Blood flow distribution in an anatomically detailed arterial network model: criteria and algorithms

A computational study of the connection between coronary revascularization and cardio-cerebral hemodynamics

BACKGROUND AND OBJECTIVE

Some patients experience life-threatening strokes during coronary revascularization. Despite its clinical importance, few numerical studies have investigated the impact of coronary revascularization on cardio-cerebral hemodynamics. This study aims to address this gap by evaluating the effects of eliminating coronary stenosis on both coronary and cerebral blood flow using patient-specific simulations.

METHODS

A patient-specific cardio-cerebral arterial network with a 70% stenosis in the left main coronary artery was reconstructed, and computational fluid dynamics were employed to evaluate the effects of eliminating coronary stenosis. The three-dimensional time-dependent incompressible Navier-Stokes equations were discretized using a stabilized P1-P1 Galerkin finite element method and an implicit second-order backward differentiation formula. A regional blood flow distribution model, coupled with a lumped Windkessel model, was applied at the outlet boundaries. The 3D pulsatile blood flow was solved using a parallel solver based on a scalable Newton-Krylov-Schwarz algorithm, enabling fast and efficient simulations.

RESULTS

Coronary revascularization significantly improved myocardial blood flow, increasing the coronary fractional flow reserve from 0.742 to 0.904, indicating enhanced myocardial perfusion. However, cerebral hemodynamics were negatively affected, with a 2.49% reduction in blood flow through the main cerebral artery, suggesting an elevated risk of cerebral ischemia. The proposed computational framework demonstrated good parallel scalability across thousands of processor cores.

CONCLUSIONS

This study highlights the dual impact of coronary revascularization, improving myocardial perfusion while potentially elevating the risk of cerebral ischemic complications. The efficient computational approach provides a valuable tool for evaluating cardio-cerebral hemodynamics in patient-specific settings, making it suitable for complex and time-intensive simulations.

Semi-automated pipeline for generating personalised cerebrovascular models

Subject-specific cerebrovascular models predict individual unmeasurable vessel haemodynamics using principles of physics, assumed constitutive laws, and measurement-deduced boundary conditions. However, the process of generating these models can be time-consuming, which is a barrier for use in time-sensitive clinical applications. In this work, we developed a semi-automated pipeline to generate anatomically and functionally personalised 0D cerebrovascular models from vasculature geometry and blood flow data. The pipeline extracts the vessel connectivity and geometric parameters from vessel segmentation to automatically generate a bond graph-based (linear and time-dependent) model of subject vasculature. Then, using a neurofuzzy control scheme, the peripheral resistances of the model are calibrated to minimise the discrepancy between measured and predicted blood flow distributions. We validated the pipeline by generating subject-specific models of the Circle of Willis (CoW) for 10 cases and compared haemodynamic predictions against acquired 4D flow MRI data. The results showed a relative error of 0.25 ± 0.66 % for flow and 13.87 ± 18.24 % for pulsatility, with a higher error for smaller vessels. We then demonstrated a use case of the model by simulating the blood flow redistribution during vascular occlusion for different CoW geometries. The results highlighted the benefit of a completely connected CoW to redistribute flow. The modular nature and rapid model generation time of this pipeline make it a promising tool for research and clinical use, where the type and structure of data are variable, and computing resources may be limited.

Closed-loop baroreflex model with biophysically detailed afferent pathway

In this work, we couple a lumped-parameter closed-loop model of the cardiovascular system with a physiologically-detailed mathematical description of the baroreflex afferent pathway. The model features a classical Hodgkin-Huxley current-type model for the baroreflex afferent limb (primary neuron) and for the second-order neuron in the central nervous system. The pulsatile arterial wall distension triggers a frequency-modulated sequence of action potentials at the afferent neuron. This signal is then integrated at the brainstem neuron model. The efferent limb, representing the sympathetic and parasympathetic nervous system, is described as a transfer function acting on heart and blood vessel model parameters in order to control arterial pressure. Three in silico experiments are shown here: a step increase in the aortic pressure to evaluate the functionality of the reflex arch, a hemorrhagic episode and an infusion simulation. Through this model, it is possible to study the biophysical dynamics of the ionic currents proposed for the afferent limb components of the baroreflex during the cardiac cycle, and the way in which currents dynamics affect the cardiovascular function. Moreover, this system can be further developed to study in detail each baroreflex loop component, helping to unveil the mechanisms involved in the cardiovascular afferent information processing.

Non-invasive assessment of stroke volume and cardiovascular parameters based on peripheral pressure waveform

Cardiovascular diseases are the leading cause of death globally, making the development of non-invasive and simple-to-use tools that bring insights into the state of the cardiovascular system of utmost importance. We investigated the possibility of using peripheral pulse wave recordings to estimate stroke volume (SV) and subject-specific parameters describing the selected properties of the cardiovascular system. Peripheral pressure waveforms were recorded in the radial artery using applanation tonometry (SphygmoCor) in 35 hemodialysis (HD) patients and 14 healthy subjects. The pressure waveforms were then used to estimate subject-specific parameters of a mathematical model of pulse wave propagation coupled with the elastance-based model of the left ventricle. Bioimpedance cardiography measurements (PhysioFlow) were performed to validate the model-estimated SV. Mean absolute percentage error between the simulated and measured pressure waveforms was 4.0% and 2.8% for the HD and control group, respectively. We obtained a moderate correlation between the model-estimated and bioimpedance-based SV (r = 0.57, p<0.05, and r = 0.58, p<0.001, for the control group and HD patients, respectively). We also observed a correlation between the estimated end-systolic elastance of the left ventricle and the peripheral systolic pressure in both HD patients (r = 0.84, p<0.001) and the control group (r = 0.70, p<0.01). These preliminary results suggest that, after additional validation and possibly further refinement to increase accuracy, the proposed methodology could support non-invasive assessment of stroke volume and selected heart function parameters and vascular properties. Importantly, the proposed method could be potentially implemented in the existing devices measuring peripheral pressure waveforms.

An anatomically detailed arterial-venous network model. Cerebral and coronary circulation

In recent years, several works have addressed the problem of modeling blood flow phenomena in veins, as a response to increasing interest in modeling pathological conditions occurring in the venous network and their connection with the rest of the circulatory system. In this context, one-dimensional models have proven to be extremely efficient in delivering predictions in agreement with in-vivo observations. Pursuing the increase of anatomical accuracy and its connection to physiological principles in haemodynamics simulations, the main aim of this work is to describe a novel closed-loop Anatomically-Detailed Arterial-Venous Network (ADAVN) model. An extremely refined description of the arterial network consisting of 2,185 arterial vessels is coupled to a novel venous network featuring high level of anatomical detail in cerebral and coronary vascular territories. The entire venous network comprises 189 venous vessels, 79 of which drain the brain and 14 are coronary veins. Fundamental physiological mechanisms accounting for the interaction of brain blood flow with the cerebro-spinal fluid and of the coronary circulation with the cardiac mechanics are considered. Several issues related to the coupling of arterial and venous vessels at the microcirculation level are discussed in detail. Numerical simulations are compared to patient records published in the literature to show the descriptive capabilities of the model. Furthermore, a local sensitivity analysis is performed, evidencing the high impact of the venous circulation on main cardiovascular variables.

Automated model calibration with parallel MCMC: Applications for a cardiovascular system model

Computational physiological models continue to increase in complexity, however, the task of efficiently calibrating the model to available clinical data remains a significant challenge. One part of this challenge is associated with long calibration times, which present a barrier for the routine application of model-based prediction in clinical practice. Another aspect of this challenge is the limited available data for the unique calibration of complex models. Therefore, to calibrate a patient-specific model, it may be beneficial to verify that task-specific model predictions have acceptable uncertainty, rather than requiring all parameters to be uniquely identified. We have developed a pipeline that reduces the set of fitting parameters to make them structurally identifiable and to improve the efficiency of a subsequent Markov Chain Monte Carlo (MCMC) analysis. MCMC was used to find the optimal parameter values and to determine the confidence interval of a task-specific prediction. This approach was demonstrated on numerical experiments where a lumped parameter model of the cardiovascular system was calibrated to brachial artery cuff pressure, echocardiogram volume measurements, and synthetic cerebral blood flow data that approximates what can be obtained from 4D-flow MRI data. This pipeline provides a cerebral arterial pressure prediction that may be useful for determining the risk of hemorrhagic stroke. For a set of three patients, this pipeline successfully reduced the parameter set of a cardiovascular system model from 12 parameters to 8–10 structurally identifiable parameters. This enabled a significant (>4×) efficiency improvement in determining confidence intervals on predictions of pressure compared to performing a naive MCMC analysis with the full parameter set. This demonstrates the potential that the proposed pipeline has in helping address one of the key challenges preventing clinical application of such models. Additionally, for each patient, the MCMC approach yielded a 95% confidence interval on systolic blood pressure prediction in the middle cerebral artery smaller than ±10 mmHg (±1.3 kPa). The proposed pipeline exploits available high-performance computing parallelism to allow straightforward automation for general models and arbitrary data sets, enabling automated calibration of a parameter set that is specific to the available clinical data with minimal user interaction.

High-resolution cerebral blood flow simulation with a domain decomposition method and verified by the TCD measurement

BACKGROUND

An efficient and accurate blood flow simulation can be useful for understanding many vascular diseases. Accurately resolving the blood flow velocity based on patient-specific geometries and model parameters is still a major challenge because of complex geomerty and turbulence issues. In addition, obtaining results in a short amount of computing time is important so that the simulation can be used in the clinical environment. In this work, we present a parallel scalable method for the patient-specific blood flow simulation with focuses on its parallel performance study and clinical verification.

METHODS

We adopt a fully implicit unstructured finite element method for a patient-specific simulation of blood flow in a full precerebral artery. The 3D artery is constructed from MRI images, and a parallel Newton-Krylov method preconditioned with a two-level domain decomposition method is adopted to solve the large nonlinear system discretized from the time-dependent 3D Navier-Stokes equations in the artery with an integral outlet boundary condition. The simulated results are verified using the clinical data measured by transcranial Doppler ultrasound, and the parallel performance of the algorithm is studied on a supercomputer.

RESULTS

The simulated velocity matches the clinical measured data well. Other simulated blood flow parameters, such as pressure and wall shear stress, are within reasonable ranges. The results show that the parallel algorithm scales up to 2160 processors with a 49% parallel efficiency for solving a problem with over 20 million unstructured elements on a supercomputer. For a standard cerebral blood flow simulation case with approximately 4 million finite elements, the calculation of one cardiac cycle can be finished within one hour with 1000 processors.

CONCLUSION

The proposed method is able to perform high-resolution 3D blood flow simulations in a patient-specific full precerebral artery within an acceptable time, and the simulated results are comparable with the clinical measured data, which demonstrates its high potential for clinical applications.

A computationally efficient physiologically comprehensive 3D-0D closed-loop model of the heart and circulation

Computer models of cardiac electro-mechanics (EM) show promise as an effective means for the quantitative analysis of clinical data and, potentially, for predicting therapeutic responses. To realize such advanced applications methodological key challenges must be addressed. Enhanced computational efficiency and robustness is crucial to facilitate, within tractable time frames, model personalization, the simulation of prolonged observation periods under a broad range of conditions, and physiological completeness encompassing therapy-relevant mechanisms is needed to endow models with predictive capabilities beyond the mere replication of observations. Here, we introduce a universal feature-complete cardiac EM modeling framework that builds on a flexible method for coupling a 3D model of bi-ventricular EM to the physiologically comprehensive 0D CircAdapt model representing atrial mechanics and closed-loop circulation. A detailed mathematical description is given and efficiency, robustness, and accuracy of numerical scheme and solver implementation are evaluated. After parameterization and stabilization of the coupled 3D-0D model to a limit cycle under baseline conditions, the model's ability to replicate physiological behaviors is demonstrated, by simulating the transient response to alterations in loading conditions and contractility, as induced by experimental protocols used for assessing systolic and diastolic ventricular properties. Mechanistic completeness and computational efficiency of this novel model render advanced applications geared towards predicting acute outcomes of EM therapies feasible.

Parallel generation of extensive vascular networks with application to an archetypal human kidney model

Given the relevance of the inextricable coupling between microcirculation and physiology, and the relation to organ function and disease progression, the construction of synthetic vascular networks for mathematical modelling and computer simulation is becoming an increasingly broad field of research. Building vascular networks that mimic in vivo morphometry is feasible through algorithms such as constrained constructive optimization (CCO) and variations. Nevertheless, these methods are limited by the maximum number of vessels to be generated due to the whole network update required at each vessel addition. In this work, we propose a CCO-based approach endowed with a domain decomposition strategy to concurrently create vascular networks. The performance of this approach is evaluated by analysing the agreement with the sequentially generated networks and studying the scalability when building vascular networks up to 200 000 vascular segments. Finally, we apply our method to vascularize a highly complex geometry corresponding to the cortex of a prototypical human kidney. The technique presented in this work enables the automatic generation of extensive vascular networks, removing the limitation from previous works. Thus, we can extend vascular networks (e.g. obtained from medical images) to pre-arteriolar level, yielding patient-specific whole-organ vascular models with an unprecedented level of detail.

Integrated cardiorespiratory system model with short timescale control mechanisms

A compartmental model of the cardiorespiratory system featuring pulsatile blood flow and gas transport, as well as closed loop mechanisms of cardiorespiratory regulation is presented. Short timescale regulatory action includes baroreflex, peripheral and central chemoreflex feedback. The cardiorespiratory model is composed by compartments to describe blood flow and gas exchange in the major systemic and pulmonic regions. The control systems include formulations to afferent activity of arterial baroreceptor and peripheral and central chemoreceptors. Simulations described here include situations of hypoxia, hypercapnia, and hemorrhage. The overall responses of our simulations agree with physiological (experimental) and theoretical data. Our results suggest that the present model could be used to further understand the interplay among major regulatory mechanisms in the functioning of the cardiovascular and respiratory systems in cases of normal and abnormal physiological conditions.

A physiologically realistic virtual patient database for the study of arterial haemodynamics

This study creates a physiologically realistic virtual patient database (VPD), representing the human arterial system, for the primary purpose of studying the effects of arterial disease on haemodynamics. A low dimensional representation of an anatomically detailed arterial network is outlined, and a physiologically realistic posterior distribution for its parameters constructed through the use of a Bayesian approach. This approach combines both physiological/geometrical constraints and the available measurements reported in the literature. A key contribution of this work is to present a framework for including all such available information for the creation of virtual patients (VPs). The Markov Chain Monte Carlo (MCMC) method is used to sample random VPs from this posterior distribution, and the pressure and flow-rate profiles associated with each VP computed through a physics based model of pulse wave propagation. This combination of the arterial network parameters (representing a virtual patient) and the haemodynamics waveforms of pressure and flow-rates at various locations (representing functional response and potential measurements that can be acquired in the virtual patient) makes up the VPD. While 75,000 VPs are sampled from the posterior distribution, 10,000 are discarded as the initial burn-in period of the MCMC sampler. A further 12,857 VPs are subsequently removed due to the presence of negative average flow-rate, reducing the VPD to 52,143. Due to undesirable behaviour observed in some VPs-asymmetric under- and over-damped pressure and flow-rate profiles in left and right sides of the arterial system-a filter is proposed to remove VPs showing such behaviour. Post application of the filter, the VPD has 28,868 subjects. It is shown that the methodology is appropriate by comparing the VPD statistics to those reported in literature across real populations. Generally, a good agreement between the two is found while respecting physiological/geometrical constraints.

Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. Intravascular pressure and velocity are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix.

Sex and the G Protein-Coupled Estrogen Receptor Impact Vascular Stiffness

[Figure: see text].

Uncertainty quantification implementations in human hemodynamic flows

BACKGROUND AND OBJECTIVE

Human hemodynamic modeling is usually influenced by uncertainties occurring from a considerable unavailability of information linked to the boundary conditions and the physical properties used in the numerical models. Calculating the effect of these uncertainties on the numerical findings along the cardiovascular system is a demanding process due to the complexity of the morphology of the body and the area dynamics. To cope with all these difficulties, Uncertainty Quantification (UQ) methods seem to be an ideal tool.

RESULTS

This study focuses on analyzing and summarizing some of the recent research efforts and directions of implementing UQ in human hemodynamic flows by analyzing 139 research papers. Initially, the suitability of applying this approach is analyzed and demonstrated. Then, an overview of the most significant research work in various fields of biomedical hemodynamic engineering is presented. Finally, it is attempted to identify any possible forthcoming directions for research and methodological progress of UQ in biomedical sciences.

CONCLUSION

This review concludes that by finding the best statistical methods and parameters to represent the propagated uncertainties, while achieving a good interpretation of the interaction between input-output, is crucial for implementing UQ in biomedical sciences.

A simple coronary blood flow model to study the collateral flow index

In this work, we present a novel modeling framework to investigate the effects of collateral circulation into the coronary blood flow physiology. A prototypical model of the coronary tree, integrated with the concept of Collateral Flow Index (CFI), is employed to gain insight about the role of model parameters associated with the collateral circuitry, which results in physically-realizable solutions for specific CFI data. Then, we discuss the mathematical feasibility of pressure-derived CFI, anatomical implications and practical considerations involving the estimation of model parameters in collateral connections. A sensitivity analysis is carried out, and the investigation of the impact of the collateral circulation on FFR values is also addressed.

Adaptive constrained constructive optimisation for complex vascularisation processes

Mimicking angiogenetic processes in vascular territories acquires importance in the analysis of the multi-scale circulatory cascade and the coupling between blood flow and cell function. The present work extends, in several aspects, the Constrained Constructive Optimisation (CCO) algorithm to tackle complex automatic vascularisation tasks. The main extensions are based on the integration of adaptive optimisation criteria and multi-staged space-filling strategies which enhance the modelling capabilities of CCO for specific vascular architectures. Moreover, this vascular outgrowth can be performed either from scratch or from an existing network of vessels. Hence, the vascular territory is defined as a partition of vascular, avascular and carriage domains (the last one contains vessels but not terminals) allowing one to model complex vascular domains. In turn, the multi-staged space-filling approach allows one to delineate a sequence of biologically-inspired stages during the vascularisation process by exploiting different constraints, optimisation strategies and domain partitions stage by stage, improving the consistency with the architectural hierarchy observed in anatomical structures. With these features, the aDaptive CCO (DCCO) algorithm proposed here aims at improving the modelled network anatomy. The capabilities of the DCCO algorithm are assessed with a number of anatomically realistic scenarios.

Numerical Simulation of Blood Flows in Patient-specific Abdominal Aorta with Primary Organs

The abdominal aorta is the largest artery in the abdominal cavity that supplies blood flows to vital organs through the complex visceral arterial branches, including the celiac trunk (the liver, stomach, spleen, etc.), the renal arteries (the kidneys) and the superior and inferior mesenteric arteries (the small and large intestine, pancreas, etc.). An accurate simulation of blood flows in this network of arteries is important for the understanding of the hemodynamics in various organs of healthy and diseased patients, but the computational cost is very high. As a result, most researchers choose to focus on a portion of the artery or use a low-dimensional approximation of the artery. In the present work, we introduce a parallel algorithm for the modeling of pulsatile flows in the abdominal aorta with branches to the primary organs, and an organ-based two-level method for calculating the resistances for the outflow boundary conditions. With this highly parallel approach, the simulation of the blood flow for a cardiac cycle of the anatomically detailed aorta can be obtained within a few hours, and the blood distribution to organs including liver, spleen and kidneys are also computed with certain accuracy. Moreover, we discuss the significant hemodynamic differences resulted from the influence of the peripheral branches. In addition, we examine the accuracy of the results with respect to the mesh size and time-step size and show the high parallel scalability of the proposed algorithm with up to 3000 processor cores.

Particle Image Velocimetry Measurements of the Flow-Diverting Effects of a New Generation of the eCLIPs Implant for the Treatment of Intracranial Bifurcation Aneurysms

Flow diverters (FDs) for the endovascular treatment of intracranial aneurysms are effective for sidewall aneurysms, but their use at a bifurcation is problematic because FDs only partially cover the aneurysm neck and impede flow into a daughter branch; they are thus not employed routinely in this anatomy. eCLIPs was developed as a non-tubular implant to completely cover the neck of an aneurysm and serve as a coil retention device necessary for the adequate treatment of wide-neck bifurcation aneurysms. eCLIPs has shown some flow diversion effects in bifurcation anatomy but not equal to those exhibited by clinically accepted flow diverters in sidewall anatomy. A new generation of eCLIPs implant, the eCLIPs bifurcation flow diverter (eBFD), with higher metal coverage, was developed to achieve a similar flow diversion as a Pipeline Embolization Device (PED), a prototypical FD. Particle image velocimetry was used to capture the fluid dynamics and velocity reduction within silicone aneurysm replicas. A circulatory mimicking loop was developed to circulate the flow through the silicone models. All generations of eCLIPs implants had some flow-diverting effect, with increasing metal coverage density of the implant proportionately increasing the flow diversion effect. The eBFD, with a metal density of 35%, showed greater flow diversion than PED, with 30% metal density, for bifurcation anatomy. The eBFD showed similar reduction of flow in a bifurcation anatomy to PED in a sidewall, both sufficient to permit early thrombosis of the aneurysm. Thus, the eBFD can potentially provide sufficient flow diversion for the treatment of bifurcation aneurysms to avoid adjunctive coiling.

Comparison of arterial wave intensity analysis by pressure-velocity and diameter-velocity methods in a virtual population of adult subjects

Pressure-velocity-based analysis of arterial wave intensity gives clinically relevant information about the performance of the heart and vessels, but its utility is limited because accurate pressure measurements can only be obtained invasively. Diameter-velocity-based wave intensity can be obtained noninvasively using ultrasound; however, due to the nonlinear relationship between blood pressure and arterial diameter, the two wave intensities might give disparate clinical indications. To test the magnitude of the disagreement, we have generated an age-stratified virtual population to investigate how the two dominant nonlinearities viscoelasticity and strain-stiffening cause the two formulations to differ. We found strong agreement between the pressure-velocity and diameter-velocity methods, particularly for the systolic wave energy, the ratio between systolic and diastolic wave heights, and older subjects. The results are promising regarding the introduction of noninvasive wave intensities in the clinic.

Computational feasibility of simulating changes in blood flow through whole-organ vascular networks from radiation injury

Vasculature is necessary to the healthy function of most tissues. In radiation therapy, injury of the vasculature can have both beneficial and detrimental effects, such as tumor starvation, cardiac fibrosis, and white-matter necrosis. These effects are caused by changes in blood flow due to the vascular injury. Previously, research has focused on simulating the radiation injury of vasculature in small volumes of tissue, ignoring the systemic effects of local damage on blood flow. Little is known about the computational feasibility of simulating the radiation injury to whole-organ vascular networks. The goal of this study was to test the computational feasibility of simulating the dose deposition to a whole-organ vascular network and the resulting change in blood flow. To do this, we developed an amorphous track-structure model to transport radiation and combined this with existing methods to model the vasculature and blood flow rates. We assessed the algorithm's computational scalability, execution time, and memory usage. The data demonstrated it is computationally feasible to calculate the radiation dose and resulting changes in blood flow from 2 million protons to a network comprising 8.5 billion blood vessels (approximately the number in the human brain) in 87 hours using a 128-node cluster. Furthermore, the algorithm demonstrated both strong and weak scalability, meaning that additional computational resources can reduce the execution time further. These results demonstrate, for the first time, that it is computationally feasible to calculate radiation dose deposition in whole-organ vascular networks. These findings provide key insights into the computational aspects of modeling whole-organ radiation damage. Modeling the effects radiation has on vasculature could prove useful in the study of radiation effects on tissues, organs, and organisms.

Personalization of electro-mechanical models of the pressure-overloaded left ventricle: fitting of Windkessel-type afterload models

Computer models of left ventricular (LV) electro-mechanics (EM) show promise as a tool for assessing the impact of increased afterload upon LV performance. However, the identification of unique afterload model parameters and the personalization of EM LV models remains challenging due to significant clinical input uncertainties. Here, we personalized a virtual cohort of N = 17 EM LV models under pressure overload conditions. A global-local optimizer was developed to uniquely identify parameters of a three-element Windkessel (Wk3) afterload model. The sensitivity of Wk3 parameters to input uncertainty and of the EM LV model to Wk3 parameter uncertainty was analysed. The optimizer uniquely identified Wk3 parameters, and outputs of the personalized EM LV models showed close agreement with clinical data in all cases. Sensitivity analysis revealed a strong dependence of Wk3 parameters on input uncertainty. However, this had limited impact on outputs of EM LV models. A unique identification of Wk3 parameters from clinical data appears feasible, but it is sensitive to input uncertainty, thus depending on accurate invasive measurements. By contrast, the EM LV model outputs were less sensitive, with errors of less than 8.14% for input data errors of 10%, which is within the bounds of clinical data uncertainty. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

The Effects of Cerebral Vasospasm on Cerebral Blood Flow and the Effects of Induced Hypertension: A Mathematical Modelling Study

Background

Induced hypertension has been used to promote cerebral blood flow under vasospastic conditions although there is no randomised clinical trial to support its use. We sought to mathematically model the effects of vasospasm on the cerebral blood flow and the effects of induced hypertension.

Methods

The Anatomically Detailed Arterial Network (ADAN) model is employed as the anatomical substrate in which the cerebral blood flow is simulated as part of the simulation of the whole body arterial circulation. The pressure drop across the spastic vessel is modelled by inserting a specific constriction model within the corresponding vessel in the ADAN model. We altered the degree of vasospasm, the length of the vasospastic segment, the location of the vasospasm, the pressure (baseline mean arterial pressure [MAP] 90 mm Hg, hypertension MAP 120 mm Hg, hypotension), and the presence of collateral supply.

Results

Larger decreases in cerebral flow were seen for diffuse spasm and more severe vasospasm. The presence of collateral supply could maintain cerebral blood flow, but only if the vasospasm did not occur distal to the collateral. Induced hypertension caused an increase in blood flow in all scenarios, but did not normalise blood flow even in the presence of moderate vasospasm (30%). Hypertension in the presence of a complete circle of Willis had a marginally greater effect on the blood flow, but did not normalise flow.

Conclusion

Under vasospastic condition, cerebral blood flow varies considerably. Hypertension can raise the blood flow, but it is unable to restore cerebral blood flow to baseline.

Blood Flow Contributions to Cancer Metastasis

The distribution patterns of cancer metastasis depend on a sequence of steps involving adhesion molecules and on mechanical and geometrical effects related to blood circulation, but how much each of these two aspects contributes to the metastatic spread of a specific tumor is still unknown. Here we address this question by simulating cancer cell trajectories in a high-resolution humanoid model of global blood circulation, including stochastic adhesion events, and comparing the results with the location of metastasis recorded in thousands of human autopsies for seven different solid tumors, including lung, prostate, pancreatic and colorectal cancers, showing that on average 40% of the variation in the metastatic distribution can be attributed to blood circulation. Our humanoid model of circulating tumor cells allows us to predict the metastatic spread in specific realistic conditions and can therefore guide precise therapeutic interventions to fight metastasis.

Personalized Hemodynamic Modeling of the Human Cardiovascular System: A Reduced-Order Computing Model

OBJECTIVE

Personalization of hemodynamic modeling plays a crucial role in functional prediction of the cardiovascular system (CVS). While reduced-order models of one-dimensional (1D) blood vessel models with zero-dimensional (0D) blood vessel and heart models have been widely recognized to be an effective tool for reasonably estimating the hemodynamic functions of the whole CVS, practical personalized models are still lacking. In this paper, we present a novel 0-1D coupled, personalized hemodynamic model of the CVS that can predict both pressure waveforms and flow velocities in arteries.

METHODS

We proposed a methodology by combining the multiscale CVS model with the Levenberg-Marquardt optimization algorithm for effectively solving an inverse problem based on measured blood pressure waveforms. Hemodynamic characteristics including brachial arterial pressure waveforms, artery diameters, stroke volumes, and flow velocities were measured noninvasively for 62 volunteers aged from 20 to 70 years for developing and validating the model.

RESULTS

The estimated arterial stiffness shows a physiologically realistic distribution. The model-fitted individual pressure waves have an averaged mean square error (MSE) of 7.1 mmHg²; simulated blood flow velocity waveforms in carotid artery match ultrasound measurements well, achieving an average correlation coefficient of 0.911.

CONCLUSION

The model is efficient, versatile, and capable of obtaining well-fitting individualized pressure waveforms while reasonably predicting flow waveforms.

SIGNIFICANCE

The proposed methodology of personalized hemodynamic modeling may therefore facilitate individualized patient-specific assessment of both physiological and pathological functions of the CVS.

A novel porous media-based approach to outflow boundary resistances of 1D arterial blood flow models

In this paper we introduce a novel method for prescribing terminal boundary conditions in one-dimensional arterial flow networks. This is carried out by coupling the terminal arterial vessel with a poro-elastic tube, representing the flow resistance offered by microcirculation. The performance of the proposed porous media-based model has been investigated through several different numerical examples. First, we investigate model parameters that have a profound influence on the flow and pressure distributions of the system. The simulation results have been compared against the waveforms generated by three elements (RCR) Windkessel model. The proposed model is also integrated into a realistic arterial tree, and the results obtained have been compared against experimental data at different locations of the network. The accuracy and simplicity of the proposed model demonstrates that it can be an excellent alternative for the existing models.

Mathematical synthesis of the cortical circulation for the whole mouse brain-part I. theory and image integration

Microcirculation plays a significant role in cerebral metabolism and blood flow control, yet explaining and predicting functional mechanisms remains elusive because it is difficult to make physiologically accurate mathematical models of the vascular network. As a precursor to the human brain, this paper presents a computational framework for synthesizing anatomically accurate network models for the cortical blood supply in mouse. It addresses two critical deficiencies in cerebrovascular modeling. At the microscopic length scale of individual capillaries, we present a novel synthesis method for building anatomically consistent capillary networks with loops and anastomoses (=microcirculatory closure). This overcomes shortcomings in existing algorithms which are unable to create closed circulatory networks. A second critical innovation allows the incorporation of detailed anatomical features from image data into vascular growth. Specifically, computed tomography and two photon laser scanning microscopy data are input into the novel synthesis algorithm to build the cortical circulation for the entire mouse brain in silico. Computer predictions of blood flow and oxygen exchange executed on synthetic large-scale network models are expected to elucidate poorly understood functional mechanisms of the cerebral circulation.

A multi-scale model of the coronary circulation applied to investigate transmural myocardial flow

Distribution of blood flow in myocardium is a key determinant of the localization and severity of myocardial ischemia under impaired coronary perfusion conditions. Previous studies have extensively demonstrated the transmural difference of ischemic vulnerability. However, it remains incompletely understood how transmural myocardial flow is regulated under in vivo conditions. In the present study, a computational model of the coronary circulation was developed to quantitatively evaluate the sensitivity of transmural flow distribution to various cardiovascular and hemodynamic factors. The model was further incorporated with the flow autoregulatory mechanism to simulate the regulation of myocardial flow in the presence of coronary artery stenosis. Numerical tests demonstrated that heart rate (HR), intramyocardial tissue pressure (P_im ), and coronary perfusion pressure (P_per ) were the major determinant factors for transmural flow distribution (evaluated by the subendocardial-to-subepicardial (endo/epi) flow ratio) and that the flow autoregulatory mechanism played an important compensatory role in preserving subendocardial perfusion against reduced P_per . Further analysis for HR variation-induced hemodynamic changes revealed that the rise in endo/epi flow ratio accompanying HR decrease was attributable not only to the prolongation of cardiac diastole relative to systole, but more predominantly to the fall in P_im . Moreover, it was found that P_im and P_per interfered with each other with respect to their influence on transmural flow distribution. These results demonstrate the interactive effects of various cardiovascular and hemodynamic factors on transmural myocardial flow, highlighting the importance of taking into account patient-specific conditions in the explanation of clinical observations.

Assessment of boundary conditions for CFD simulation in human carotid artery

Computational fluid dynamics (CFD) is an increasingly used method for investigation of hemodynamic parameters and their alterations under pathological conditions, which are important indicators for diagnosis of cardiovascular disease. In hemodynamic simulation models, the employment of appropriate boundary conditions (BCs) determines the computational accuracy of the CFD simulation in comparison with pressure and velocity measurements. In this study, we have first assessed the influence of inlet boundary conditions on hemodynamic CFD simulations. We selected two typical patients suspected of carotid artery disease, with mild stenosis and severe stenosis. Both patients underwent digital subtraction angiography (DSA), magnetic resonance angiography, and the invasive pressure guide wire measured pressure profile. We have performed computational experiments to (1) study the hemodynamic simulation outcomes of distributions of wall shear stress, pressure, pressure gradient and (2) determine the differences in hemodynamic performances caused by inlet BCs derived from DSA and Womersley analytical solution. Our study has found that the difference is related to the severity of the stenosis; the greater the stenosis, the more the difference ensues. Further, in our study, the two typical subjects with invasively measured pressure profile and thirty subjects with ultrasound Doppler velocimeter (UDV) measurement served as the criteria to evaluate the hemodynamic outcomes of wall shear stress, pressure, pressure gradient and velocity due to different outlet BCs based on the Windkessel model, structured-tree model, and fully developed flow model. According to the pressure profiles, the fully developed model appeared to have more fluctuations compared with the other two models. The Windkessel model had more singularities before convergence. The three outlet BCs models also showed good correlation with the UDV measurement, while the Windkessel model appeared to be slightly better ([Formula: see text]). The structured-tree model was seen to have the best performance in terms of available computational cost and accuracy. The results of our numerical simulation and the good correlation with the computed pressure and velocity with their measurements have highlighted the effectiveness of CFD simulation in patient-specific human carotid artery with suspected stenosis.

Quantification of near-wall hemodynamic risk factors in large-scale cerebral arterial trees

Detailed hemodynamic analysis of blood flow in pathological segments close to aneurysm and stenosis has provided physicians with invaluable information about the local flow patterns leading to vascular disease. However, these diseases have both local and global effects on the circulation of the blood within the cerebral tree. The aim of this paper is to demonstrate the importance of extending subject-specific hemodynamic simulations to the entire cerebral arterial tree with hundreds of bifurcations and vessels, as well as evaluate hemodynamic risk factors and waveform shape characteristics throughout the cerebral arterial trees. Angioarchitecture and in vivo blood flow measurement were acquired from healthy subjects and in cases with symptomatic intracranial aneurysm and stenosis. A global map of cerebral arterial blood flow distribution revealed regions of low to high hemodynamic risk that may significantly contribute to the development of intracranial aneurysms or atherosclerosis. Comparison of pre-intervention and post-intervention of pathological cases further shows large angular phase shift (~33.8°), and an augmentation of the peak-diastolic velocity. Hemodynamic indexes of waveform analysis revealed on average a 16.35% reduction in the pulsatility index after treatment from lesion site to downstream distal vessels. The lesion regions not only affect blood flow streamlines of the proximal sites but also generate pulse wave shift and disturbed flow in downstream vessels. This network effect necessitates the use of large-scale simulation to visualize both local and global effects of pathological lesions.

Bond Graph Model of Cerebral Circulation: Toward Clinically Feasible Systemic Blood Flow Simulations

We propose a detailed CellML model of the human cerebral circulation that runs faster than real time on a desktop computer and is designed for use in clinical settings when the speed of response is important. A lumped parameter mathematical model, which is based on a one-dimensional formulation of the flow of an incompressible fluid in distensible vessels, is constructed using a bond graph formulation to ensure mass conservation and energy conservation. The model includes arterial vessels with geometric and anatomical data based on the ADAN circulation model. The peripheral beds are represented by lumped parameter compartments. We compare the hemodynamics predicted by the bond graph formulation of the cerebral circulation with that given by a classical one-dimensional Navier-Stokes model working on top of the whole-body ADAN model. Outputs from the bond graph model, including the pressure and flow signatures and blood volumes, are compared with physiological data.

Determinant Factors for Arterial Hemodynamics in Hypertension: Theoretical Insights From a Computational Model-Based Study

Hypertension is a well-documented predictive factor for cardiovascular events. Clinical studies have extensively demonstrated the differential hemodynamic consequences of various antihypertensive drugs, but failed to clearly elucidate the underlying mechanisms due to the difficulty in performing a quantitative deterministic analysis based on clinical data that carry confounding information stemming from interpatient differences and the nonlinearity of cardiovascular hemodynamics. In the present study, a multiscale model of the cardiovascular system was developed to quantitatively investigate the relationships between hemodynamic variables and cardiovascular properties under hypertensive conditions, aiming to establish a theoretical basis for assisting in the interpretation of clinical observations or optimization of therapy. Results demonstrated that heart period, central arterial stiffness, and arteriolar radius were the major determinant factors for blood pressures and flow pulsatility indices both in large arteries and in the microcirculation. These factors differed in the degree and the way in which they affect hemodynamic variables due to their differential effects on wave reflections in the vascular system. In particular, it was found that the hemodynamic effects of varying arteriolar radius were considerably influenced by the state of central arterial stiffness, and vice versa, which implied the potential of optimizing antihypertensive treatment by selecting proper drugs based on patient-specific cardiovascular conditions. When analyzed in relation to clinical observations, the simulated results provided mechanistic explanations for the beneficial pressure-lowering effects of vasodilators as compared to β-blockers, and highlighted the significance of monitoring and normalizing arterial stiffness in the treatment of hypertension.

Subject-specific pulse wave propagation modeling: Towards enhancement of cardiovascular assessment methods

Cardiovascular diseases are the leading cause of death worldwide. Pulse wave analysis (PWA) technique, which reconstructs and analyses aortic pressure waveform based on non-invasive peripheral pressure recording, became an important bioassay for cardiovascular assessment in a general population. The aim of our study was to establish a pulse wave propagation modeling framework capable of matching clinical PWA data from healthy individuals on a per-subject basis. Radial pressure profiles from 20 healthy individuals (10 males, 10 females), with mean age of 42 ± 10 years, were recorded using applanation tonometry (SphygmoCor, AtCor Medical, Australia) and used to estimate subject-specific parameters of mathematical model of blood flow in the system of fifty-five arteries. The model was able to describe recorded pressure profiles with high accuracy (mean absolute percentage error of 1.87 ± 0.75%) when estimating only 6 parameters for each subject. Cardiac output (CO) and stroke volume (SV) have been correctly identified by the model as lower in females than males (CO of 3.57 ± 0.54 vs. 4.18 ± 0.72 L/min with p-value < 0.05; SV of 49.5 ± 10.1 vs. 64.2 ± 16.8 ml with p-value = 0.076). Moreover, the model identified age related changes in the heart function, i.e. that the cardiac output at rest is maintained with age (r = 0.23; p-value = 0.32) despite the decreasing heart rate (r = −0.49; p-value < 0.05), because of the increase in stroke volume (r = 0.46; p-value < 0.05). Central PWA indices derived from recorded waveforms strongly correlated with those obtained using corresponding model-predicted radial waves (r > 0.99 and r > 0.97 for systolic (SP) and diastolic (DP) pressures, respectively; r > 0.77 for augmentation index (AI); all p—values < 0.01). Model-predicted central waveforms, however, had higher SP than those reconstructed by PWA using recorded radial waves (5.6 ± 3.3 mmHg on average). From all estimated subject-specific parameters only the time to the peak of heart ejection profile correlated with clinically measured AI. Our study suggests that the proposed model may serve as a tool to computationally investigate virtual patient scenarios mimicking different cardiovascular abnormalities. Such a framework can augment our understanding and help with the interpretation of PWA results.

Arterial pulse attenuation prediction using the decaying rate of a pressure wave in a viscoelastic material model

The present study examines the possibility of attenuating blood pulses by means of introducing prosthetic viscoelastic materials able to absorb energy and damp such pulses. Vascular prostheses made of polymeric materials modify the mechanical properties of blood vessels. The effect of these materials on the blood pulse propagation remains to be fully understood. Several materials for medical applications, such as medical polydimethylsiloxane or polytetrafluoroethylene, show viscoelastic behavior, modifying the original vessel stiffness and affecting the propagation of blood pulses. This study focuses on the propagation of pressure waves along a pipe with viscoelastic materials using the Maxwell and the Zener models. An expression of exponential decay has been obtained for the Maxwell material model and also for low viscous coefficient values in the Zener model. For relatively high values of the viscous term in the Zener model, the steepest part of the pulse can be damped quickly, leaving a smooth, slowly decaying wave. These mathematical models are critical to tailor those materials used in cardiovascular implants to the mechanical environment they are confronted with to repair or improve blood vessel function.

Assessment of reduced-order unscented Kalman filter for parameter identification in 1-dimensional blood flow models using experimental data

This work presents a detailed investigation of a parameter estimation approach on the basis of the reduced-order unscented Kalman filter (ROUKF) in the context of 1-dimensional blood flow models. In particular, the main aims of this study are (1) to investigate the effects of using real measurements versus synthetic data for the estimation procedure (i.e., numerical results of the same in silico model, perturbed with noise) and (2) to identify potential difficulties and limitations of the approach in clinically realistic applications to assess the applicability of the filter to such setups. For these purposes, the present numerical study is based on a recently published in vitro model of the arterial network, for which experimental flow and pressure measurements are available at few selected locations. To mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Young's modulus and wall thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis on the basis of the generalized sensitivity function, comparing then the results owith the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.

An implicit solver for 1D arterial network models

In this study, the 1D blood flow equations are solved using a newly proposed enhanced trapezoidal rule method (ETM), which is an extension to the simplified trapezoidal rule method. At vessel junctions, the conservation of mass and conservation of total pressure are held as system constraints using Lagrange multipliers that can be physically interpreted as external flow rates. The ETM scheme is compared with published arterial network benchmark problems and a dam break problem. Strengths of the ETM scheme include being simple to implement, intuitive connection to lumped parameter models, and no restrictive stability criteria such as the Courant-Friedrichs-Lewy (CFL) number. The ETM scheme does not require the use of characteristics at vessel junctions, or for inlet and outlet boundary conditions. The ETM forms an implicit system of equations, which requires only one global solve per time step for pressure, followed by flow rate update on the elemental system of equations; thus, no iterations are required per time step. Consistent results are found for all benchmark cases, and for a 56-vessel arterial network problem, it gives very satisfactory solutions at a spatial and time discretization that results in a maximum CFL of 3, taking 4.44 seconds per cardiac cycle. By increasing the time step and element size to produce a maximum CFL number of 15, the method takes only 0.39 second per cardiac cycle with only a small compromise on accuracy.

Blood pressure gradients in cerebral arteries: a clue to pathogenesis of cerebral small vessel disease

Rationale

The role of hypertension in cerebral small vessel disease is poorly understood. At the base of the brain (the ‘vascular centrencephalon’), short straight arteries transmit blood pressure directly to small resistance vessels; the cerebral convexity is supplied by long arteries with many branches, resulting in a drop in blood pressure. Hypertensive small vessel disease (lipohyalinosis) causes the classically described lacunar infarctions at the base of the brain; however, periventricular white matter intensities (WMIs) seen on MRI and WMI in subcortical areas over the convexity, which are often also called ‘lacunes’, probably have different aetiologies.

Objectives

We studied pressure gradients from proximal to distal regions of the cerebral vasculature by mathematical modelling.

Methods and results

Blood flow/pressure equations were solved in an Anatomically Detailed Arterial Network (ADAN) model, considering a normotensive and a hypertensive case. Model parameters were suitably modified to account for structural changes in arterial vessels in the hypertensive scenario. Computations predict a marked drop in blood pressure from large and medium-sized cerebral vessels to cerebral peripheral beds. When blood pressure in the brachial artery is 192/113 mm Hg, the pressure in the small arterioles of the posterior parietal artery bed would be only 117/68 mm Hg. In the normotensive case, with blood pressure in the brachial artery of 117/75 mm Hg, the pressure in small parietal arterioles would be only 59/38 mm Hg.

Conclusion

These findings have important implications for understanding small vessel disease. The marked pressure gradient across cerebral arteries should be taken into account when evaluating the pathogenesis of small WMIs on MRI. Hypertensive small vessel disease, affecting the arterioles at the base of the brain should be distinguished from small vessel disease in subcortical regions of the convexity and venous disease in the periventricular white matter.

Transversally enriched pipe element method (TEPEM): An effective numerical approach for blood flow modeling

In this work, we present a novel approach tailored to approximate the Navier-Stokes equations to simulate fluid flow in three-dimensional tubular domains of arbitrary cross-sectional shape. The proposed methodology is aimed at filling the gap between (cheap) one-dimensional and (expensive) three-dimensional models, featuring descriptive capabilities comparable with the full and accurate 3D description of the problem at a low computational cost. In addition, this methodology can easily be tuned or even adapted to address local features demanding more accuracy. The numerical strategy employs finite (pipe-type) elements that take advantage of the pipe structure of the spatial domain under analysis. While low order approximation is used for the longitudinal description of the physical fields, transverse approximation is enriched using high order polynomials. Although our application of interest is computational hemodynamics and its relevance to pathological dynamics like atherosclerosis, the approach is quite general and can be applied in any internal fluid dynamics problem in pipe-like domains. Numerical examples covering academic cases as well as patient-specific coronary arterial geometries demonstrate the potentialities of the developed methodology and its performance when compared against traditional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.

A computational framework to characterize and compare the geometry of coronary networks

This work presents a computational framework to perform a systematic and comprehensive assessment of the morphometry of coronary arteries from in vivo medical images. The methodology embraces image segmentation, arterial vessel representation, characterization and comparison, data storage, and finally analysis. Validation is performed using a sample of 48 patients. Data mining of morphometric information of several coronary arteries is presented. Results agree to medical reports in terms of basic geometric and anatomical variables. Concerning geometric descriptors, inter-artery and intra-artery correlations are studied. Data reported here can be useful for the construction and setup of blood flow models of the coronary circulation. Finally, as an application example, similarity criterion to assess vasculature likelihood based on geometric features is presented and used to test geometric similarity among sibling patients. Results indicate that likelihood, measured through geometric descriptors, is stronger between siblings compared with non-relative patients. Copyright © 2016 John Wiley & Sons, Ltd.

A high-order local time stepping finite volume solver for one-dimensional blood flow simulations: application to the ADAN model

In recent years, the complexity of vessel networks for one-dimensional blood flow models has significantly increased, because of enhanced anatomical detail or automatic peripheral vasculature generation, for example. This fact, along with the application of these models in uncertainty quantification and parameter estimation poses the need for extremely efficient numerical solvers. The aim of this work is to present a finite volume solver for one-dimensional blood flow simulations in networks of elastic and viscoelastic vessels, featuring high-order space-time accuracy and local time stepping (LTS). The solver is built on (i) a high-order finite volume type numerical scheme, (ii) a high-order treatment of the numerical solution at internal vertexes of the network, often called junctions, and (iii) an accurate LTS strategy. The accuracy of the proposed methodology is verified by empirical convergence tests. Then, the resulting LTS scheme is applied to arterial networks of increasing complexity and spatial scale heterogeneity, with a number of one-dimensional segments ranging from a few tens up to several thousands and vessel lengths ranging from less than a millimeter up to tens of centimeters, in order to evaluate its computational cost efficiency. The proposed methodology can be extended to any other hyperbolic system for which network applications are relevant. Copyright © 2016 John Wiley & Sons, Ltd.

Roadmap for cardiovascular circulation model

Computational models of many aspects of the mammalian cardiovascular circulation have been developed. Indeed, along with orthopaedics, this area of physiology is one that has attracted much interest from engineers, presumably because the equations governing blood flow in the vascular system are well understood and can be solved with well-established numerical techniques. Unfortunately, there have been only a few attempts to create a comprehensive public domain resource for cardiovascular researchers. In this paper we propose a roadmap for developing an open source cardiovascular circulation model. The model should be registered to the musculo-skeletal system. The computational infrastructure for the cardiovascular model should provide for near real-time computation of blood flow and pressure in all parts of the body. The model should deal with vascular beds in all tissues, and the computational infrastructure for the model should provide links into CellML models of cell function and tissue function. In this work we review the literature associated with 1D blood flow modelling in the cardiovascular system, discuss model encoding standards, software and a model repository. We then describe the coordinate systems used to define the vascular geometry, derive the equations and discuss the implementation of these coupled equations in the open source computational software OpenCMISS. Finally, some preliminary results are presented and plans outlined for the next steps in the development of the model, the computational software and the graphical user interface for accessing the model.

A Novel Analytical Approach to Pulsatile Blood Flow in the Arterial Network

Haemodynamic simulations using one-dimensional (1-D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. We propose a novel linear 1-D dynamical theory of blood flow in networks of flexible vessels that is based on a generalized Darcy's model and for which a full analytical solution exists in frequency domain. We assess the accuracy of this formulation in a series of benchmark test cases for which computational 1-D and 3-D solutions are available. Accordingly, we calculate blood flow and pressure waves, and velocity profiles in the human common carotid artery, upper thoracic aorta, aortic bifurcation, and a 20-artery model of the aorta and its larger branches. Our analytical solution is in good agreement with the available solutions and reproduces the main features of pulse waveforms in networks of large arteries under normal physiological conditions. Our model reduces computational time and provides a new approach for studying arterial pulse wave mechanics; e.g.,  the analyticity of our model allows for a direct identification of the role played by physical properties of the cardiovascular system on the pressure waves.

Computational modeling of blood flow steal phenomena caused by subclavian stenoses

The study of steal mechanisms caused by vessel obstructions is of the utmost importance to gain understanding about their pathophysiology, as well as to improve diagnosis and management procedures. The goal of this work is to perform a computational study to gain insight into the hemodynamic forces that drive blood flow steal mechanisms caused by subclavian artery stenosis. Such condition triggers a flow disorder known as subclavian steal. When this occurs in patients with internal thoracic artery anastomosed to the coronary vessels, the phenomenon includes a coronary-subclavian steal. True steal can exist in cases of increased arm blood flow, potentially resulting in neurological complications and, in the case of coronary-subclavian steal, graft function failure. In this context, the anatomically detailed arterial network (ADAN) model is employed to simulate subclavian steal and coronary-subclavian steal phenomena. Model results are verified by comparison with published data. It is concluded that this kind of model allows us to effectively address complex hemomdynamic phenomena occurring in clinical practice. More specifically, in the studied conditions it is observed that a regional brain steal occurs, primarily affecting the posterior circulation, not fully compensated by the anterior circulation. In the case of patients with coronary revascularization, it is concluded that there is a large variability in graft hemodynamic environments, which physically explain both the success of the procedure in cases of severe occlusive disease, and the reason for graft dysfunction in mildly stenosed left anterior descending coronary artery, due to alternating graft flow waveform signatures.

Comparison of the Windkessel model and structured-tree model applied to prescribe outflow boundary conditions for a one-dimensional arterial tree model

One-dimensional (1D) modeling is a widely adopted approach for studying wave propagation phenomena in the arterial system. Despite the frequent use of the Windkessel (WK) model to prescribe outflow boundary conditions for 1D arterial tree models, it remains unclear to what extent the inherent limitation of the WK model in describing wave propagation in distal vasculatures affect hemodynamic variables simulated at the arterial level. In the present study, a 1D model of the arterial tree was coupled respectively with a WK boundary model and a structured-tree (ST) boundary model, yielding two types of arterial tree models. The effective resistances, compliances and inductances of the WK and ST boundary models were matched to facilitate quantitative comparisons. Obtained results showed that pressure/flow waves simulated by the two models were comparable in the aorta, whereas, their discrepancies increased towards the periphery. Wave analysis revealed that the differences in reflected waves generated by the boundary models were the major sources of pressure wave discrepancies observed in large arteries. Additional simulations performed under aging conditions demonstrated that arterial stiffening with age enlarged the discrepancies, but with the effects being partly counteracted by physiological aortic dilatation with age. These findings suggest that the method adopted for modeling the outflow boundary conditions has considerable influence on the performance of a 1D arterial tree model, with the extent of influence varying with the properties of the arterial system.

A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling

Haemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of three-dimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes.

Reducing the number of parameters in 1D arterial blood flow modeling: less is more for patient-specific simulations

Patient-specific one-dimensional (1D) blood flow modeling requires estimating model parameters from available clinical data, ideally acquired noninvasively. The larger the number of arterial segments in a distributed 1D model, the greater the number of input parameters that need to be estimated. We investigated the effect of a reduction in the number of arterial segments in a given distributed 1D model on the shape of the simulated pressure and flow waveforms. This is achieved by systematically lumping peripheral 1D model branches into windkessel models that preserve the net resistance and total compliance of the original model. We applied our methodology to a model of the 55 larger systemic arteries in the human and to an extended 67-artery model that contains the digital arteries that perfuse the fingers. Results show good agreement in the shape of the aortic and digital waveforms between the original 55-artery (67-artery) and reduced 21-artery (37-artery) models. Reducing the number of segments also enables us to investigate the effect of arterial network topology (and hence reflection sites) on the shape of waveforms. Results show that wave reflections in the thoracic aorta and renal arteries play an important role in shaping the aortic pressure and flow waves and in generating the second peak of the digital pressure and flow waves. Our novel methodology is important to simplify the computational domain while maintaining the precision of the numerical predictions and to assess the effect of wave reflections.

An anatomically detailed arterial network model for one-dimensional computational hemodynamics

Simulation platforms are increasingly becoming complementary tools for cutting-edge cardiovascular research. The interplay among structural properties of the arterial wall, morphometry, anatomy, wave propagation phenomena, and ultimately, cardiovascular diseases continues to be poorly understood. Accurate models are powerful tools to shed light on these open problems. We developed an anatomically detailed computational model of the arterial vasculature to conduct 1-D blood flow simulations to serve as simulation infrastructure to aid cardiovascular research. An average arterial vasculature of a man was outlined in 3-D space to serve as geometrical substrate for the mathematical model. The architecture of this model comprises almost every arterial vessel acknowledged in the medical/anatomical literature, with a resolution down to the luminal area of perforator arteries. Over 2000 arterial vessels compose the model. Anatomical, physiological, and mechanical considerations were employed for the set up of model parameters and to determine criteria for blood flow distribution. Computational fluid dynamics was used to simulate blood flow and wave propagation phenomena in such arterial network. A sensitivity analysis was developed to unveil the contributions of model parameters to the conformation of the pressure waveforms. In addition, parameters were modified to target model to a patient-specific scenario. On the light of the knowledge domain, we conclude that the present model features excellent descriptive and predictive capabilities in both patient-generic and patient-specific cases, presenting a new step toward integrating an unprecedented anatomical description, morphometric, and simulations data to help in understanding complex arterial blood flow phenomena and related cardiovascular diseases.