Xianfeng Song[1], Keith L. March[2], Sima Setayeshgar[1] 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 Pericardial Delivery Mathematical Modeling The key processes Substrate transport across boundary layer between pericardial sac and myocardium, described by the parameter a which is the permeability of the peri/epicardium boundary Substrate diffusion in the myocardium, described by the effective diffusion constant DT Substrate washout through the vascular and lymphatic capillaries, described by the rate k Comparison with experiment Discussion Two possible mechanisms can increase the effective diffusion constant! Transport via Intramural Vasculature Diffusion in Active Viscoelastic Media A typical volume for human pericardial sac is 10-15ml 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[1,2]. Comparison of experimentally measured concentration profiles in tissue with simulation results from the model using the best fitted parameters: Each slice corresponds to 0.4 mm. Epi Endo 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 [1]Verrier VL, et al., Circulation (1998) 98:2331-2333. [2]Stoll HP, et al., Clin Cardiol (1999) 22(Suppl-I): I-10-I-16 R1 = 2.5cm R2 = 3.5cm Vperi= 10ml - 40ml Pericardial sac: R2 – R3 Myocardium: R1 – R2 Chamber: 0 – R1 Idealized Spherical Geometry 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.  Experimental Methods CT(x,T) = i CiT(x,T) x: depth in tissue Experimental subjects: juvenile farm pigs Radiotracer method to determine the spatial concentration profile from gamma radiation rate, using radio-iodinated test agents Insulin-like Growth Factor (125I-IGF, MW: 7734 Da) Basic Fibroblast Growth Factor (125I-bFGF, MW: 18000 Da) Initial concentration delivered to the pericardial sac at t=0 200 or 2000 g in 10 ml of injectate Harvesting at t=1h or 24h after delivery At t = T (1h or 24h), sac fluid is distilled: CP(T) Tissue strips are submerged in liquid nitrogen to fix concentration. Cylindrical transmyocardial specimens are sectioned into slices: CiT(x,T) Experimental procedure Typical chi-square surfaces (in this case, for IGF_2000_24h), showing distinct minima giving the optimal fit parameters (D, k , a). 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 Effective Diffusion,D* in Tortuous Media Stokes-Einstein relation D: diffusion constant R: hydrodynamic radius u: viscosity T: temperature Diffusion in tortuous medium D*: effective diffusion constant D: diffusion constant in fluid l: tortuosity For myocardium, l= 2.11. (from M. Suenson, D.R. Richmond, J.B. Bassingthwaighte, “Diffusion of sucrose, sodium, and water in ventricular myocardium, American Joural of Physiology,” 227(5), 1974 ) 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 cm2sec-1, 10 to 50 times larger 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: DT = (9±3) x 10-6 cm2s-1 bFGF: DT = (6±3) x 10-6 cm2s-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. Result 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. The best fit parameters for each group of experiments. Numerical values for DT, k, a consistent for IGF, bFGF to within experimental errors