Analysis of non-linear pulsatile blood flow in arteries

Some aspects of the problem of computation of non-linear pulsatile blood flow in large arteries are investigated, in the context of the computational method developed by Ling and Atabek (1972). As examples, the following aspects are considered: stability of the computations; representation of higher-frequency components of the flow; effects of keeping or omitting non-linear terms in the equations; effects of varying the dimensionless parameters of the problem. The computational method is extended to include effects of viscoelasticity of arterial walls.     

Modeling Imatinib-Treated Chronic Myelogenous Leukemia: Reducing the Complexity of Agent-Based Models

We develop a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia. Our model is based on replacing the recent agent-based model of Roeder et al. (Nat. Med. 12(10):1181–1184, 2006) by a system of deterministic difference equations. These difference equations describe the time-evolution of clusters of individual agents that are grouped by discretizing the state space. Hence, unlike standard agent-base models, the complexity of our model is independent of the number of agents, which allows to conduct simulation studies with a realistic number of cells. This approach also allows to directly evaluate the expected steady states of the system. The results of our numerical simulations show that our model replicates the averaged behavior of the original Roeder model with a significantly reduced computational cost. Our general approach can be used to simplify other similar agent-based models. In particular, due to the reduced computational complexity of our technique, one can use it to conduct sensitivity studies of the parameters in large agent-based systems.     

Parallel Computational Subunits in Dentate Granule Cells Generate Multiple Place Fields

A fundamental question in understanding neuronal computations is how dendritic events influence the output of the neuron. Different forms of integration of neighbouring and distributed synaptic inputs, isolated dendritic spikes and local regulation of synaptic efficacy suggest that individual dendritic branches may function as independent computational subunits. In the present paper, we study how these local computations influence the output of the neuron. Using a simple cascade model, we demonstrate that triggering somatic firing by a relatively small dendritic branch requires the amplification of local events by dendritic spiking and synaptic plasticity. The moderately branching dendritic tree of granule cells seems optimal for this computation since larger dendritic trees favor local plasticity by isolating dendritic compartments, while reliable detection of individual dendritic spikes in the soma requires a low branch number. Finally, we demonstrate that these parallel dendritic computations could contribute to the generation of multiple independent place fields of hippocampal granule cells.     

MOLKERN: A library of software components for molecular modeling programs

The library of software components MOLKERN is designed to construct efficient programs (with linear computational complexity) for modeling, optimization, and analysis of spatial structures of proteins, cofactors, ligands, and their complexes and computation of their physical properties. The interactions between atoms and molecules are taken into account within the force field method. The library is realized in C++ using static polymorphism technology. The library utilizes STL and BOOST libraries and contains a number of software components for parallel computation using the MPI technology. The library can be used under Windows and Linux.

     

Evolution of dispersal distance

The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. Ezoe (J Theor Biol 190:287–293, 1998) and Levin and Muller-Landau (Evol Ecol Res 2:409–435, 2000) approached this problem using models that were essentially deterministic, and used calculus to find optimal dispersal parameters. Here we follow Hiebeler (Theor Pop Biol 66:205–218, 2004) and use a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems is done by simulation or nonrigorous methods such as pair approximation. Here, we use machinery developed by Cox et al. (Voter model perturbations and reaction diffusion equations 2011) to rigorously and explicitly compute evolutionarily stable strategies.     

Using fuzzy numbers to propagate uncertainty in matrix-based LCI

Background, aim, and scope  Analysis of uncertainties plays a vital role in the interpretation of life cycle assessment findings. Some of these uncertainties arise from parametric data variability in life cycle inventory analysis. For instance, the efficiencies of manufacturing processes may vary among different industrial sites or geographic regions; or, in the case of new and unproven technologies, it is possible that prospective performance levels can only be estimated. Although such data variability is usually treated using a probabilistic framework, some recent work on the use of fuzzy sets or possibility theory has appeared in the literature. The latter school of thought is based on the notion that not all data variability can be properly described in terms of frequency of occurrence. In many cases, it is necessary to model the uncertainty associated with the subjective degree of plausibility of parameter values. Fuzzy set theory is appropriate for such uncertainties. However, the computations required for handling fuzzy quantities has not been fully integrated with the formal matrix-based life cycle inventory analysis (LCI) described by Heijungs and Suh (2002). Materials and methods  This paper integrates computations with fuzzy numbers into the matrix-based LCI computational model described in the literature. The approach uses fuzzy numbers to propagate the data variability in LCI calculations, and results in fuzzy distributions of the inventory results. The approach is developed based on similarities with the fuzzy economic input–output (EIO) model proposed by Buckley (Eur J Oper Res 39:54–60, 1989). Results  The matrix-based fuzzy LCI model is illustrated using three simple case studies. The first case shows how fuzzy inventory results arise in simple systems with variability in industrial efficiency and emissions data. The second case study illustrates how the model applies for life cycle systems with co-products, and thus requires the inclusion of displaced processes. The third case study demonstrates the use of the method in the context of comparing different carbon sequestration technologies. Discussion  These simple case studies illustrate the important features of the model, including possible computational issues that can arise with larger and more complex life cycle systems. Conclusions  A fuzzy matrix-based LCI model has been proposed. The model extends the conventional matrix-based LCI model to allow for computations with parametric data variability represented as fuzzy numbers. This approach is an alternative or complementary approach to interval analysis, probabilistic or Monte Carlo techniques. Recommendations and perspectives  Potential further work in this area includes extension of the fuzzy model to EIO-LCA models and to life cycle impact assessment (LCIA); development of hybrid fuzzy-probabilistic approaches; and integration with life cycle-based optimization or decision analysis. Additional theoretical work is needed for modeling correlations of the variability of parameters using interacting or correlated fuzzy numbers, which remains an unresolved computational issue. Furthermore, integration of the fuzzy model into LCA software can also be investigated.     

Computing the ankle-brachial index with parallel computational fluid dynamics

The ankle-brachial index (ABI), a ratio of arterial blood pressure in the ankles and upper arms, is used to diagnose and monitor circulatory conditions such as coarctation of the aorta and peripheral artery disease. Computational simulations of the ABI can potentially determine the parameters that produce an ABI indicative of ischemia or other abnormalities in blood flow. However, 0- and 1-D computational methods are limited in describing a 3-D patient-derived geometry. Thus, we present a massively parallel framework for computational fluid dynamics (CFD) simulations in the full arterial system. Using the lattice Boltzmann method to solve the Navier–Stokes equations, we employ highly parallelized and scalable methods to generate the simulation domain and efficiently distribute the computational load among processors. For the first time, we compute an ABI with 3-D CFD. In this proof-of-concept study, we investigate the dependence of ABI on the presence of stenoses, or narrowed regions of the arteries, by directly modifying the arterial geometry. As a result, our framework enables the computation a hemodynamic factor characterizing flow at the scale of the full arterial system, in a manner that is extensible to patient-specific imaging data and holds potential for treatment planning.     

Prefrontal cortical activity predicts the occurrence of nonlocal hippocampal representations during spatial navigation

The receptive field of a neuron describes the regions of a stimulus space where the neuron is consistently active. Sparse spiking outside of the receptive field is often considered to be noise, rather than a reflection of information processing. Whether this characterization is accurate remains unclear. We therefore contrasted the sparse, temporally isolated spiking of hippocampal CA1 place cells to the consistent, temporally adjacent spiking seen within their spatial receptive fields (“place fields”). We found that isolated spikes, which occur during locomotion, are strongly phase coupled to hippocampal theta oscillations and transiently express coherent nonlocal spatial representations. Further, prefrontal cortical activity is coordinated with and can predict the occurrence of future isolated spiking events. Rather than local noise within the hippocampus, sparse, isolated place cell spiking reflects a coordinated cortical–hippocampal process consistent with the generation of nonlocal scenario representations during active navigation.

This study of active navigation shows that, rather than being local noise within the hippocampus, sparse, isolated place cell spiking reflects a coordinated cortical-hippocampal process consistent with the generation of non-local scenario representations.     

The repeated replacement method: a pure lagrangian meshfree method for computational fluid dynamics

In this paper we describe the repeated replacement method (RRM), a new meshfree method for computational fluid dynamics (CFD). RRM simulates fluid flow by modeling compressible fluids' tendency to evolve towards a state of constant density, velocity, and pressure. To evolve a fluid flow simulation forward in time, RRM repeatedly "chops out" fluid from active areas and replaces it with new "flattened" fluid cells with the same mass, momentum, and energy. We call the new cells "flattened" because we give them constant density, velocity, and pressure, even though the chopped-out fluid may have had gradients in these primitive variables. RRM adaptively chooses the sizes and locations of the areas it chops out and replaces. It creates more and smaller new cells in areas of high gradient, and fewer and larger new cells in areas of lower gradient. This naturally leads to an adaptive level of accuracy, where more computational effort is spent on active areas of the fluid, and less effort is spent on inactive areas. We show that for common test problems, RRM produces results similar to other high-resolution CFD methods, while using a very different mathematical framework. RRM does not use Riemann solvers, flux or slope limiters, a mesh, or a stencil, and it operates in a purely Lagrangian mode. RRM also does not evaluate numerical derivatives, does not integrate equations of motion, and does not solve systems of equations.     

A simple method for measuring force, velocity and power output during squat jump

Our aim was to clarify the relationship between power output and the different mechanical parameters influencing it during squat jumps, and to further use this relationship in a new computation method to evaluate power output in field conditions. Based on fundamental laws of mechanics, computations were developed to express force, velocity and power generated during one squat jump. This computation method was validated on eleven physically active men performing two maximal squat jumps. During each trial, mean force, velocity and power were calculated during push-off from both force plate measurements and the proposed computations. Differences between the two methods were not significant and lower than 3% for force, velocity and power. The validity of the computation method was also highlighted by Bland and Altman analyses and linear regressions close to the identity line (P<0.001). The low coefficients of variation between two trials demonstrated the acceptable reliability of the proposed method. The proposed computations confirmed, from a biomechanical analysis, the positive relationship between power output, body mass and jump height, hitherto only shown by means of regression-based equations. Further, these computations pointed out that power also depends on push-off vertical distance. The accuracy and reliability of the proposed theoretical computations were in line with those observed when using laboratory ergometers such as force plates. Consequently, the proposed method, solely based on three simple parameters (body mass, jump height and push-off distance), allows to accurately evaluate force, velocity and power developed by lower limbs extensor muscles during squat jumps in field conditions.     

Comments on Bartolino et al. (2011): limits of cumulative relative frequency distribution curves for hotspot identification

The recent paper by Bartolino et al. (Popul Ecol 53:351–359, 2011) presents a new method to objectively select hotspots using cumulative relative frequency distribution (CRFD) curves. This method is presented as being independent from the selection of any threshold and, therefore, less arbitrary than traditional approaches. We argue that this method, albeit mathematically sound, is based on likewise arbitrary decisions regarding threshold selection. Specifically, the use of the CRFD curve approach requires the occurrence of two criteria for the method to be applied correctly: the selection of a 45° tangent to the curve, and the need to consider the highest relative value of the study parameter corresponding to a 45° slope tangent to the curve. Using two case studies (dealing with species richness and abundance of a particular species), we demonstrate that these two criteria are really unrelated to the underlying causes that shape the spatial pattern of the phenomena under study, but rather related to sampling design and spatial scale; hence, one could likewise use different but valid criteria. Consequently, the CRFD curve approach is based on the selection of a pre-defined threshold that has little, if any, ecological justification, and that heavily influences the final hotspot selection. Therefore, we conclude that the CRFD curve approach itself is not necessarily better and more objective than any of the global methods typically used for hotspot identification. Indeed, mathematical and/or statistical approaches should not be viewed as a panacea to solve conservation problems, but rather used in combination with biological, practical, economic and social considerations.     

Modelling attention in individual cells leads to a system with realistic saccade behaviours

Single cell recordings in monkey inferior temporal cortex (IT) and area V4 during visual search tasks indicate that modulation of responses by the search target object occurs in the late portion of the cell’s sensory response (Chelazzi et al. in J Neurophysiol 80:2918–2940, 1998; Cereb Cortex 11:761–772, 2001) whereas attention to a spatial location influences earlier responses (Luck et al. in J Neurophysiol 77:24–42, 1997). Previous computational models have not captured differences in the latency of these attentional effects and yet the more protracted development of the object-based effect could have implications for behaviour. We present a neurodynamic biased competition model of visual attention in which we aimed to model the timecourse of spatial and object-based attention in order to simulate cellular responses and saccade onset times observed in monkey recordings. In common with other models, a top-down prefrontal signal, related to the search target, biases activity in the ventral visual stream. However, we conclude that this bias signal is more complex than modelled elsewhere: the latency of object-based effects in V4 and IT, and saccade onset, can be accurately simulated when the target object feedback bias consists of a sensory response component in addition to a mnemonic response. These attentional effects in V4 and IT cellular responses lead to a system that is able to produce search scan paths similar to those observed in monkeys and humans, with attention being guided to locations containing behaviourally relevant stimuli. This work demonstrates that accurate modelling of the timecourse of single cell responses can lead to biologically realistic behaviours being demonstrated by the system as a whole.     

The Morphostatic Limit for a Model of Skeletal Pattern Formation in the Vertebrate Limb

A recently proposed mathematical model of a “core” set of cellular and molecular interactions present in the developing vertebrate limb was shown to exhibit pattern-forming instabilities and limb skeleton-like patterns under certain restrictive conditions, suggesting that it may authentically represent the underlying embryonic process (Hentschel et al., Proc. R. Soc. B 271, 1713–1722, 2004). The model, an eight-equation system of partial differential equations, incorporates the behavior of mesenchymal cells as “reactors,” both participating in the generation of morphogen patterns and changing their state and position in response to them. The full system, which has smooth solutions that exist globally in time, is nonetheless highly complex and difficult to handle analytically or numerically. According to a recent classification of developmental mechanisms (Salazar-Ciudad et al., Development 130, 2027–2037, 2003), the limb model of Hentschel et al. is “morphodynamic,” since differentiation of new cell types occurs simultaneously with cell rearrangement. This contrasts with “morphostatic” mechanisms, in which cell identity becomes established independently of cell rearrangement. Under the hypothesis that development of some vertebrate limbs employs the core mechanism in a morphostatic fashion, we derive in an analytically rigorous fashion a pair of equations representing the spatiotemporal evolution of the morphogen fields under the assumption that cell differentiation relaxes faster than the evolution of the overall cell density (i.e., the morphostatic limit of the full system). This simple reaction–diffusion system is unique in having been derived analytically from a substantially more complex system involving multiple morphogens, extracellular matrix deposition, haptotaxis, and cell translocation. We identify regions in the parameter space of the reduced system where Turing-type pattern formation is possible, which we refer to as its “Turing space.” Obtained values of the parameters are used in numerical simulations of the reduced system, using a new Galerkin finite element method, in tissue domains with nonstandard geometry. The reduced system exhibits patterns of spots and stripes like those seen in developing limbs, indicating its potential utility in hybrid continuum-discrete stochastic modeling of limb development. Lastly, we discuss the possible role in limb evolution of selection for increasingly morphostatic developmental mechanisms.     

Mathematical Model for Two Germline Stem Cells Competing for Niche Occupancy

In the Drosophila germline stem cell ovary niche, two stem cells compete with each other for niche occupancy to maintain stem cell quality by ensuring that differentiated stem cells are rapidly pushed out the niche and replenished by normal ones (Jin et al. in Cell Stem Cell 2:39–49, 2008). To gain a deeper understanding of this biological phenomenon, we have derived a mathematical model for explaining the physical interactions between two stem cells. The model is a system of two nonlinear first order and one second order differential equations coupled with E-cadherins expression levels. The model can explain the dynamics of the competition process of two germline stem cells and may help to reveal missing information obtained from experimental results. The model predicts several qualitative features in the competition process, which may help to design rational experiments for a better understanding of the stem cell competition process.     

A One-dimensional Finite Element Method for Simulation-based Medical Planning for Cardiovascular Disease

We have previously described a new approach to planning treatments for cardiovascular disease, Simulation-Based Medical Planning, whereby a physician utilizes computational tools to construct and evaluate a combined anatomic/physiologic model to predict the outcome of alternative treatment plans for an individual patient. Current systems for Simulation-Based Medical Planning utilize finite element methods to solve the time-dependent, three-dimensional equations governing blood flow and provide detailed data on blood flow distribution, pressure gradients and locations of flow recirculation, low wall shear stress and high particle residence. However, these methods are computationally expensive and often require hours of time on parallel computers. This level of computation is necessary for obtaining detailed information about blood flow, but likely is unnecessary for obtaining information about mean flow rates and pressure losses. We describe, herein, a space-time finite element method for solving the one-dimensional equations of blood flow. This method is applied to compute flow rate and pressure in a single segment model, a bifurcation, an idealized model of the abdominal aorta, in three alternate treatment plans for a case of aorto-iliac occlusive disease and in a vascular bypass graft. All of these solutions were obtained in less than 5 min of computation time on a personal computer.     

Towards a more general understanding of the algorithmic utility of recurrent connections

Lateral and recurrent connections are ubiquitous in biological neural circuits. Yet while the strong computational abilities of feedforward networks have been extensively studied, our understanding of the role and advantages of recurrent computations that might explain their prevalence remains an important open challenge. Foundational studies by Minsky and Roelfsema argued that computations that require propagation of global information for local computation to take place would particularly benefit from the sequential, parallel nature of processing in recurrent networks. Such “tag propagation” algorithms perform repeated, local propagation of information and were originally introduced in the context of detecting connectedness, a task that is challenging for feedforward networks. Here, we advance the understanding of the utility of lateral and recurrent computation by first performing a large-scale empirical study of neural architectures for the computation of connectedness to explore feedforward solutions more fully and establish robustly the importance of recurrent architectures. In addition, we highlight a tradeoff between computation time and performance and construct hybrid feedforward/recurrent models that perform well even in the presence of varying computational time limitations. We then generalize tag propagation architectures to propagating multiple interacting tags and demonstrate that these are efficient computational substrates for more general computations of connectedness by introducing and solving an abstracted biologically inspired decision-making task. Our work thus clarifies and expands the set of computational tasks that can be solved efficiently by recurrent computation, yielding hypotheses for structure in population activity that may be present in such tasks.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier–Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field.

Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

Adding Adhesion to a Chemical Signaling Model for Somite Formation

Somites are condensations of mesodermal cells that form along the two sides of the neural tube during early vertebrate development. They are one of the first instances of a periodic pattern, and give rise to repeated structures such as the vertebrae. A number of theories for the mechanisms underpinning somite formation have been proposed. For example, in the “clock and wavefront” model (Cooke and Zeeman in J. Theor. Biol. 58:455–476, 1976), a cellular oscillator coupled to a determination wave progressing along the anterior-posterior axis serves to group cells into a presumptive somite. More recently, a chemical signaling model has been developed and analyzed by Maini and coworkers (Collier et al. in J. Theor. Biol. 207:305–316, 2000; Schnell et al. in C. R. Biol. 325:179–189, 2002; McInerney et al. in Math. Med. Biol. 21:85–113, 2004), with equations for two chemical regulators with entrained dynamics. One of the chemicals is identified as a somitic factor, which is assumed to translate into a pattern of cellular aggregations via its effect on cell–cell adhesion. Here, the authors propose an extension to this model that includes an explicit equation for an adhesive cell population. They represent cell adhesion via an integral over the sensing region of the cell, based on a model developed previously for adhesion driven cell sorting (Armstrong et al. in J. Theor. Biol. 243:98–113, 2006). The expanded model is able to reproduce the observed pattern of cellular aggregates, but only under certain parameter restrictions. This provides a fuller understanding of the conditions required for the chemical model to be applicable. Moreover, a further extension of the model to include separate subpopulations of cells is able to reproduce the observed differentiation of the somite into separate anterior and posterior halves. N.J. Armstrong was supported by a Doctoral Training Account Studentship from EPSRC. K.J. Painter and J.A. Sherratt were supported in part by Integrative Cancer Biology Program Grant CA113004 from the US National Institute of Health and in part by BBSRC grant BB/D019621/1 for the Centre for Systems Biology at Edinburgh.