Mathematical model of wall shear stress-dependent vasomotor response based on physiological mechanisms

A comprehensive physics-based model for the brachial Artery's full flow mediated dilation (FMD) response observed during the FMD test

In this study, a physics-based model is developed to describe the entire flow mediated dilation (FMD) response. A parameter quantifying the arterial wall's tendency to recover arises from the model, thereby providing a more elaborate description of the artery's physical state, in concert with other parameters characterizing mechanotransduction and structural aspects of the arterial wall. The arterial diameter's behavior throughout the full response is successfully reproduced by the model. Experimental FMD response data were obtained from healthy volunteers. The model's parameters are then adjusted to yield the closest match to the observed experimental response, hence delivering the parameter values pertaining to each subject. This study establishes a foundation based on which future potential clinical applications can be introduced, where endothelial function and general cardiovascular health are inexpensively and noninvasively quantified.

High resolution simulation of basilar artery infarct and flow within the circle of Willis

On a global scale, cerebro- and cardiovascular diseases have long been one of the leading causes of death and disability and their prevalence appears to be increasing in recent times. Understanding potential biomarkers and risk factors will help to identify individuals potentially at risk of suffering an ischemic stroke. However, the widely variable construction of the cerebral vasculature makes it difficult to provide a specific assessment without the knowledge of a patient's physiology. In this paper we use the 3D blood flow simulator HemeLB to study flow within three common structural variations of the circle of Willis during and in the moments after a blockage of the basilar artery. This tool, based on the lattice Boltzmann method, allows the 3D flow entering the basilar artery to be finely controlled to replicate the cessation of blood feeding this particular vessel-we demonstrate this with several examples including a sudden halt to flow and a gradual loss of flow over three heartbeat cycles. In this work we start with an individualised 3D representation of a full circle of Willis and then construct two further domains by removing the left or right posterior communicating arteries from this geometry. Our results indicate how, and how quickly, the circle of Willis is able to redistribute flow following such a stroke. Due to the choice of infarct, the greatest reduction in flow was observed in the posterior cerebral arteries where flow was reduced by up to 70% in some cases. The high resolution domains used in this study permit the velocity magnitude and wall shear stress to be analysed at key points during and following the stroke. The model we present here indicates how personalised vessels are required to provide the best insight into stroke risk for a given individual.

On the Ca2＋ elevation in vascular endothelial cells due to inositol trisphosphate-sensitive store receptors activation: A data-driven modeling approach

Agonist-induced Ca2+ signaling is essential for the regulation of many vital functions in endothelial cells (ECs). A broad range of stimuli elevate the cytosolic Ca2＋ concentration by promoting a pathway mediated by inositol 1,4,5 trisphosphate (IP3) which causes Ca2＋ release from intracellular stores. Despite its importance, there are very few studies focusing on the quantification of such dynamics in the vascular endothelium. Here, by using data from isolated ECs, we established a minimalistic modeling framework able to quantitatively capture the main features (averaged over a cell population) of the cytosolic Ca2＋ response to different IP3 stimulation levels. A suitable description of Ca2＋-regulatory function of inositol 1,4,5 trisphosphate receptors (IP3Rs) and corresponding parameter space are identified by comparing the different model variants against experimental mean population data. The same approach is used to numerically assess the relevance of cytosolic Ca2＋ buffering, as well as Ca2＋ store IP3-sensitivity in the overall cell dynamics. The variability in the dynamics' features observed across the population can be explained (at least in part) through variation of certain model parameters (such as buffering capacity or Ca2＋ store sensitivity to IP3). The results, in terms of experimental fitting and validation, support the proposed minimalistic model as a reference framework for the quantification of the EC Ca2＋ dynamics induced by IP3Rs activation.

Modeling Reactive Hyperemia to Better Understand and Assess Microvascular Function: A Review of Techniques

Reactive hyperemia is a well-established technique for the non-invasive evaluation of the peripheral microcirculatory function, measured as the magnitude of limb re-perfusion after a brief period of ischemia. Despite widespread adoption by researchers and clinicians alike, many uncertainties remain surrounding interpretation, compounded by patient-specific confounding factors (such as blood pressure or the metabolic rate of the ischemic limb). Mathematical modeling can accelerate our understanding of the physiology underlying the reactive hyperemia response and guide in the estimation of quantities which are difficult to measure experimentally. In this work, we aim to provide a comprehensive guide for mathematical modeling techniques that can be used for describing the key phenomena involved in the reactive hyperemia response, alongside their limitations and advantages. The reported methodologies can be used for investigating specific reactive hyperemia aspects alone, or can be combined into a computational framework to be used in (pre-)clinical settings.

Delivery of Nitric Oxide in the Cardiovascular System: Implications for Clinical Diagnosis and Therapy

Nitric oxide (NO) is a key molecule in cardiovascular homeostasis and its abnormal delivery is highly associated with the occurrence and development of cardiovascular disease (CVD). The assessment and manipulation of NO delivery is crucial to the diagnosis and therapy of CVD, such as endothelial dysfunction, atherosclerotic progression, pulmonary hypertension, and cardiovascular manifestations of coronavirus (COVID-19). However, due to the low concentration and fast reaction characteristics of NO in the cardiovascular system, clinical applications centered on NO delivery are challenging. In this tutorial review, we first summarized the methods to estimate the in vivo NO delivery process, based on computational modeling and flow-mediated dilation, to assess endothelial function and vulnerability of atherosclerotic plaque. Then, emerging bioimaging technologies that have the potential to experimentally measure arterial NO concentration were discussed, including Raman spectroscopy and electrochemical sensors. In addition to diagnostic methods, therapies aimed at controlling NO delivery to regulate CVD were reviewed, including the NO release platform to treat endothelial dysfunction and atherosclerosis and inhaled NO therapy to treat pulmonary hypertension and COVID-19. Two potential methods to improve the effectiveness of existing NO therapy were also discussed, including the combination of NO release platform and computational modeling, and stem cell therapy, which currently remains at the laboratory stage but has clinical potential for the treatment of CVD.

A validated reduced-order dynamic model of nitric oxide regulation in coronary arteries

Nitric Oxide (NO) provides myocardial oxygen demands of the heart during exercise and cardiac pacing and also prevents cardiovascular diseases such as atherosclerosis and platelet adhesion and aggregation. However, the direct in vivo measurement of NO in coronary arteries is still challenging. To address this matter, a mathematical model of dynamic changes of calcium and NO concentration in the coronary artery was developed for the first time. The model is able to simulate the effect of NO release in coronary arteries and its impact on the hemodynamics of the coronary arterial tree and also to investigate the vasodilation effects of arteries during cardiac pacing. For these purposes, flow rate, time-averaged wall shear stress, dilation percent, NO concentration, and Calcium (Ca2+) concentration within coronary arteries were obtained. In addition, the impact of hematocrit on the flow rate of the coronary artery was studied. It was seen that the behavior of flow rate, wall shear stress, and Ca2+ is biphasic, but the behavior of NO concentration and the dilation percent is triphasic. Also, by increasing the Hematocrit, the blood flow reduces slightly. The results were compared with several experimental measurements to validate the model qualitatively and quantitatively. It was observed that the presented model is well capable of predicting the behavior of arteries after releasing NO during cardiac pacing. Such a study would be a valuable tool to understand the mechanisms underlying vessel damage, and thereby to offer insights for the prevention or treatment of cardiovascular diseases.

Flow-mediated dilation analysis coupled with nitric oxide transport to enhance the assessment of endothelial function

Flow-mediated dilation (FMD), mainly mediated by nitric oxide (NO), aims to assess the shear-induced endothelial function, which is widely quantified by the relative change in arterial diameter after dilation (FMD%). However, FMD% is affected by individual differences in blood pressure, blood flow, and arterial diameter. To reduce these differences and enhance the assessment of FMD to endothelial function, we continuously measured not only the brachial artery diameter and blood flow with ultrasound but also blood pressure with noninvasive monitor during standard FMD test. We further constructed an analytical model of FMD coupled with NO transport, blood flow, and arterial deformation. Combining the time-averaged and peak values of arterial diameter, blood flow, and pressure, and the modeling, we assumed the artery was completely healthy and calculated an ideally expected FMD% (eFMD%). Then, we expressed the fractional flow-mediated dilation (FFMD%) for the ratio of measured FMD% (mFMD%) to eFMD%. Furthermore, using the continuous waveforms of arterial diameter, blood flow, and pressure, the endothelial characteristic parameter (ϵ) was calculated, which describes the function of the endothelium to produce NO and ranges from 1 to 0 representing the endothelial function from healthiness to complete loss. We found that the mFMD% and eFMD% between the young age (n = 5, 21.2 ± 1.8 yr) and middle age group (n = 5, 34.0 ± 2.1 yr) have no significant difference (P = 0.222, P = 0.385). In contrast, the FFMD% (P = 0.008) and ϵ (P = 0.007) both show significant differences. Therefore, the fractional flow-mediated dilation (FFMD%) and the endothelial characteristic parameter (ϵ) may have the potential for specifically diagnosing the endothelial function.NEW & NOTEWORTHY FMD% is affected by various factors, which limits its ability to assess the endothelial function. We developed an analytical model of FMD process coupled with nitric oxide based on the mathematical modeling and physiological measurements. Two model-derived indicators (FFMD% and ϵ) were introduced based on the modeling. Our results indicated that FFMD% and ϵ may have the potential to distinguish the endothelial function between the young- and middle age groups.

An in silico simulation of flow-mediated dilation reveals that blood pressure and other factors may influence the response independent of endothelial function

Endothelial dysfunction is thought to underpin atherosclerotic cardiovascular disease. The most widely used in vivo test of endothelial function is flow-mediated dilation (FMD). However, the results of FMD may be subject to some confounding factors that are not fully understood. We investigated potential biophysical confounding factors that could cause a disassociation between FMD and true endothelial cell shear stress response (the release of endothelium-dependent relaxing factors in response to wall shear stress). Arterial hemodynamics during FMD was simulated using a novel computational modeling approach. The model included an endothelial response function relating changes in wall shear stress to changes in local vascular stiffness in the arm arteries and accounted for vascular stiffening with increasing blood pressure. The hemodynamic effects of cuff inflation and deflation were modeled by prescribing intraluminal arterial pressure changes and peripheral vasodilation. Evolution of arterial diameter and flow velocity during FMD was assessed by comparison against in vivo data. Our model revealed that vasoconstriction occurring immediately after cuff deflation is independent of endothelial response function and entirely caused by the change in transmural pressure along conduit arteries. Moreover, for the same endothelial response function model, FMD values increased exponentially with increasing peak flow velocity, decreased linearly with increasing arterial stiffness at a rate of 0.95%/MPa, and increased linearly with increasing central blood pressure at a rate of 0.22%/mmHg. Dependence of FMD on confounding factors, such as arterial stiffness and blood pressure, suggests that the current FMD test may not reflect the true endothelial cell response.NEW & NOTEWORTHY First, a novel computational model simulating arterial hemodynamics during flow-mediated dilation (FMD) was proposed. Second, the model was used to explain why FMD may be influenced by endothelium-independent factors, showing that FMD results are 1) partly masked by the vasoconstriction due to the change in transmural pressure and 2) affected by physiological factors (i.e., arterial stiffness and arterial blood pressure) that are difficult to eliminate due to their multiple interactions.

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.

Modeling of pulsatile flow-dependent nitric oxide regulation in a realistic microvascular network

Hemodynamic pulsatility has been reported to regulate microcirculatory function. To quantitatively assess the impact of flow pulsatility on the microvasculature, a mathematical model was first developed to simulate the regulation of NO production by pulsatile flow in the microcirculation. Shear stress and pressure pulsatility were selected as regulators of endothelial NO production and NO-dependent vessel dilation as feedback to control microvascular hemodynamics. The model was then applied to a real microvascular network of the rat mesentery consisting of 546 microvessels. As compared to steady flow conditions, pulsatile flow increased the average NO concentration in arterioles from 256.8±93.1nM to 274.8±101.1nM (P<0.001), with a corresponding increase in vessel dilation by approximately 7% from 27.5±10.6% to 29.4±11.4% (P<0.001). In contrast, NO concentration and vessel size showed a far lesser increase (about 1.7%) in venules under pulsatile flow as compared to steady flow conditions. Network perfusion and flow heterogeneity were improved under pulsatile flow conditions, and vasodilation within the network was more sensitive to heart rate changes than pulse pressure amplitude. The proposed model simulates the role of flow pulsatility in the regulation of a complex microvascular network in terms of NO concentration and hemodynamics under varied physiological conditions.

Exposure-response modeling of flow-mediated dilation provides an unbiased and informative measure of endothelial function

The brachial artery flow-mediated dilation (FMD) test is the most widely utilized method to evaluate endothelial function noninvasively in humans by calculating the percent change in diameter (FMD%). However, the underutilized velocity and diameter time course data, coupled with confounding influences in shear exposure, noise, and upward bias, make the FMD test less desirable. In this study, we developed an exposure-response, model-based approach that not only quantifies FMD based on the rich velocity and diameter data, it overcomes previously acknowledged challenges. FMD data were obtained from 15 apparently healthy participants, each exposed to four different cuff occlusion durations. The velocity response following cuff release was described by an exponential model with two parameters defining peak velocity and rate of decay. Shear exposure derived from velocity was used to drive the diameter response model, which consists of additive constriction and dilation terms. Three parameters describing distinct aspects of the vascular response to shear (magnitude of the initial constriction response, and magnitude and time constant of the dilation response) were estimated for both the individuals and population. These parameters are independent of shear exposure. Thus this approach produces identifiable and physiologically meaningful parameters that may provide additional information for comparing differences between experimental groups or over time, and provides a means to completely account for shear exposure.NEW & NOTEWORTHY While flow-mediated dilation (FMD) is a valuable tool for evaluating endothelial function, analytical challenges include confounding influences of shear exposure, upward bias, and underutilization of rich time course data collected during FMD testing. We have developed an exposure-response, model-based approach that quantifies endothelial function based on the velocity and diameter data and fully accounts for shear exposure. It produces physiologically meaningful parameters that may provide useful information for comparing differences between experimental groups or over time.