Numerical Results

Mathematical ModelingKey Biophysical ProcessesSubstrate 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 DT

Substrate washout through the vascular and lymphatic capillaries, described by the rate k

Governing Equations / 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 DiscussionTwo possible mechanisms can increase the effective diffusion constant:

1) Transport via Intramural Vasculature

2) 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, Bloomington, [2]Indiana University School of Medicine, Indianapolis

ConclusionsModel accounting for effective diffusion, washout and finite permeability of peri/epi boundary is consistent with experiments despite its simplicity, allowing quantitative determination of numerical values for physical parameters.

We show:

Comparison with Experiment

Diffusion in Tortuous MediaStokes-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*. [*] M. Suenson, D.R. Richmond, J.B. Bassingthwaighte, 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!

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 only recently been proven to be experimentally feasible[1,2].

R1 = 2.5cmR2 = 3.5cmVperi= 10ml - 40ml

Pericardial sac: R2 – R3

Myocardium: R1 – R2

Chamber: 0 – R1

Idealized Spherical Geometry

Best fit parameters for each group of experiments: Numerical values for (DT, k, are consistent for IGF, bFGF to within experimental errors.

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

Heart tissue is a porous medium consisting of extracellular space and muscle fibers. The extracellular space is made up 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. 

Typical volume for the human pericardial sac is

10-15ml.

GoalEstablish a minimal biophysical model for drug penetration in the myocardium using this mode of delivery and extract numerical values for the governing parameters by comparison with experimental data.

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.

Typical surfaces showing distinct minima giving the optimal fit parameters (DT, k , ).

[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.

ExperimentsThe experiments were performed on juvenile farms pigs using the radiotracer method to determine the concentration of radio-iodinated test agents in the tissue from the rate of radioactive decay. These agents, 125I-IGF (MW=7734 Da) and 125bFGF (MW=18000 Da), are relevant therapeutic growth factors. Different initial amounts (200 and 2000 g 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.

CT(x,t) = i CiT(x,t)

x: depth in tissue

Samples were taken from the pericardial sac fluid, giving CP(t). Tissue strips were excised and fixed in liquid nitrogen. Cylindrical transmyocardial specimens were sectioned into slices as shown, giving CT(x,t), where x is the thickness through the tissue. We focus on the data obtained from the left ventricle only, and average CT

i(x,t) obtained at different (i=1, …, 9) spatial locations to obtain a single concentration profile CT(x,t).

i

IGF_2000_24h

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.