1. 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:
[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: Da)
Basic Fibroblast Growth Factor (125I-bFGF, MW: 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 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