Smeared multiscale finite element model for electrophysiology and ionic transport in biological tissue.

Basic functions of living organisms are governed by the nervous system through bidirectional signals transmitted from the brain to neural networks. These signals are similar to electrical waves. In electrophysiology the goal is to study the electrical properties of biological cells and tissues, and the transmission of signals. From a physics perspective, there exists a field of electrical potential within the living body, the nervous system, extracellular space and cells. Electrophysiological problems can be investigated experimentally and also theoretically by developing appropriate mathematical or computational models. Due to the enormous complexity of biological systems, it would be almost impossible to establish a detailed computational model of the electrical field, even for only a single organ (e.g. heart), including the entirety of cells comprising the neural network. In order to make computational models feasible for practical applications, we here introduce the concept of smeared fields, which represents a generalization of the previously formulated multiscale smeared methodology for mass transport in blood vessels, lymph, and tissue. We demonstrate the accuracy of the smeared finite element computational models for the electric field in numerical examples. The electrical field is further coupled with ionic mass transport within tissue composed of interstitial spaces extracellularly and by cytoplasm and organelles intracellularly. The proposed methodology, which couples electrophysiology and molecular ionic transport, is applicable to a variety of biological systems.

Correction function for accuracy improvement of the Composite Smeared Finite Element for diffusive transport in biological tissue systems.

Modeling of drug transport within capillaries and tissue remains a challenge, especially in tumors and cancers where the capillary network exhibits extremely irregular geometry. Recently introduced Composite Smeared Finite Element (CSFE) provides a new methodology of modeling complex convective and diffusive transport in the capillary-tissue system. The basic idea in the formulation of CSFE is in dividing the FE into capillary and tissue domain, coupled by 1D connectivity elements at each node. Mass transport in capillaries is smeared into continuous fields of pressure and concentration by introducing the corresponding Darcy and diffusion tensors. Despite theoretically correct foundation, there are still differences in the overall mass transport to (and from) tissue when comparing smeared model and a true 3D model. The differences arise from the fact that the smeared model cannot take into account the detailed non-uniform pressure and concentration distribution in the vicinity of capillaries. We introduced a field of correction function for diffusivity through the capillary walls of smeared models, in order to have the same mass accumulation in tissue as in case of true 3D models. The parameters of the numerically determined correction function are: ratio of thickness and diameter of capillary wall, ratio of diffusion coefficient in capillary wall and surrounding tissue; and volume fraction of capillaries within tissue domain. Partitioning at the capillary wall - blood interface can also be included. It was shown that the correction function is applicable to complex configurations of capillary networks, providing improved accuracy of our robust smeared models in computer simulations of real transport problems, such as in tumors or human organs.

Progression-dependent transport heterogeneity of breast cancer liver metastases as a factor in therapeutic resistance.

Metastatic disease is a major cause of mortality in cancer patients. While many drug delivery strategies for anticancer therapeutics have been developed in preclinical studies of primary tumors, the drug delivery properties of metastatic tumors have not been sufficiently investigated. Therapeutic efficacy hinges on efficient drug permeation into the tumor microenvironment, which is known to be heterogeneous thus potentially making drug permeation heterogeneous, also. In this study, we have identified that 4 T1 liver metastases, treated with pegylated liposomal doxorubicin, have unfavorable and heterogeneous transport of doxorubicin. Our drug extravasation results differ greatly from analogous studies with 4 T1 tumors growing in the primary site. A probabilistic tumor population model was developed to estimate drug permeation efficiency and drug kinetics of liver metastases by integrating the transport and structural properties of tumors and delivered drugs. The results demonstrate significant heterogeneity in metastases with regard to transport properties of doxorubicin within the same animal model, and even within the same organ. These results also suggest that the degree of heterogeneity depends on the stage of tumor progression and that differences in transport properties can define transport-based tumor phenotypes. These findings may have valuable clinical implications by illustrating that therapeutic agents can permeate and eliminate metastases of "less resistant" transport phenotypes, while sparing tumors with more "resistant" transport properties. We anticipate that these results could challenge the current paradigm of drug delivery into metastases, highlight potential caveats for therapies that may alter tumor perfusion, and deepen our understanding of the emergence of drug transport-based therapeutic resistance.

Multiscale smeared finite element model for mass transport in biological tissue: From blood vessels to cells and cellular organelles.

One of the basic and vital processes in living organisms is mass exchange, which occurs on several levels: it goes from blood vessels to cells and organelles within cells. On that path, molecules, as oxygen, metabolic products, drugs, etc. Traverse different macro and micro environments - blood, extracellular/intracellular space, and interior of organelles; and also biological barriers such as walls of blood vessels and membranes of cells and organelles. Many aspects of this mass transport remain unknown, particularly the biophysical mechanisms governing drug delivery. The main research approach relies on laboratory and clinical investigations. In parallel, considerable efforts have been directed to develop computational tools for additional insight into the intricate process of mass exchange and transport. Along these lines, we have recently formulated a composite smeared finite element (CSFE) which is composed of the smeared continuum pressure and concentration fields of the capillary and lymphatic system, and of these fields within tissue. The element offers an elegant and simple procedure which opens up new lines of inquiry and can be applied to large systems such as organs and tumors models. Here, we extend this concept to a multiscale scheme which concurrently couples domains that span from large blood vessels, capillaries and lymph, to cell cytosol and further to organelles of nanometer size. These spatial physical domains are coupled by the appropriate connectivity elements representing biological barriers. The composite finite element has "degrees of freedom" which include pressures and concentrations of all compartments of the vessels-tissue assemblage. The overall model uses the standard, measurable material properties of the continuum biological environments and biological barriers. It can be considered as a framework into which we can incorporate various additional effects (such as electrical or biochemical) for transport through membranes or within cells. This concept and the developed FE software within our package PAK offers a computational tool that can be applied to whole-organ systems, while also including specific domains such as tumors. The solved examples demonstrate the accuracy of this model and its applicability to large biological systems.

Mass release curves as the constitutive curves for modeling diffusive transport within biological tissue.

In diffusion governed by Fick's law, the diffusion coefficient represents the phenomenological material parameter and is, in general, a constant. In certain cases of diffusion through porous media, the diffusion coefficient can be variable (i.e. non-constant) due to the complex process of solute displacements within microstructure, since these displacements depend on porosity, internal microstructural geometry, size of the transported particles, chemical nature, and physical interactions between the diffusing substance and the microstructural surroundings. In order to provide a simple and general approach of determining the diffusion coefficient for diffusion through porous media, we have introduced mass release curves as the constitutive curves of diffusion. The mass release curve for a selected direction represents cumulative mass (per surface area) passed in that direction through a small reference volume, in terms of time. We have developed a methodology, based on numerical Finite Element (FE) and Molecular Dynamics (MD) methods, to determine simple mass release curves of solutes through complex media from which we calculate the diffusion coefficient. The diffusion models take into account interactions between solute particles and microstructural surfaces, as well as hydrophobicity (partitioning). We illustrate the effectiveness of our approach on several examples of complex composite media, including an imaging-based analysis of diffusion through pancreatic cancer tissue. The presented work offers an insight into the role of mass release curves in describing diffusion through porous media in general, and further in case of complex composite media such as biological tissue.

A composite smeared finite element for mass transport in capillary systems and biological tissue.

One of the key processes in living organisms is mass transport occurring from blood vessels to tissues for supplying tissues with oxygen, nutrients, drugs, immune cells, and - in the reverse direction - transport of waste products of cell metabolism to blood vessels. The mass exchange from blood vessels to tissue and vice versa occurs through blood vessel walls. This vital process has been investigated experimentally over centuries, and also in the last decades by the use of computational methods. Due to geometrical and functional complexity and heterogeneity of capillary systems, it is however not feasible to model in silico individual capillaries (including transport through the walls and coupling to tissue) within whole organ models. Hence, there is a need for simplified and robust computational models that address mass transport in capillary-tissue systems. We here introduce a smeared modeling concept for gradient-driven mass transport and formulate a new composite smeared finite element (CSFE). The transport from capillary system is first smeared to continuous mass sources within tissue, under the assumption of uniform concentration within capillaries. Here, the fundamental relation between capillary surface area and volumetric fraction is derived as the basis for modeling transport through capillary walls. Further, we formulate the CSFE which relies on the transformation of the one-dimensional (1D) constitutive relations (for transport within capillaries) into the continuum form expressed by Darcy's and diffusion tensors. The introduced CSFE is composed of two volumetric parts - capillary and tissue domains, and has four nodal degrees of freedom (DOF): pressure and concentration for each of the two domains. The domains are coupled by connectivity elements at each node. The fictitious connectivity elements take into account the surface area of capillary walls which belongs to each node, as well as the wall material properties (permeability and partitioning). The overall FE model contains geometrical and material characteristics of the entire capillary-tissue system, with physiologically measurable parameters assigned to each FE node within the model. The smeared concept is implemented into our implicit-iterative FE scheme and into FE package PAK. The first three examples illustrate accuracy of the CSFE element, while the liver and pancreas models demonstrate robustness of the introduced methodology and its applicability to real physiological conditions.

A multiscale MD-FE model of diffusion in composite media with internal surface interaction based on numerical homogenization procedure.

Mass transport by diffusion within composite materials may depend not only on internal microstructural geometry, but also on the chemical interactions between the transported substance and the material of the microstructure. Retrospectively, there is a gap in methods and theory to connect material microstructure properties with macroscale continuum diffusion characteristics. Here we present a new hierarchical multiscale model for diffusion within composite materials that couples material microstructural geometry and interactions between diffusing particles and the material matrix. This model, which bridges molecular dynamics (MD) and the finite element (FE) method, is employed to construct a continuum diffusion model based on a novel numerical homogenization procedure. The procedure is general and robust for evaluating constitutive material parameters of the continuum model. These parameters include the traditional bulk diffusion coefficients and, additionally, the distances from the solid surface accounting for surface interaction effects. We implemented our models to glucose diffusion through the following two geometrical/material configurations: tightly packed silica nanospheres, and a complex fibrous structure surrounding nanospheres. Then, rhodamine 6G diffusion analysis through an aga-rose gel network was performed, followed by a model validation using our experimental results. The microstructural model, numerical homogenization and continuum model offer a new platform for modeling and predicting mass diffusion through complex biological environment and within composite materials that are used in a wide range of applications, like drug delivery and nanoporous catalysts.

Interfacial effects on nanoconfined diffusive mass transport regimes.

A hierarchical multiscale modeling approach, incorporating molecular dynamics and finite element techniques, is used to study parametrically diffusion regimes through nanoconfined fluid. Novel parameters that determine the character of the diffusion regime and diffusion kinetics within the nanoscale confined fluids is established by exploring diffusion where the interface effects at the solid surface are important. New diffusion transport characteristics are established when nanochannel confining dimension approaches 3-4 sizes of diffusing molecules, which also marks peripheries of the non-fickian transport regime.

Confinement effects on monosaccharide transport in nanochannels.

Transport theories based on the continuum hypothesis may not be appropriate at the nanoscale in view of surface effects. We employed molecular dynamics simulations to study the effects of confinement and concentration on diffusive transport of glucose in silica nanochannels (10 nm or smaller). We found that glucose modifies the electrical properties of nanochannels and that, below 5 nm in channel height, glucose adsorption and diffusivity are significantly reduced. With increasing concentration, the diffusivity is reduced linearly in the bulk, while it is reduced nonlinearly at the interface. The effective diffusivity reduction is related to the interface thickness, which can be 2-4 nm depending on concentration, and has an unexpected reduction at low concentrations. Results suggest that nanochannels present a one-dimensional cage environment that affects diffusivity in a fashion similar to cage-breaking diffusion. Our simulation results, consistent with the experimental observations presented here, suggest that nanoconfinement is the essential cause of the observed altered fluid diffusive transport, not accounted for by classical theories, because of coupling of confinement and concentration effects.

Nanochannel systems for personalized therapy and laboratory diagnostics.

Significant recent progress has been made in the development of microfabricated nanofluidic devices for use in the biomedical sciences. Novel nanotechnological approaches have been explored in view of a more individualized medical approach. Much of the development has been fuelled by the advantages derived from utilizing nanoscale phenomena to manipulate fluid samples or mediate drug delivery. As such, we present a comprehensive review of nanochannel technologies, highlighting their potential for diagnostic and therapeutic applications.

Molecular modelling of K(ATP) channel blockers-ADP/ ATP carrier interactions.

The modelling of molecule-molecule interactions has been widely accepted as a tool for drug discovery and development studies. However, this powerful technique is unappreciated in physiological and biochemical studies, where it could be extremely useful for understanding the mechanisms of action of various compounds in cases when experimental data are controversial due to complexity of the investigated systems. In this study, based on the biochemical data suggesting involvement of mitochondrial ADP/ATP carrier in K+ and H+ transport to mitochondrial matrix molecular modelling is applied to elucidate the possible interactions between the ADP/ATP carrier and its putative ligands--K(ATP) channel blockers glybenclamide, tolbutamide and 5-hydroxydecanoate. Results revealed that K(ATP) channel blockers could bind to the specific location proximal to H1, H4, H5 and H6 transmembrane helices within the cavity of the ADP/ ATP carrier. Analysis of the predicted binding site suggests that K(ATP) channel blockers could interfere with both the ADP/ATP translocation and possible cation flux through the ADP/ATP carrier, and supports the hypothesis that the ADP/ATP carrier is a target of K(ATP) channel modulators.

Probing Aspergillus niger glucose oxidase with pentacyanoferrate(III) aza- and thia-complexes.

Complexes of pentacyanoferrate(III) and biologically relevant ligands, such as pyridine, pyrazole, imidazole, histidine, and other aza- and thia-heterocycles, were synthesized. Their spectral, electrochemical properties, electron exchange constants, electronic structure parameters, and reactivity with glucose oxidase from Aspergillus niger were determined. The formation of the complexes following ammonia replacement by the ligands was associated with the appearance of a new band of absorbance in the visible spectrum. The constants of the complexes formation calculated at a ligand-pentacyanoferrate(III) concentrations ratio of 10:1, were 7.5 x 10(-5), 7.7 x 10(-5), and 1.8 x 10(-3) s(-1) for benzotriazole, benzimidazole, and aminothiazole ligands, respectively. The complexes showed quasi-reversible redox conversion at a glassy carbon electrode. The redox potential of the complexes spanned the potential range from 70 to 240 mV vs. saturated calomel electrode (SCE) at pH7.2. For most of the complexes self-exchange constants (k(11)) were similar to or larger than that of hexacyanoferrate(III) (ferricyanide). The complexes containing pyridine derivatives and thia-heterocyclic ligands held a lower value of k(11) than that of ferricyanide. All complexes reacted with reduced glucose oxidase at pH7.2. The reactivity of the complex containing pyrazole was the largest in comparison to the rest of the complexes. Correlations between the complexes' reactivity and both the free energy of reaction and k(11) shows that the reactivity of pentacyanoferrates obeys the principles of Marcus's electron transfer theory. The obtained data suggest that large negative charges of the complexes decrease their reactivity.

Heme peroxidase clothing and inhibition with polyphenolic substances revealed by molecular modeling.

Molecular modeling techniques were applied to study oligomeric derivatives of phenols, which are produced during peroxidase-catalyzed oxidation. The interaction of substrates and oligomers with Arthromyces ramosus peroxidase (ARP) was analyzed by docking and molecular dynamics methods. The most possible interaction site of oligomers is the active center of the peroxidase. The affinity of oligomers increases with increasing length of oligomers. However, the complexed oligomers produce non-productive complexes with the peroxidase. Molecular dynamics studies showed that oligomer-peroxidase complexes are stable. It seems likely that strong and stable, but non-productive docking of the oligomers determinates peroxidase inhibition during the reaction by preventing the access of regular substrates to the active center of the enzyme.

A role of proton transfer in peroxidase-catalyzed process elucidated by substrates docking calculations.

BACKGROUND: Previous kinetic investigations of fungal-peroxidase catalyzed oxidation of N-aryl hydroxamic acids (AHAs) and N-aryl-N-hydroxy urethanes (AHUs) revealed that the rate of reaction was independent of the formal redox potential of substrates. Moreover, the oxidation rate was 3-5 orders of magnitude less than for oxidation of physiological phenol substrates, though the redox potential was similar. RESULTS: To explain the unexpectedly low reactivity of AHAs and AHUs we made ab initio calculations of the molecular structure of the substrates following in silico docking in the active center of the enzyme. CONCLUSIONS: AHAs and AHUs were docked at the distal side of heme in the sites formed by hydrophobic amino acid residues that retarded a proton transfer and finally the oxidation rate. The analogous phenol substrates were docked at different sites permitting fast proton transfer in the relay of distal His and water that helped fast substrate oxidation.

Kinetics and thermodynamics of peroxidase- and laccase-catalyzed oxidation of N-substituted phenothiazines and phenoxazines.

Steady-state and single-turnover kinetics for the oxidation of the N-substituted phenothiazines (PTs) and phenoxazines (POs) catalyzed by fungal Coprinus cinereus peroxidase and Polyporus pinsitus laccase were investigated at pH 4-10. In the case of peroxidase, an apparent bimolecular rate constant (expressed as k(cat)/K(m)) varied from 1 x10(7)M(-1)s(-1) to 2.6 x 108 M(-1)s(-1) at pH 7.0. The constants for PO oxidation were higher in comparison to PT. pH dependence revealed two or three ionizable groups with pKa values of 4.9-5.7 and 7.7-9.7 that significantly affected the activity of peroxidase. Single-turnover experiments showed that the limiting step of PT oxidation was reduction of compound II and second-order rate constants were obtained which were consistent with the constants at steady-state conditions. Laccase-catalyzed PT and PO oxidation rates were lower; apparent bimolecular rate constants varied from 1.8x 10(5) M(-1) s(-1) to 2.0 x 10(7) M(-1) s(-1) at pH 5.3. PO constants were higher in comparison to PT, as was the case with peroxidase. The dependence of the apparent bimolecular constants of compound II or copper type 1 reduction, in the case of peroxidase or laccase, respectively, was analyzed in the framework of the Marcus outer-sphere electron-transfer theory. Peroxidase-catalyzed reactions with PT, as well as PO, fitted the same hyperbolic dependence with a maximal oxidation rate of 1.6 x 10(8)M(-1)s(-1) and a reorganization energy of 0.30 eV. The respective parameters for laccase were 5.0 x 10(7) M(-1) s(-1) and 0.29 eV.