Mixed uncertainty analysis on pumping by peristaltic hearts using Dempster-Shafer theory.

In this paper, we introduce the numerical strategy for mixed uncertainty propagation based on probability and Dempster-Shafer theories, and apply it to the computational model of peristalsis in a heart-pumping system. Specifically, the stochastic uncertainty in the system is represented with random variables while epistemic uncertainty is represented using non-probabilistic uncertain variables with belief functions. The mixed uncertainty is propagated through the system, resulting in the uncertainty in the chosen quantities of interest (QoI, such as flow volume, cost of transport and work). With the introduced numerical method, the uncertainty in the statistics of QoIs will be represented using belief functions. With three representative probability distributions consistent with the belief structure, global sensitivity analysis has also been implemented to identify important uncertain factors and the results have been compared between different peristalsis models. To reduce the computational cost, physics constrained generalized polynomial chaos method is adopted to construct cheaper surrogates as approximations for the full simulation.

Passive concentration dynamics incorporated into the library IB2d, a two-dimensional implementation of the immersed boundary method.

In this paper, we present an open-source software library that can be used to numerically simulate the advection and diffusion of a chemical concentration or heat density in a viscous fluid where a moving, elastic boundary drives the fluid and acts as a source or sink. The fully-coupled fluid-structure interaction problem of an elastic boundary in a viscous fluid is solved using Peskin's immersed boundary method. The addition or removal of the concentration or heat density from the boundary is solved using an immersed boundary-like approach in which the concentration is spread from the immersed boundary to the fluid using a regularized delta function. The concentration or density over time is then described by the advection-diffusion equation and numerically solved. This functionality has been added to our software library,IB2d, which provides an easy-to-use immersed boundary method in two dimensions with full implementations in MATLAB and Python. We provide four examples that illustrate the usefulness of the method. A simple rubber band that resists stretching and absorbs and releases a chemical concentration is simulated as a first example. Complete convergence results are presented for this benchmark case. Three more biological examples are presented: (1) an oscillating row of cylinders, representative of an idealized appendage used for filter-feeding or sniffing, (2) an oscillating plate in a background flow is considered to study the case of heat dissipation in a vibrating leaf, and (3) a simplified model of a pulsing soft coral where carbon dioxide is taken up and oxygen is released as a byproduct from the moving tentacles. This method is applicable to a broad range of problems in the life sciences, including chemical sensing by antennae, heat dissipation in plants and other structures, the advection-diffusion of morphogens during development, filter-feeding by marine organisms, and the release of waste products from organisms in flows.

Exploring the sensitivity in jellyfish locomotion under variations in scale, frequency, and duty cycle.

Jellyfish have been called one of the most energy-efficient animals in the world due to the ease in which they move through their fluid environment, by product of their bell kinematics coupled with their morphological, muscular, material properties. We investigated jellyfish locomotion by conducting in silico comparative studies and explored swimming performance across different fluid scales (i.e., Reynolds Number), bell contraction frequencies, and contraction phase kinematics (duty cycle) for a jellyfish with a fineness ratio of 1 (ratio of bell height to bell diameter). To study these relationships, an open source implementation of the immersed boundary method was used (IB2d) to solve the fully coupled fluid-structure interaction problem of a flexible jellyfish bell in a viscous fluid. Thorough 2D parameter subspace explorations illustrated optimal parameter combinations in which give rise to enhanced swimming performance. All performance metrics indicated a higher sensitivity to bell actuation frequency than fluid scale or duty cycle, via Sobol sensitivity analysis, on a higher performance parameter subspace. Moreover, Pareto-like fronts were identified in the overall performance space involving the cost of transport and forward swimming speed. Patterns emerged within these performance spaces when highlighting different parameter regions, which complemented the global sensitivity results. Lastly, an open source computational model for jellyfish locomotion is offered to the science community that can be used as a starting place for future numerical experimentation.

Hopscotching jellyfish: combining different duty cycle kinematics can lead to enhanced swimming performance.

Jellyfish (Medusozoa) have been deemed the most energy-efficient animals in the world. Their bell morphology and relatively simple nervous systems make them attractive to robotocists. Although, the science community has devoted much attention to understanding their swimming performance, there is still much to be learned about the jet propulsive locomotive gait displayed by prolate jellyfish. Traditionally, computational scientists have assumed uniform duty cycle kinematics when computationally modeling jellyfish locomotion. In this study we used fluid-structure interaction modeling to determine possible enhancements in performance from shuffling different duty cycles together across multiple Reynolds numbers and contraction frequencies. Increases in speed and reductions in cost of transport were observed as high as 80% and 50%, respectively. Generally, the net effects were greater for cases involving lower contraction frequencies. Overall, robust duty cycle combinations were determined that led to enhanced or impeded performance.

Diving into a Simple Anguilliform Swimmer's Sensitivity.

Computational models of aquatic locomotion range from modest individual simple swimmers in 2D to sophisticated 3D multi-swimmer models that attempt to parse collective behavioral dynamics. Each of these models contain a multitude of model input parameters to which its outputs are inherently dependent, that is, various performance metrics. In this work, the swimming performance's sensitivity to parameters is investigated for an idealized, simple anguilliform swimming model in 2D. The swimmer considered here propagates forward by dynamically varying its body curvature, similar to motion of a Caenorhabditis elegans. The parameter sensitivities were explored with respect to the fluid scale (Reynolds number), stroke (undulation) frequency, as well as a kinematic parameter controlling the velocity and acceleration of each upstroke and downstroke. The input Reynolds number and stroke frequencies sampled were from [450, 2200] and [1, 3] Hz, respectively. In total, 5000 fluid-structure interaction simulations were performed, each with a unique parameter combination selected via a Sobol sequence, in order to conduct global sensitivity analysis. Results indicate that the swimmer's performance is most sensitive to variations in its stroke frequency. Trends in swimming performance were discovered by projecting the performance data onto particular 2D subspaces. Pareto-like optimal fronts were identified. This work is a natural extension of the parameter explorations of the same model from Battista in 2020.

Uncertainty quantification reveals the physical constraints on pumping by peristaltic hearts.

Most biological functional systems are complex, and this complexity is a fundamental driver of diversity. Because input parameters interact in complex ways, a holistic understanding of functional systems is key to understanding how natural selection produces diversity. We present uncertainty quantification (UQ) as a quantitative analysis tool on computational models to study the interplay of complex systems and diversity. We investigate peristaltic pumping in a racetrack circulatory system using a computational model and analyse the impact of three input parameters (Womersley number, compression frequency, compression ratio) on flow and the energetic costs of circulation. We employed two models of peristalsis (one that allows elastic interactions between the heart tube and fluid and one that does not), to investigate the role of elastic interactions on model output. A computationally cheaper surrogate of the input parameter space was created with generalized polynomial chaos expansion to save computational resources. Sobol indices were then calculated based on the generalized polynomial chaos expansion and model output. We found that all flow metrics were highly sensitive to changes in compression ratio and insensitive to Womersley number and compression frequency, consistent across models of peristalsis. Elastic interactions changed the patterns of parameter sensitivity for energetic costs between the two models, revealing that elastic interactions are probably a key physical metric of peristalsis. The UQ analysis created two hypotheses regarding diversity: favouring high flow rates (where compression ratio is large and highly conserved) and minimizing energetic costs (which avoids combinations of high compression ratios, high frequencies and low Womersley numbers).

Swimming Through Parameter Subspaces of a Simple Anguilliform Swimmer.

Computational scientists have investigated swimming performance across a multitude of different systems for decades. Most models depend on numerous model input parameters and performance is sensitive to those parameters. In this article, parameter subspaces are qualitatively identified in which there exists enhanced swimming performance for an idealized, simple swimming model that resembles a Caenorhabditis elegans, an organism that exhibits an anguilliform mode of locomotion. The computational model uses the immersed boundary method to solve the fluid-interaction system. The 1D swimmer propagates itself forward by dynamically changing its preferred body curvature. Observations indicate that the swimmer's performance appears more sensitive to fluid scale and stroke frequency, rather than variations in the velocity and acceleration of either its upstroke or downstroke as a whole. Pareto-like optimal fronts were also identified within the data for the cost of transport and swimming speed. While this methodology allows one to locate robust parameter subspaces for desired performance in a straight-forward manner, it comes at the cost of simulating orders of magnitude more simulations than traditional fluid-structure interaction studies.

A semi-automated finite difference mesh creation method for use with immersed boundary software IB2d and IBAMR.

Numerous fluid-structure interaction problems in biology have been investigated using the immersed boundary method. The advantage of this method is that complex geometries, e.g., internal or external morphology, can easily be handled without the need to generate matching grids for both the fluid and the structure. Consequently, the difficulty of modeling the structure lies often in discretizing the boundary of the complex geometry (morphology). Both commercial and open source mesh generators for finite element methods have long been established; however, the traditional immersed boundary method is based on a finite difference discretization of the structure. Here we present a software library for obtaining finite difference discretizations of boundaries for direct use in the 2D immersed boundary method. This library provides tools for extracting such boundaries as discrete mesh points from digital images. We give several examples of how the method can be applied that include passing flow through the veins of insect wings, within lymphatic capillaries, and around starfish using open-source immersed boundary software.

Modeling the Prescription Opioid Epidemic.

Opioid addiction has become a global epidemic and a national health crisis in recent years, with the number of opioid overdose fatalities steadily increasing since the 1990s. In contrast to the dynamics of a typical illicit drug or disease epidemic, opioid addiction has its roots in legal, prescription medication-a fact which greatly increases the exposed population and provides additional drug accessibility for addicts. In this paper, we present a mathematical model for prescription drug addiction and treatment with parameters and validation based on data from the opioid epidemic. Key dynamics considered include addiction through prescription, addiction from illicit sources, and treatment. Through mathematical analysis, we show that no addiction-free equilibrium can exist without stringent control over how opioids are administered and prescribed, in which case we estimate that the epidemic would cease to be self-sustaining. Numerical sensitivity analysis suggests that relatively low states of endemic addiction can be obtained by primarily focusing on medical prevention followed by aggressive treatment of remaining cases-even when the probability of relapse from treatment remains high. Further empirical study focused on understanding the rate of illicit drug dependence versus overdose risk, along with the current and changing rates of opioid prescription and treatment, would shed significant light on optimal control efforts and feasible outcomes for this epidemic and drug epidemics in general.

Vortex Dynamics in Trabeculated Embryonic Ventricles.

Proper heart morphogenesis requires a delicate balance between hemodynamic forces, myocardial activity, morphogen gradients, and epigenetic signaling, all of which are coupled with genetic regulatory networks. Recently both in vivo and in silico studies have tried to better understand hemodynamics at varying stages of veretebrate cardiogenesis. In particular, the intracardial hemodynamics during the onset of trabeculation is notably complex-the inertial and viscous fluid forces are approximately equal at this stage and small perturbations in morphology, scale, and steadiness of the flow can lead to significant changes in bulk flow structures, shear stress distributions, and chemical morphogen gradients. The immersed boundary method was used to numerically simulate fluid flow through simplified two-dimensional and stationary trabeculated ventricles of 72, 80, and 120 h post fertilization wild type zebrafish embryos and ErbB2-inhibited embryos at seven days post fertilization. A 2D idealized trabeculated ventricular model was also used to map the bifurcations in flow structure that occur as a result of the unsteadiness of flow, trabeculae height, and fluid scale ( R e ). Vortex formation occurred in intertrabecular regions for biologically relevant parameter spaces, wherein flow velocities increased. This indicates that trabecular morphology may alter intracardial flow patterns and hence ventricular shear stresses and morphogen gradients. A potential implication of this work is that the onset of vortical (disturbed) flows can upregulate Notch1 expression in endothelial cells in vivo and hence impacts chamber morphogenesis, valvulogenesis, and the formation of the trabeculae themselves. Our results also highlight the sensitivity of cardiac flow patterns to changes in morphology and blood rheology, motivating efforts to obtain spatially and temporally resolved chamber geometries and kinematics as well as the careful measurement of the embryonic blood rheology. The results also suggest that there may be significant changes in shear signalling due to morphological and mechanical variation across individuals and species.

Fluid dynamics in heart development: effects of hematocrit and trabeculation.

Recent in vivo experiments have illustrated the importance of understanding the haemodynamics of heart morphogenesis. In particular, ventricular trabeculation is governed by a delicate interaction between haemodynamic forces, myocardial activity, and morphogen gradients, all of which are coupled to genetic regulatory networks. The underlying haemodynamics at the stage of development in which the trabeculae form is particularly complex, given the balance between inertial and viscous forces. Small perturbations in the geometry, scale, and steadiness of the flow can lead to changes in the overall flow structures and chemical morphogen gradients, including the local direction of flow, the transport of morphogens, and the formation of vortices. The immersed boundary method was used to solve the two-dimensional fluid-structure interaction problem of fluid flow moving through a two chambered heart of a zebrafish (Danio rerio), with a trabeculated ventricle, at 96 hours post fertilization (hpf). Trabeculae heights and hematocrit were varied, and simulations were conducted for two orders of magnitude of Womersley number, extending beyond the biologically relevant range (0.2-12.0). Both intracardial and intertrabecular vortices formed in the ventricle for biologically relevant parameter values. The bifurcation from smooth streaming flow to vortical flow depends upon the trabeculae geometry, hematocrit, and Womersley number, $Wo$. This work shows the importance of hematocrit and geometry in determining the bulk flow patterns in the heart at this stage of development.

IB2d: a Python and MATLAB implementation of the immersed boundary method.

The development of fluid-structure interaction (FSI) software involves trade-offs between ease of use, generality, performance, and cost. Typically there are large learning curves when using low-level software to model the interaction of an elastic structure immersed in a uniform density fluid. Many existing codes are not publicly available, and the commercial software that exists usually requires expensive licenses and may not be as robust or allow the necessary flexibility that in house codes can provide. We present an open source immersed boundary software package, IB2d, with full implementations in both MATLAB and Python, that is capable of running a vast range of biomechanics models and is accessible to scientists who have experience in high-level programming environments. IB2d contains multiple options for constructing material properties of the fiber structure, as well as the advection-diffusion of a chemical gradient, muscle mechanics models, and artificial forcing to drive boundaries with a preferred motion.

A Mathematical Model and MATLAB Code for Muscle-Fluid-Structure Simulations.

This article provides models and code for numerically simulating muscle-fluid-structure interactions (FSIs). This work was presented as part of the symposium on Leading Students and Faculty to Quantitative Biology through Active Learning at the society-wide meeting of the Society for Integrative and Comparative Biology in 2015. Muscle mechanics and simple mathematical models to describe the forces generated by muscular contractions are introduced in most biomechanics and physiology courses. Often, however, the models are derived for simplifying cases such as isometric or isotonic contractions. In this article, we present a simple model of the force generated through active contraction of muscles. The muscles' forces are then used to drive the motion of flexible structures immersed in a viscous fluid. An example of an elastic band immersed in a fluid is first presented to illustrate a fully-coupled FSI in the absence of any external driving forces. In the second example, we present a valveless tube with model muscles that drive the contraction of the tube. We provide a brief overview of the numerical method used to generate these results. We also include as Supplementary Material a MATLAB code to generate these results. The code was written for flexibility so as to be easily modified to many other biological applications for educational purposes.