Multiscale FEM modeling of vascular tone: from membrane currents to vessel mechanics

Using a Human Circulation Mathematical Model to Simulate the Effects of Hemodialysis and Therapeutic Hypothermia

Background: We developed a hemodynamic mathematical model of human circulation coupled to a virtual hemodialyzer. The model was used to explore mechanisms underlying our clinical observations involving hemodialysis. Methods: The model consists of whole body human circulation, baroreflex feedback control, and a hemodialyzer. Four model populations encompassing baseline, dialysed, therapeutic hypothermia treated, and simultaneous dialysed with hypothermia were generated. In all populations atrial fibrillation and renal failure as co-morbidities, and exercise as a treatment were simulated. Clinically relevant measurables were used to quantify the effects of each in silico experiment. Sensitivity analysis was used to uncover the most relevant parameters. Results: Relative to baseline, the modelled dialysis increased the population mean diastolic blood pressure by 5%, large vessel wall shear stress by 6%, and heart rate by 20%. Therapeutic hypothermia increased systolic blood pressure by 3%, reduced large vessel shear stress by 15%, and did not affect heart rate. Therapeutic hypothermia reduced wall shear stress by 15% in the aorta and 6% in the kidneys, suggesting a potential anti-inflammatory benefit. Therapeutic hypothermia reduced cardiac output under atrial fibrillation by 12% and under renal failure by 20%. Therapeutic hypothermia and exercise did not affect dialyser function, but increased water removal by approximately 40%. Conclusions: This study illuminates some mechanisms of the action of therapeutic hypothermia. It also suggests clinical measurables that may be used as surrogates to diagnose underlying diseases such as atrial fibrillation.

Numerical simulation of heat induced flow-mediated dilation of blood vessels

Local heat can accelerate the blood circulation and induce the vasodilatation. Investigators reported that local heat causes an increase in skin blood flow consisting of two phases. The first is solely sensory neural, and the second is nitric oxide mediated. However, the mechanism underlying the skin blood flow response to local heating are complex and poorly understood. The mechanisms behind these two phases are deduced to be linked by flow-mediated dilation. In this study, the variation of the blood flow and the blood vessel diameter are monitored during local heating. According to the dynamic blood flow, the theoretical model of flow mediated dilation involving the key agents production and transportation was first used to study vasodilatation process during heating, and the variations of blood vessel was obtained. Finally, accurate distributions of the nitric oxide, calcium and myosin concentrations in the arterial wall were found during autoregulation. We evaluated the time course of the blood vessel changing and verified the fact that the second increase in blood flow is the result of flow dilation mediation. The effects of dilation of blood vessel were also analyzed.

Can endothelial hemoglobin-α regulate nitric oxide vasodilatory signaling?

We used mathematical modeling to investigate nitric oxide (NO)-dependent vasodilatory signaling in the arteriolar wall. Detailed continuum cellular models of calcium (Ca²⁺) dynamics and membrane electrophysiology in smooth muscle and endothelial cells (EC) were coupled with models of NO signaling and biotransport in an arteriole. We used this theoretical approach to examine the role of endothelial hemoglobin-α (Hbα) as a modulator of NO-mediated myoendothelial feedback, as previously suggested in Straub et al. (Nature 491: 473-477, 2012). The model considers enriched expression of inositol 1,4,5-triphosphate receptors (IP₃Rs), endothelial nitric oxide synthase (eNOS) enzyme, Ca²⁺-activated potassium (K_Ca) channels and Hbα in myoendothelial projections (MPs) between the two cell layers. The model suggests that NO-mediated myoendothelial feedback is plausible if a significant percentage of eNOS is localized within or near the myoendothelial projection. Model results show that the ability of Hbα to regulate the myoendothelial feedback is conditional to its colocalization with eNOS near MPs at concentrations in the high nanomolar range (>0.2 μM or 24,000 molecules). Simulations also show that the effect of Hbα observed in in vitro experimental studies may overestimate its contribution in vivo, in the presence of blood perfusion. Thus, additional experimentation is required to quantify the presence and spatial distribution of Hbα in the EC, as well as to test that the strong effect of Hbα on NO signaling seen in vitro, translates also into a physiologically relevant response in vivo.NEW & NOTEWORTHY Mathematical modeling shows that although regulation of nitric oxide signaling by hemoglobin-α (Hbα) is plausible, it is conditional to its presence in significant concentrations colocalized with endothelial nitric oxide synthase in myoendothelial projections. Additional experimentation is required to test that the strong effect of Hbα seen in vitro translates into a physiologically relevant response in vivo.

Stochastic model of endothelial TRPV4 calcium sparklets: effect of bursting and cooperativity on EDH

We examined the endothelial transient receptor vanilloid 4 (TRPV4) channel's vasodilatory signaling using mathematical modeling. The model analyzes experimental data by Sonkusare and coworkers on TRPV4-induced endothelial Ca(2+) events (sparklets). A previously developed continuum model of an endothelial and a smooth muscle cell coupled through microprojections was extended to account for the activity of a TRPV4 channel cluster. Different stochastic descriptions for the TRPV4 channel flux were examined using finite-state Markov chains. The model also took into consideration recent evidence for the colocalization of intermediate-conductance calcium-activated potassium channels (IKCa) and TRPV4 channels near the microprojections. A single TRPV4 channel opening resulted in a stochastic localized Ca(2+) increase in a small region (i.e., few μm(2) area) close to the channel. We predict micromolar Ca(2+) increases lasting for the open duration of the channel sufficient for the activation of low-affinity endothelial KCa channels. Simulations of a cluster of four TRPV4 channels incorporating burst and cooperative gating kinetics provided quantal Ca(2+) increases (i.e., steps of fixed amplitude), similar to the experimentally observed Ca(2+) sparklets. These localized Ca(2+) events result in endothelium-derived hyperpolarization (and SMC relaxation), with magnitude that depends on event frequency. The gating characteristics (bursting, cooperativity) of the TRPV4 cluster enhance Ca(2+) spread and the distance of KCa channel activation. This may amplify the EDH response by the additional recruitment of distant KCa channels.

Role of microprojections in myoendothelial feedback--a theoretical study

We investigated the role of myoendothelial projections (MPs) in endothelial cell (EC) feedback response to smooth muscle cell (SMC) stimulation using mathematical modelling. A previously developed compartmental EC-SMC model is modified to include MPs as subcellular compartments in the EC. The model is further extended into a 2D continuum model using a finite element method (FEM) approach and electron microscopy images to account for MP geometry. The EC and SMC are coupled via non-selective myoendothelial gap junctions (MEGJs) which are located on MPs and allow exchange of Ca(2+), K(+), Na(+) and Cl(-) ions and inositol 1,4,5-triphosphate (IP3). Models take into consideration recent evidence for co-localization of intermediate-conductance calcium-activated potassium channels (IKCa) and IP3 receptors (IP3Rs) in the MPs. SMC stimulation causes an IP3-mediated Ca(2+) transient in the MPs with limited global spread in the bulk EC. A hyperpolarizing feedback generated by the localized IKCa channels is transmitted to the SMC via MEGJs. MEGJ resistance (Rgj) and the density of IKCa and IP3R in the projection influence the extent of EC response to SMC stimulation. The predicted Ca(2+) transients depend also on the volume and geometry of the MP. We conclude that in the myoendothelial feedback response to SMC stimulation, MPs are required to amplify the SMC initiated signal. Simulations suggest that the signal is mediated by IP3 rather than Ca(2+) diffusion and that a localized rather than a global EC Ca(2+) mobilization is more likely following SMC stimulation.

Estimation of shear-stress-induced endothelial nitric oxide production from flow-mediated dilation

Vascular endothelial function assessment yields important diagnostic and prognostic information on patients with (or at risk of) cardiovascular diseases. Flow-mediated dilation (FMD) is the most widely used noninvasive method for assessing the endothelial function. In this method, the magnitude of FMD is used as a surrogate marker for the endothelial function. However, because vasodilation is affected by several other factors as well, the details of how this marker represents the endothelial function are not fully understood. Previously, we developed a mathematical model for vasodilation with intra- and intercellular pathways. In this study, we estimated the shear-stress-induced endothelial nitric oxide production from FMD measurements using the model. The results suggested that the accuracy of the estimated endothelial function obeys the characteristics of the shear stress added to the vascular wall. Furthermore, we showed that the FMD response describes not only the endothelial function but also vascular wall characteristics.

Intercellular communication in the vascular wall: a modeling perspective

Movement of ions (Ca(2+) , K(+) , Na(+) , and Cl(-) ) and second messenger molecules like inositol 1, 4, 5-trisphosphate inside and in between different cells is the basis of many signaling mechanisms in the microcirculation. In spite of the vast experimental efforts directed toward evaluation of these fluxes, it has been a challenge to establish their roles in many essential microcirculatory phenomena. Recently, detailed theoretical models of calcium dynamics and plasma membrane electrophysiology have emerged to assist in the quantification of these intra and intercellular fluxes and enhance understanding of their physiological importance. This perspective reviews selected models relevant to estimation of such intra and intercellular ionic and second messenger fluxes and prediction of their relative significance to a variety of vascular phenomena, such as myoendothelial feedback, conducted responses, and vasomotion.