A Coupled Lumped-Parameter and Distributed Network Model for Cerebral Pulse-Wave Hemodynamics

Modeling cerebrovascular responses to assess the impact of the collateral circulation following middle cerebral artery occlusion

OBJECTIVE

An improved understanding of the role of the leptomeningeal collateral circulation in blood flow compensation following middle cerebral artery (MCA) occlusion can contribute to more effective treatment development for ischemic stroke. The present study introduces a model of the cerebral circulation to predict cerebral blood flow and tissue oxygenation following MCA occlusion.

METHODS

The model incorporates flow regulation mechanisms based on changes in pressure, shear stress, and metabolic demand. Oxygen saturation in cerebral vessels and tissue is calculated using a Krogh cylinder model. The model is used to assess the effects of changes in oxygen demand and arterial pressure on cerebral blood flow and oxygenation after MCA occlusion.

RESULTS

An increase from five to 11 leptomeningeal collateral vessels was shown to increase the oxygen saturation in the region distal to the occlusion by nearly 100%. Post-occlusion, the model also predicted a loss of autoregulation and a decrease in flow to the ischemic territory as oxygen demand was increased; these results were consistent with data from experiments that induced cerebral ischemia.

CONCLUSIONS

This study highlights the importance of leptomeningeal collaterals following MCA occlusion and reinforces the idea that lower oxygen demand and higher arterial pressure improve conditions of flow and oxygenation.

Intracranial pressure-flow relationships in traumatic brain injury patients expose gaps in the tenets of models and pressure-oriented management

Background

The protocols and therapeutic guidance established for treating traumatic brain injuries (TBI) in neurointensive care focus on managing cerebral blood flow (CBF) and brain tissue oxygenation based on pressure signals. The decision support process relies on assumed relationships between cerebral perfusion pressure (CPP) and blood flow, pressure-flow relationships (PFRs), and shares this framework of assumptions with mathematical intracranial hemodynamic models. These foundational assumptions are difficult to verify, and their violation can impact clinical decision-making and model validity.

Method

A hypothesis- and model-driven method for verifying and understanding the foundational intracranial hemodynamic PFRs is developed and applied to a novel multi-modality monitoring dataset.

Results

Model analysis of joint observations of CPP and CBF validates the standard PFR when autoregulatory processes are impaired as well as unmodelable cases dominated by autoregulation. However, it also identifies a dynamical regime -or behavior pattern- where the PFR assumptions are wrong in a precise, data-inferable way due to negative CPP-CBF coordination over long timescales. This regime is of both clinical and research interest: its dynamics are modelable under modified assumptions while its causal direction and mechanistic pathway remain unclear.

Conclusions

Motivated by the understanding of mathematical physiology, the validity of the standard PFR can be assessed a) directly by analyzing pressure reactivity and mean flow indices (PRx and Mx) or b) indirectly through the relationship between CBF and other clinical observables. This approach could potentially help personalize TBI care by considering intracranial pressure and CPP in relation to other data, particularly CBF. The analysis suggests a threshold using clinical indices of autoregulation jointly generalizes independently set indicators to assess CA functionality. These results support the use of increasingly data-rich environments to develop more robust hybrid physiological-machine learning models.

Multi-Dimensional Modeling of Cerebral Hemodynamics: A Systematic Review

The Circle of Willis (CoW) describes the arterial system in the human brain enabling the neurovascular blood supply. Neurovascular diseases like intracranial aneurysms (IAs) can occur within the CoW and carry the risk of rupture, which can lead to subarachnoid hemorrhage. The assessment of hemodynamic information in these pathologies is crucial for their understanding regarding detection, diagnosis and treatment. Multi-dimensional in silico approaches exist to evaluate these hemodynamics based on patient-specific input data. The approaches comprise low-scale (zero-dimensional, one-dimensional) and high-scale (three-dimensional) models as well as multi-scale coupled models. The input data can be derived from medical imaging, numerical models, literature-based assumptions or from measurements within healthy subjects. Thus, the most realistic description of neurovascular hemodynamics is still controversial. Within this systematic review, first, the models of the three scales (0D, 1D, 3D) and second, the multi-scale models, which are coupled versions of the three scales, were discussed. Current best practices in describing neurovascular hemodynamics most realistically and their clinical applicablility were elucidated. The performance of 3D simulation entails high computational expenses, which could be reduced by analyzing solely the region of interest in detail. Medical imaging to establish patient-specific boundary conditions is usually rare, and thus, lower dimensional models provide a realistic mimicking of the surrounding hemodynamics. Multi-scale coupling, however, is computationally expensive as well, especially when taking all dimensions into account. In conclusion, the 0D-1D-3D multi-scale approach provides the most realistic outcome; nevertheless, it is least applicable. A 1D-3D multi-scale model can be considered regarding a beneficial trade-off between realistic results and applicable performance.

Approaches to vascular network, blood flow, and metabolite distribution modeling in brain tissue

The cardiovascular system plays a key role in the transport of nutrients, ensuring a continuous supply of all cells of the body with the metabolites necessary for life. The blood supply to the brain is carried out by the large arteries located on its surface, which branch into smaller arterioles that penetrate the cerebral cortex and feed the capillary bed, thereby forming an extensive branching network. The formation of blood vessels is carried out via vasculogenesis and angiogenesis, which play an important role in both embryo and adult life. The review presents approaches to modeling various aspects of both the formation of vascular networks and the construction of the formed arterial tree. In addition, a brief description of models that allows one to study the blood flow in various parts of the circulatory system and the spatiotemporal metabolite distribution in brain tissues is given. Experimental study of these issues is not always possible due to both the complexity of the cardiovascular system and the mechanisms through which the perfusion of all body cells is carried out. In this regard, mathematical models are a good tool for studying hemodynamics and can be used in clinical practice to diagnose vascular diseases and assess the need for treatment.

First blood: An efficient, hybrid one- and zero-dimensional, modular hemodynamic solver

Low-dimensional (1D or 0D) models can describe the whole human blood circulation, for example, 1D distributed parameter model for the arterial network and 0D concentrated models for the heart or other organs. This paper presents a combined 1D-0D solver, called first_blood, that solves the governing equations of fluid dynamics to model low-dimensional hemodynamic effects. An extended method of characteristics is applied here to solve the momentum, and mass conservation equations and the viscoelastic wall model equation, mimicking the material properties of arterial walls. The heart and the peripheral lumped models are solved with a general zero-dimensional (0D) nonlinear solver. The model topology can be modular, that is, first_blood can solve any 1D-0D hemodynamic model. To demonstrate the applicability of first_blood, the human arterial system, the heart and the peripherals are modelled using the solver. The simulation time of a heartbeat takes around 2 s, that is, first_blood requires only twice the real-time for the simulation using an average PC, which highlights the computational efficiency. The source code is available on GitHub, that is, it is open source. The model parameters are based on the literature suggestions and on the validation of output data to obtain physiologically relevant results.

Uncertainty quantification in cerebral circulation simulations focusing on the collateral flow: Surrogate model approach with machine learning

Collateral circulation in the circle of Willis (CoW), closely associated with disease mechanisms and treatment outcomes, can be effectively investigated using one-dimensional–zero-dimensional hemodynamic simulations. As the entire cardiovascular system is considered in the simulation, it captures the systemic effects of local arterial changes, thus reproducing collateral circulation that reflects biological phenomena. The simulation facilitates rapid assessment of clinically relevant hemodynamic quantities under patient-specific conditions by incorporating clinical data. During patient-specific simulations, the impact of clinical data uncertainty on the simulated quantities should be quantified to obtain reliable results. However, as uncertainty quantification (UQ) is time-consuming and computationally expensive, its implementation in time-sensitive clinical applications is considered impractical. Therefore, we constructed a surrogate model based on machine learning using simulation data. The model accurately predicts the flow rate and pressure in the CoW in a few milliseconds. This reduced computation time enables the UQ execution with 100 000 predictions in a few minutes on a single CPU core and in less than a minute on a GPU. We performed UQ to predict the risk of cerebral hyperperfusion (CH), a life-threatening condition that can occur after carotid artery stenosis surgery if collateral circulation fails to function appropriately. We predicted the statistics of the postoperative flow rate increase in the CoW, which is a measure of CH, considering the uncertainties of arterial diameters, stenosis parameters, and flow rates measured using the patients’ clinical data. A sensitivity analysis was performed to clarify the impact of each uncertain parameter on the flow rate increase. Results indicated that CH occurred when two conditions were satisfied simultaneously: severe stenosis and when arteries of small diameter serve as the collateral pathway to the cerebral artery on the stenosis side. These findings elucidate the biological aspects of cerebral circulation in terms of the relationship between collateral flow and CH.

The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model

In this paper, least squares homotopy perturbation is presented as a straightforward and accurate method to compute approximate analytical solutions for systems of ordinary differential equations. The method is employed to solve a problem related to a laminar flow of a viscous fluid in a semi-porous channel, which may be used to model the blood flow through a blood vessel, taking into account the effects of a magnetic field. The numerical computations show that the method is both easy to use and very accurate compared to the other methods previously used to solve the given problem.

Parameter estimation to study the immediate impact of aortic cross-clamping using reduced order models

Aortic cross-clamping is a common strategy during vascular surgery, however, its instantaneous impact on hemodynamics is unknown. We, therefore, developed two numerical models to estimate the immediate impact of aortic clamping on the vascular properties. To assess the validity of the models, we recorded continuous invasive pressure signals during abdominal aneurysm repair surgery, immediately before and after clamping. The first model is a zero-dimensional (0D) three-element Windkessel model, which we coupled to a gradient-based parameter estimation algorithm to identify patient-specific parameters such as vascular resistance and compliance. We found a 10% increase in the total resistance and a 20% decrease in the total compliance after clamping. The second model is a nine-artery network corresponding to an average human body in which we solved the one-dimensional (1D) blood flow equations. With a similar parameter estimation method and using the results from the 0D model, we identified the resistance boundary conditions of the 1D network. Determining the patient-specific total resistance and the distribution of peripheral resistances through the parameter estimation process was sufficient for the 1D model to accurately reproduce the impact of clamping on the pressure waveform. Both models gave an accurate description of the pressure wave and had a high correlation (R² > .95) with experimental blood pressure data.

Intracranial collateral circulation and its role in neurovascular pathology

Background

Collateral circulation is a vascular network which maintains the blood flow after the partial blockage of primary vascular pathways. This acts as potential vascular supplementary system and plays important role in the cerebral ischemia.

Main body

Collateral circulation has implications in the management especially related to cerebral endovascular treatment and thrombolytic therapy. It is considered as subsidiary network of vascular channels, which is highly variable. Insufficient arterial supply is due to the hemodynamic compromise because of thromboembolism. Apart from the collaterals, there is additional existence of a group of vessels known as venous collaterals. Their function is variable and they contribute to the augmentation of venous drainage in venous ischemias. Various pharmacological interventions are used to modulate the collaterals, these can prove to be a complementary alternative to the invasive intracerebral interventions.

Conclusions

The aim of this review article is to highlight the importance of cerebral collateral circulation and to discuss the various available pharmacological alternatives available and their current relevance in the management of various neurovascular pathologies.

Clinical Decision Support for Traumatic Brain Injury: Identifying a Framework for Practical Model-Based Intracranial Pressure Estimation at Multihour Timescales

BACKGROUND

The clinical mitigation of intracranial hypertension due to traumatic brain injury requires timely knowledge of intracranial pressure to avoid secondary injury or death. Noninvasive intracranial pressure (nICP) estimation that operates sufficiently fast at multihour timescales and requires only common patient measurements is a desirable tool for clinical decision support and improving traumatic brain injury patient outcomes. However, existing model-based nICP estimation methods may be too slow or require data that are not easily obtained.

OBJECTIVE

This work considers short- and real-time nICP estimation at multihour timescales based on arterial blood pressure (ABP) to better inform the ongoing development of practical models with commonly available data.

METHODS

We assess and analyze the effects of two distinct pathways of model development, either by increasing physiological integration using a simple pressure estimation model, or by increasing physiological fidelity using a more complex model. Comparison of the model approaches is performed using a set of quantitative model validation criteria over hour-scale times applied to model nICP estimates in relation to observed ICP.

RESULTS

The simple fully coupled estimation scheme based on windowed regression outperforms a more complex nICP model with prescribed intracranial inflow when pulsatile ABP inflow conditions are provided. We also show that the simple estimation data requirements can be reduced to 1-minute averaged ABP summary data under generic waveform representation.

CONCLUSIONS

Stronger performance of the simple bidirectional model indicates that feedback between the systemic vascular network and nICP estimation scheme is crucial for modeling over long intervals. However, simple model reduction to ABP-only dependence limits its utility in cases involving other brain injuries such as ischemic stroke and subarachnoid hemorrhage. Additional methodologies and considerations needed to overcome these limitations are illustrated and discussed.

Cardiovascular deconditioning during long-term spaceflight through multiscale modeling

Human spaceflight has been fascinating man for centuries, representing the intangible need to explore the unknown, challenge new frontiers, advance technology, and push scientific boundaries further. A key area of importance is cardiovascular deconditioning, that is, the collection of hemodynamic changes-from blood volume shift and reduction to altered cardiac function-induced by sustained presence in microgravity. A thorough grasp of the 0G adjustment point per se is important from a physiological viewpoint and fundamental for astronauts' safety and physical capability on long spaceflights. However, hemodynamic details of cardiovascular deconditioning are incomplete, inconsistent, and poorly measured to date; thus a computational approach can be quite valuable. We present a validated 1D-0D multiscale model to study the cardiovascular response to long-term 0G spaceflight in comparison to the 1G supine reference condition. Cardiac work, oxygen consumption, and contractility indexes, as well as central mean and pulse pressures were reduced, augmenting the cardiac deconditioning scenario. Exercise tolerance of a spaceflight traveler was found to be comparable to an untrained person with a sedentary lifestyle. At the capillary-venous level significant waveform alterations were observed which can modify the regular perfusion and average nutrient supply at the cellular level. The present study suggests special attention should be paid to future long spaceflights which demand prompt physical capacity at the time of restoration of partial gravity (e.g., Moon/Mars landing). Since spaceflight deconditioning has features similar to accelerated aging understanding deconditioning mechanisms in microgravity are also relevant to the understanding of aging physiology on the Earth.

Morphological changes of intracranial pressure quantifies vasodilatory effect of verapamil to treat cerebral vasospasm

INTRODUCTION

After aneurysmal subarachnoid hemorrhage (SAH), both proximal and distal cerebral vasospasm can contribute to the development of delayed cerebral ischemia. Intra-arterial (IA) vasodilators are a mainstay of treatment for distal arterial vasospasm, but no methods of assessing the efficacy of interventions in real time have been established.

OBJECTIVE

To introduce a new method for continuous intraprocedural assessment of endovascular treatment for cerebral vasospasm.

METHODS

The premise of our approach was that distal cerebral arterial changes induce a consistent pattern in the morphological changes of intracranial pressure (ICP) pulse. This premise was demonstrated using a published algorithm in previous papers. In this study, we applied the algorithm to calculate the likelihood of cerebral vasodilation (VDI) and cerebral vasoconstriction (VCI) from intraprocedural ICP signals that are synchronized with injection of the IA vasodilator, verapamil. Cerebral blood flow velocities (CBFVs) on bilateral cerebral arteries were studied before and after IA therapy.

RESULTS

192 recordings of patients with SAH were reviewed, and 27 recordings had high-quality ICP waveforms. The VCI was significantly lower after the first verapamil injection (0.47±0.017) than VCI at baseline (0.49±0.020, p<0.001). A larger dose of injected verapamil resulted in a larger and longer VDI increase. CBFV of the middle cerebral artery increases across the days before the injection of verapamil and decreases after IA therapy.

CONCLUSION

This study provides preliminary validation of an algorithm for continuous assessment of distal cerebral arterial changes in response to IA vasodilator infusion in patients with vasospasm and aneurysmal SAH.

Data-Augmented Modeling of Intracranial Pressure

Precise management of patients with cerebral diseases often requires intracranial pressure (ICP) monitoring, which is highly invasive and requires a specialized ICU setting. The ability to noninvasively estimate ICP is highly compelling as an alternative to, or screening for, invasive ICP measurement. Most existing approaches for noninvasive ICP estimation aim to build a regression function that maps noninvasive measurements to an ICP estimate using statistical learning techniques. These data-based approaches have met limited success, likely because the amount of training data needed is onerous for this complex applications. In this work, we discuss an alternative strategy that aims to better utilize noninvasive measurement data by leveraging mechanistic understanding of physiology. Specifically, we developed a Bayesian framework that combines a multiscale model of intracranial physiology with noninvasive measurements of cerebral blood flow using transcranial Doppler. Virtual experiments with synthetic data are conducted to verify and analyze the proposed framework. A preliminary clinical application study on two patients is also performed in which we demonstrate the ability of this method to improve ICP prediction.

Computational Fluid Dynamics and Additive Manufacturing to Diagnose and Treat Cardiovascular Disease

Noninvasive engineering models are now being used for diagnosing and planning the treatment of cardiovascular disease. Techniques in computational modeling and additive manufacturing have matured concurrently, and results from simulations can inform and enable the design and optimization of therapeutic devices and treatment strategies. The emerging synergy between large-scale simulations and 3D printing is having a two-fold benefit: first, 3D printing can be used to validate the complex simulations, and second, the flow models can be used to improve treatment planning for cardiovascular disease. In this review, we summarize and discuss recent methods and findings for leveraging advances in both additive manufacturing and patient-specific computational modeling, with an emphasis on new directions in these fields and remaining open questions.

The cerebral collateral circulation: Relevance to pathophysiology and treatment of stroke

The brain's collateral circulation consists of arterial anastomotic channels capable of providing nutrient perfusion to brain regions whose normal sources of flow have become compromised, as occurs in acute ischemic stroke. Modern CT-based neuroimaging is capable of providing detailed information as to collateral extent and sufficiency and is complemented by magnetic resonance-based methods. In the present era of standard-of-care IV thrombolysis for acute ischemic stroke, and following the recent therapeutic successes of randomized clinical trials of acute endovascular intervention, the sufficiency of the collateral circulation has been convincingly established as a key factor influencing the likelihood of successful reperfusion and favorable clinical outcome. This article reviews the features of the brain's collateral circulation; methods for its evaluation in the acute clinical setting; the relevance of collateral circulation to prognosis in acute ischemic stroke; the specific insights into the collateral circulation learned from recent trials of endovascular intervention; and the major influence of genetic factors. Finally, we emphasize the need to develop therapeutic approaches to augment collateral perfusion as an adjunctive strategy to be employed along with, or prior to, thrombolysis and endovascular interventions, and we highlight the possible potential of inhaled nitric oxide, albumin, and other approaches. This article is part of the Special Issue entitled 'Cerebral Ischemia'.

Numerical Investigation of Vasospasm Detection by Extracranial Blood Velocity Ratios

BACKGROUND

Early diagnosis of vasospasm following subarachnoid hemorrhage can prevent cerebral ischemia and improve neurological outcomes. This study numerically evaluates the relevance of extracranial blood velocity indices to detect vasospasm.

METHODS

A numerical model of cerebral blood flow was used to evaluate the hemodynamics associated with anterior and posterior vasospasm under normal and impaired cerebral autoregulation conditions. Extracranial blood velocities at the carotid and vertebral arteries and their ratios between ipsilateral and contralateral, anterior and posterior, and downstream and upstream arteries were monitored during vasospasm progression.

RESULTS

For current clinical indices that track blood velocities at vasospastic arterial segments using transcranial Doppler (TCD), we observed that velocities increased initially and then decreased with vasospasm progression. This nonmonotonic behavior can lead to false-negative decisions in moderate to severe vasospasm. Alternatively, volumetric flow decreased monotonically at the affected arteries, leading to blood velocities upstream of the vasospastic artery also decreasing monotonically. Based on this principle, we demonstrate that velocity ratios between the carotid and vertebral arteries may better identify moderate to severe vasospasm and improve sensitivity and specificity of vasospasm detection.

CONCLUSION

The velocity indices proposed in this study may enable new or improved noninvasive diagnosis of vasospasm using extracranial Doppler ultrasound. Compared to current clinical indices, the new indices may improve the handling of (1) scenarios of severe vasospasm or impaired cerebral autoregulation, (2) systemic changes in blood pressure and cardiac output, (3) vasospasm occurring in arteries distal to the cerebral circle region, and (4) cases with insufficient acoustic bone window for TCD. The results provide a concrete basis for future clinical evaluation of extracranial indices for vasospasm detection.

A LUMPED PARAMETER MODEL OF CARDIOVASCULAR SYSTEM WITH PULSATING HEART FOR DIAGNOSTIC STUDIES

Mathematical modeling of cardiovascular system provides an ability to study hemodynamics and to predict the results of treatment based on individual anatomical and physiological data of patients. However, the presently developed models of cardiovascular system have a limitation on use in clinical practice due to their physical and computational complexities. The aim of this study is to derive a lumped parameter model of cardiovascular system with pulsating heart in which all parameters have a physically based quantitative value and can be identified using clinical methods. For development of a cardiovascular system model the chamber analog was used which describes whole cardiovascular system as a set of elastic chambers. The proposed model consists of systemic and pulmonary circulation, four-chamber heart and four valves. The description of heart is based on a four-element representation of a cardiac muscle. The reverse blood flow via valves is considered. The accuracy of the derived model was evaluated by comparing the data of numerical simulation with experimental data. The limitations of the model were discussed as well as possible applications of the model were suggested. The proposed lumped parameter model can be used to support clinicians in their decisions in treating cardiovascular disorders.