A computational model of in vitro angiogenesis based on extracellular matrix fibre orientation

Recent interest in the process of vascularisation within the biomedical community has motivated numerous new research efforts focusing on the process of angiogenesis. Although the role of chemical factors during angiogenesis has been well documented, the role of mechanical factors, such as the interaction between angiogenic vessels and the extracellular matrix, remains poorly understood. In vitro methods for studying angiogenesis exist; however, measurements available using such techniques often suffer from limited spatial and temporal resolutions. For this reason, computational models have been extensively employed to investigate various aspects of angiogenesis. This paper outlines the formulation and validation of a simple and robust computational model developed to accurately simulate angiogenesis based on length, branching and orientation morphometrics collected from vascularised tissue constructs. Microvessels were represented as a series of connected line segments. The morphology of the vessels was determined by a linear combination of the collagen fibre orientation, the vessel density gradient and a random walk component. Excellent agreement was observed between computational and experimental morphometric data over time. Computational predictions of microvessel orientation within an anisotropic matrix correlated well with experimental data. The accuracy of this modelling approach makes it a valuable platform for investigating the role of mechanical interactions during angiogenesis.     

A homogenized constrained mixture (and mechanical analog) model for growth and remodeling of soft tissue

Most mathematical models of the growth and remodeling of load-bearing soft tissues are based on one of two major approaches: a kinematic theory that specifies an evolution equation for the stress-free configuration of the tissue as a whole or a constrained mixture theory that specifies rates of mass production and removal of individual constituents within stressed configurations. The former is popular because of its conceptual simplicity, but relies largely on heuristic definitions of growth; the latter is based on biologically motivated micromechanical models, but suffers from higher computational costs due to the need to track all past configurations. In this paper, we present a temporally homogenized constrained mixture model that combines advantages of both classical approaches, namely a biologically motivated micromechanical foundation, a simple computational implementation, and low computational cost. As illustrative examples, we show that this approach describes well both cell-mediated remodeling of tissue equivalents in vitro and the growth and remodeling of aneurysms in vivo. We also show that this homogenized constrained mixture model suggests an intimate relationship between models of growth and remodeling and viscoelasticity. That is, important aspects of tissue adaptation can be understood in terms of a simple mechanical analog model, a Maxwell fluid (i.e., spring and dashpot in series) in parallel with a “motor element” that represents cell-mediated mechanoregulation of extracellular matrix. This analogy allows a simple implementation of homogenized constrained mixture models within commercially available simulation codes by exploiting available models of viscoelasticity.     

Machine learning for accelerating process-based computation of land biogeochemical cycles

Global change ecology nowadays embraces ever-growing large observational datasets (big-data) and complex mathematical models that track hundreds of ecological processes (big-model). The rapid advancement of the big-data-big-model has reached its bottleneck: high computational requirements prevent further development of models that need to be integrated over long time-scales to simulate the distribution of ecosystems carbon and nutrient pools and fluxes. Here, we introduce a machine-learning acceleration (MLA) tool to tackle this grand challenge. We focus on the most resource-consuming step in terrestrial biosphere models (TBMs): the equilibration of biogeochemical cycles (spin-up), a prerequisite that can take up to 98% of the computational time. Through three members of the ORCHIDEE TBM family part of the IPSL Earth System Model, including versions that describe the complex interactions between nitrogen, phosphorus and carbon that do not have any analytical solution for the spin-up, we show that an unoptimized MLA reduced the computation demand by 77%–80% for global studies via interpolating the equilibrated state of biogeochemical variables for a subset of model pixels. Despite small biases in the MLA-derived equilibrium, the resulting impact on the predicted regional carbon balance over recent decades is minor. We expect a one-order of magnitude lower computation demand by optimizing the choices of machine learning algorithms, their settings, and balancing the trade-off between quality of MLA predictions and need for TBM simulations for training data generation and bias reduction. Our tool is agnostic to gridded models (beyond TBMs), compatible with existing spin-up acceleration procedures, and opens the door to a wide variety of future applications, with complex non-linear models benefit most from the computational efficiency.     

Mechanistic,Mathematical Model to Predict the Dynamics of Tissue Genesis in Bone Defects via Mechanical Feedback and Mediation of Biochemical Factors

The link between mechanics and biology in the generation and the adaptation of bone has been well studied in context of skeletal development and fracture healing. Yet, the prediction of tissue genesis within - and the spatiotemporal healing of - postnatal defects, necessitates a quantitative evaluation of mechano-biological interactions using experimental and clinical parameters. To address this current gap in knowledge, this study aims to develop a mechanistic mathematical model of tissue genesis using bone morphogenetic protein (BMP) to represent of a class of factors that may coordinate bone healing. Specifically, we developed a mechanistic, mathematical model to predict the dynamics of tissue genesis by periosteal progenitor cells within a long bone defect surrounded by periosteum and stabilized via an intramedullary nail. The emergent material properties and mechanical environment associated with nascent tissue genesis influence the strain stimulus sensed by progenitor cells within the periosteum. Using a mechanical finite element model, periosteal surface strains are predicted as a function of emergent, nascent tissue properties. Strains are then input to a mechanistic mathematical model, where mechanical regulation of BMP-2 production mediates rates of cellular proliferation, differentiation and tissue production, to predict healing outcomes. A parametric approach enables the spatial and temporal prediction of endochondral tissue regeneration, assessed as areas of cartilage and mineralized bone, as functions of radial distance from the periosteum and time. Comparing model results to histological outcomes from two previous studies of periosteum-mediated bone regeneration in a common ovine model, it was shown that mechanistic models incorporating mechanical feedback successfully predict patterns (spatial) and trends (temporal) of bone tissue regeneration. The novel model framework presented here integrates a mechanistic feedback system based on the mechanosensitivity of periosteal progenitor cells, which allows for modeling and prediction of tissue regeneration on multiple length and time scales. Through combination of computational, physical and engineering science approaches, the model platform provides a means to test new hypotheses in silico and to elucidate conditions conducive to endogenous tissue genesis. Next generation models will serve to unravel intrinsic differences in bone genesis by endochondral and intramembranous mechanisms.     

A mechano-regulatory bone-healing model incorporating cell-phenotype specific activity

Phenomenological computational models of tissue regeneration and bone healing have been only partially successful in predicting experimental observations. This may be a result of simplistic modeling of cellular activity. Furthermore, phenomenological models are limited when considering the effects of combined physical and biological interventions. In this study, a new model of cell and tissue differentiation, using a more mechanistic approach, is presented and applied to fracture repair. The model directly couples cellular mechanisms to mechanical stimulation during bone healing and is based on the belief that the cells act as transducers during tissue regeneration. In the model, the cells within the matrix proliferate, differentiate, migrate, and produce extracellular matrix, all at cell-phenotype specific rates, based on the mechanical stimulation they experience. The model is assembled from coupled partial differentiation equations, which are solved using a newly developed finite element formulation. The evolution of four cell types, i.e. mesenchymal stem cells, fibroblasts, chondrocytes and osteoblasts, and the production of extracellular matrices of fibrous tissue, cartilage and bone are calculated. The material properties of the tissues are iteratively updated based on actual amounts of extracellular matrix in material elements at progressive time points. A two-dimensional finite element model of a long bone osteotomy was used to evaluate the model's potential. The additional value of the presented model and the importance of including cell-phenotype specific activities when modeling tissue differentiation and bone healing, were demonstrated by comparing the predictions with phenomenological models. The model's capacity was established by showing that it can correctly predict several aspects of bone healing, including cell and tissue distributions during normal fracture healing. Furthermore, it was able to predict experimentally established alterations due to excessive mechanical stimulation, periosteal stripping and impaired effects of cartilage remodeling.     

Method for generation of homogeneous multicellular tumor spheroids applicable to a wide variety of cell types

Multicellular tumor spheroids (MCTS) are used as organotypic models of normal and solid tumor tissue. Traditional techniques for generating MCTS, such as growth on nonadherent surfaces, in suspension, or on scaffolds, have a number of drawbacks, including the need for manual selection to achieve a homogeneous population and the use of nonphysiological matrix compounds. In this study we describe a mild method for the generation of MCTS, in which individual spheroids form in hanging drops suspended from a microtiter plate. The method has been successfully applied to a broad range of cell lines and shows nearly 100% efficiency (i.e., one spheroid per drop). Using the hepatoma cell line, HepG2, the hanging drop method generated well-rounded MCTS with a narrow size distribution (coefficient of variation [CV] 10% to 15%, compared with 40% to 60% for growth on nonadherent surfaces). Structural analysis of HepG2 and a mammary gland adenocarcinoma cell line, MCF-7, composed spheroids, revealed highly organized, three-dimensional, tissue-like structures with an extensive extracellular matrix. The hanging drop method represents an attractive alternative for MCTS production, because it is mild, can be applied to a wide variety of cell lines, and can produce spheroids of a homogeneous size without the need for sieving or manual selection. The method has applications for basic studies of physiology and metabolism, tumor biology, toxicology, cellular organization, and the development of bioartificial tissue.     

PSysCal: a parallel tool for calibration of ecosystem models

The methods used for ecosystem modelling are generally based on differential equations. Nowadays, new computational models based on concurrent processing of multiple agents (multi-agents) or the simulation of biological processes with the Population Dynamic P-System models (PDPs) are gaining importance. These models have significant advantages over traditional models, such as high computational efficiency, modularity and its ability to model the interaction between different biological processes which operate concurrently. By this, they are becoming useful for simulating complex dynamic ecosystems, untreatable with classical techniques. On the other hand, the main counterpart of P-System models is the need for calibration. The model parameters represent the field measurements taken by experts. However, the exact values of some of these parameters are unknown and experts define a numerical interval of possible values. Therefore, it is necessary to perform a calibration process to fit the best value of each interval. When the number of unknown parameters increases, the calibration process becomes computationally complex and storage requirements increase significantly. In this paper, we present a parallel tool (PSysCal) for calibrating next generation PDP models. The results shown that the calibration time is reduced exponentially with the amount of computational resources. However, the complexity of the calibration process and a limitation in the number of available computational resources make the calibration process intractable for large models. To solve this, we propose a heuristic technique (PSysCal+H). The results show that this technique significantly reduces the computational cost, it being practical for solving large model instances even with limited computational resources.     

Considerations for the use of cellular electrophysiology models within cardiac tissue simulations

The use of mathematical models to study cardiac electrophysiology has a long history, and numerous cellular scale models are now available, covering a range of species and cell types. Their use to study emergent properties in tissue is also widespread, typically using the monodomain or bidomain equations coupled to one or more cell models. Despite the relative maturity of this field, little has been written looking in detail at the interface between the cellular and tissue-level models. Mathematically this is relatively straightforward and well-defined. There are however many details and potential inconsistencies that need to be addressed, in order to ensure correct operation of a cellular model within a tissue simulation. This paper will describe these issues and how to address them.Simply having models available in a common format such as CellML is still of limited utility, with significant manual effort being required to integrate these models within a tissue simulation. We will thus also discuss the facilities available for automating this in a consistent fashion within Chaste, our robust and high-performance cardiac electrophysiology simulator.It will be seen that a common theme arising is the need to go beyond a representation of the model mathematics in a standard language, to include additional semantic information required in determining the model’s interface, and hence to enhance interoperability. Such information can be added as metadata, but agreement is needed on the terms to use, including development of appropriate ontologies, if reliable automated use of CellML models is to become common.     

The Impact of Macroscopic Epistasis on Long-Term Evolutionary Dynamics

Genetic interactions can strongly influence the fitness effects of individual mutations, yet the impact of these epistatic interactions on evolutionary dynamics remains poorly understood. Here we investigate the evolutionary role of epistasis over 50,000 generations in a well-studied laboratory evolution experiment in Escherichia coli. The extensive duration of this experiment provides a unique window into the effects of epistasis during long-term adaptation to a constant environment. Guided by analytical results in the weak-mutation limit, we develop a computational framework to assess the compatibility of a given epistatic model with the observed patterns of fitness gain and mutation accumulation through time. We find that a decelerating fitness trajectory alone provides little power to distinguish between competing models, including those that lack any direct epistatic interactions between mutations. However, when combined with the mutation trajectory, these observables place strong constraints on the set of possible models of epistasis, ruling out many existing explanations of the data. Instead, we find that the data are consistent with a “two-epoch” model of adaptation, in which an initial burst of diminishing-returns epistasis is followed by a steady accumulation of mutations under a constant distribution of fitness effects. Our results highlight the need for additional DNA sequencing of these populations, as well as for more sophisticated models of epistasis that are compatible with all of the experimental data.     

Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms

Biomechanical factors play fundamental roles in the natural history of abdominal aortic aneurysms (AAAs) and their responses to treatment. Advances during the past two decades have increased our understanding of the mechanics and biology of the human abdominal aorta and AAAs, yet there remains a pressing need for considerable new data and resulting patient-specific computational models that can better describe the current status of a lesion and better predict the evolution of lesion geometry, composition, and material properties and thereby improve interventional planning. In this paper, we briefly review data on the structure and function of the human abdominal aorta and aneurysmal wall, past models of the mechanics, and recent growth and remodeling models. We conclude by identifying open problems that we hope will motivate studies to improve our computational modeling and thus general understanding of AAAs.     

Tumour-stromal interactions in acid-mediated invasion: A mathematical model

It is well established that the tumour microenvironment can both promote and suppress tumour growth and invasion, however, most mathematical models of invasion view the normal tissue as inhibiting tumour progression via immune modulation or spatial constraint. In particular, the production of acid by tumour cells and the subsequent creation of a low extracellular pH environment has been explored in several ‘acid-mediated tumour invasion’ models where the acidic environment facilitates normal cell death and permits tumour invasion. In this paper, we extend the acid-invasion model developed by Gatenby and Gawlinski (1996) to include both the competitive and cooperative interactions between tumour and normal cells, by incorporating the influence of extracellular matrix and protease production at the tumour-stroma interface. Our model predicts an optimal level of tumour acidity which produces both cell death and matrix degradation. Additionally, very aggressive tumours prevent protease production and matrix degradation by excessive normal cell destruction, leading to an acellular (but matrix filled) gap between the tumour and normal tissue, a feature seen in encapsulated tumours. These results sugest, counterintuitively, that increasing tumour acidity may, in some cases, prevent tumour invasion.     

Network modeling of signal transduction: establishing the global view

Embryonic development and adult tissue homeostasis are controlled through activation of intracellular signal transduction pathways by extracellular growth factors. In the past, signal transduction has largely been regarded as a linear process. However, more recent data from large-scale and high-throughput experiments indicate that there is extensive cross-talk between individual signaling cascades leading to the notion of a signaling network. The behavior of such complex networks cannot be predicted by simple intuitive approaches but requires sophisticated models and computational simulations. The purpose of such models is to generate experimentally testable hypotheses and to find explanations for unexpected experimental results. Here, we discuss the need for, and the future impact of, mathematical models for exploring signal transduction in different biological contexts such as for example development.     

A Probabilistic Treatment of Phylogeny and Sequence Alignment

Carrying out simultaneous tree-building and alignment of sequence data is a difficult computational task, and the methods currently available are either limited to a few sequences or restricted to highly simplified models of alignment and phylogeny. A method is given here for overcoming these limitations by Bayesian sampling of trees and alignments simultaneously. The method uses a standard substitution matrix model for residues together with a hidden Markov model structure that allows affine gap penalties. It escapes the heavy computational burdens of other models by using an approximation called the ``*' rule, which replaces missing data by a sum over all possible values of variables. The behavior of the model is demonstrated on test sets of globins. Received: 25 May 1998 / Accepted: 8 December 1998     

Predicting development, metabolism and secondary production for the invertebrate predator Bythotrephes

1. Rates of embryonic and post-embryonic development for Bythotrephes cederstroemi from Lakes Erie, Huron and Michigan are represented almost equally well by three empirical models across water temperatures ranging from about 12–22 °C, but at lower temperatures two of the competing models fail and an exponential development rate model proves most robust.
2. Clutch masses of parthenogenic females can greatly exceed the tissue mass of the mother. Clutch size is strongly correlated with the mass of reproductive adults, accounting for over 90% of the variation among individuals. Hence, the mass gain from neonate to reproductive adult can be estimated directly from clutch size.
3. Tissue stoichiometries, respiration quotients and stoichiometries of C and N metabolism were determined experimentally, extending the predictions of existing respiration and growth models.
4. A predictive model for growth and production by the invertebrate predator has advantages over previous model formulations owing to our expanded calibration data base. The model is presented in a modular design that is easily upgraded as additional calibration data become available.     

Synthesis rates and binding kinetics of matrix products in engineered cartilage constructs using chondrocyte-seeded agarose gels

Large-sized cartilage constructs suffer from inhomogeneous extracellular matrix deposition due to insufficient nutrient availability. Computational models of nutrient consumption and tissue growth can be utilized as an efficient alternative to experimental trials to optimize the culture of large constructs; models require system-specific growth and consumption parameters. To inform models of the [bovine chondrocyte]−[agarose gel] system, total synthesis rate (matrix accumulation rate+matrix release rate) and matrix retention fractions of glycosaminoglycans (GAG), collagen, and cartilage oligomeric matrix protein (COMP) were measured either in the presence (continuous or transient) or absence of TGF-β3 supplementation. TGF-β3's influences on pyridinoline content and mechanical properties were also measured. Reversible binding kinetic parameters were characterized using computational models. Based on our recent nutrient supplementation work, we measured glucose consumption and critical glucose concentration for tissue growth to computationally simulate the culture of a human patella-sized tissue construct, reproducing the experiment of Hung et al. (2003). Transient TGF-β3 produced the highest GAG synthesis rate, highest GAG retention ratio, and the highest binding affinity; collagen synthesis was elevated in TGF-β3 supplementation groups over control, with the highest binding affinity observed in the transient supplementation group; both COMP synthesis and retention were lower than those for GAG and collagen. These results informed the modeling of GAG deposition within a large patella construct; this computational example was similar to the previous experimental results without further adjustments to modeling parameters. These results suggest that these nutrient consumption and matrix synthesis models are an attractive alternative for optimizing the culture of large-sized constructs.     

A novel micro-to-macro approach for cardiac tissue mechanics

For studying cardiac mechanics, hyperelastic anisotropic computational models have been developed which require the tissue anisotropic and hyperelastic parameters. These parameters are obtained by tissue samples mechanically testing. The validity of such parameters are limited to the specific tissue sample only. They are not adaptable for pathological tissues commonly associated with tissue microstructure alterations. To investigate cardiac tissue mechanics, a novel approach is proposed to model hyperelasticity and anisotropy. This approach is adaptable to various tissue microstructural constituent’s distributions in normal and pathological tissues. In this approach, the tissue is idealized as composite material consisting of cardiomyocytes distributed in extracellular matrix (ECM). The major myocardial tissue constituents are mitochondria and myofibrils while the main ECM’s constituents are collagen fibers and fibroblasts. Accordingly, finite element simulations of uniaxial and equibiaxial tests of normal and infarcted tissue samples with known amounts of these constituents were conducted, leading to corresponding tissue stress–strain data that were fitted to anisotropic/hyperelastic models. The models were validated where they showed good agreement characterized by maximum average stress-strain errors of 16.17 and 10.01% for normal and infarcted cardiac tissue, respectively. This demonstrate the effectiveness of the proposed models in accurate characterization of healthy and pathological cardiac tissues.     

From sarcomere to cell: An efficient algorithm for linking mathematical models of muscle contraction

Two classes of mathematical framework have previously been developed to model active tension generation in contracting muscle. Cross-bridge models of muscle are biophysically based but computationally expensive to solve, and thus unsuitable for embedding in spatially distributed continuum representations. Fading memory models are computationally efficient but provide limited biophysical insight. In this study a novel computational method is proposed for coupling these two frameworks such that biophysical events can be determined and computational tractability maintained. Within the cross-bridge model, the functional forms of the distribution of cross-bridges, as a function of strain in each state, are approximated using the distribution moment approach. Using the variables of area, mean and standard deviation of each distribution, analytic expressions are developed to calculate the temporal dynamics of stiffness, tension and energy. A root finding method is employed to adjust the variables such that the temporal dynamics of the cross-bridge model match those of an equivalent fading memory model. The method is demonstrated for sinusoidal perturbations in length at two frequencies, with an approximate 30-fold increase in computational efficiency over a conventional technique for finding a solution to the cross-bridge model.     

Prediction of enteric methane emissions from cattle

Agriculture has a key role in food production worldwide and it is a major component of the gross domestic product of several countries. Livestock production is essential for the generation of high quality protein foods and the delivery of foods in regions where animal products are the main food source. Environmental impacts of livestock production have been examined for decades, but recently emission of methane from enteric fermentation has been targeted as a substantial greenhouse gas source. The quantification of methane emissions from livestock on a global scale relies on prediction models because measurements require specialized equipment and may be expensive. The predictive ability of current methane emission models remains poor. Moreover, the availability of information on livestock production systems has increased substantially over the years enabling the development of more detailed methane prediction models. In this study, we have developed and evaluated prediction models based on a large database of enteric methane emissions from North American dairy and beef cattle. Most probable models of various complexity levels were identified using a Bayesian model selection procedure and were fitted under a hierarchical setting. Energy intake, dietary fiber and lipid proportions, animal body weight and milk fat proportion were identified as key explanatory variables for predicting emissions. Models here developed substantially outperformed models currently used in national greenhouse gas inventories. Additionally, estimates of repeatability of methane emissions were lower than the ones from the literature and multicollinearity diagnostics suggested that prediction models are stable. In this context, we propose various enteric methane prediction models which require different levels of information availability and can be readily implemented in national greenhouse gas inventories of different complexity levels. The utilization of such models may reduce errors associated with prediction of methane and allow a better examination and representation of policies regulating emissions from cattle.     

Replicating and breaking models: good for you and good for ecology

There are two major limitations to the potential of computational models in ecology for producing general insights: their design is path‐dependent, reflecting different underlying questions, assumptions, and data, and there is too little robustness analysis exploring where the model mechanisms explaining certain observations break down. We here argue that both limitations could be overcome if modellers in ecology would more often replicate existing models, try to break the models, and explore modifications. Replication comprises the re‐implementation of an existing model and the replication of its results. Breaking models means to identify under what conditions the mechanisms represented in a model can no longer explain observed phenomena. The benefits of replication include less effort being spent to enter the iterative stage of model development and having more time for systematic robustness analysis. A culture of replication would lead to increased credibility, coherence and efficiency of computational modelling and thereby facilitate theory development.