Lumped parameter models for two-ventricle and healthy and failing extracardiac Fontan circulations

Fontan circulations are surgical strategies to treat infants born with single ventricle physiology. Clinical and mathematical definitions of Fontan failure are lacking, and understanding is needed of parameters indicative of declining physiologies. Our objective is to develop lumped parameter models of two-ventricle and single-ventricle circulations. These models, their mathematical formulations and a proof of existence of periodic solutions are presented. Sensitivity analyses are performed to identify key parameters. Systemic venous and systolic left ventricular compliances and systemic capillary and pulmonary venous resistances are identified as key parameters. Our models serve as a framework to study the differences between two-ventricle and single-ventricle physiologies and healthy and failing Fontan circulations.

A mathematical model of cardiovascular dynamics for the diagnosis and prognosis of hemorrhagic shock

A variety of mathematical models of the cardiovascular system have been suggested over several years in order to describe the time-course of a series of physiological variables (i.e. heart rate, cardiac output, arterial pressure) relevant for the compensation mechanisms to perturbations, such as severe haemorrhage. The current study provides a simple but realistic mathematical description of cardiovascular dynamics that may be useful in the assessment and prognosis of hemorrhagic shock. The present work proposes a first version of a differential-algebraic equations model, the model dynamical ODE model for haemorrhage (dODEg). The model consists of 10 differential and 14 algebraic equations, incorporating 61 model parameters. This model is capable of replicating the changes in heart rate, mean arterial pressure and cardiac output after the onset of bleeding observed in four experimental animal preparations and fits well to the experimental data. By predicting the time-course of the physiological response after haemorrhage, the dODEg model presented here may be of significant value for the quantitative assessment of conventional or novel therapeutic regimens. The model may be applied to the prediction of survivability and to the determination of the urgency of evacuation towards definitive surgical treatment in the operational setting.

Classification under uncertainty: data analysis for diagnostic antibody testing

Formulating accurate and robust classification strategies is a key challenge of developing diagnostic and antibody tests. Methods that do not explicitly account for disease prevalence and uncertainty therein can lead to significant classification errors. We present a novel method that leverages optimal decision theory to address this problem. As a preliminary step, we develop an analysis that uses an assumed prevalence and conditional probability models of diagnostic measurement outcomes to define optimal (in the sense of minimizing rates of false positives and false negatives) classification domains. Critically, we demonstrate how this strategy can be generalized to a setting in which the prevalence is unknown by either (i) defining a third class of hold-out samples that require further testing or (ii) using an adaptive algorithm to estimate prevalence prior to defining classification domains. We also provide examples for a recently published SARS-CoV-2 serology test and discuss how measurement uncertainty (e.g. associated with instrumentation) can be incorporated into the analysis. We find that our new strategy decreases classification error by up to a decade relative to more traditional methods based on confidence intervals. Moreover, it establishes a theoretical foundation for generalizing techniques such as receiver operating characteristics by connecting them to the broader field of optimization.

Tear film dynamics with blinking and contact lens motion

We develop a lubrication theory-based mathematical model that describes the dynamics of a tear film during blinking and contact lens (CL) wear. The model extends previous work on pre-corneal tear film dynamics during blinking by coupling the partial differential equation for tear film thickness to a dynamic model for CL motion. We explore different models for eyelid motion and also account for possible voluntary and involuntary globe (eyeball) rotation that may accompany blinking. Boundary conditions for mass flux at the eyelids are also adapted to account for the presence and motion of the CL. Our predictions for CL motion compare reasonably with existing data. Away from the eyelids the pre-lens tear film (PrLTF) is shifted, relative to its pre-corneal counterpart, in the direction of CL motion. Near the eyelids, the inflow/outflow of fluid under the eyelids also influences the PrLTF profile. We also compare our PrLTF dynamics to existing in vivo tear film thickness measurements.

Models for plasma kinetics during simultaneous therapeutic plasma exchange and extracorporeal membrane oxygenation

This paper focuses on the derivation and simulation of mathematical models describing new plasma fraction in blood for patients undergoing simultaneous extracorporeal membrane oxygenation and therapeutic plasma exchange. Models for plasma exchange with either veno-arterial or veno-venous extracorporeal membrane oxygenation are considered. Two classes of models are derived for each case, one in the form of an algebraic delay equation and another in the form of a system of delay differential equations. In special cases, our models reduce to single compartment ones for plasma exchange that have been validated with experimental data (Randerson et al., 1982, Artif. Organs, 6, 43–49). We also show that the algebraic differential equations are forward Euler discretizations of the delay differential equations, with timesteps equal to transit times through model compartments. Numerical simulations are performed to compare different model types, to investigate the impact of plasma device port switching on the efficiency of the exchange process, and to study the sensitivity of the models to their parameters.

A mathematical model for bleb regulation in zebrafish primordial germ cells

Blebs are cell protrusions generated by local membrane–cortex detachments followed by expansion of the plasma membrane. Blebs are formed by some migrating cells, e.g. primordial germ cells of the zebrafish. While blebs occur randomly at each part of the membrane in unpolarized cells, a polarization process guarantees the occurrence of blebs at a preferential site and thereby facilitates migration toward a specified direction. Little is known about the factors involved in the controlled and directed bleb generation, yet recent studies revealed the influence of an intracellular flow and the stabilizing role of the membrane–cortex linker molecule Ezrin. Based on this information, we develop and analyse a coupled bulk-surface model describing a potential cellular mechanism by which a bleb could be induced at a controlled site. The model rests upon intracellular Darcy flow and a diffusion–advection–reaction system, describing the temporal evolution from a homogeneous to a strongly anisotropic Ezrin distribution. We prove the well-posedness of the mathematical model and show that simulations qualitatively correspond to experimental observations, suggesting that indeed the interaction of an intracellular flow with membrane proteins can be the cause of the Ezrin redistribution accompanying bleb formation.

Calculating prescription rates and addiction probabilities for the four most commonly prescribed opioids and evaluating their impact on addiction using compartment modelling

In 2016, more than 11 million Americans abused prescription opioids. The National Institute on Drug Abuse considers the opioid crisis a national addiction epidemic, as an increasing number of people are affected each year. Using the framework developed in mathematical modelling of infectious diseases, we create and analyse a compartmental opioid-abuse model consisting of a system of ordinary differential equations. Since $40\%$ of opioid overdoses are caused by prescription opioids, our model includes prescription compartments for the four most commonly prescribed opioids, as well as for the susceptible, addicted and recovered populations. While existing research has focused on drug abuse models in general and opioid models with one prescription compartment, no previous work has been done comparing the roles that the most commonly prescribed opioids have had on the crisis. By combining data from the Substance Abuse and Mental Health Services Administration (which tracked the proportion of people who used or misused one of the four individual opioids) with data from the Centers of Disease Control and Prevention (which counted the total number of prescriptions), we estimate prescription rates and probabilities of addiction for the four most commonly prescribed opioids. Additionally, we perform a sensitivity analysis and reallocate prescriptions to determine which opioid has the largest impact on the epidemic. Our results indicate that oxycodone prescriptions are both the most likely to lead to addiction and have the largest impact on the size of the epidemic, while hydrocodone prescriptions had the smallest impact.

Action potential propagation and block in a model of atrial tissue with myocyte–fibroblast coupling

The electrical coupling between myocytes and fibroblasts and the spacial distribution of fibroblasts within myocardial tissues are significant factors in triggering and sustaining cardiac arrhythmias, but their roles are poorly understood. This article describes both direct numerical simulations and an asymptotic theory of propagation and block of electrical excitation in a model of atrial tissue with myocyte–fibroblast coupling. In particular, three idealized fibroblast distributions are introduced: uniform distribution, fibroblast barrier and myocyte strait—all believed to be constituent blocks of realistic fibroblast distributions. Primary action potential biomarkers including conduction velocity, peak potential and triangulation index are estimated from direct simulations in all cases. Propagation block is found to occur at certain critical values of the parameters defining each idealized fibroblast distribution, and these critical values are accurately determined. An asymptotic theory proposed earlier is extended and applied to the case of a uniform fibroblast distribution. Biomarker values are obtained from hybrid analytical-numerical solutions of coupled fast-time and slow-time periodic boundary value problems and compare well to direct numerical simulations. The boundary of absolute refractoriness is determined solely by the fast-time problem and is found to depend on the values of the myocyte potential and on the slow inactivation variable of the sodium current ahead of the propagating pulse. In turn, these quantities are estimated from the slow-time problem using a regular perturbation expansion to find the steady state of the coupled myocyte–fibroblast kinetics. The asymptotic theory gives a simple analytical expression that captures with remarkable accuracy the block of propagation in the presence of fibroblasts.