1. Result Mathematical Modeling The key processes  Substrate transport across boundary layer between pericardial sac and myocardium, described by the parameter  which is the permeability of the peri/epicardium boundary  Substrate diffusion in the myocardium, described by the effective diffusion constant D T  Substrate washout through the vascular and lymphatic capillaries, described by the rate k Governing Equations and Boundary Conditions Governing equation in myocardium (diffusion + washout) CT: concentration of agent in tissue DT: effective diffusion constant in tissue k: washout rate Pericardial sac as a drug reservoir (well-mixed and no washout): drug number conservation Boundary condition: drug current at peri/epicardial boundary Pericardial Delivery Experiments The experiments were performed on juvenile farms pigs using the radiotracer method to determine the concentration of radio- iodinated test agents in the tissue from rate of radioactive decay. These agents, IGF and bFGF, are relevant therapeutic growth factors. Different initial amounts (200 and 2000 micrograms in an injectate volume of 10 ml) were delivered to the pericardial space of an anesthetisized animal at t=0. At t=1 hour or t=24 hours, the heart was harvested. Discussion (cont.) Transport via Intramural Vasculature Diffusion in Active Viscoelastic Media Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac: Insights From Physical Modeling Xianfeng Song [1], Keith L. March [2], Sima Setayeshgar [1] [1] Department of Physics, Indiana University, [2] IUPUI Medical School Conclusion  Model accounting for effective diffusion and washout is consistent with experiments despite its simplicity.  Quantitative determination of numerical values for physical parameters  Effective diffusion constant IGF: D T = (9±3) x 10 -6 cm 2 s -1 bFGF: D T = (6±3) x 10-6 cm 2 s -1  Washout rate IGF: k = (8±3) x 10 -4 s -1 bFGF: k = (9±3) x 10 -4 s -1  Peri-epicardial boundary permeability IGF: a = (2.7±0.8) x 10 -6 cm s -1 bFGF: a = (6.0±1.6) x 10 -6 cm s -1  Enhanced effective diffusion, allowing for improved transport  Feasibility of computational studies of amount and time course of pericardial drug delivery to cardiac tissue, using experimentally derived values for physical parameters. Comparison with experiment C T (x,T) =  i C i T (x,T) x: depth in tissue Effective Diffusion,D* in Tortuous Media Stokes-Einstein relation D: diffusion constant R: hydrodynamic radius  : viscosity T: temperature Diffusion in tortuous medium D*: effective diffusion constant D: diffusion constant in fluid : tortuosity For myocardium,  = 2.11. Numerical estimates for diffusion constants IGF : D ~ 4 x 10-7 cm2s-1 bFGF: D ~ 3 x 10-7 cm2s-1 Our fitted values are in order of 10 -6 - 10 -5 cm 2 sec -1, 10 to 50 times larger Epi Endo The pericardial sac is a fluid-filled self- contained space surrounding the heart. As such, it can be potentially used therapeutically as a “drug reservoir” to deliver anti-arrhythmic and gene therapeutic agents to coronary vasculature and myocardium. This has recently been proved to be experimentally feasible. Samples were taken from the pericardial sac fluid, giving C P (T). Tissue strips were excised and fixed in liquid nitrogen. Cylindrical transmyocardial specimens were sectioned into slices as shown, giving C T (x,T), where x is the thickness through the tissue. We focus on the data obtained from the left ventricle only, and average C T i (x,t) obtained at different (total of 9) spatial locations to obtain a single concentration profile C T (x, T). R 1 = 2.5cm R 2 = 3.5cm V peri = 10ml - 40ml Pericardial sac: R 2 – R 3 Myocardium: R 1 – R 2 Chamber: 0 – R 1 Idealized Spherical Geometry This is an example of the data showing the concentration of IGF at 24hr through the thickness of the tissue and the resulting fit for an initial delivery amount of 2000 micrograms. We have included only 10 slices in the fits since the concentration below this point was at the background. The Chi-square surface as a function of alpha and k (for example) clearly showing a minimum. Fit Results: The best parameters for each group of experiment. Drug permeates into vasculature from extracellular space at high concentration and permeates out of the vasculature into the extracellular space at low concentration, thereby increasing the effective diffusion constant in the tissue Heart tissue is a porous medium consisting of extracellular space and muscle fibers. The extracellular space consists of an incompressible fluid (mostly water) and collagen. Expansion and contraction of the fiber bundles and sheets leads to changes in pore size at the tissue level and therefore mixing of the extracellular volume. This effective "stirring" results in larger diffusion constants. A typical volume for human pericardial sac is 10-15ml Our Goal Our goals are to establish a minimal physical model for drug penetration in the myocardium using this mode of delivery and to extract numerical values for the governing parameters by comparison with experimental data. Discussion Contradiction? NO! Two possible mechanisms can increase the effective diffusion constant!