William W. van Osdol

Finite element modeling of coupled diffusion with partitioning in transdermal drug delivery.

The finite element method is employed to simulate two-dimensional (axisymmetric) drug diffusion from a finite drug reservoir into the skin. The numerical formulation is based on a general mathematical model for multicomponent nonlinear diffusion that takes into account the coupling effects between the different components. The presence of several diffusing components is crucial, as many transdermal drug delivery formulations contain one or more permeation enhancers in addition to the drug. The coupling between the drug and permeation enhancer(s) results in nonlinear diffusion with concentration-dependent diffusivities of the various components. The framework is suitable for modeling both linear and nonlinear, single- and multicomponent diffusions, however, as it reduces to the correct formulation simply by setting the relevant parameters to zero. In addition, we show that partitioning of the penetrants from the reservoir into the skin can be treated in a straightforward manner in this framework using the mixed method. Partitioning at interface boundaries poses some difficulty with the standard finite element method as it creates a discontinuity in the concentration variable at the interface. To our knowledge, nonlinear (concentration-dependent) partitioning in diffusion problems has not been treated numerically before, and we demonstrate that nonlinear partitioning may have an important role in the effect of permeation enhancers. The mixed method that we adopt includes the flux at the interface explicitly in the formulation, allowing the modeling of concentration-dependent partitioning of the permeants between the reservoir and the skin as well as constant (linear) partitioning. The result is a versatile finite element framework suitable for modeling both linear and nonlinear diffusions in heterogeneous media where the diffusivities and partition coefficients may vary in each subregion.

Multiscale Modeling Framework of Transdermal Drug Delivery

This study addresses the modeling of transdermal diffusion of drugs to better understand the permeation of molecules through the skin, especially the stratum corneum, which forms the main permeation barrier to percutaneous permeation. In order to ensure reproducibility and predictability of drug permeation through the skin and into the body, a quantitative understanding of the permeation barrier properties of the stratum corneum (SC) is crucial. We propose a multiscale framework of modeling the multicomponent transdermal diffusion of molecules. The problem is divided into subproblems of increasing length scale: microscopic, mesoscopic, and macroscopic. First, the microscopic diffusion coefficient in the lipid bilayers of the SC is found through molecular dynamics (MD) simulations. Then, a homogenization procedure is performed over a model unit cell of the heterogeneous SC, resulting in effective diffusion parameters. These effective parameters are the macroscopic diffusion coefficients for the homogeneous medium that is “equivalent” to the heterogeneous SC, and thus can be used in finite element simulations of the macroscopic diffusion process. The resulting drug flux through the skin shows very reasonable agreement to experimental data.

Finite element model of antibody penetration in a prevascular tumor nodule embedded in normal tissue

We have developed a pharmacokinetic model for monoclonal antibodies (mAb) to aid in investigating protocols for targeting small primary tumors or sites of metastatic disease. The model describes the uptake of systemically-administered antibody by a prevascular spherical tumor nodule embedded in normal tissue. The model incorporates plasma kinetics, transcapillary transport, interstitial diffusion, binding reactions, and lymphatic clearance. Antigen internalization can easily be incorporated. Simulations obtained from a three-dimensional finite element analysis are used to assess errors in predictions from earlier models in which the influence of the normal tissue was collapsed into a boundary condition at the tumor surface. The model employing a Dirichlet boundary condition substantially overpredicted the mean total tumor mAb concentration at all times. Although the model with a concentration-dependent flux (composite) boundary condition underpredicted mAb concentration, the discrepancy with finite element results is only notable at early times. Sensitivity analyses were performed on mAb dose and on the coefficients for mAb diffusion in the tissue regions, since reported antibody diffusivity values have varied over 30-fold. The results of the study suggest that mAb diffusivity and mAb binding site density in tumors should have major influences on optimizing doses and scheduling of mAb administration in tumor targeting protocols.

Mechanical and Microstructural Properties of Stratum Corneum

Abstract : A mechanics approach is presented to study the intercellular delamination resistance and mechanical behavior of stratum comeum (SC) tissue in the direction normal to the skin surface. The effects of temperature and hydration on debonding behavior were also explored. Such understanding, which includes the relationship of mechanical behavior to the underlying SC cellular structure, is essential for emerging transdermal drug delivery technologies. Fracture mechanics-based cantilever-beam specimens were used to determine reproducibly the energy release rates to quantify the cohesive strength of human SC. The debond resistance of fully hydrated SC was found to decrease with increasing temperature, while dehydrated SC exhibited a more complex variation with temperature. Stress-separation tests showed that fracture energies and peak separation stresses decreased with increasing temperature and hydration, although the SC modulus varied only marginally with temperature and hydration. Results are described in terms of microstructural changes associated with hydrophilic regions and intercellular lipid phase transitions.

Antibody Penetration into a Spherical Prevascular Tumor Nodule Embedded in Normal Tissue

AbstractA finite-element (FE) method is used to numerically solve a pharmacokinetic model that describes the uptake of systemically administered antibody (mAb) in a prevascular spherical tumor nodule embedded in normal tissue. The model incorporates plasma kinetics, transcapillary transport, lymphatic clearance, interstitial diffusion in both the normal tissue and tumor, and binding reactions. We use results from the FE analysis to assess previous predictions that employed either a Dirichlet boundary condition (b.c.), or an approximate, composite (Dirichlet and Neumann) b.c. at the tumor surface. We find that the Dirichlet b.c. significantly overpredicted the mean total tumor mAb concentration. In contrast, the composite b.c. yielded good agreement with FE predictions, except at early times. We also used the FE model to investigate the influence of the approximately 30-fold difference in the values of mAb diffusion coefficient measured by Clauss and Jain (Cancer Res. 50:3487–3492, 1990) and Berk [et_al.] (Proc. Natl. Acad. Sci. U.S.A. 94:1785–1790, 1997). For low diffusivity, diffusional resistance slows both mAb uptake by and efflux from the tumor. For high diffusivity at the same mAb dose, more rapid uptake produces earlier and higher peak mAb levels in the tumor, while the efflux rate is limited by the dissociation of the mAb–tumor antigen complex. The differences in spatial and temporal variation in mAb concentration between low and high diffusivities are of sufficient magnitude to be experimentally observable, particularly at short times after antibody administration.

Computational models of antibody-based tumor imaging and treatment protocols.

AbstractWe present improved computational models for investigating monoclonal antibody-based protocols for diagnostic imaging and therapy of solid tumors. Our earlier models used a boundary condition (Dirichlet) that specified concentrations of diffusing molecular species at the interface between a prevascular tumor nodule and surrounding normal tissue. Here we introduce a concentration-dependent flux boundary condition with finite rates of diffusion in the normal tissue. We then study the effects of this new condition on the tumors temporal uptake and spatial distribution of radiolabeled targeting agents. We compare these results to ones obtained with the Dirichlet boundary condition and also conduct parameter sensitivity analyses. Introducing finite diffusivity for any molecular species in normal tissue retards its delivery to and removal from the tumor nodule. Effects are protocol- and dose regimen-dependent; generally, however, mean radionuclide concentration and tumor-to-blood ratio declined, whereas relative exposure and mean residence time increased. Finite diffusivity exacerbates the negative effects of antigen internalization. Also, the sensitivity analyses show that mean concentration and tumor-to-blood ratio are quite sensitive to transcapillary permeability and lymphatic efflux values, yet relatively insensitive to precise values of diffusion coefficients. Our analysis underscores that knowledge of antigen internalization rates and doses required to saturate antigen in the tumor will be important for exploiting antibody-based imaging and treatment approaches.

In Situ Forming Systems (Depots)

In situ forming systems transform into semi-solids (or viscous masses) upon injection and provide sustained release of pharmacological agents, including small molecule drugs, peptides and proteins. These formulations can be utilized for systemic or site specific delivery and generally comprise a polymer or carrier and a solvent. This chapter reviews examples including the Atrigel (PLGA + water miscible solvent), SABERTM (sucrose acetate isobutyrate + solvent), ALZAMER (PLGA + solvent) and ReGel systems (PLGA/PEG copolymer + water), among others. In vivo delivery durations for in situ forming systems range from days to months.