A three-dimensional multiscale model for the prediction of thrombus growth under flow with single-platelet resolution

Modeling thrombus growth in pathological flows allows evaluation of risk under patient-specific pharmacological, hematological, and hemodynamical conditions. We have developed a 3D multiscale framework for the prediction of thrombus growth under flow on a spatially resolved surface presenting collagen and tissue factor (TF). The multiscale framework is composed of four coupled modules: a Neural Network (NN) that accounts for platelet signaling, a Lattice Kinetic Monte Carlo (LKMC) simulation for tracking platelet positions, a Finite Volume Method (FVM) simulator for solving convection-diffusion-reaction equations describing agonist release and transport, and a Lattice Boltzmann (LB) flow solver for computing the blood flow field over the growing thrombus. A reduced model of the coagulation cascade was embedded into the framework to account for TF-driven thrombin production. The 3D model was first tested against in vitro microfluidics experiments of whole blood perfusion with various antiplatelet agents targeting COX-1, P₂Y₁, or the IP receptor. The model was able to accurately capture the evolution and morphology of the growing thrombus. Certain problems of 2D models for thrombus growth (artifactual dendritic growth) were naturally avoided with realistic trajectories of platelets in 3D flow. The generalizability of the 3D multiscale solver enabled simulations of important clinical situations, such as cylindrical blood vessels and acute flow narrowing (stenosis). Enhanced platelet-platelet bonding at pathologically high shear rates (e.g., von Willebrand factor unfolding) was required for accurately describing thrombus growth in stenotic flows. Overall, the approach allows consideration of patient-specific platelet signaling and vascular geometry for the prediction of thrombotic episodes.     

A multiscale model for avascular tumor growth

The desire to understand tumor complexity has given rise to mathematical models to describe the tumor microenvironment. We present a new mathematical model for avascular tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion, and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter, and inhibitor concentrations). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. Based on the simulations, we predict: 1), the microenvironmental conditions required for tumor cell survival; and 2), growth promoters and inhibitors have diffusion coefficients in the range between 10(-6) and 10(-7) cm2/h, corresponding to molecules of size 80-90 kDa. Using the same parameters, the model also accurately predicts spheroid growth curves under different external nutrient supply conditions.     

A multiscale model for red blood cell mechanics

The objective of this article is the derivation of a continuum model for mechanics of red blood cells via multiscale analysis. On the microscopic level, we consider realistic discrete models in terms of energy functionals defined on networks/lattices. Using concepts of Γ-convergence, convergence results as well as explicit homogenisation formulae are derived. Based on a characterisation via energy functionals, appropriate macroscopic stress–strain relationships (constitutive equations) can be determined. Further, mechanical moduli of the derived macroscopic continuum model are directly related to microscopic moduli. As a test case we consider optical tweezers experiments, one of the most common experiments to study mechanical properties of cells. Our simulations of the derived continuum model are based on finite element methods and account explicitly for membrane mechanics and its coupling with bulk mechanics. Since the discretisation of the continuum model can be chosen freely, rather than it is given by the topology of the microscopic cytoskeletal network, the approach allows a significant reduction of computational efforts. Our approach is highly flexible and can be generalised to many other cell models, also including biochemical control.     

A mathematical model of alveolar gas exchange in partial liquid ventilation

In partial liquid ventilation (PLV), perfluorocarbon (PFC) acts as a diffusion barrier to gas transport in the alveolar space since the diffusivities of oxygen and carbon dioxide in this medium are four orders of magnitude lower than in air. Therefore convection in the PFC layer resulting from the oscillatory motions of the alveolar sac during ventilation can significantly affect gas transport. For example, a typical value of the Péclet number in air ventilation is Pe approximately 0.01, whereas in PLV it is Pe approximately 20. To study the importance of convection, a single terminal alveolar sac is modeled as an oscillating spherical shell with gas, PFC, tissue and capillary blood compartments. Differential equations describing mass conservation within each compartment are derived and solved to obtain time periodic partial pressures. Significant partial pressure gradients in the PFC layer and partial pressure differences between the capillary and gas compartments (P(C)-Pg) are found to exist. Because Pe> 1, temporal phase differences are found to exist between P(C)-Pg and the ventilatory cycle that cannot be adequately described by existing non-convective models of gas exchange in PLV The mass transfer rate is nearly constant throughout the breath when Pe>1, but when Pe<1 nearly 100% of the transport occurs during inspiration. A range of respiratory rates (RR), including those relevant to high frequency oscillation (HFO) +PLV, tidal volumes (V(T)) and perfusion rates are studied to determine the effect of heterogeneous distributions of ventilation and perfusion on gas exchange. The largest changes in P(C)O2 and P(C)CO2 occur at normal and low perfusion rates respectively as RR and V(T) are varied. At a given ventilation rate, a low RR-high V(T) combination results in higher P(C)O2, lower P(C)CO2 and lower (P(C)-Pg) than a high RR-low V(T) one.     

A multiscale model for multiple platelet aggregation in shear flow

Biomechanics and Modeling in Mechanobiology - We developed a multiscale model for simulating aggregation of multiple, free-flowing platelets in low–intermediate shear viscous flow, in which...     

A mathematical model of pulmonary gas exchange under inflammatory stress

During a severe local or systemic inflammatory response, immune mediators target lung tissue. This process may lead to acute lung injury and impaired diffusion of gas molecules. Although several mathematical models of gas exchange have been described, none simulate acute lung injury following inflammatory stress. In view of recent laboratory and clinical progress in the understanding of the pathophysiology of acute lung injury, such a mathematical model would be useful. We first derived a partial differential equations model of gas exchange on a small physiological unit of the lung (≈25 alveoli), which we refer to as a respiratory unit (RU). We next developed a simple model of the acute inflammatory response and implemented its effects within a RU, creating a single RU model. Linking multiple RUs with various ventilation/perfusion ratios and taking into account pulmonary venous blood remixing yielded our lung-scale model. Using the lung-scale model, we explored the predicted effects of inflammation on ventilation/perfusion distribution and the resulting pulmonary venous partial pressure oxygen level during systemic inflammatory stresses. This model represents a first step towards the development of anatomically faithful models of gas exchange and ventilation under a broad range of local and systemic inflammatory stimuli resulting in acute lung injury, such as infection and mechanical strain of lung tissue.     

A model evaluation of estimates of breath-to-breath alveolar gas exchange

A mathematical model has been implemented for evaluation of methods for estimating breath-to-breath alveolar gas exchange during exercise in humans. This model includes a homogeneous alveolar gas exchange compartment, a dead space compartment, and tissue spaces for CO2 (alveolar and dead space). The dead space compartment includes a mixing portion surrounded by tissue and an unmixed (slug flow) portion which is partitioned between anatomical and apparatus contributions. A random sinusoidal flow pattern generates a breath-to-breath variation in pulmonary stores. The Auchincloss algorithm for estimating alveolar gas exchange (Auchincloss et al., J. Appl. Physiol. 21: 810-818, 1966) was applied to the model, and the results were compared with the simulated gas exchange. This comparison indicates that a compensation for changes in pulmonary stores must include factors for alveolar gas concentration change as well as alveolar volume change and thus implies the use of end-tidal measurements. Although this algorithm yields reasonable estimates of breath-to-breath alveolar gas exchange, it does not yield a "true" indirect measurement because of inherent error in the estimation of a homogeneous alveolar gas concentration at the end of expiration.     

A multiscale model for plant invasion through allelopathic suppression

In the present paper, we propose and study by numerical simulations a multiscale model for plant invasion based on allelopathic suppression in a homogeneous environment. The negative effects on seed production and germination, establishment and mortality of native plants generated by the root-secreted alien phytotoxin constitute the basic mechanism contributing to invasiveness. We obtained the invasion patterns, their success probabilities, the time evolution of plant populations, the gyration radius and the border roughness of the invaded region. As an important result, it was observed that, in addition to the phytotoxin nature (synthesis and degradation rates, diffusivity and phytotoxic threshold), invasive patterns and invasion success depend on the kind of native plants present in the area. In fact, both success and invasion speed decrease in the presence of resistant native plants. Also, self-affine invasion fronts are smooth (Hurst exponent H = 1) in the absence of resistant plants, but are rough (H ≠ 1) on the contrary. Furthermore, if the resistant native species are randomly distributed on the landscape, the invasion front exhibits long-range correlations (H ～ 0.76), while its border is anti-correlated (H ～ 0.20), if resistant plants are distributed in patches. Finally, the cluster size distribution functions of resistant plants are exponentials with characteristic cluster sizes increasing in time.     

Three-dimensional gas exchange pathways in pome fruit characterized by synchrotron x-ray computed tomography

Our understanding of the gas exchange mechanisms in plant organs critically depends on insights in the three-dimensional (3-D) structural arrangement of cells and voids. Using synchrotron radiation x-ray tomography, we obtained for the first time high-contrast 3-D absorption images of in vivo fruit tissues of high moisture content at 1.4-microm resolution and 3-D phase contrast images of cell assemblies at a resolution as low as 0.7 microm, enabling visualization of individual cell morphology, cell walls, and entire void networks that were previously unknown. Intercellular spaces were always clear of water. The apple (Malus domestica) cortex contains considerably larger parenchyma cells and voids than pear (Pyrus communis) parenchyma. Voids in apple often are larger than the surrounding cells and some cells are not connected to void spaces. The main voids in apple stretch hundreds of micrometers but are disconnected. Voids in pear cortex tissue are always smaller than parenchyma cells, but each cell is surrounded by a tight and continuous network of voids, except near brachyssclereid groups. Vascular and dermal tissues were also measured. The visualized network architecture was consistent over different picking dates and shelf life. The differences in void fraction (5.1% for pear cortex and 23.0% for apple cortex) and in gas network architecture helps explain the ability of tissues to facilitate or impede gas exchange. Structural changes and anisotropy of tissues may eventually lead to physiological disorders. A combined tomography and internal gas analysis during growth are needed to make progress on the understanding of void formation in fruit.     

A replication model for sister-chromatid exchange

A model for the production of sister-chromatid exchanges is presented, based on the idea that double-strand breaks are generated at junctions between a completely duplicated replicon cluster and a partially duplicated replicon cluster. Agents that induce absolute blocks to DNA fork displacement will cause this condition to persist longer than normal, whereas agents that inhibit initiation of whole clusters will rarely cause it at all. During the blunt-end repair of the double-strand breaks, sister-chromatid exchange would be initiated when daughter strands of a duplicated cluster recombine with the parental strands of the partially replicated cluster. When the latter finishes replication, sister-chromatid exchange would be completed.     

A multiscale theoretical model for diffusive mass transfer in cellular biological media

An integrated methodology is developed for the theoretical analysis of solute transport and reaction in cellular biological media, such as tissues, microbial flocs, and biofilms. First, the method of local spatial averaging with a weight function is used to establish the equation which describes solute conservation at the cellular biological medium scale, starting with a continuum-based formulation of solute transport at finer spatial scales. Second, an effective-medium model is developed for the self-consistent calculation of the local diffusion coefficient in the cellular biological medium, including the effects of the structural heterogeneity of the extra-cellular space and the reversible adsorption to extra-cellular polymers. The final expression for the local effective diffusion coefficient is: D(Abeta)=lambda(beta)D(Aupsilon), where D(Aupsilon) is the diffusion coefficient in water, and lambda(beta) is a function of the composition and fundamental geometric and physicochemical system properties, including the size of solute molecules, the size of extra-cellular polymer fibers, and the mass permeability of the cell membrane. Furthermore, the analysis sheds some light on the function of the extra-cellular hydrogel as a diffusive barrier to solute molecules approaching the cell membrane, and its implications on the transport of chemotherapeutic agents within a cellular biological medium. Finally, the model predicts the qualitative trend as well as the quantitative variability of a large number of published experimental data on the diffusion coefficient of oxygen in cell-entrapping gels, microbial flocs, biofilms, and mammalian tissues.     

A model of the gas exchange response ofPicea abies to habitat conditions

Databases describing branch gas exchange ofPicea abies L. at two montane forest sites, Lägeren, Switzerland (National Forschungsprojekt 14 of the Schweizerische Nationalfonds) and Oberwarmensteinach, Germany (Bayerische Forschungsgruppe Forsttoxikologie), were analyzed in conjunction with a physiologically based model. Parameter estimates for describing carboxylase kinetics, electron transport, and stomatal function were derived, utilizing information from both single factor dependencies and diurnal time course measurements of gas exchange. Data subsets were used for testing the model at the branch level. Most of the observed variation in gas exchange characteristics can be explained with the model, while a number of systematic errors remain unexplained. Factors seen as contributing to the unexplained residual variation and not included in the model are light acclimation, degree of damage in adjustment to pollutant deposition, needle age, and cold stress effects. Nevertheless, a set of parameter values has been obtained for general application with spruce, e.g., for use in calculating canopy flux rates and to aid in planning of focused leaf and canopy level experiments. The value of the model for estimating fluxes between the forest and the atmosphere must be evaluated together with measurements at the stand level.     

A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis

We present a novel computational model for maladaptive cardiac growth in which kinematic changes of the cardiac chambers are attributed to alterations in cytoskeletal architecture and in cellular morphology. We adopt the concept of finite volume growth characterized through the multiplicative decomposition of the deformation gradient into an elastic part and a growth part. The functional form of its growth tensor is correlated to sarcomerogenesis, the creation and deposition of new sarcomere units. In response to chronic volume-overload, an increased diastolic wall strain leads to the addition of sarcomeres in series, resulting in a relative increase in cardiomyocyte length, associated with eccentric hypertrophy and ventricular dilation. In response to chronic pressure-overload, an increased systolic wall stress leads to the addition of sacromeres in parallel, resulting in a relative increase in myocyte cross sectional area, associated with concentric hypertrophy and ventricular wall thickening. The continuum equations for both forms of maladaptive growth are discretized in space using a nonlinear finite element approach, and discretized in time using the implicit Euler backward scheme. We explore a generic bi-ventricular heart model in response to volume- and pressure-overload to demonstrate how local changes in cellular morphology translate into global alterations in cardiac form and function.