Predicting the effects of dATP on cardiac contraction using multiscale modeling of the sarcomere.

2'-deoxy-ATP (dATP) is a naturally occurring small molecule that has shown promise as a therapeutic because it significantly increases cardiac myocyte force development even at low dATP/ATP ratios. To investigate mechanisms by which dATP alters myosin crossbridge dynamics, we used Brownian dynamics simulations to calculate association rates between actin and ADP- or dADP-bound myosin. These rates were then directly incorporated in a mechanistic Monte Carlo Markov Chain model of cooperative sarcomere contraction. A unique combination of increased powerstroke and detachment rates was required to match experimental steady-state and kinetic data for dATP force production in rat cardiac myocytes when the myosin attachment rate in the model was constrained by the results of a Brownian dynamics simulation. Nearest-neighbor cooperativity was seen to contribute to, but not fully explain, the steep relationship between dATP/ATP ratio and steady-state force-development observed at lower dATP concentrations. Dynamic twitch simulations performed using measured calcium transients as inputs showed that the effects of dATP on the crossbridge alone were not sufficient to explain experimentally observed enhancement of relaxation kinetics by dATP treatment. Hence, dATP may also affect calcium handling even at low concentrations. By enabling the effects of dATP on sarcomere mechanics to be predicted, this multi-scale modeling framework may elucidate the molecular mechanisms by which dATP can have therapeutic effects on cardiac contractile dysfunction.

Quantification of uncertainty in a new network model of pulmonary arterial adventitial fibroblast pro-fibrotic signalling.

Here, we present a novel network model of the pulmonary arterial adventitial fibroblast (PAAF) that represents seven signalling pathways, confirmed to be important in pulmonary arterial fibrosis, as 92 reactions and 64 state variables. Without optimizing parameters, the model correctly predicted 80% of 39 results of input-output and inhibition experiments reported in 20 independent papers not used to formulate the original network. Parameter uncertainty quantification (UQ) showed that this measure of model accuracy is robust to changes in input weights and half-maximal activation levels (EC50), but is more affected by uncertainty in the Hill coefficient (n), which governs the biochemical cooperativity or steepness of the sigmoidal activation function of each state variable. Epistemic uncertainty in model structure, due to the reliance of some network components and interactions on experiments using non-PAAF cell types, suggested that this source of uncertainty had a smaller impact on model accuracy than the alternative of reducing the network to only those interactions reported in PAAFs. UQ highlighted model parameters that can be optimized to improve prediction accuracy and network modules where there is the greatest need for new experiments. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Quantification of model and data uncertainty in a network analysis of cardiac myocyte mechanosignalling.

Cardiac myocytes transduce changes in mechanical loading into cellular responses via interacting cell signalling pathways. We previously reported a logic-based ordinary differential equation model of the myocyte mechanosignalling network that correctly predicts 78% of independent experimental results not used to formulate the original model. Here, we use Monte Carlo and polynomial chaos expansion simulations to examine the effects of uncertainty in parameter values, model logic and experimental validation data on the assessed accuracy of that model. The prediction accuracy of the model was robust to parameter changes over a wide range being least sensitive to uncertainty in time constants and most affected by uncertainty in reaction weights. Quantifying epistemic uncertainty in the reaction logic of the model showed that while replacing 'OR' with 'AND' reactions greatly reduced model accuracy, replacing 'AND' with 'OR' reactions was more likely to maintain or even improve accuracy. Finally, data uncertainty had a modest effect on assessment of model accuracy. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

An audit of uncertainty in multi-scale cardiac electrophysiology models.

Models of electrical activation and recovery in cardiac cells and tissue have become valuable research tools, and are beginning to be used in safety-critical applications including guidance for clinical procedures and for drug safety assessment. As a consequence, there is an urgent need for a more detailed and quantitative understanding of the ways that uncertainty and variability influence model predictions. In this paper, we review the sources of uncertainty in these models at different spatial scales, discuss how uncertainties are communicated across scales, and begin to assess their relative importance. We conclude by highlighting important challenges that continue to face the cardiac modelling community, identifying open questions, and making recommendations for future studies. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Considering discrepancy when calibrating a mechanistic electrophysiology model.

Uncertainty quantification (UQ) is a vital step in using mathematical models and simulations to take decisions. The field of cardiac simulation has begun to explore and adopt UQ methods to characterize uncertainty in model inputs and how that propagates through to outputs or predictions; examples of this can be seen in the papers of this issue. In this review and perspective piece, we draw attention to an important and under-addressed source of uncertainty in our predictions-that of uncertainty in the model structure or the equations themselves. The difference between imperfect models and reality is termed model discrepancy, and we are often uncertain as to the size and consequences of this discrepancy. Here, we provide two examples of the consequences of discrepancy when calibrating models at the ion channel and action potential scales. Furthermore, we attempt to account for this discrepancy when calibrating and validating an ion channel model using different methods, based on modelling the discrepancy using Gaussian processes and autoregressive-moving-average models, then highlight the advantages and shortcomings of each approach. Finally, suggestions and lines of enquiry for future work are provided. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

A Stochastic Multiscale Model of Cardiac Thin Filament Activation Using Brownian-Langevin Dynamics.

We use Brownian-Langevin dynamics principles to derive a coarse-graining multiscale myofilament model that can describe the thin-filament activation process during contraction. The model links atomistic molecular simulations of protein-protein interactions in the thin-filament regulatory unit to sarcomere-level activation dynamics. We first calculate the molecular interaction energy between tropomyosin and actin surface using Brownian dynamics simulations. This energy profile is then generalized to account for the observed tropomyosin transitions between its regulatory stable states. The generalized energy landscape then served as a basis for developing a filament-scale model using Langevin dynamics. This integrated analysis, spanning molecular to thin-filament scales, is capable of tracking the events of the tropomyosin conformational changes as it moves over the actin surface. The tropomyosin coil with flexible overlap regions between adjacent tropomyosins is represented in the model as a system of coupled stochastic ordinary differential equations. The proposed multiscale approach provides a more detailed molecular connection between tropomyosin dynamics, the trompomyosin-actin interaction-energy landscape, and the generated force by the sarcomere.

A hybrid Windkessel-Womersley model for blood flow in arteries.

A hybrid Windkessel-Womersley (WK-W) coupled mathematical model for the study of pulsatile blood flow in the arterial system is proposed in this article. The model consists of the Windkessel-type proximal and distal compartments connected by a tube to represent the aorta. The blood flow in the aorta is described by the Womersley solution of the simplified Navier-Stokes equations. In addition, we defined a 6-elements Windkessel model (WK6) in which the blood flow in the connecting tube is modeled by the one-dimensional unsteady Bernoulli equation. Both models have been applied and validated using several aortic pressure and flow rate data acquired from different species such as, humans, dogs and pigs. The results have shown that, both models were able to accurately reconstruct arterial input impedance, however, only the WK-W model was able to calculate the radial distribution of the axial velocity in the aorta and consequently the model predicts the time-varying wall shear stress, and frictional pressure drop during the cardiac cycle more accurately. Additionally, the hybrid WK-W model has the capability to predict the pulsed wave velocity, which is also not possible to obtain when using the classical Windkessel models. Moreover, the values of WK-W model parameters have found to fall in the physiologically realistic range of values, therefore it seems that this hybrid model shows a great potential to be used in clinical practice, as well as in the basic cardiovascular mechanics research.

Modeling predicts a connection between sinus vortex effects and aortic compliance.

We submit this letter in order to clarify some methodological issues and concerns raised by Spronck et al. (2015) related to our mathematical modeling of aortic valve dynamics during systole Aboelkassem et al. (2015). It is important to note at the outset that these clarifications do not impact the simulation output or conclusions we originally reported in that paper. However, Spronck et al. have led us to recognize some unfortunate omissions and minor typographical errors in the methods portion of our report that, once corrected, will allow others to more easily reproduce and understand our results.

Acute Optogenetic Modulation of Cardiac Twitch Dynamics Explored Through Modeling.

Optogenetic approaches allow cellular membrane potentials to be perturbed by light. When applied to muscle cells, mechanical events can be controlled through a process that could be termed "optomechanics." Besides functioning as an optical on/off switch, we hypothesized that optomechanical control could include the ability to manipulate the strength and duration of contraction events. To explore this possibility, we constructed an electromechanical model of the human ventricular cardiomyocyte while adding a representation of channelrhodopsin-2 (ChR2), a light-activated channel commonly used in optogenetics. Two hybrid stimulus protocols were developed that combined light-based stimuli with traditional electrical current (all-or-none) excitation. The first protocol involved delivery of a subthreshold optical stimulus followed 50-90 ms later by an electrical stimulus. The result was a graded inhibition of peak cellular twitch force in concert with a prolongation of the intracellular Ca2+ transient. The second protocol was comprised of an electrical stimulus followed by a long light pulse (250-350 ms) that acted to prolong the cardiac action potential (AP). This created a pulse duration-dependent prolongation of the intracellular Ca2+ transient that in turn altered the rate of muscle relaxation without changing peak twitch force. These results illustrate the feasibility of acute, optomechanical manipulation of cardiomyocyte contraction and suggest that this approach could be used to probe the dynamic behavior of the cardiac sarcomere without altering its intrinsic properties. Other experimentally meaningful stimulus protocols could be designed by making use of the optomechanical cardiomyocyte model presented here.

Contributions of Ca2+-Independent Thin Filament Activation to Cardiac Muscle Function.

Although Ca2+ is the principal regulator of contraction in striated muscle, in vitro evidence suggests that some actin-myosin interaction is still possible even in its absence. Whether this Ca2+-independent activation (CIA) occurs under physiological conditions remains unclear, as does its potential impact on the function of intact cardiac muscle. The purpose of this study was to investigate CIA using computational analysis. We added a structurally motivated representation of this phenomenon to an existing myofilament model, which allowed predictions of CIA-dependent muscle behavior. We found that a certain amount of CIA was essential for the model to reproduce reported effects of nonfunctional troponin C on myofilament force generation. Consequently, those data enabled estimation of ΔGCIA, the energy barrier for activating a thin filament regulatory unit in the absence of Ca2+. Using this estimate of ΔGCIA as a point of reference (∼7 kJ mol(-1)), we examined its impact on various aspects of muscle function through additional simulations. CIA decreased the Hill coefficient of steady-state force while increasing myofilament Ca2+ sensitivity. At the same time, CIA had minimal effect on the rate of force redevelopment after slack/restretch. Simulations of twitch tension show that the presence of CIA increases peak tension while profoundly delaying relaxation. We tested the model's ability to represent perturbations to the Ca2+ regulatory mechanism by analyzing twitch records measured in transgenic mice expressing a cardiac troponin I mutation (R145G). The effects of the mutation on twitch dynamics were fully reproduced by a single parameter change, namely lowering ΔGCIA by 2.3 kJ mol(-1) relative to its wild-type value. Our analyses suggest that CIA is present in cardiac muscle under normal conditions and that its modulation by gene mutations or other factors can alter both systolic and diastolic function.

Mathematical modeling of aortic valve dynamics during systole.

We have derived a mathematical model describing aortic valve dynamics and blood flow during systole. The model presents a realistic coupling between aortic valve dynamics, sinus vortex local pressure, and variations in the systemic vascular resistance. The coupling is introduced by using Hill׳s classical semi-spherical vortex model and an aortic pressure-area compliance constitutive relationship. The effects of introducing aortic sinus eddy vortices and variable systemic vascular resistance on overall valve opening-closing dynamics, left ventricular pressure, aortic pressure, blood flow rate, and aortic orifice area are examined. In addition, the strength of the sinus vortex is coupled explicitly to the valve opening angle, and implicitly to the aortic orifice area in order to predict how vortex strength varies during the four descriptive phases of aortic valve motion (fast-opening, fully-opening, slow-closing, and fast-closing). Our results compare favorably with experimental observations and the model reproduces well-known phenomena corresponding to aortic valve function such as the dicrotic notch and retrograde flow at end systole. By invoking a more complete set of physical phenomena, this new model will enable representation of pathophysiological conditions such as aortic valve stenosis or insufficiency, making it possible to predict their integrated effects on cardiac load and systemic hemodynamics.

Modeling virus transport and dynamics in viscous flow medium.

In this paper, we developed a mathematical model to simulate virus transport through a viscous background flow driven by the natural pumping mechanism. Two types of respiratory pathogens viruses (SARS-Cov-2 and Influenza-A) are considered in this model. The Eulerian-Lagrangian approach is adopted to examine the virus spread in axial and transverse directions. The Basset-Boussinesq-Oseen equation is considered to study the effects of gravity, virtual mass, Basset force, and drag forces on the viruses transport velocity. The results indicate that forces acting on the spherical and non-spherical particles during the motion play a significant role in the transmission process of the viruses. It is observed that high viscosity is responsible for slowing the virus transport dynamics. Small sizes of viruses are found to be highly dangerous and propagate rapidly through the blood vessels. Furthermore, the present mathematical model can help to better understand the viruses spread dynamics in a blood flow.

Recent technologies in cardiac imaging.

Cardiac imaging allows physicians to view the structure and function of the heart to detect various heart abnormalities, ranging from inefficiencies in contraction, regulation of volumetric input and output of blood, deficits in valve function and structure, accumulation of plaque in arteries, and more. Commonly used cardiovascular imaging techniques include x-ray, computed tomography (CT), magnetic resonance imaging (MRI), echocardiogram, and positron emission tomography (PET)/single-photon emission computed tomography (SPECT). More recently, even more tools are at our disposal for investigating the heart's physiology, performance, structure, and function due to technological advancements. This review study summarizes cardiac imaging techniques with a particular interest in MRI and CT, noting each tool's origin, benefits, downfalls, clinical application, and advancement of cardiac imaging in the near future.

Human Hypertension Blood Flow Model Using Fractional Calculus.

The blood flow dynamics in human arteries with hypertension disease is modeled using fractional calculus. The mathematical model is constructed using five-element lumped parameter arterial Windkessel representation. Fractional-order capacitors are used to represent the elastic properties of both proximal large arteries and distal small arteries measured from the heart aortic root. The proposed fractional model offers high flexibility in characterizing the arterial complex tree network. The results illustrate the validity of the new model and the physiological interpretability of the fractional differentiation order through a set of validation using human hypertensive patients. In addition, the results show that the fractional-order modeling approach yield a great potential to improve the understanding of the structural and functional changes in the large and small arteries due to hypertension disease.

Sea urchin sperm exploit extremum seeking control to find the egg.

Sperm cells perform extremely demanding tasks with minimal capabilities. The cells must quickly navigate in a noisy environment to find an egg within a short time window for successful fertilization without any global positioning information. Many research efforts have been dedicated to derive mathematical principles that explain their superb navigation strategy. Here we show that the navigation strategy of sea urchin sperm, also known as helical klinotaxis, is a natural implementation of a well-established adaptive control paradigm known as extremum seeking. This bridge between control theory and the biology of taxis in microorganisms is expected to deepen our understanding of the process. For example, the formulation leads to a coarse-grained model of the signaling pathway that offers new insights on the peculiar switching-like behavior between high- and low-gain steering modes observed in sea urchin sperm. Moreover, it may guide engineers in developing bioinspired miniaturized robots with minimal sensors.

Particle swarm optimizer for arterial blood flow models.

BACKGROUND AND OBJECTIVE: Mathematical modeling and computational simulations of arterial blood flow network can offer an insilico platform for both diagnostics and therapeutic phases of patients that suffer from cardiac diseases. These models are normally complex and involve many unknown parameters. For physiological relevance, these parameters should be optimized using in-vivo human/animal data sets. The main goal of this work is to develop an efficient, yet an accurate optimization algorithm to compute parameters in the arterial blood flow models. METHODS: The particle swarm optimization (PSO) method is proposed herein for the first time, as an accurate algorithm that applies to computing parameters in the Windkessel type model of blood flow in the arterial system. We begin by defining a 6-element Windkessel (WK6) arterial flow model, which is then implemented and validated using multiple flow rate and aortic pressure measurements obtained from different subjects including dogs, pigs and humans. The parameters in the model are obtained using the PSO technique which minimizes the pressure root mean square (P-RMS) error between the computed and the measured aortic pressure waveform. RESULTS: Model parameters obtained using the proposed PSO method were able to recover the pressure waveform in the aorta during the cardiac cycle for both healthy and diseased species (animals/humans). The PSO method provides an accurate approach to solve this challenging multi-dimensional parameter identification problem. The results obtained by PSO algorithm was compared with the classical gradient-based, namely the non-linear square fit (NLSF) algorithm. CONCLUSIONS: The results indicate that the PSO method offers alternative and accurate method to find optimal physiological parameters involved in the Windkessel model for the study of arterial blood flow network. The PSO method has performed better than the NLSF approach as depicted from the P-RMS calculations. Finally, we believe that the PSO method offers a great potential and could be used for many other biomedicine optimization problems.

Towards Characterization of the Complex and Frequency-dependent Arterial Compliance based on Fractional-order Capacitor.

Arterial compliance is a vital determinant of the ventriculo-arterial coupling dynamic. Its variation is detrimental to cardiovascular functions and associated with heart diseases. Accordingly, assessment and measurement of arterial compliance are essential in the diagnosis and treatment of chronic arterial insufficiency. Recently, experimental and theoretical studies have recognized the power of fractional calculus to perceive viscoelastic blood vessel structure and biomechanical properties. This paper presents five fractional-order model representations to describe the dynamic relationship between the aortic blood pressure input and blood volume. Each configuration incorporates a fractional-order capacitor element (FOC) to lump the apparent arterial compliance's complex and frequency dependence properties. FOC combines both resistive and capacitive attributes within a unified component, which can be controlled through the fractional differentiation order factor, α. Besides, the equivalent capacitance of FOC is by its very nature frequency-dependent, compassing the complex properties using only a few numbers of parameters. The proposed representations have been compared with generalized integer-order models of arterial compliance. Both models have been applied and validated using different aortic pressure and flow rate data acquired from various species such as humans, pigs, and dogs. The results have shown that the fractional-order framework is able to accurately reconstruct the dynamic of the complex and frequency-dependent apparent compliance dynamic and reduce the complexity. It seems that this new paradigm confers a prominent potential to be adopted in clinical practice and basic cardiovascular mechanics research.

Tropomyosin dynamics during cardiac muscle contraction as governed by a multi-well energy landscape.

The dynamic oscillations of tropomyosin molecules in the azimuthal direction over the surface of the actin filament during thin filament activation are studied here from an energy landscape perspective. A mathematical model based on principles from nonlinear dynamics and chaos theory is derived to describe these dynamical motions. In particular, an energy potential with three wells is proposed to govern the tropomyosin oscillations between the observed regulatory positions observed during muscle contraction, namely the blocked "B", closed "C" and open "M" states. Based on the variations in both the frequency and amplitude of the environmental (surrounding the thin filament system) driving tractions, such as the electrostatic, hydrophobic, and Ca2+-dependent forces, the tropomyosin movements are shown to be complex; they can change from being simple harmonic oscillations to being fully chaotic. Three cases (periodic, period-2, and chaotic patterns) are presented to showcase the different possible dynamic responses of tropomyosin sliding over the actin filament. A probability density function is used as a statistical measure to calculate the average residence time spanned out by the tropomyosin molecule when visiting each (B, C, M) equilibrium state. The results were found to depend strongly on the energy landscape profile and its featured barriers, which normally govern the transitions between the B-C-M states during striated muscle activation.

Multiscale Models of Cardiac Muscle Biophysics and Tissue Remodeling in Hypertrophic Cardiomyopathies.

Myocardial hypertrophy is the result of sustained perturbations to the mechanical and/or neurohormonal homeostasis of cardiac cells and is driven by integrated, multiscale biophysical and biochemical processes that are currently not well defined. In this brief review, we highlight recent computational and experimental models of cardiac hypertrophy that span mechanisms from the molecular level to the tissue level. Specifically, we focus on: (i) molecular-level models of the structural dynamics of sarcomere proteins in hypertrophic hearts, (ii) cellular-level models of excitation-contraction coupling and mechanosensitive signaling in disease-state myocytes, and (iii) organ-level models of myocardial growth kinematics and predictors thereof. Finally, we discuss how spanning these scales and combining multiple experimental/computational models will provide new information about the processes governing hypertrophy and potential methods to prevent or reverse them.

Stokes flow patterns induced by a single cardiac cell.

Stokes flow motions induced by a beating single cardiac cell (cardiomyocyte) are obtained numerically using the method of fundamental solutions (MFS). A two-dimensional meshfree-Stokeslets computational framework is used to solve the Stokes governing equations around an isolated cardiomyocyte. An approximate beating kinematical model is derived and used to approximate the cell-length shortening over a complete cardiac cycle. The induced flow patterns have been found to be characterized by the presence of counter-rotating vortices at both cell's edges. These vortical flow structures are clearly shown by rendering the velocity streamlines. The static pressure contours are also calculated at different time snapshots during both contraction and relaxation phases of the beating motion. The pressure signal is calculated at a point in the neighborhood of cell surface to capture the induced normal stress (traction) by the cell morphological motions to the surrounding fluid medium. The presented results have shown that, cells with a slightly different shortening/beating profile can induce different flow field. This implies that, each cell is characterized by a unique flow pattern "signature", which potentially can be correlated to the sub-cellular excitation-contraction processes of cardiac cells.