Mathematical Modeling and Simulation of Human Motion Using a 3-Dimentional, Multi-Segment Coupled Pendulum System

The use of mathematical models to investigate the dynamics of human movement relies on two approaches: forward dynamics and inverse dynamics. In my investigation a new modeling approach called the Boundary Method will be used. This method addresses some of the disadvantages of both the forward and the inverse approach. The method yields as output both a set of potential movement solutions to a given motor task and the net muscular impulses required to produce those movements. The input to the boundary method is a finite and adjustable number of critical target body configurations. In each phase of the motion that occurs between two contiguous target configurations the equations of motion are solved in the forward direction as a two point ballistic boundary value problem. In the limit as the number of specified target configurations increases the boundary method approaches a stable algorithm for doing inverse dynamics.

As a continuation of my master’s thesis work, I am interested in using the boundary method in studying and analyzing different motor tasks of athletes with sport injuries or subjects with disabilities.

Local growth factor delivery to accelerating impaired osseous repair

Osteoporoses and impaired bone formation / regeneration have been shown to be a long term systemic effects of diabetes mellitus and other hormonal deficiencies in animal and clinical studies.

Recent advances in understanding the biology of osteogenesis have opened new avenues for enhancing bone repair via the use of growth factors and cytokines as potential therapeutic agents. The short biological half lives of growth factors and cytokines may impose sever restrains on their clinical usefulness, therefore, convenient methods must be developed for obtaining sustained therapeutic concentrations of the appropriate factors locally around the fracture site. The key growth factors characterized as being present in the fracture site are TGF-?1, TGF-?2, BMP-2, BMP-3, BMP-4, and BMP-7 (OP-1), PDGF, and acidic and basic FGF (FGF-1 and FGF-2).

Only a few studies have explored the effects of locally applied growth factor upon normal and impaired fracture healing. Different approaches were used to achieve local growth factor delivery, including the use of mechanical external pumps, implantable biodegradable scaffolds, and gene transfer.

The aim of my research is to investigate calcium salts (tri-calcium phosphate and calcium sulfate) and other bone biocompatible materials as possible delivery vehicles for key growth factors and cytokines to accelerate impaired osseous repair. The hypothesis is that these bone-biocompatible carriers will cause a prolonged growth factor/cytokine release that will normalize not only early but also late parameters of impaired osseous repair. The outcomes of this research will advance the fracture repair biotechnology and have a translational potential for clinical application. The cost-benefit of the outcomes is substantial especially in reducing hospitalization time and reducing medication cost.

Tissue engineering and cell based therapy to accelerate impaired osseous repair

New techniques have been developed, many derived from biotechnology, that enhance and expand the use of human cells and tissues as therapeutic products. These new techniques hold the promise that some day it will provide therapies for many serious medical conditions.

With the increasing numbers of trauma, cancer cases and diabetes mellitus cases every year, it is important to explore and introduce the use of new therapeutic technologies, i.e. biotechnology, tissue engineering, and cell based therapy in the health care system. In the United States the value of the collective cell therapy market was estimated to be $26.6 billion in 2005. Projections for 2010 and 2015 expected the number to become $56.2 billion and 96.3 billion respectively. (Cell Therapy-Technologies, Markets, and Companies).

Orthopaedic surgeons are challenged every day to accelerate healing, reduce cost, and reduce hospitalization time. Several approaches have been utilized to elicit the formation of bone in segmental defects and facilitate the healing process. These approaches have included the implantation of osteoconductive extracellular scaffolds and the implantation of bone morphogenetic proteins in various matrices. Another concept is based upon ex vivo expansion of pluripotent mesenchymal stem cells (MSC) loaded onto a carrier system. 

The MSC are self renewing pluripotent progenitor cells that have been isolated from the whole marrow of chicks, mice, rats, rabbits, goats, and humans. These cells have the capability of differentiating into osteoblasts, chondrocytes, adipocytes, tenocytes, and myoblasts. Potential advantages of this strategy consist of decreased need for massive cellular proliferation and osteoblast progenitor cell chemotaxis into the defect as well as development of appropriate signaling for early bone formation in the graft site. The role of MSC in bone regeneration and formation continues to be defined, and manipulation of MSC has resulted in new therapeutic strategies.

In various fracture healing models, diabetes correlated with a reduction in fracture callus cellular proliferation, collagen synthesis, and biomechanical properties. The mechanism by which diabetes impairs fracture healing is unknown. In normal fracture healing a blood clot forms at the fracture site that entraps platelets within the fibrin matrix. The platelet ?-granules act as a reservoir of critical early growth factors [1]. Degranulation of the ?-granules releases multiple growth factors, including platelet derived growth factor (PDGF) and transforming growth factor b (TGF-?), into the fracture site.  Previous studies in diabetic rats, however, have demonstrated that at early time points after fracture (2, 4 and 7 days) there are reduced levels of PDGF, TGF-?, insulin-like growth factor I (IGF-I) and vascular endothelial growth factor (VEGF) present [2, 3] and a corresponding decrease in cellular proliferation at the fracture site.

The aim of my research is to investigate ex vivo expansion of MSC in accelerating impaired osseous repair. The hypothesis behind this research is that the introduction of ex vivo expansions of MSC would mitigate against compromised healing in a diabetic fracture model.

Modeling and Simulating the Effects of Local Growth Factors on Osseous Repair

Fracture healing is a complex process that involves the sequential recruitment of cells and the specific temporal expression of genes essential for bone repair. While the process by which fracture repair occurs is well describe, relatively little is understood about the coordinate regulation of events leading to successful repair. Furthermore, even less is understood about how the process can fail, leading to cases of delayed union, nonunion, and pseudoarthrosis. Local growth factors are believed to play an integral role in the regulation of bone formation, resorption and remodeling and are expressed at different times during fracture healing.

Based on experimental findings in rats, the aim of my research will be the development of a mathematical model that investigates the effect of local growth factors on normal and impaired fracture healing.

Faced with the task of understanding a complex system, it is often useful to extract its most essential features and use them to create a simplified representation of the system, or a ‘model’ of the system. A model allows one to observe more closely the behavior of the system and to make predictions regarding its performance under altered input conditions and different system parameters [4].

In vivo and in vitro experiments have demonstrated that cell activity during bone tissue morphogenesis is initiated and tightly regulated by locally produced growth factors and matrix proteins [5]. Although in secondary fracture repair a variable mechanical and angiogenic environment may influence the development of cartilage and bone tissue, tissue differentiation is initiated and largely regulated by autocrine and paracrine growth factors. These factors constitute promising therapeutic agents for complicated fractures, distraction osteogenesis and for other clinical skeleton conditions such as osteoporosis. Before their use, however, which requires precisely determining growth factor dosage, treatment duration and delivery method, a better understanding of their regulatory mechanisms is needed. 

Mathematical description may help us understand the regulatory mechanisms involved in bone regeneration and provides a framework to design experiments and understand pathological conditions.

Most of the early theoretical models correlate tissue histology to the mechanical environment [6] [7] [8]. However, these models did not account for growth factor concentration involved in the repair process. Moreover, these models solve for discrete time points only and fail in predicting a continuous spatio-temporal progression of the healing process.

Based on the hypothesis that the spatio-temporal production of local growth factors in the fracture callus determines where cartilage or bone tissue forms, the main objective of my project is to develop a mathematical framework to simulate the effects of critical local growth factors on cell migration, differentiation, and proliferation in the fracture callus. Furthermore, the long term objective would be combining the contribution of both the mechanical environment and the role of local growth factors in one mathematical framework that will describe the spatio-temporal progression of normal and impaired osseous healing.

The short term objectives would be:

1)                              Modeling and simulating Mesenchymal Stem Cells (MSCs) proliferation in the fracture callus.

2)                              Modeling and simulating Mesenchymal Stem Cells (MSCs) migration in the fracture callus.

3)                              Modeling and simulating Mesenchymal Stem Cells (MSCs) differentiation into Osteoblast linage in the fracture callus.

4)                              Modeling and simulating Mesenchymal Stem Cells (MSCs) differentiation into Chondryocyte linage in the fracture callus.

Reference:

1.         Slater M, Patava J, Kingham K, et al. 1995. Involvement of platelets in stimulating osteogenic activity. J Orthop Res 13(5): p. 655-63.

2.         Beam HA, O'Connor JP, Parsons JR, et al. Reductions in Early Growth Factors in Diabetic Fracture Callus. in Orthopaedic Research Society. 2002. Dallas, TX.

3.         Gandhi A, Doumas C, O'Connor JP, et al. 2006. The effects of local platelet rich plasma delivery on diabetic fracture healing. Bone 38(4): p. 540-6.

4.         Prendergast PJ. 1997. Finite element models in tissue mechanics and orthopaedic implant design. Clin Biomech (Bristol, Avon) 12(6): p. 343-366.

5.         Bailon-Plaza A and van der Meulen MC. 2001. A mathematical framework to study the effects of growth factor influences on fracture healing. J Theor Biol 212(2): p. 191-209.

6.         Ament C and Hofer EP. 2000. A fuzzy logic model of fracture healing. J Biomech 33(8): p. 961-8.

7.         Blenman PR, Carter DR, and Beaupre GS. 1989. Role of mechanical loading in the progressive ossification of a fracture callus. J Orthop Res 7(3): p. 398-407.

8.         Claes LE and Heigele CA. 1999. Magnitudes of local stress and strain along bony surfaces predict the course and type of fracture healing. J Biomech 32(3): p. 255-66.