Result

Mathematical ModelingThe 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 DT

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 DiscussionTwo possible mechanisms can increase the effective diffusion constant!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 constantIGF: DT = (9±3) x 10-6 cm2s-1

bFGF: DT = (6±3) x 10-6 cm2s-1

Washout rateIGF: k = (8±3) x 10-4 s-1

bFGF: k = (9±3) x 10-4 s-1

Peri-epicardial boundary permeabilityIGF: 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

Effective Diffusion,D* 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. (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

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].

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

Pericardial sac: R2 – R3

Myocardium: R1 – R2

Chamber: 0 – R1

Idealized Spherical Geometry

The best fit parameters for each group of experiments. Numerical values for DT, k, 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 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 GoalOur 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.

Experimental Methods

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

x: depth in tissue

Experimental subjects: juvenile farm pigsRadiotracer method to determine the spatial

concentration profile from gamma radiation rate, using radio-iodinated test agentsInsulin-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=0200 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: Ci

T(x,T)

Experimental procedure

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 chi-square surfaces (in this case, for IGF_2000_24h), showing distinct minima giving the optimal fit parameters (D, 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