Applications of Dynamical Systems in Biology and Medicine

Practically, all chemotherapeutic agents lead to drug resistance. Clinically, it is a challenge to determine whether resistance arises prior to, or as a result of, cancer therapy. Further, a number of different intracellular and microenvironmental factors have been correlated with the emergence of drug resistance. With the goal of better understanding drug resistance and its connection with the tumor microenvironment, we have developed a hybrid discrete-continuous mathematical model. In this model, cancer cells described through a particle-spring approach respond to dynamically changing oxygen and DNA damaging drug concentrations described through partial differential equations. We thoroughly explored the behavior of our self-calibrated model under the following common conditions: a fixed layout of the vasculature, an identical initial configuration of cancer cells, the same mechanism of drug action, and one mechanism of cellular response to the drug. We considered one set of simulations in which drug resistance existed prior to the start of treatment, and another set in which drug resistance is acquired in response to treatment. This allows us to compare how both kinds of resistance influence the spatial and temporal dynamics of the developing tumor, and its clonal diversity. We show that both pre-existing and acquired resistance can give rise to three biologically distinct parameter regimes: successful tumor eradication, reduced effectiveness of drug during the course of treatment (resistance), and complete treatment failure. When a drug resistant tumor population forms from cells that acquire resistance, we find that the spatial component of our model (the microenvironment) has a significant impact on the transient and long-term tumor behavior. On the other hand, when a resistant tumor population forms from pre-existing resistant cells, the microenvironment only has a minimal transient impact on treatment response. Finally, we present evidence that the microenvironmental niches of low drug/sufficient oxygen and low drug/low oxygen play an important role in tumor cell survival and tumor expansion. This may play role in designing new therapeutic agents or new drug combination schedules.

In this paper we utilize the method of regularized Stokeslets to explore flow fields induced by ‘carpets’ of rotating flagella. We model each flagellum as a rigid, rotating helix attached to a wall, and study flows around both a single helix and a small patch of multiple helices. To test our numerical method and gain intuition about flows induced by a single rotating helix, we first perform a numerical time-reversibility experiment. Next, we investigate the hypothesis put forth in (Darnton et al., Biophys J 86, 1863–1870, 2004) that a small number of rotating flagella could produce “whirlpools” and “rivers” a small distance above them. Using our model system, we are able to produce “whirlpools” and “rivers” when the helices are rotating out of phase. Finally, to better understand the transport of microscale loads by flagellated microorganisms, we model a fully coupled helix-vesicle system by placing a finite-sized vesicle held together by elastic springs in fluid near one or two rotating helices. We compare the trajectories of the vesicle and a tracer particle initially placed at the centroid of vesicle and find that the two trajectories can diverge significantly within a short amount of time. Interestingly, the divergent behavior is extremely sensitive to the initial position within the fluid.

A mathematical model of renal hemodynamics is developed in this study to investigate autoregulation in the rat kidney under physiological and pathophysiological conditions. The model simulates the blood supply to a nephron via the afferent arteriole, the filtration of blood through the glomerulus, and the transport of water and ions in the thick ascending limb of the short loop of Henle. The afferent arteriole exhibits the myogenic response, which induces changes in vascular smooth muscle tone in response to hydrostatic pressure variations. Chloride transport is simulated along the thick ascending limb, and the concentration of chloride at the macula densa provides the signal for the constriction or dilation of the afferent arteriole via tubuloglomerular feedback (TGF). With this configuration, the model predicts a stable glomerular filtration rate within a physiological range of perfusion pressure (60–180 mmHg). The contribution of TGF to overall blood flow autoregulation in the kidney is significant only within a narrow band of perfusion pressure values. Simulations of renal autoregulation under conditions of diabetes mellitus yield a > 60% increase in glomerular filtration rate, due in large part to the impairment of the voltage-gated Ca²⁺ channels of the afferent arteriole smooth muscle cells.

The formation of a thrombus (commonly referred to as a blood clot) can potentially pose a severe health risk to an individual, particularly when a thrombus is large enough to impede blood flow. If an individual is considered to be at risk for forming a thrombus, he/she may be prophylactically treated with anticoagulant medication such as warfarin. When an individual is treated with warfarin, a blood test that measures clotting times must be performed. The test yields a number known as the International Normalized Ratio (INR). The INR test must be performed on an individual on a regular basis (e.g., monthly) to ensure that warfarin’s anticoagulation action is targeted appropriately. In this work, we explore the conditions under which an injury-induced thrombus may form in vivo even when the in vitro test shows the appropriate level of anticoagulation action by warfarin. We extend previous models to describe the in vitro clotting time test, as well as thrombus formation in vivo with warfarin treatments. We present numerical simulations that compare scenarios in which warfarin doses and flow rates are modified within biological ranges. Our results indicate that traditional INR measurements may not accurately reflect in vivo clotting times.

We investigate the clustering dynamics of a network of inhibitory interneurons, where each neuron is connected to some set of its neighbors. We use phase model analysis to study the existence and stability of cluster solutions. In particular, we describe cluster solutions which exist for any type of oscillator, coupling and connectivity. We derive conditions for the stability of these solutions in the case where each neuron is coupled to its two nearest neighbors on each side. We apply our analysis to show that changing the connection weights in the network can change the stability of solutions in the inhibitory network. Numerical simulations of the full network model confirm and supplement our theoretical analysis. Our results support the hypothesis that cluster solutions may be related to the formation of neural assemblies.

In this paper we construct a mathematical model of human sleep–wake regulation with thermoregulation and temperature effects. Simulations of this model show features previously presented in experimental data such as elongation of duration and number of REM bouts across the night as well as the appearance of awakenings due to deviations in body temperature from thermoneutrality. This model helps to demonstrate the importance of temperature in the sleep cycle. Further modifications of the model to include more temperature effects on other aspects of sleep regulation such as sleep and REM latency are discussed.

Melanopsin is an unusual vertebrate photopigment that, in mammals, is expressed in a small subset of intrinsically photosensitive retinal ganglion cells (ipRGCs), whose signaling has been implicated in non-image forming vision, regulating such functions as circadian rhythms, pupillary light reflex, and sleep. The biochemical cascade underlying the light response in ipRGCs has not yet been fully elucidated. We developed a stochastic model of the hypothesized melanopsin phototransduction cascade and illustrated that the stochastic model can qualitatively reproduce experimental results under several different conditions. The model allows us to probe various mechanisms in the phototransduction cascade in a way that is not currently experimentally feasible.

Intermittent Preventive Treatment (IPT) is a malaria control strategy in which vulnerable asymptomatic individuals are given a full curative dose of an antimalarial medication at specified intervals, regardless of whether they are infected with malaria or not. A mathematical model is developed to explore the effect of IPT use on the malaria prevalence and control under different scenarios. The model includes both drug-sensitive and drug-resistant strains of the parasite as well as interactions between human hosts and mosquitoes. The basic reproduction numbers for both strains and the invasion reproduction numbers are computed and used to examine the role of IPT on the development of resistant infections. Numerical simulations are performed to examine the effect of treatment of symptomatic infections and IPT on the prevalence levels of both strains. The model results suggest that the schedule of IPT may have an important influence on the prevalence of resistant infections as well as the total infections of both strains. Moreover, the extent to which IPT may influence the development of resistant strains depends also on the half-life of the drug used. A sensitivity and uncertainty analysis indicates the model outcomes are most sensitive to several model parameters including the reduction factor of transmission for the resistant strain, rate of immunity loss, and the clearance rate of sensitive infections.