Dr. Ascenzi can reduce the variation of specimen parameters, thus producing more accurate models in her studies. She creates models using Maple with data collected from her experiments and from imaging obtained by collaborators.

The collaborators of Dr. Ascenzi who have made this modeling possible are Mariasevera Dicomite, Eve Donnelly, Cornelia Farnum, Jaya Gill, Alexandre Lomovtsev, Plamen Mitov, and J. Michael Kabo.