Dynamics of cerebral blood flow regulation explained using a lumped parameter model

Global sensitivity analysis with multifidelity Monte Carlo and polynomial chaos expansion for vascular haemodynamics

Computational models of the cardiovascular system are increasingly used for the diagnosis, treatment, and prevention of cardiovascular disease. Before being used for translational applications, the predictive abilities of these models need to be thoroughly demonstrated through verification, validation, and uncertainty quantification. When results depend on multiple uncertain inputs, sensitivity analysis is typically the first step required to separate relevant from unimportant inputs, and is key to determine an initial reduction on the problem dimensionality that will significantly affect the cost of all downstream analysis tasks. For computationally expensive models with numerous uncertain inputs, sample-based sensitivity analysis may become impractical due to the substantial number of model evaluations it typically necessitates. To overcome this limitation, we consider recently proposed Multifidelity Monte Carlo estimators for Sobol' sensitivity indices, and demonstrate their applicability to an idealized model of the common carotid artery. Variance reduction is achieved combining a small number of three-dimensional fluid-structure interaction simulations with affordable one- and zero-dimensional reduced-order models. These multifidelity Monte Carlo estimators are compared with traditional Monte Carlo and polynomial chaos expansion estimates. Specifically, we show consistent sensitivity ranks for both bi- (1D/0D) and tri-fidelity (3D/1D/0D) estimators, and superior variance reduction compared to traditional single-fidelity Monte Carlo estimators for the same computational budget. As the computational burden of Monte Carlo estimators for Sobol' indices is significantly affected by the problem dimensionality, polynomial chaos expansion is found to have lower computational cost for idealized models with smooth stochastic response.

Hemodynamic effects of pulsatile frequency of right ventricular assist device (RVAD) on pulmonary perfusion: a simulation study

Right ventricular assist devices (RVADs) have been extensively used to provide hemodynamic support for patients with end-stage right heart (RV) failure. However, conventional in-parallel RVADs can lead to an elevation of pulmonary artery (PA) pressure, consequently increasing the right ventricular (RV) afterload, which is unfavorable for the relaxation of cardiac muscles and reduction of valve complications. The aim of this study is to investigate the hemodynamic effects of the pulsatile frequency of the RVAD on pulmonary artery. Firstly, a mathematical model incorporating heart, systemic circulation, pulmonary circulation, and RVAD is developed to simulate the cardiovascular system. Subsequently, the frequency characteristics of the pulmonary circulation system are analyzed, and the calculated results demonstrate that the pulsatile frequency of the RVAD has a substantive impact on the pulmonary artery pressure. Finally, to verify the analysis results, the hemodynamic effects of the pulsatile frequency of the RVAD on pulmonary artery are compared under diffident support modes. It is found that the pulmonary artery pressure decreases by approximately 6% when the pulsatile frequency changes from 1 to 3 Hz. The increased pulsatile frequency of RA-PA support mode may facilitate the opening of the pulmonary valve, while the RV-PA support mode can more effectively reduce the load of RV. This work provides a useful method to decrease the pulmonary artery pressure during the RVAD supports and may be beneficial for improving myocardial function in patients with end-stage right heart failure, especially those with pulmonary hypertension.

Deep-learning-based real-time individualization for reduce-order haemodynamic model

The reduced-order lumped parameter model (LPM) has great computational efficiency in real-time numerical simulations of haemodynamics but is limited by the accuracy of patient-specific computation. This study proposed a method to achieve the individual LPM modeling with high accuracy to improve the practical clinical applicability of LPM. Clinical data was collected from two medical centres comprising haemodynamic indicators from 323 individuals, including brachial artery pressure waveforms, cardiac output data, and internal carotid artery flow waveforms. The data were expanded to 5000 synthesised cases that all fell within the physiological range of each indicator. LPM of the human blood circulation system was established. A double-path neural network (DPNN) was designed to input the waveforms of each haemodynamic indicator and their key features and then output the individual parameters of the LPM, which was labelled using a conventional optimization algorithm. Clinically collected data from the other 100 cases were used as the test set to verify the accuracy of the individual LPM parameters predicted by DPNN. The results show that DPNN provided good convergence in the training process. In the test set, compared with clinical measurements, the mean differences between each haemodynamic indicator and the estimate calculated by the individual LPM based on the DPNN were about 10 %. Furthermore, DPNN prediction only takes 4 s for 100 cases. The DPNN proposed in this study permits real-time and accurate individualization of LPM's. When facing medical issues involving haemodynamics, it lays the foundation for patient-specific numerical simulation, which may be beneficial for potential clinical application.

Differential sensitivities to blood pressure variations in internal carotid and intracranial arteries: a numerical approach to stroke prediction

Stroke remains a global health concern, necessitating early prediction for effective management. Atherosclerosis-induced internal carotid and intra cranial stenosis contributes significantly to stroke risk. This study explores the relationship between blood pressure and stroke prediction, focusing on internal carotid artery (ICA) branches: middle cerebral artery (MCA), anterior cerebral artery (ACA), and their role in hemodynamics. Computational fluid dynamics (CFD) informed by the Windkessel model were employed to simulate patient-specific ICA models with introduced stenosis. Central to our investigation is the impact of stenosis on blood pressure, flow velocity, and flow rate across these branches, incorporating Fractional Flow Reserve (FFR) analysis. Results highlight differential sensitivities to blood pressure variations, with M1 branch showing high sensitivity, ACA moderate, and M2 minimal. Comparing blood pressure fluctuations between ICA and MCA revealed heightened sensitivity to potential reverse flow compared to ICA and ACA comparisons, emphasizing MCA's role. Blood flow adjustments due to stenosis demonstrated intricate compensatory mechanisms. FFR emerged as a robust predictor of stenosis severity, particularly in the M2 branch. In conclusion, this study provides comprehensive insights into hemodynamic complexities within major intracranial arteries, elucidating the significance of blood pressure variations, flow attributes, and FFR in stenosis contexts. Subject-specific data integration enhances model reliability, aiding stroke risk assessment and advancing cerebrovascular disease understanding.

Hemodynamics of vascular shunts: trends, challenges, and prospects

Vascular bypass surgery takes a significant place in the treatment of vascular disease. According to various assessments, this type of surgery is associated with almost 20 % of all vascular surgery episodes (up to 23 % according to the Federal Neurosurgical Center of Novosibirsk). Even though the problem of using of vascular grafts is obvious and natural, many problems associated with them are not still elucidated. From the mechanics' point of view, a vascular bypass is a converging or diverging tee, and the functioning of such structures still does not have strict mathematical formulations and proofs in the general case, which forces many researchers to solve specific engineering problems associated with shunting. Mathematical modeling, which is the gold standard for virtual simulations of industrial and medical problems, faces great difficulties and limitations in solving problems for vascular bypasses. Complications in the treatment of the vascular disease may follow the difficulties in mathematical modeling, and the price can be a cardiac arrest or a stroke. This work is devoted to the main aspects of the medical application of vascular bypasses and their functioning as a mechanical system, as well the mathematical aspects of their possible setup.

Blood Pressure Measurement: From Cuff-Based to Contactless Monitoring

Blood pressure (BP) determines whether a person has hypertension and offers implications as to whether he or she could be affected by cardiovascular disease. Cuff-based sphygmomanometers have traditionally provided both accuracy and reliability, but they require bulky equipment and relevant skills to obtain precise measurements. BP measurement from photoplethysmography (PPG) signals has become a promising alternative for convenient and unobtrusive BP monitoring. Moreover, the recent developments in remote photoplethysmography (rPPG) algorithms have enabled new innovations for contactless BP measurement. This paper illustrates the evolution of BP measurement techniques from the biophysical theory, through the development of contact-based BP measurement from PPG signals, and to the modern innovations of contactless BP measurement from rPPG signals. We consolidate knowledge from a diverse background of academic research to highlight the importance of multi-feature analysis for improving measurement accuracy. We conclude with the ongoing challenges, opportunities, and possible future directions in this emerging field of research.

Baroreflex control model for cardiovascular system subjected to postural changes under normal and orthostatic conditions

Baroreflex dysfunction is one of the common causes associated with the cardiovascular system. The buffering capability and baroreflex gain influences large variation in blood pressure for short term control. For regulating the blood pressure, an integrated analytical model for baroreflex control along with the cardiovascular system is presented to study the complex interactions between autonomic nervous system and cardiovascular system. In the proposed model, the autonomic nervous system utilizes sympathetic and parasympathetic nerve activities. This comprises a heart modeled by Mulier's approach, systemic vasculature, baroreceptor sensor using stress-strain based Voigt model and Hodgkin-Huxley based autonomic nervous control. This model can handle the distribution of total blood volume changes under the influence of gravity upon postural changes by means of short term baroreflex control. The simulation is carried out for the integrated model along with (i) non pulsatile and (ii) pulsatile model of heart. The proposed model is validated for supine to standing position under influence of gravity. To show the efficiency of the proposed model, the simulation is carried out further for (i) postural changes like supine to standing and standing to supine under normal condition and (ii) Orthostatic hypotension and hypertension conditions. Also the robustness of the proposed pulsatile model is tested by introducing disturbance signal in mean arterial pressure under various postural changes.

Modeling of Cerebral Blood Flow Autoregulation Using Mathematical Control Theory

A mathematical model of cerebral blood flow in the form of a dynamical system is studied. The cerebral blood flow autoregulation modeling problem is treated as a nonlinear control problem and the potential and applicability of the nonlinear control theory techniques are analyzed in this respect. It is shown that the cerebral hemodynamics model in question is differentially flat. Then, the integrator backstepping approach combined with barrier Lyapunov functions is applied to construct the control laws that recover the cerebral autoregulation performance of a healthy human. Simulation results confirm the good performance and flexibility of the suggested cerebral blood flow autoregulation design. The conducted research should enrich our understanding of the mathematics behind the cerebral blood flow autoregulation mechanisms and medical treatments to compensate for impaired cerebral autoregulation, e.g., in preterm infants.

A nonlinear multi-scale model for blood circulation in a realistic vascular system

In the last decade, numerical models have become an increasingly important tool in biological and medical science. Numerical simulations contribute to a deeper understanding of physiology and are a powerful tool for better diagnostics and treatment. In this paper, a nonlinear multi-scale model framework is developed for blood flow distribution in the full vascular system of an organ. We couple a quasi one-dimensional vascular graph model to represent blood flow in larger vessels and a porous media model to describe flow in smaller vessels and capillary bed. The vascular model is based on Poiseuille's Law, with pressure correction by elasticity and pressure drop estimation at vessels' junctions. The porous capillary bed is modelled as a two-compartment domain (artery and venous) using Darcy's Law. The fluid exchange between the artery and venous capillary bed compartments is defined as blood perfusion. The numerical experiments show that the proposed model for blood circulation: (i) is closely dependent on the structure and parameters of both the larger vessels and of the capillary bed, and (ii) provides a realistic blood circulation in the organ. The advantage of the proposed model is that it is complex enough to reliably capture the main underlying physiological function, yet highly flexible as it offers the possibility of incorporating various local effects. Furthermore, the numerical implementation of the model is straightforward and allows for simulations on a regular desktop computer.

Epidural Oscillating Cardiac-Gated Intracranial Implant Modulates Cerebral Blood Flow

BACKGROUND

We have previously reported a method and device capable of manipulating ICP pulsatility while minimally effecting mean ICP.

OBJECTIVE

To test the hypothesis that different modulations of the intracranial pressure (ICP) pulse waveform will have a differential effect on cerebral blood flow (CBF).

METHODS

Using an epidural balloon catheter attached to a cardiac-gated oscillating pump, 13 canine subjects underwent ICP waveform manipulation comparing different sequences of oscillation in successive animals. The epidural balloon was implanted unilaterally superior to the Sylvian sulcus. Subjects underwent ICP pulse augmentation, reduction and inversion protocols, directly comparing time segments of system activation and deactivation. ICP and CBF were measured bilaterally along with systemic pressure and heart rate. CBF was measured using both thermal diffusion, and laser doppler probes.

RESULTS

The activation of the cardiac-gate balloon implant resulted in an ipsilateral/contralateral ICP pulse amplitude increase with augmentation (217%/202% respectively, P < .0005) and inversion (139%/120%, P < .0005). The observed changes associated with the ICP mean values were smaller, increasing with augmentation (23%/31%, P < .0001) while decreasing with inversion (7%/11%, P = .006/.0003) and reduction (4%/5%, P < .0005). CBF increase was observed for both inversion and reduction protocols (28%/7.4%, P < .0001/P = .006 and 2.4%/1.3%, P < .0001/P = .003), but not the augmentation protocol. The change in CBF was correlated with ICP pulse amplitude and systolic peak changes and not with change in mean ICP or systemic variables (heart rate, arterial blood pressure).

CONCLUSION

Cardiac-gated manipulation of ICP pulsatility allows the study of intracranial pulsatile dynamics and provides a potential means of altering CBF.

Retrograde branched extension limb assembling stent of pararenal abdominal aortic aneurysm: A longitudinal hemodynamic analysis for stent graft migration

PURPOSE

Pararenal abdominal aortic aneurysms (PRAAAs) are a life-threatening disease, and hemodynamic analysis may provide greater insight into the effectiveness and long-term outcomes of endovascular aneurysm repair (EVAR). However, the lack of patient-specific boundary conditions on the periphery compromises the accuracy. Windkessel (WK) boundary conditions coupled to hemodynamic follow-up models of a PRAAA patient, aims to provide insights into the link between hemodynamics and poor prognosis.

METHOD

One PRAAA patient underwent EVAR and reintervention after one branch of stent-graft (SG) had migrated. Totally five computational follow-up models were studied. Patient-specific flow data acquired via ultrasound were used to define the boundary conditions in the ascending aorta and the following three branches. Coupled zero-dimensional WK models representing the distal vasculature were used to define the outlet boundary conditions under the abdomen.

RESULTS

Flow divisions of the main SG branches were 40.7% and 24.7%, respectively. Time-averaged wall shear stress and oscillatory shear index (OSI) increased at the junction connected the SG branch and the stent leading to the right common iliac artery (RCIA) where the stent migrated. The OSI and relative residence time (RRT) value in superior mesenteric artery increased notably after the migration, the RRT continuously increased following the reintervention.

CONCLUSION

Unbalanced flow, resulting in locally high-speed flow, high WSS and OSI might significantly affect stent stability. Results suggest that diameters and interconnection design of stents in complex cases should take the flow division into consideration and computational simulations might be considered as a tool for intervention protocol design.

A parallel non-nested two-level domain decomposition method for simulating blood flows in cerebral artery of stroke patient

Numerical simulation of blood flows in patient-specific arteries can be useful for the understanding of vascular diseases, as well as for surgery planning. In this paper, we simulate blood flows in the full cerebral artery of stroke patients. To accurately resolve the flow in this rather complex geometry with stenosis is challenging and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in pre- and/or post-surgery planning. For this purpose, we introduce a highly scalable, parallel non-nested two-level domain decomposition method for the three-dimensional unsteady incompressible Navier-Stokes equations with an impedance outlet boundary condition. The problem is discretized with a stabilized finite element method on unstructured meshes in space and a fully implicit method in time, and the large nonlinear systems are solved by a preconditioned parallel Newton-Krylov method with a two-level Schwarz method. The key component of the method is a non-nested coarse problem solved using a subset of processor cores and its solution is interpolated to the fine space using radial basis functions. To validate and verify the proposed algorithm and its highly parallel implementation, we consider a case with available clinical data and show that the computed result matches with the measured data. Further numerical experiments indicate that the proposed method works well for realistic geometry and parameters of a full size cerebral artery of an adult stroke patient on a supercomputers with thousands of processor cores.

A patient-specific modelling method of blood circulatory system for the numerical simulation of enhanced external counterpulsation

Lumped parameter model (LPM) is a common numerical model for hemodynamic simulation of human's blood circulatory system. The numerical simulation of enhanced external counterpulsation (EECP) is a typical biomechanical simulation process based on the LPM of blood circulatory system. In order to simulate patient-specific hemodynamic effects of EECP and develop best treatment strategy for each individual, this study developed an optimization algorithm to individualize LPM elements. Physiological data from 30 volunteers including approximate aortic pressure, cardiac output, ankle pressure and carotid artery flow were clinically collected as optimization objectives. A closed-loop LPM was established for the simulation of blood circulatory system. Aiming at clinical data, a sensitivity analysis for each element was conducted to identify the significant ones. We improved the traditional simulated annealing algorithm to iteratively optimize the sensitive elements. To verify the accuracy of the patient-specific model, 30 samples of simulated data were compared with clinical measurements. In addition, an EECP experiment was conducted on a volunteer to verify the applicability of the optimized model for the simulation of EECP. For these 30 samples, the optimization results show a slight difference between clinical data and simulated data. The average relative root mean square error is lower than 5%. For the subject of EECP experiment, the relative error of hemodynamic responses during EECP is lower than 10%. This slight error demonstrated a good state of optimization. The optimized modeling algorithm can effectively individualize the LPM for blood circulatory system, which is significant to the numerical simulation of patient-specific hemodynamics.

A coupled one dimension and transmission line model for arterial flow simulation

A broad choice of numerical schemes and methods currently exists for blood flow simulations. The results rely critically on the prescription of boundary conditions. The outflow boundary condition for a one-dimensional (1D) flow solver is usually prescribed via a Windkessel or lumped parameter model. The weakness of such an approach is the determination of the parameters. In the present work, we use an alternative approach, that is, a reflection coefficient (RC), to lumped parameter models for distal boundary conditions. With such a RC, the number of parameters required is reduced to one. We derive the theoretical foundation for the RC. Specifically, we couple a transmission line theory for peripheral resistance with a 1D arterial flow solver. We apply this method to a healthy and a stenosed virtual aorta, and show this method can reproduce some subtle features in arterial pressure propagation, such as the steepened pressure waveform and the reflection from the stenosed site. In summary, the RC parameter has strong physical implications in the theory of wave propagation and may be used in flow simulations where reflections need to be explicitly modeled. NOVELTY STATEMENT: A novel coupled one-dimensional-transimission line model has been developed in this work with detailed implementations. Only one outflow boundary condition, that is, the refection coefficient is required in the model. Reflections for a pulse wave from aortic terminals as well as from a stenotic site are numerically simulated.

Simulations of blood as a suspension predicts a depth dependent hematocrit in the circulation throughout the cerebral cortex

Recent advances in modeling oxygen supply to cortical brain tissue have begun to elucidate the functional mechanisms of neurovascular coupling. While the principal mechanisms of blood flow regulation after neuronal firing are generally known, mechanistic hemodynamic simulations cannot yet pinpoint the exact spatial and temporal coordination between the network of arteries, arterioles, capillaries and veins for the entire brain. Because of the potential significance of blood flow and oxygen supply simulations for illuminating spatiotemporal regulation inside the cortical microanatomy, there is a need to create mathematical models of the entire cerebral circulation with realistic anatomical detail. Our hypothesis is that an anatomically accurate reconstruction of the cerebrocirculatory architecture will inform about possible regulatory mechanisms of the neurovascular interface. In this article, we introduce large-scale networks of the murine cerebral circulation spanning the Circle of Willis, main cerebral arteries connected to the pial network down to the microcirculation in the capillary bed. Several multiscale models were generated from state-of-the-art neuroimaging data. Using a vascular network construction algorithm, the entire circulation of the middle cerebral artery was synthesized. Blood flow simulations indicate a consistent trend of higher hematocrit in deeper cortical layers, while surface layers with shorter vascular path lengths seem to carry comparatively lower red blood cell (RBC) concentrations. Moreover, the variability of RBC flux decreases with cortical depth. These results support the notion that plasma skimming serves a self-regulating function for maintaining uniform oxygen perfusion to neurons irrespective of their location in the blood supply hierarchy. Our computations also demonstrate the practicality of simulating blood flow for large portions of the mouse brain with existing computer resources. The efficient simulation of blood flow throughout the entire middle cerebral artery (MCA) territory is a promising milestone towards the final aim of predicting blood flow patterns for the entire brain.

The brain’s astonishing cognitive capacity depends on the coordination between neurons and the cerebral circulation, a system known as the neurovascular unit. The spatial and temporal coupling between these two networks is the object of intense research. However, the concise anatomical description of the cerebral circulation has so far been intractable. This paper introduces a methodology for the in silico creation of realistic models for the entire cerebral circulation. This innovation incorporates topological data from several neuroimaging modalities covering three lengths scales as input into a computer algorithm, which assembles anatomically accurate circulatory networks. When simulating blood flow as red blood cells suspended in plasma for experimental and synthetic cortical network models, we discovered that red blood cells tend to be more concentrated in deeper layers of the cortex compared to the surface. RBC fluxes are more homogenous in deeper layers. The phenomenon of depth dependent red blood cell supply supports the notion that the intricate architecture of the cortical microcirculation serves a self-regulating function to maintain uniform oxygen perfusion to neurons. We also demonstrate the practicality of predicting blood flow patterns for the entire brain with existing computer power.

An effective model of cerebrovascular pressure reactivity and blood flow autoregulation

Understanding cerebral blood flow dynamics is crucial for the care of patients at risk of poor cerebral perfusion. We describe an effective model of cerebral hemodynamics designed to reveal important macroscopic features of cerebral blood flow without having to resolve the detailed microvasculature of the brain. Based on principles of fluid and elastic dynamics and vascular pressure-reactivity, the model quantifies the physical means by which the vasculature executes autoregulatory reflexes. We demonstrate that the frequency response of the proposed model matches experimental measurements and explains the influence of mechanical factors on the autoregulatory performance. Analysis of the model indicates the existence of an optimal mean arterial pressure which minimizes the sensitivity of the flow to changes in perfusion pressure across the frequency spectrum of physiological oscillations. We highlight the simplicity of the model and its potential to improve monitoring of brain perfusion via real-time computational simulations of cerebro- and cardio-vascular interventions.

Arterial Pulsations cannot Drive Intramural Periarterial Drainage: Significance for Aβ Drainage

Alzheimer's Disease (AD) is the most common form of dementia and to date there is no cure or efficient prophylaxis. The cognitive decline correlates with the accumulation of amyloid-β (Aβ) in the walls of capillaries and arteries. Our group has demonstrated that interstitial fluid and Aβ are eliminated from the brain along the basement membranes of capillaries and arteries, the intramural periarterial drainage (IPAD) pathway. With advancing age and arteriosclerosis, the stiffness of arterial walls, this pathway fails in its function and Aβ accumulates in the walls of arteries. In this study we tested the hypothesis that arterial pulsations drive IPAD and that a valve mechanism ensures the net drainage in a direction opposite to that of the blood flow. This hypothesis was tested using a mathematical model of the drainage mechanism. We demonstrate firstly that arterial pulsations are not strong enough to produce drainage velocities comparable to experimental observations. Secondly, we demonstrate that a valve mechanism such as directional permeability of the IPAD pathway is necessary to achieve a net reverse flow. The mathematical simulation results are confirmed by assessing the pattern of IPAD in mice using pulse modulators, showing no significant alteration of IPAD. Our results indicate that forces other than the cardiac pulsations are responsible for efficient IPAD.

Virtual endovascular treatment of intracranial aneurysms: models and uncertainty

Virtual endovascular treatment models (VETMs) have been developed with the view to aid interventional neuroradiologists and neurosurgeons to pre-operatively analyze the comparative efficacy and safety of endovascular treatments for intracranial aneurysms. Based on the current state of VETMs in aneurysm rupture risk stratification and in patient-specific prediction of treatment outcomes, we argue there is a need to go beyond personalized biomechanical flow modeling assuming deterministic parameters and error-free measurements. The mechanobiological effects associated with blood clot formation are important factors in therapeutic decision making and models of post-treatment intra-aneurysmal biology and biochemistry should be linked to the purely hemodynamic models to improve the predictive power of current VETMs. The influence of model and parameter uncertainties associated to each component of a VETM is, where feasible, quantified via a random-effects meta-analysis of the literature. This allows estimating the pooled effect size of these uncertainties on aneurysmal wall shear stress. From such meta-analyses, two main sources of uncertainty emerge where research efforts have so far been limited: (1) vascular wall distensibility, and (2) intra/intersubject systemic flow variations. In the future, we suggest that current deterministic computational simulations need to be extended with strategies for uncertainty mitigation, uncertainty exploration, and sensitivity reduction techniques. WIREs Syst Biol Med 2017, 9:e1385. doi: 10.1002/wsbm.1385 For further resources related to this article, please visit the WIREs website.

Personalized blood flow computations: A hierarchical parameter estimation framework for tuning boundary conditions

We propose a hierarchical parameter estimation framework for performing patient-specific hemodynamic computations in arterial models, which use structured tree boundary conditions. A calibration problem is formulated at each stage of the hierarchical framework, which seeks the fixed point solution of a nonlinear system of equations. Common hemodynamic properties, like resistance and compliance, are estimated at the first stage in order to match the objectives given by clinical measurements of pressure and/or flow rate. The second stage estimates the parameters of the structured trees so as to match the values of the hemodynamic properties determined at the first stage. A key feature of the proposed method is that to ensure a large range of variation, two different structured tree parameters are personalized for each hemodynamic property. First, the second stage of the parameter estimation framework is evaluated based on the properties of the outlet boundary conditions in a full body arterial model: the calibration method converges for all structured trees in less than 10 iterations. Next, the proposed framework is successfully evaluated on a patient-specific aortic model with coarctation: only six iterations are required for the computational model to be in close agreement with the clinical measurements used as objectives, and overall, there is a good agreement between the measured and computed quantities. Copyright © 2016 John Wiley & Sons, Ltd.

Modelling confounding effects from extracerebral contamination and systemic factors on functional near-infrared spectroscopy

Haemodynamics-based neuroimaging is widely used to study brain function. Regional blood flow changes characteristic of neurovascular coupling provide an important marker of neuronal activation. However, changes in systemic physiological parameters such as blood pressure and concentration of CO2 can also affect regional blood flow and may confound haemodynamics-based neuroimaging. Measurements with functional near-infrared spectroscopy (fNIRS) may additionally be confounded by blood flow and oxygenation changes in extracerebral tissue layers. Here we investigate these confounds using an extended version of an existing computational model of cerebral physiology, ‘BrainSignals’. Our results show that confounding from systemic physiological factors is able to produce misleading haemodynamic responses in both positive and negative directions. By applying the model to data from previous fNIRS studies, we demonstrate that such potentially deceptive responses can indeed occur in at least some experimental scenarios. It is therefore important to record the major potential confounders in the course of fNIRS experiments. Our model may then allow the observed behaviour to be attributed among the potential causes and hence reduce identification errors.

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Confounding of fNIRS haemoglobin signals is simulated using a computational model.

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Model is extended to simulate scalp haemodynamics.

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Changes in blood pressure and CO₂ can mimic and mask functional activation.

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Experimental recording of systemic factors is recommended to aid interpretation.

Cerebral blood flow and autoregulation: current measurement techniques and prospects for noninvasive optical methods

Cerebral blood flow (CBF) and cerebral autoregulation (CA) are critically important to maintain proper brain perfusion and supply the brain with the necessary oxygen and energy substrates. Adequate brain perfusion is required to support normal brain function, to achieve successful aging, and to navigate acute and chronic medical conditions. We review the general principles of CBF measurements and the current techniques to measure CBF based on direct intravascular measurements, nuclear medicine, X-ray imaging, magnetic resonance imaging, ultrasound techniques, thermal diffusion, and optical methods. We also review techniques for arterial blood pressure measurements as well as theoretical and experimental methods for the assessment of CA, including recent approaches based on optical techniques. The assessment of cerebral perfusion in the clinical practice is also presented. The comprehensive description of principles, methods, and clinical requirements of CBF and CA measurements highlights the potentially important role that noninvasive optical methods can play in the assessment of neurovascular health. In fact, optical techniques have the ability to provide a noninvasive, quantitative, and continuous monitor of CBF and autoregulation.

A Highly Similar Mathematical Model for Cerebral Blood Flow Velocity in Geriatric Patients with Suspected Cerebrovascular Disease

Cerebral blood flow velocity(CBFV) is an important parameter for study of cerebral hemodynamics. However, a simple and highly similar mathematical model has not yet been established for analyzing CBFV. To alleviate this issue, through TCD examination in 100 geriatric patients with suspected cerebrovascular disease (46 males and 54 females), we established a representative eighth-order Fourier function V_x(t) that simulates the CBFV. The measured TCD waveforms were compared to those derived from V_x(t), an illustrative Kolmogorov-Smirnov test was employed to determine the validity. The results showed that the TCD waves could been reconstructed for patients with different CBFVs by implementing their variable heart rates and the formulated maximum/minimum of V_x(t). Comparisons between derived and measured TCD waveforms suggest that the two waveforms are very similar. The results confirm that CBFV can be well-modeled through an eighth-order Fourier function. This function V_x(t) can be used extensively for a prospective study of cerebral hemodynamics in geriatric patients with suspected cerebrovascular disease.

Assessing intracranial vascular compliance using dynamic arterial spin labeling

Vascular compliance (VC) is an important marker for a number of cardiovascular diseases and dementia, which is typically assessed in the central and peripheral arteries indirectly by quantifying pulse wave velocity (PWV), and/or pulse pressure waveform. To date, very few methods are available for the quantification of intracranial VC. In the present study, a novel MRI technique for in-vivo assessment of intracranial VC was introduced, where dynamic arterial spin labeling (ASL) scans were synchronized with the systolic and diastolic phases of the cardiac cycle. VC is defined as the ratio of change in arterial cerebral blood volume (ΔCBV) and change in arterial pressure (ΔBP). Intracranial VC was assessed in different vascular components using the proposed dynamic ASL method. Our results show that VC mainly occurs in large arteries, and gradually decreases in small arteries and arterioles. The comparison of intracranial VC between young and elderly subjects shows that aging is accompanied by a reduction of intracranial VC, in good agreement with the literature. Furthermore, a positive association between intracranial VC and cerebral perfusion measured using pseudo-continuous ASL with 3D GRASE MRI was observed independent of aging effects, suggesting loss of VC is associated with a decline in perfusion. Finally, a significant positive correlation between intracranial and central (aortic arch) VC was observed using an ungated phase-contrast 1D projection PWV technique. The proposed dynamic ASL method offers a promising approach for assessing intracranial VC in a range of cardiovascular diseases and dementia.

Outflow boundary conditions for blood flow in arterial trees

In the modeling of the pulse wave in the systemic arterial tree, it is necessary to truncate small arterial crowns representing the networks of small arteries and arterioles. Appropriate boundary conditions at the truncation points are required to represent wave reflection effects of the truncated arterial crowns. In this work, we provide a systematic method to extract parameters of the three-element Windkessel model from the impedance of a truncated arterial tree or from experimental measurements of the blood pressure and flow rate at the inlet of the truncated arterial crown. In addition, we propose an improved three-element Windkessel model with a complex capacitance to accurately capture the fundamental-frequency time lag of the reflection wave with respect to the incident wave. Through our numerical simulations of blood flow in a single artery and in a large arterial tree, together with the analysis of the modeling error of the pulse wave in large arteries, we show that both a small truncation radius and the complex capacitance in the improved Windkessel model play an important role in reducing the modeling error, defined as the difference in dynamics induced by the structured tree model and the Windkessel models.

Applicability of the polynomial chaos expansion method for personalization of a cardiovascular pulse wave propagation model

Patient-specific modeling requires model personalization, which can be achieved in an efficient manner by parameter fixing and parameter prioritization. An efficient variance-based method is using generalized polynomial chaos expansion (gPCE), but it has not been applied in the context of model personalization, nor has it ever been compared with standard variance-based methods for models with many parameters. In this work, we apply the gPCE method to a previously reported pulse wave propagation model and compare the conclusions for model personalization with that of a reference analysis performed with Saltelli's efficient Monte Carlo method. We furthermore differentiate two approaches for obtaining the expansion coefficients: one based on spectral projection (gPCE-P) and one based on least squares regression (gPCE-R). It was found that in general the gPCE yields similar conclusions as the reference analysis but at much lower cost, as long as the polynomial metamodel does not contain unnecessary high order terms. Furthermore, the gPCE-R approach generally yielded better results than gPCE-P. The weak performance of the gPCE-P can be attributed to the assessment of the expansion coefficients using the Smolyak algorithm, which might be hampered by the high number of model parameters and/or by possible non-smoothness in the output space.

Fundamental relationships between blood pressure and cerebral blood flow in humans

Cerebral blood flow responses to transient blood pressure challenges are frequently attributed to cerebral autoregulation (CA), yet accumulating evidence indicates vascular properties like compliance are also influential. We hypothesized that middle cerebral blood velocity (MCAv) dynamics during or following a transient blood pressure perturbation can be accurately explained by the windkessel mechanism. Eighteen volunteers underwent blood pressure manipulations, including bilateral thigh-cuff deflation and sit-to-stand maneuvers under normocapnic and hypercapnic (5% CO2) conditions. Pressure-flow recordings were analyzed using a windkessel analysis approach that partitions the frequency-dependent resistance and compliance contributions to MCAv dynamics. The windkessel was typically able to explain more than 50% of the MCAv variance, as indicated by R2 values for both the flow recovery and postrecovery phase. The most consistent predictors of MCAv dynamics under the control condition were the windkessel capacitive gain and high-frequency resistive gain. However, there were significant interindividual variations in the composition of windkessel predictors. Hypercapnia consistently reduced the capacitive gain and enhanced the low-frequency (0.04–0.20 Hz) resistive gain for both thigh-cuff deflation and sit-to-stand trials. These findings indicate that 1) MCAv dynamics during acute transient hypotension challenges are dominated by cerebrovascular windkessel properties independent of CA; 2) there is significant heterogeneity in windkessel properties between individuals; and 3) hemodynamic effects of hypercapnia during transient blood pressure challenges primarily reflect changes in windkessel properties rather than pure CA impairment.

Mathematical modelling of cerebral blood circulation and cerebral autoregulation: towards preventing intracranial hemorrhages in preterm newborns

Impaired cerebral autoregulation leads to fluctuations in cerebral blood flow, which can be especially dangerous for immature brain of preterm newborns. In this paper, two mathematical models of cerebral autoregulation are discussed. The first one is an enhancement of a vascular model proposed by Piechnik et al. We extend this model by adding a polynomial dependence of the vascular radius on the arterial blood pressure and adjusting the polynomial coefficients to experimental data to gain the autoregulation behavior. Moreover, the inclusion of a Preisach hysteresis operator, simulating a hysteretic dependence of the cerebral blood flow on the arterial pressure, is tested. The second model couples the blood vessel system model by Piechnik et al. with an ordinary differential equation model of cerebral autoregulation by Ursino and Lodi. An optimal control setting is proposed for a simplified variant of this coupled model. The objective of the control is the maintenance of the autoregulatory function for a wider range of the arterial pressure. The control can be interpreted as the effect of a medicament changing the cerebral blood flow by, for example, dilation of blood vessels. Advanced numerical methods developed by the authors are applied for the numerical treatment of the control problem.

MODELING OF CORONARY ARTERY BALLOON-ANGIOPLASTY USING EQUIVALENT ELECTRICAL CIRCUIT

Arterial thrombosis and atherosclerosis result in chronic total or partial occlusion of the coronary artery. Coronary artery disease (CAD) destroys some parts of the heart muscle tissue and is the leading cause of human deaths in the industrialized world. In this study, cardiovascular system is simulated by 42 compartments and then a coronary set (including artery, venous, myocardium and capillaries) is added to the model. Each vessel is modeled by three parameters, such as resistor, capacitor and inductor. These three parameters are variable with respect to the radius of the vessel. In this paper, first of all, aortic and coronary flow under healthy condition is studied. The obtained results are in complete agreement with experimental outcomes. Then cardiovascular system behavior in coronary artery stenosis condition is investigated. Finally, the effect of intra-coronary balloon pump on heart attack risk and also on stabilization of patient's emergency condition by mathematical simulation is analyzed. The results of modeling show that the balloon pumping of coronary artery is an advantageous way in rendering primary cure to patients. The proposed model, in addition, has implications for investigation of effects of different diseases on the cardiovascular system. It also has the potential to model different treatment methods on heart's performance and, as a result, recommend new methods in order to cure variant cardiovascular diseases.

Blood pressure regulation IX: cerebral autoregulation under blood pressure challenges

Cerebral autoregulation (CA) is integral to the delicate process of maintaining stable cerebral perfusion and brain tissue oxygenation against changes in arterial blood pressure. The last four decades has seen dramatic advances in understanding CA physiology, and the role that CA might play in the causation and progression of disease processes that affect the cerebral circulation such as stroke. However, the translation of these basic scientific advances into clinical practice has been limited by the maintenance of old constructs and because there are persistent gaps in our understanding of how this vital vascular mechanism should be quantified. In this review, we re-evaluate relevant studies that challenge established paradigms about how the cerebral perfusion pressure and blood flow are related. In the context of blood pressure being a major haemodynamic challenge to the cerebral circulation, we conclude that: (1) the physiological properties of CA remain inconclusive, (2) many extant methods for CA characterisation are based on simplistic assumptions that can give rise to misleading interpretations, and (3) robust evaluation of CA requires thorough consideration not only of active vasomotor function, but also the unique properties of the intracranial environment.

Blood pressure regulation IX: cerebral autoregulation under blood pressure challenges

Cerebral autoregulation (CA) is integral to the delicate process of maintaining stable cerebral perfusion and brain tissue oxygenation against changes in arterial blood pressure. The last four decades has seen dramatic advances in understanding CA physiology, and the role that CA might play in the causation and progression of disease processes that affect the cerebral circulation such as stroke. However, the translation of these basic scientific advances into clinical practice has been limited by the maintenance of old constructs and because there are persistent gaps in our understanding of how this vital vascular mechanism should be quantified. In this review, we re-evaluate relevant studies that challenge established paradigms about how the cerebral perfusion pressure and blood flow are related. In the context of blood pressure being a major haemodynamic challenge to the cerebral circulation, we conclude that: (1) the physiological properties of CA remain inconclusive, (2) many extant methods for CA characterisation are based on simplistic assumptions that can give rise to misleading interpretations, and (3) robust evaluation of CA requires thorough consideration not only of active vasomotor function, but also the unique properties of the intracranial environment.

Model-based noninvasive estimation of intracranial pressure from cerebral blood flow velocity and arterial pressure

Intracranial pressure (ICP) is affected in many neurological conditions. Clinical measurement of pressure on the brain currently requires placing a probe in the cerebrospinal fluid compartment, the brain tissue, or other intracranial space. This invasiveness limits the measurement to critically ill patients. Because ICP is also clinically important in conditions ranging from brain tumors and hydrocephalus to concussions, noninvasive determination of ICP would be desirable. Our model-based approach to continuous estimation and tracking of ICP uses routinely obtainable time-synchronized, noninvasive (or minimally invasive) measurements of peripheral arterial blood pressure and blood flow velocity in the middle cerebral artery (MCA), both at intra-heartbeat resolution. A physiological model of cerebrovascular dynamics provides mathematical constraints that relate the measured waveforms to ICP. Our algorithm produces patient-specific ICP estimates with no calibration or training. Using 35 hours of data from 37 patients with traumatic brain injury, we generated ICP estimates on 2665 nonoverlapping 60-beat data windows. Referenced against concurrently recorded invasive parenchymal ICP that varied over 100 millimeters of mercury (mmHg) across all records, our estimates achieved a mean error (bias) of 1.6 mmHg and SD of error (SDE) of 7.6 mmHg. For the 1673 data windows over 22 hours in which blood flow velocity recordings were available from both the left and the right MCA, averaging the resulting bilateral ICP estimates reduced the bias to 1.5 mmHg and SDE to 5.9 mmHg. This accuracy is already comparable to that of some invasive ICP measurement methods in current clinical use.

A nonlinear dynamic approach reveals a long-term stroke effect on cerebral blood flow regulation at multiple time scales

Cerebral autoregulation (CA) is an important vascular control mechanism responsible for relatively stable cerebral blood flow despite changes of systemic blood pressure (BP). Impaired CA may leave brain tissue unprotected against potentially harmful effects of BP fluctuations. It is generally accepted that CA is less effective or even inactive at frequencies >∼0.1 Hz. Without any physiological foundation, this concept is based on studies that quantified the coupling between BP and cerebral blood flow velocity (BFV) using transfer function analysis. This traditional analysis assumes stationary oscillations with constant amplitude and period, and may be unreliable or even invalid for analysis of nonstationary BP and BFV signals. In this study we propose a novel computational tool for CA assessment that is based on nonlinear dynamic theory without the assumption of stationary signals. Using this method, we studied BP and BFV recordings collected from 39 patients with chronic ischemic infarctions and 40 age-matched non-stroke subjects during baseline resting conditions. The active CA function in non-stroke subjects was associated with an advanced phase in BFV oscillations compared to BP oscillations at frequencies from ∼0.02 to 0.38 Hz. The phase shift was reduced in stroke patients even at > = 6 months after stroke, and the reduction was consistent at all tested frequencies and in both stroke and non-stroke hemispheres. These results provide strong evidence that CA may be active in a much wider frequency region than previously believed and that the altered multiscale CA in different vascular territories following stroke may have important clinical implications for post-stroke recovery. Moreover, the stroke effects on multiscale cerebral blood flow regulation could not be detected by transfer function analysis, suggesting that nonlinear approaches without the assumption of stationarity are more sensitive for the assessment of the coupling of nonstationary physiological signals.

Cerebral autoregulation is an important mechanism that regulates blood supply to brain tissue to match metabolic demands during daily activities. Impaired cerebral autoregulation increases the dependence of cerebral blood flow on systemic blood pressure, and is associated with fatal outcomes in patients after brain injury and acute ischemic stroke. Reliable and noninvasive assessment of cerebral autoregulation is still a major challenge in medical diagnostics and clinic studies, mainly because blood pressure and flow are intrinsically nonstationary (possessing complex oscillations/fluctuations with varying amplitude and frequency) while traditional methods for assessment of the pressure-flow dependence assume stationary signals. We propose a new computational technique that is based on nonlinear theories without the assumption of stationary signals. This technique allows us to detect the degradation of cerebral autoregulation in patients with mild ischemic stroke even at >6 months after the insult. The degradation was present in both stroke and non-stroke sides and was accompanied by an altered pressure-flow interaction over a wide range of frequencies from 0.02–0.38 Hz. Our results challenges the traditionally accepted functional region of autoregulation (<∼0.1 Hz). The observed long-term influences of stroke highlight the importance of reliable monitoring of cerebral blood flow regulation for the management and daily care of stroke patients.

Direct numerical simulation of transitional flow in a patient-specific intracranial aneurysm

In experiments turbulence has previously been shown to occur in intracranial aneurysms. The effects of turbulence induced oscillatory wall stresses could be of great importance in understanding aneurysm rupture. To investigate the effects of turbulence on blood flow in an intracranial aneurysm, we performed a high resolution computational fluid dynamics (CFD) simulation in a patient specific middle cerebral artery (MCA) aneurysm using a realistic, pulsatile inflow velocity. The flow showed transition to turbulence just after peak systole, before relaminarization occurred during diastole. The turbulent structures greatly affected both the frequency of change of wall shear stress (WSS) direction and WSS magnitude, which reached a maximum value of 41.5Pa. The recorded frequencies were predominantly in the range of 1-500Hz. The current study confirms, through properly resolved CFD simulations that turbulence can occur in intracranial aneurysms.

Contribution of arterial Windkessel in low-frequency cerebral hemodynamics during transient changes in blood pressure

The Windkessel properties of the vasculature are known to play a significant role in buffering arterial pulsations, but their potential importance in dampening low-frequency fluctuations in cerebral blood flow has not been clearly examined. In this study, we quantitatively assessed the contribution of arterial Windkessel (peripheral compliance and resistance) in the dynamic cerebral blood flow response to relatively large and acute changes in blood pressure. Middle cerebral artery flow velocity (MCA(V); transcranial Doppler) and arterial blood pressure were recorded from 14 healthy subjects. Low-pass-filtered pressure-flow responses (<0.15 Hz) during transient hypertension (intravenous phenylephrine) and hypotension (intravenous sodium nitroprusside) were fitted to a two-element Windkessel model. The Windkessel model was found to provide a superior goodness of fit to the MCA(V) responses during both hypertension and hypotension (R² = 0.89 ± 0.03 and 0.85 ± 0.05, respectively), with a significant improvement in adjusted coefficients of determination (P < 0.005) compared with the single-resistance model (R² = 0.62 ± 0.06 and 0.61 ± 0.08, respectively). No differences were found between the two interventions in the Windkessel capacitive and resistive gains, suggesting similar vascular properties during pressure rise and fall episodes. The results highlight that low-frequency cerebral hemodynamic responses to transient hypertension and hypotension may include a significant contribution from the mechanical properties of vasculature and, thus, cannot solely be attributed to the active control of vascular tone by cerebral autoregulation. The arterial Windkessel should be regarded as an important element of dynamic cerebral blood flow modulation during large and acute blood pressure perturbation.

Modeling the cardiovascular-respiratory control system: data, model analysis, and parameter estimation

Several key areas in modeling the cardiovascular and respiratory control systems are reviewed and examples are given which reflect the research state of the art in these areas. Attention is given to the interrelated issues of data collection, experimental design, and model application including model development and analysis. Examples are given of current clinical problems which can be examined via modeling, and important issues related to model adaptation to the clinical setting.

The influence of contrast agent injection on physiological flow in the circle of Willis

X-ray videodensitometry allows in vivo flow measurements from gradients in contrast agent concentration. However, the injection of contrast agent alters the flow to be measured. Here, the temporal, spatial, and inter-patient variability of the response to injection are examined. To this purpose, an injection is prescribed in the internal carotid in a 1D wave propagation model of the arterial circulation. Although the resulting effect of injection is constant over a cardiac cycle, the response does vary with the location within the cerebral circulation and the geometry of the circle of Willis. At the injection site, the injection partly suppresses the incoming blood flow, such that the distal flow is increased by approximately 10%. This corresponds to approximately 20% of the injection rate added to the blood flow during injection, depending on the vascular geometry. In the communicating arteries, the flow direction is reversed during injection. Since the measured flow is not equal to the physiological blood flow, the effect of injection should be taken into account when deriving the flow from travelling contrast agent.

The complex interaction between ocular perfusion pressure and ocular blood flow - relevance for glaucoma

Glaucoma is an optic neuropathy of unknown origin. The most important risk factor for the disease is an increased intraocular pressure (IOP). Reducing IOP is associated with reduced progression in glaucoma. Several recent large scale trials have indicated that low ocular perfusion pressure (OPP) is a risk factor for the incidence, prevalence and progression of the disease. This is a strong indicator that vascular factors are involved in the pathogenesis of the disease, a hypothesis that was formulated 150 years ago. The relation between OPP and blood flow to the posterior pole of the eye is, however, complex, because of a phenomenon called autoregulation. Autoregulatory processes attempt to keep blood flow constant despite changes in OPP. Although autoregulation has been observed in many experiments in the ocular vasculature the mechanisms underlying the vasodilator and vasoconstrictor responses in face of changes in OPP remain largely unknown. There is, however, recent evidence that the human choroid regulates its blood flow better during changes in blood pressure induced by isometric exercise than during changes in IOP induced by a suction cup. This may have consequences for our understanding of glaucoma, because it indicates that blood flow regulation is strongly dependent not only on OPP, but also on the level of IOP itself. Indeed there is data indicating that reduction of IOP by pharmacological intervention improves optic nerve head blood flow regulation independently of an ocular vasodilator effect.

Toward integrated management of cerebral aneurysms

In the last few years, some of the visionary concepts behind the virtual physiological human began to be demonstrated on various clinical domains, showing great promise for improving healthcare management. In the current work, we provide an overview of image- and biomechanics-based techniques that, when put together, provide a patient-specific pipeline for the management of intracranial aneurysms. The derivation and subsequent integration of morphological, morphodynamic, haemodynamic and structural analyses allow us to extract patient-specific models and information from which diagnostic and prognostic descriptors can be obtained. Linking such new indices with relevant clinical events should bring new insights into the processes behind aneurysm genesis, growth and rupture. The development of techniques for modelling endovascular devices such as stents and coils allows the evaluation of alternative treatment scenarios before the intervention takes place and could also contribute to the understanding and improved design of more effective devices. A key element to facilitate the clinical take-up of all these developments is their comprehensive validation. Although a number of previously published results have shown the accuracy and robustness of individual components, further efforts should be directed to demonstrate the diagnostic and prognostic efficacy of these advanced tools through large-scale clinical trials.

A cerebrovascular response model for functional neuroimaging including dynamic cerebral autoregulation

Functional neuroimaging techniques such as functional magnetic resonance imaging (fMRI) and near-infrared spectroscopy (NIRS) can be used to isolate an evoked response to a stimulus from significant background physiological fluctuations. Data analysis approaches typically use averaging or linear regression to remove this physiological baseline with varying degrees of success. Biophysical model-based analysis of the functional hemodynamic response has also been advanced previously with the Balloon and Windkessel models. In the present work, a biophysical model of systemic and cerebral circulation and gas exchange is applied to resting state NIRS neuroimaging data from 10 human subjects. The model further includes dynamic cerebral autoregulation, which modulates the cerebral arteriole compliance to control cerebral blood flow. This biophysical model allows for prediction, from noninvasive blood pressure measurements, of the background hemodynamic fluctuations in the systemic and cerebral circulations. Significantly higher correlations with the NIRS data were found using the biophysical model predictions compared to blood pressure regression and compared to transfer function analysis (multifactor ANOVA, p<0.0001). This finding supports the further development and use of biophysical models for removing baseline activity in functional neuroimaging analysis. Future extensions of this work could model changes in cerebrovascular physiology that occur during development, aging, and disease.

Dynamic pressure-flow relationship of the cerebral circulation during acute increase in arterial pressure

The physiological mechanism(s) for the regulation of the dynamic pressure-flow relationship of the cerebral circulation are not well understood. We studied the effects of acute cerebral vasoconstriction on the transfer function between spontaneous changes in blood pressure (BP) and cerebral blood flow velocity (CBFV) in 13 healthy subjects (30 +/- 7 years). CBFV was measured in the middle cerebral artery using transcranial Doppler. BP was increased stepwise with phenylephrine infusion at 0.5, 1.0 and 2.0 microg kg(-1) min(-1). Phenylephrine increased BP by 11, 23 and 37% from baseline, while CBFV increased (11%) only with the highest increase in BP. Cerebrovascular resistance index (BP/CBFV) increased progressively by 6, 17 and 23%, demonstrating effective steady-state autoregulation. Transfer function gain at the low frequencies (LF, 0.07-0.20 Hz) was reduced by 15, 14 and 14%, while the phase was reduced by 10, 17 and 31%. A similar trend of changes was observed at the high frequencies (HF, 0.20-0.35 Hz), but gain and phase remained unchanged at the very low frequencies (VLF, 0.02-0.07 Hz). Windkessel model simulation suggests that increases in steady-state cerebrovascular resistance and/or decreases in vascular compliance during cerebral vasoconstriction contribute to the changes in gain and phase. These findings suggest that changes in steady-state cerebrovascular resistance and/or vascular compliance modulate the dynamic pressure-flow relationship at the low and high frequencies, while dynamic autoregulation is likely to be dominant at the very low frequencies. Thus, oscillations in CBFV are modulated not only by dynamic autoregulation, but also by changes in steady-state cerebrovascular resistance and/or vascular compliance.

Nonlinear assessment of cerebral autoregulation from spontaneous blood pressure and cerebral blood flow fluctuations

Cerebral autoregulation (CA) is an most important mechanism responsible for the relatively constant blood flow supply to brain when cerebral perfusion pressure varies. Its assessment in nonacute cases has been relied on the quantification of the relationship between noninvasive beat-to-beat blood pressure (BP) and blood flow velocity (BFV). To overcome the nonstationary nature of physiological signals such as BP and BFV, a computational method called multimodal pressure-flow (MMPF) analysis was recently developed to study the nonlinear BP-BFV relationship during the Valsalva maneuver (VM). The present study aimed to determine (i) whether this method can estimate autoregulation from spontaneous BP and BFV fluctuations during baseline rest conditions; (ii) whether there is any difference between the MMPF measures of autoregulation based on intra-arterial BP (ABP) and based on cerebral perfusion pressure (CPP); and (iii) whether the MMPF method provides reproducible and reliable measure for noninvasive assessment of autoregulation. To achieve these aims, we analyzed data from existing databases including: (i) ABP and BFV of 12 healthy control, 10 hypertensive, and 10 stroke subjects during baseline resting conditions and during the Valsalva maneuver, and (ii) ABP, CPP, and BFV of 30 patients with traumatic brain injury (TBI) who were being paralyzed, sedated, and ventilated. We showed that autoregulation in healthy control subjects can be characterized by specific phase shifts between BP and BFV oscillations during the Valsalva maneuver, and the BP-BFV phase shifts were reduced in hypertensive and stroke subjects (P < 0.01), indicating impaired autoregulation. Similar results were found during baseline condition from spontaneous BP and BFV oscillations. The BP-BFV phase shifts obtained during baseline and during VM were highly correlated (R > 0.8, P < 0.0001), showing no statistical difference (paired-t test P > 0.47). In TBI patients there were strong correlations between phases of ABP and CPP oscillations (R = 0.99, P < 0.0001) and, thus, between ABP-BFV and CPP-BFV phase shifts (P < 0.0001, R = 0.76). By repeating the MMPF 4 times on data of TBI subjects, each time on a selected cycle of spontaneous BP and BFV oscillations, we showed that MMPF had better reproducibility than traditional autoregulation index. These results indicate that the MMPF method, based on instantaneous phase relationships between cerebral blood flow velocity and peripheral blood pressure, has better performance than the traditional standard method, and can reliably assess cerebral autoregulation dynamics from ambulatory blood pressure and cerebral blood flow during supine rest conditions.

Integration of cerebrovascular CO2 reactivity and chemoreflex control of breathing: mechanisms of regulation, measurement, and interpretation

Cerebral blood flow (CBF) and its distribution are highly sensitive to changes in the partial pressure of arterial CO(2) (Pa(CO(2))). This physiological response, termed cerebrovascular CO(2) reactivity, is a vital homeostatic function that helps regulate and maintain central pH and, therefore, affects the respiratory central chemoreceptor stimulus. CBF increases with hypercapnia to wash out CO(2) from brain tissue, thereby attenuating the rise in central Pco(2), whereas hypocapnia causes cerebral vasoconstriction, which reduces CBF and attenuates the fall of brain tissue Pco(2). Cerebrovascular reactivity and ventilatory response to Pa(CO(2)) are therefore tightly linked, so that the regulation of CBF has an important role in stabilizing breathing during fluctuating levels of chemical stimuli. Indeed, recent reports indicate that cerebrovascular responsiveness to CO(2), primarily via its effects at the level of the central chemoreceptors, is an important determinant of eupneic and hypercapnic ventilatory responsiveness in otherwise healthy humans during wakefulness, sleep, and exercise and at high altitude. In particular, reductions in cerebrovascular responsiveness to CO(2) that provoke an increase in the gain of the chemoreflex control of breathing may underpin breathing instability during central sleep apnea in patients with congestive heart failure and on ascent to high altitude. In this review, we summarize the major factors that regulate CBF to emphasize the integrated mechanisms, in addition to Pa(CO(2)), that control CBF. We discuss in detail the assessment and interpretation of cerebrovascular reactivity to CO(2). Next, we provide a detailed update on the integration of the role of cerebrovascular CO(2) reactivity and CBF in regulation of chemoreflex control of breathing in health and disease. Finally, we describe the use of a newly developed steady-state modeling approach to examine the effects of changes in CBF on the chemoreflex control of breathing and suggest avenues for future research.

Onset responses of ventilation and cerebral blood flow to hypercapnia in humans: rest and exercise

The respiratory and cerebrovascular reactivity to changes in arterial Pco(2) (Pa(CO(2))) is an important mechanism that maintains CO(2) or pH homeostasis in the brain. It remains unclear, however, how cerebrovascular CO(2) reactivity might influence the respiratory chemoreflex. The purpose of the present study was therefore to examine the interaction between onset responses of the respiratory chemoreflex and middle cerebral artery (MCA) mean blood velocity (V(mean)) to hypercapnia (5.0% CO(2)-40% O(2)-balance N(2)) at rest and during dynamic exercise ( approximately 1.0 l/min O(2) consumption). Each onset response was evaluated using a single-exponential regression model consisting of the response time latency [CO(2)-response delay (t(0))] and time constant (tau). At rest, t(0) and tau data indicated that the MCA V(mean) onset response was faster than the ventilatory (Ve) response (P < 0.001). In contrast, during exercise, t(0) of Ve and MCA V(mean) onset responses were decreased. In addition, despite the enhanced Pa(CO(2)) response to CO(2) administration (P = 0.014), tau of MCA V(mean) tended to increase during exercise (P = 0.054), whereas tau of Ve decreased (P = 0.015). These findings indicate that 1) at rest, faster washout of CO(2) via cerebral vasodilation results in a reduced activation of the central chemoreflex and subsequent reduced Ve onset response, and 2) during exercise, despite higher rates of increasing Pa(CO(2)), the lack of change in the onset response of cerebral blood flow and reduced washout of CO(2) may act to augment the Ve onset response.

Modeling perfusion in the cerebral vasculature

The constant perfusion of a human organ with nutrients and oxygen demands a robust regulatory mechanisms in the face of normal day-to-day pressure variations in the vasculature. The brain, in a similar manner to the heart requires this mechanism to be extremely quick acting, relative to other ways of altering perfusion such as varying systemic blood pressure, since oxygen depravation in the tissues of the brain can be tolerated for only of the order of tens of seconds before significant damage can be done. In recent years computational models, and it must be noted computer architecture have evolved to an extent where mathematicians and engineers can play a large part in discovering how the brain functions physiologically as well as investigating pathological conditions. This review will look at a number of increasingly complex computational models of blood flow to the brain and how variations in arterial geometry can influence the perfusion in the cerebral vasculature. Although these models have provided an insight into complex mechanisms the research area is densely populated with important questions that perhaps only computer models can answer. The review will indicate possible areas of investigation.

Mathematical modeling of the brain: principles and challenges

OBJECTIVE

The use of mathematics in the study of phenomena and systems of interest to medicine has become quite popular in recent years, but not much progress has been made as a result of these efforts. The aim of this article is to identify the reasons for this failure and to suggest procedures for more successful outcomes.

METHODS

We review and assess a variety of mathematical modeling procedures, from microscopic (at the level of molecular behavior) to macroscopic standpoints, from lumped-parameters to distributed-parameters approaches. Using examples that are as simple as possible, we elucidate the difference between the predictive and the explanatory powers of mathematical models, as well as the uses (and abuses) of analogy in their construction.

RESULTS

Mathematical medicine is a truly interdisciplinary area that brings together medical researchers, engineers, and applied mathematicians whose vast differences in expertise and background make collaboration difficult.

CONCLUSION

The lack of a common language and a common way of understanding what a mathematical model is, and what it can do, is identified as the main source of the slow progress to date, and constructive suggestions are made to improve the situation.

Efficacy of using mean arterial blood pressure sequence for three-element Windkessel model estimation

The three-element Windkessel model is widely used and accepted for analyzing blood flow and pressure in arterial system and cerebral circulation. In most studies, changes in mean arterial blood pressure data is used as input to estimate the model parameters. However, estimation of linear model parameters, using input-output data, requires that the input be persistently exciting. This study examined the efficacy of using mean arterial blood pressure (MABP) sequence as an input stimulus for estimating the parameters of the three-element Windkessel model. Additionally, the study explored the use of a shorter MABP data segment of 1.5 mm as compared to the commonly used 6 mm data. MABP data was obtained from 11 healthy subjects. One thousand three-element Windkessel models, with parameter values randomly selected to be within physiological range, were subjected to seven different input sequences. For each input sequence and model, the values of the model (target-parameters) were estimated. The seven input sequence were: 1) six minutes of MABP measured from subjects; 2-5) four 1.5 mm of measured MABP obtained by dividing the measured six minutes of MABP into non- overlapping contiguous segments; 6) a six-minutes of pseudo random binary sequence (PRBS) with amplitudes comparable to the MABP sequence; and 7) a 1.5 mm of PRBS sequence with amplitudes comparable to the MABP sequence. The MABP data used was randomly selected from the 11 subjects for each estimation run. The model parameter estimation method had two phases of optimization. In the first phase, the parameters were estimated and optimized using the frequency transform of the input and output. In the second phase, the values of the estimated parameters were used as initial estimates and time-domain optimization was carried out to further refine the estimates. Results from the study, comparing the estimated-parameters with the target-parameters, show that for the MABP data, there was no significant difference between using the six minutes or 1.5 mm of data for estimating the target-parameters. Also, parameters estimated from the MABP data were either equivalent or superior to the PRBS results, suggesting that changes in MABP can be used as an effective sequence for linear model estimation.