Modeling the Diffusion of TGF-β1 from a Fibrin Scaffold through Alveolar Bone Martin Barrio and Sindhu Raghunandan 10/28/2011 BENG221Modeling the Diffusion of TGF-β1 from a Fibrin Scaffold through Alveolar Bone Martin Barrio and Sindhu Raghunandan 10/28/11 2 I. Background Periodontitis is defined as infection and inflammation of the primary tissues surrounding one or more teeth [5]. While the initial stages involve the infection and inflammation of the coronal gingival tissue (or the tissue surrounding each tooth) as seen in Figure 1, more extreme cases are characterized by degradation of ligaments and bones that support teeth. The condition is perpetuated by the buildup of plaque in the coronal and apical gingival sulcus (which is the gap between the tooth and the surrounding tissue) leading to chronic and recurring infection and inflammation [6]. Once plaque reaches the interface between the gingival tissue and the alveolar bone, the bacterial cells begin to degrade the bone. A complication that can arise after this stage is that the gingival tissue cells infiltrate the gap in the alveolar bone and begin to inhibit bone regeneration ultimately leading to insufficient mechanical support for teeth. Figure 1. A visual juxtaposition of periodontal tissue and healthy tissue. Conventional treatments can be broken up into the initial and surgical categories. If initial treatments do not facilitate regression of periodontitis then more substantial measures are taken to prevent alveolar bone and tooth loss. The surgeries that are most commonly applied to these cases are open-flap debridement, osseous surgery, and guided tissue regeneration and bone grafting [4]. The goal of these surgeries is to create a space for bone and ligaments to regenerate around the tooth after periodontal surgery. Without this barrier faster growing soft tissue, such as epithelium and gingival tissue, would fill in the space around the tooth. Generally, surgical treatment and post treatment maintenance prevents tooth loss for 85% of patients [4, 7]. One avenue for guided tissue regeneration is the development of synthetic polymer scaffolds infused with bone regenerating growth factors. The properties of both the scaffold and growth factors can be controlled in order to elute a certain concentration of the growth factor into the tissue intended for regeneration. Furthermore, the use of mathematical models can be used to examine any combination of available scaffolds and growth factors to generate the most efficient delivery of growth factors to the alveolar bone. The motivation for this study is to generate a model that will plot the diffusion of the growth factor into the bone over time with hypothetical boundary conditions and initial conditions. The accuracy with which theModeling the Diffusion of TGF-β1 from a Fibrin Scaffold through Alveolar Bone Martin Barrio and Sindhu Raghunandan 10/28/11 3 model matches physiological circumstances will then be assessed and further improvements to the model will be proposed. II. Design Concept The goal of this project is to model the diffusion of transforming growth factor (TGF-β1, a cell proliferation associated growth factor) from a fibrin scaffold into the alveolar bone to regenerate degraded bone tissue due to periodontitis. The physiological system relevant to periodontitis is shown in the figures below. The image to the left shows the actual layering of the gingival tissue, alveolar bone, and mandibular canal. The simplified representation used for this project is shown to the right. . Figure2. The physiological system and the simplified model. This study assumes that the progression of periodontitis is at a point where the gingival tissue has degraded and there is a small cavity in the alveolar bone. The scaffold employed will, therefore, be placed inside of the cavity against the surface of the bone to regenerate this bone before gingival tissue infiltration. Before deriving an analytical or numerical solution, a set of assumptions were established to guide the solution. - First, the initial concentration of the growth factor was chosen to be 2mM. Since the initial concentration is tailored to the degree of regeneration required, this value was somewhat arbitrarily chosen. - Second, a study by Leddy et.al suggested that the diffusion of dextrans through various scaffolds is comparable to the diffusion of growth factors through those same scaffolds. So, by making the same assumption, the diffusivity of generic growth factors through a fibrin scaffold is approximately 115µm2/ s [3]. - Third, the thickness of the alveolar bone (the layer through which the growth factors are diffusing) is ~2.06mm [1] as indicated by another study. - It was also assumed that no other factors in the matrix or the surrounding tissues affect the diffusion or uptake of TGF-β1. While this assumption significantly limits the physiological relevance of the model, this was a necessary simplification to isolate a model of diffusion of a specific growth factor through a specific scaffold. - Fourth, looking at the scaffold and mandibular canal boundaries shown in figure 2, it was assumed that no growth factors can escape from the scaffold away from the bone (no flux boundary condition). - Fifth, we assumed that the osteocytes that make up the alveolar bone will consume the growth factor at a constant rate 1.86E-5 M/s. [2]Modeling the Diffusion of TGF-β1 from a Fibrin Scaffold through Alveolar Bone Martin Barrio and Sindhu Raghunandan 10/28/11 4 - Sixth, we assumed that the application of the scaffold to the alveolar bone can be modeled as an impulse at t = 0. - Finally, it was assumed that once the growth factors diffuse through the alveolar bone into the mandibular channel, the vessels present in that canal will instantly uptake the growth factors (zero boundary condition). III. Analytical solution Diffusion Equation Boundary and Initial Conditions ( ) ( ) ( ) ( ) = g(x) The most generic analytical solution for any system with any initial and boundary conditions can be written as follows. ( ) ∫ ( ) ( ) ∫ ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) However, based on the boundary and initial conditions as well as the nature of the source term, ( ), an appropriate Green’s function can be chosen and the above statement can be