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 patient-specific aortic valve model based on moving resistive immersed implicit surfaces

In this paper, we propose a full computational framework to simulate the hemodynamics in the aorta including the valve. Closed and open valve surfaces, as well as the lumen aorta, are reconstructed directly from medical images using new ad hoc algorithms, allowing a patient-specific simulation. The fluid dynamics problem that accounts from the movement of the valve is solved by a new 3D–0D fluid–structure interaction model in which the valve surface is implicitly represented through level set functions, yielding, in the Navier–Stokes equations, a resistive penalization term enforcing the blood to adhere to the valve leaflets. The dynamics of the valve between its closed and open position is modeled using a reduced geometric 0D model. At the discrete level, a finite element formulation is used and the SUPG stabilization is extended to include the resistive term in the Navier–Stokes equations. Then, after time discretization, the 3D fluid and 0D valve models are coupled through a staggered approach. This computational framework, applied to a patient-specific geometry and data, allows to simulate the movement of the valve, the sharp pressure jump occurring across the leaflets, and the blood flow pattern inside the aorta.     

Microtubule dynamic instability: A new model with coupled GTP hydrolysis and multistep catastrophe

A key question in understanding microtubule dynamics is how GTP hydrolysis leads to catastrophe, the switch from slow growth to rapid shrinkage. We first provide a review of the experimental and modeling literature, and then present a new model of microtubule dynamics. We demonstrate that vectorial, random, and coupled hydrolysis mechanisms are not consistent with the dependence of catastrophe on tubulin concentration and show that, although single‐protofilament models can explain many features of dynamics, they do not describe catastrophe as a multistep process. Finally, we present a new combined (coupled plus random hydrolysis) multiple‐protofilament model that is a simple, analytically solvable generalization of a single‐protofilament model. This model accounts for the observed lifetimes of growing microtubules, the delay to catastrophe following dilution and describes catastrophe as a multistep process.     

Microtubule dynamics in root hairs of Medicago truncatula

The microtubular cytoskeleton plays an important role in the development of tip-growing plant cells, but knowledge about its dynamics is incomplete. In this study, root hairs of the legume Medicago truncatula have been chosen for a detailed analysis of microtubular cytoskeleton dynamics using GFP-MBD and EB1-YFP as markers and 4D imaging. The microtubular cytoskeleton appears mainly to be composed of bundles which form tracks along which new microtubules polymerise. Polymerisation rates of microtubules are highest in the tip of growing root hairs. Treatment of root hairs with Nod factor and latrunculin B result in a twofold decrease in polymerisation rate. Nonetheless, no direct, physical interaction between the actin filament cytoskeleton and microtubules could be observed. A new picture of how the plant cytoskeleton is organised in apically growing root hairs emerges from these observations, revealing similarities with the organisation in other, non-plant, tip-growing cells.     

A one-dimensional model of water flow in soil-plant systems based on plant architecture

The estimation of root water uptake and water flow in plants is crucial to quantify transpiration and hence the water exchange between land surface and atmosphere. In particular the soil water extraction by plant roots which provides the water supply of plants is a highly dynamic and non-linear process interacting with soil transport processes that are mainly determined by the natural soil variability at different scales. To better consider this root-soil interaction we extended and further developed a finite element tree hydro-dynamics model based on the one-dimensional (1D) porous media equation. This is achieved by including in addition to the explicit three-dimensional (3D) architectural representation of the tree crown a corresponding 3D characterisation of the root system. This 1D xylem water flow model was then coupled to a soil water flow model derived also from the 1D porous media equation. We apply the new model to conduct sensitivity analysis of root water uptake and transpiration dynamics and compare the results to simulation results obtained by using a 3D model of soil water flow and root water uptake. Based on data from lysimeter experiments with young European beech trees (Fagus silvatica L.) is shown, that the model is able to correctly describe transpiration and soil water flow. In conclusion, compared to a fully 3D model the 1D porous media approach provides a computationally efficient alternative, able to reproduce the main mechanisms of plant hydro-dynamics including root water uptake from soil.     

Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh–Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.     

A model of growth and rupture of abdominal aortic aneurysm

We present here a coupled mathematical model of growth and failure of the abdominal aortic aneurysm (AAA). The failure portion of the model is based on the constitutive theory of softening hyperelasticity where the classical hyperelastic law is enhanced with a new constant indicating the maximum energy that an infinitesimal material volume can accumulate without failure. The new constant controls material failure and it can be interpreted as the average energy of molecular bonds from the microstructural standpoint. The constitutive model is compared to the data from uniaxial tension tests providing an excellent fit to the experiment. The AAA failure model is coupled with a phenomenological theory of soft tissue growth. The unified theory includes both momentum and mass balance laws coupled with the help of the constitutive equations. The microstructural alterations in the production of elastin and remodeling of collagen are reflected in the changing macroscopic parameters characterizing tissue stiffness, strength and density. The coupled theory is used to simulate growth and rupture of an idealized spherical AAA. The results of the simulation showing possible AAA ruptures in growth are reasonable qualitatively while the quantitative calibration of the model will require further clinical observations and in vitro tests. The presented model is the first where growth and rupture are coupled.     

Musculotendon lengths and moment arms for a three-dimensional upper-extremity model

Generating muscle-driven forward dynamics simulations of human movement using detailed musculoskeletal models can be computationally expensive. This is due in part to the time required to calculate musculotendon geometry (e.g., musculotendon lengths and moment arms), which is necessary to determine and apply individual musculotendon forces during the simulation. Modeling upper-extremity musculotendon geometry can be especially challenging due to the large number of multi-articular muscles and complex muscle paths. To accurately represent this geometry, wrapping surface algorithms and/or other computationally expensive techniques (e.g., phantom segments) are used. This paper provides a set of computationally efficient polynomial regression equations that estimate musculotendon length and moment arms for thirty-two (32) upper-extremity musculotendon actuators representing the major muscles crossing the shoulder, elbow and wrist joints. Equations were developed using a least squares fitting technique based on geometry values obtained from a validated public-domain upper-extremity musculoskeletal model that used wrapping surface elements (Holzbaur et al., 2005). In general, the regression equations fit well the original model values, with an average root mean square difference for all musculotendon actuators over the represented joint space of 0.39 mm (1.1% of peak value). In addition, the equations reduced the computational time required to simulate a representative upper-extremity movement (i.e., wheelchair propulsion) by more than two orders of magnitude (315 versus 2.3 s). Thus, these equations can assist in generating computationally efficient forward dynamics simulations of a wide range of upper-extremity movements.     

Unique responses of respiration,growth, and non-structural carbohydrate storage in sink tissue of conifer seedlings to an elevation gradient at timberline

The role of carbon balance, and particularly carbon sinks, to forest boundaries and climate responses is a major question in forest ecophysiology. At timberline, low-temperature limitations on carbon-sink processes of stem and especially root tissue have been implicated in hypotheses of the upper range limits to tree distributions. Studies on carbon sinks in root and stem tissue of trees at timberline typically report variation in only one carbon sink, such as either growth, respiration, or non-structural carbohydrates (NSCs). However, these carbon sinks may differ in their response to elevation. We asked how three carbon-sink processes in root and stem tissue (i.e. all tissue below the crown of needles) change in conifer seedlings growing from the lower (2450 m) to the upper (3000 m) edges of the timberline ecotone throughout their first growth season. We repeatedly measured respiration (mg⁻¹ and individual⁻¹), growth (relative growth rates [RGR] and biomass), and NSCs in root and stem tissue of Abies lasiocarpa and Pseudotsuga menziesii.RGR of root and stem tissue were less at the upper compared to lower elevation, but only for the first few weeks of the growing season. Total biomass of root and stem tissue was generally less at the upper site, apparently due to low early season RGR, but ultimately did not significantly differ between sites by the end of the growing season. Unlike growth, respiration rates (mg⁻¹) did not differ between elevations during any period of the growing season. Nevertheless, total respiration of CO₂ from root and stem tissue (individual⁻¹) was 22% less at the upper site, which was attributable to less biomass. NSCs of root and stem tissue, specifically starch, were overall greater at the upper site, particularly for A. lasiocarpa at the end of the season, which did not parallel spatiotemporal trends in growth or respiration. The differences in seasonal trends and the effects of elevation on carbon sinks indicate a degree of independence or uncoupling of growth, respiration, and NSCs of root and stem tissue, which is not commonly appreciated in hypotheses about physiological limitations for trees at timberline.     

Simulating tumor growth in confined heterogeneous environments

The holy grail of computational tumor modeling is to develop a simulation tool that can be utilized in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies. In order to develop such a predictive model, one must account for many of the complex processes involved in tumor growth. One interaction that has not been incorporated into computational models of neoplastic progression is the impact that organ-imposed physical confinement and heterogeneity have on tumor growth. For this reason, we have taken a cellular automaton algorithm that was originally designed to simulate spherically symmetric tumor growth and generalized the algorithm to incorporate the effects of tissue shape and structure. We show that models that do not account for organ/tissue geometry and topology lead to false conclusions about tumor spread, shape and size. The impact that confinement has on tumor growth is more pronounced when a neoplasm is growing close to, versus far from, the confining boundary. Thus, any clinical simulation tool of cancer progression must not only consider the shape and structure of the organ in which a tumor is growing, but must also consider the location of the tumor within the organ if it is to accurately predict neoplastic growth dynamics.     

Computational model of flow-tissue interactions in intussusceptive angiogenesis

Angiogenesis, the growth of vascular structures, is a complex biological process which has long puzzled scientists. Better physiological understanding of this phenomenon could result in many useful medical applications such as the development of new methods for cancer therapy. We report on the development of a simple computational model of micro-vascular structure formation in intussusceptive angiogenesis observed in vivo. The tissue is represented by a discrete set of basic structural entities and flow conditions within the resulting domain are obtained by solving the Navier-Stokes equations. The tissue is then remodelled according to the tangential shear stress while approximating advection by means of simple non-diffusive heuristics. The updated tissue geometry then becomes the input for the next remodelling step. The model, consisting of steady-state flow and a simple mechanistic tissue response, successfully predicts bifurcation formation and micro-vessel separation in a porous cellular medium. This opens new modelling possibilities in computational studies of the cellular transport involved in micro-vascular growth.     

Dynamic root growth and architecture responses to limiting nutrient availability: linking physiological models and experimentation

In recent years the study of root phenotypic plasticity in response to sub-optimal environmental factors and the genetic control of these responses have received renewed attention. As a path to increased productivity, in particular for low fertility soils, several applied research projects worldwide target the improvement of crop root traits both in plant breeding and biotechnology contexts. To assist these tasks and address the challenge of optimizing root growth and architecture for enhanced mineral resource use, the development of realistic simulation models is of great importance. We review this research field from a modeling perspective focusing particularly on nutrient acquisition strategies for crop production on low nitrogen and low phosphorous soils. Soil heterogeneity and the dynamics of nutrient availability in the soil pose a challenging environment in which plants have to forage efficiently for nutrients in order to maintain their internal nutrient homeostasis throughout their life cycle. Mathematical models assist in understanding plant growth strategies and associated root phenes that have potential to be tested and introduced in physiological breeding programs. At the same time, we stress that it is necessary to carefully consider model assumptions and development from a whole plant-resource allocation perspective and to introduce or refine modules simulating explicitly root growth and architecture dynamics through ontogeny with reference to key factors that constrain root growth. In this view it is important to understand negative feedbacks such as plant–plant competition. We conclude by briefly touching on available and developing technologies for quantitative root phenotyping from lab to field, from quantification of partial root profiles in the field to 3D reconstruction of whole root systems. Finally, we discuss how these approaches can and should be tightly linked to modeling to explore the root phenome.     

Multiscale Systems Analysis of Root Growth and Development: Modeling Beyond the Network and Cellular Scales

Over recent decades, we have gained detailed knowledge of many processes involved in root growth and development. However, with this knowledge come increasing complexity and an increasing need for mechanistic modeling to understand how those individual processes interact. One major challenge is in relating genotypes to phenotypes, requiring us to move beyond the network and cellular scales, to use multiscale modeling to predict emergent dynamics at the tissue and organ levels. In this review, we highlight recent developments in multiscale modeling, illustrating how these are generating new mechanistic insights into the regulation of root growth and development. We consider how these models are motivating new biological data analysis and explore directions for future research. This modeling progress will be crucial as we move from a qualitative to an increasingly quantitative understanding of root biology, generating predictive tools that accelerate the development of improved crop varieties.     

A 3-D model of tumor progression based on complex automata driven by particle dynamics

The dynamics of a growing tumor involving mechanical remodeling of healthy tissue and vasculature is neglected in most of the existing tumor models. This is due to the lack of efficient computational framework allowing for simulation of mechanical interactions. Meanwhile, just these interactions trigger critical changes in tumor growth dynamics and are responsible for its volumetric and directional progression. We describe here a novel 3-D model of tumor growth, which combines particle dynamics with cellular automata concept. The particles represent both tissue cells and fragments of the vascular network. They interact with their closest neighbors via semi-harmonic central forces simulating mechanical resistance of the cell walls. The particle dynamics is governed by both the Newtonian laws of motion and the cellular automata rules. These rules can represent cell life-cycle and other biological interactions involving smaller spatio-temporal scales. We show that our complex automata, particle based model can reproduce realistic 3-D dynamics of the entire system consisting of the tumor, normal tissue cells, blood vessels and blood flow. It can explain phenomena such as the inward cell motion in avascular tumor, stabilization of tumor growth by the external pressure, tumor vascularization due to the process of angiogenesis, trapping of healthy cells by invading tumor, and influence of external (boundary) conditions on the direction of tumor progression. We conclude that the particle model can serve as a general framework for designing advanced multiscale models of tumor dynamics and it is very competitive to the modeling approaches presented before.