Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries

The simulation of blood flow and pressure in arteries requires outflow boundary conditions that incorporate models of downstream domains. We previously described a coupled multidomain method to couple analytical models of the downstream domains with 3D numerical models of the upstream vasculature. This prior work either included pure resistance boundary conditions or impedance boundary conditions based on assumed periodicity of the solution. However, flow and pressure in arteries are not necessarily periodic in time due to heart rate variability, respiration, complex transitional flow or acute physiological changes. We present herein an approach for prescribing lumped parameter outflow boundary conditions that accommodate transient phenomena. We have applied this method to compute haemodynamic quantities in different physiologically relevant cardiovascular models, including patient-specific examples, to study non-periodic flow phenomena often observed in normal subjects and in patients with acquired or congenital cardiovascular disease. The relevance of using boundary conditions that accommodate transient phenomena compared with boundary conditions that assume periodicity of the solution is discussed.     

Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions

We present a one-dimensional (1D) fluid dynamic model that can predict blood flow and blood pressure during exercise using data collected at rest. To facilitate accurate prediction of blood flow, we developed an impedance boundary condition using morphologically derived structured trees. Our model was validated by computing blood flow through a model of large arteries extending from the thoracic aorta to the profunda arteries. The computed flow was compared against measured flow in the infrarenal (IR) aorta at rest and during exercise. Phase contrast-magnetic resonance imaging (PC-MRI) data was collected from 11 healthy volunteers at rest and during steady exercise. For each subject, an allometrically-scaled geometry of the large vessels was created. This geometry extends from the thoracic aorta to the femoral arteries and includes the celiac, superior mesenteric, renal, inferior mesenteric, internal iliac and profunda arteries. During rest, flow was simulated using measured supraceliac (SC) flow at the inlet and a uniform set of impedance boundary conditions at the 11 outlets. To simulate exercise, boundary conditions were modified. Inflow data collected during steady exercise was specified at the inlet and the outlet boundaries were adjusted as follows. The geometry of the structured trees used to compute impedance was scaled to simulate the effective change in the cross-sectional area of resistance vessels and capillaries due to exercise. The resulting computed flow through the IR aorta was compared to measured flow. This method produces good results with a mean difference between paired data to be 1.1 +/- 7 cm(3) s(- 1) at rest and 4.0 +/- 15 cm(3) s(- 1) at exercise. While future work will improve on these results, this method provides groundwork with which to predict the flow distributions in a network due to physiologic regulation.     

Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation.

Short-term cardiovascular responses to postural change from sitting to standing involve complex interactions between the autonomic nervous system, which regulates blood pressure, and cerebral autoregulation, which maintains cerebral perfusion. We present a mathematical model that can predict dynamic changes in beat-to-beat arterial blood pressure and middle cerebral artery blood flow velocity during postural change from sitting to standing. Our cardiovascular model utilizes 11 compartments to describe blood pressure, blood flow, compliance, and resistance in the heart and systemic circulation. To include dynamics due to the pulsatile nature of blood pressure and blood flow, resistances in the large systemic arteries are modeled using nonlinear functions of pressure. A physiologically based submodel is used to describe effects of gravity on venous blood pooling during postural change. Two types of control mechanisms are included: 1) autonomic regulation mediated by sympathetic and parasympathetic responses, which affect heart rate, cardiac contractility, resistance, and compliance, and 2) autoregulation mediated by responses to local changes in myogenic tone, metabolic demand, and CO(2) concentration, which affect cerebrovascular resistance. Finally, we formulate an inverse least-squares problem to estimate parameters and demonstrate that our mathematical model is in agreement with physiological data from a young subject during postural change from sitting to standing.     

Finite element models of total shoulder replacement: Application of boundary conditions

We present a one-dimensional (1D) fluid dynamic model that can predict blood flow and blood pressure during exercise using data collected at rest. To facilitate accurate prediction of blood flow, we developed an impedance boundary condition using morphologically derived structured trees. Our model was validated by computing blood flow through a model of large arteries extending from the thoracic aorta to the profunda arteries. The computed flow was compared against measured flow in the infrarenal (IR) aorta at rest and during exercise. Phase contrast-magnetic resonance imaging (PC-MRI) data was collected from 11 healthy volunteers at rest and during steady exercise. For each subject, an allometrically-scaled geometry of the large vessels was created. This geometry extends from the thoracic aorta to the femoral arteries and includes the celiac, superior mesenteric, renal, inferior mesenteric, internal iliac and profunda arteries. During rest, flow was simulated using measured supraceliac (SC) flow at the inlet and a uniform set of impedance boundary conditions at the 11 outlets. To simulate exercise, boundary conditions were modified. Inflow data collected during steady exercise was specified at the inlet and the outlet boundaries were adjusted as follows. The geometry of the structured trees used to compute impedance was scaled to simulate the effective change in the cross-sectional area of resistance vessels and capillaries due to exercise. The resulting computed flow through the IR aorta was compared to measured flow. This method produces good results with a mean difference between paired data to be 1.1 ± 7 cm³ s^? 1 at rest and 4.0 ± 15 cm³ s^? 1 at exercise. While future work will improve on these results, this method provides groundwork with which to predict the flow distributions in a network due to physiologic regulation.     

Numerical simulation of local blood flow in the carotid and cerebral arteries under altered gravity

A computational fluid dynamics (CFD) approach was presented to model the blood flows in the carotid bifurcation and the brain arteries under altered gravity. Physical models required for CFD simulation were introduced including a model for arterial wall motion due to fluid-wall interactions, a shear thinning fluid model of blood, a vascular bed model for outflow boundary conditions, and a model for autoregulation mechanism. The three-dimensional unsteady incompressible Navier-Stokes equations coupled with these models were solved iteratively using the pseudocompressibility method and dual time stepping. Gravity source terms were added to the Navier-Stokes equations to take the effect of gravity into account. For the treatment of complex geometry, a chimera overset grid technique was adopted to obtain connectivity between arterial branches. For code validation, computed results were compared with experimental data for both steady-state and time-dependent flows. This computational approach was then applied to blood flows through a realistic carotid bifurcation and two Circle of Willis models, one using an idealized geometry and the other using an anatomical data set. A three-dimensional Circle of Willis configuration was reconstructed from subject-specific magnetic resonance images using an image segmentation method. Through the numerical simulation of blood flow in two model problems, namely, the carotid bifurcation and the brain arteries, it was observed that the altered gravity has considerable effects on arterial contraction/dilatation and consequent changes in flow conditions.     

Study of the collateral capacity of the circle of Willis of patients with severe carotid artery stenosis by 3D computational modeling

This numerical study aims to investigate the capacity of the circle of Wills (CoW) to provide collateral blood supply for patients with unilateral carotid arterial stenosis. The basic 3D geometry of the CoW was reconstructed based on a magnetic resonance angiogram of a normal human subject. A total of 52 computational fluid dynamics simulations were performed for four geometry configurations of the CoW with an artificially inserted axisymmetric stenosis of different luminal area reductions in an internal carotid artery (ICA) under a variety of boundary conditions. The CoW geometric configurations included (a) a normal CoW with all communicating arteries; (b) as model (a) but with enlarged communicating arterial diameters; (c) as (a) but with the ipsilateral posterior communicating artery missing, and (d) as (c) but with enlarged communicating arteries. It is found that the blood perfusion pressure drop between the ipsilateral ICA and the middle cerebral artery (MCA) only becomes significant when the degree of stenosis is greater than 86%. The cerebral autoregulation range varied significantly between the different CoW configurations for the severe stenosis cases. Without causing the flow rates to decrease at the efferent arterial ends, the mean perfusion pressure in the ipsilateral ICA can drop from 100 to 73, 67, 92 and 84mmHg for the CoW models (a)-(d) with 96% luminal area reduction stenosis, respectively. The additional pathways are able to raise the ipsilateral MCA pressure significantly without reducing the total flow perfusion. Cerebral autoregulation effects were not directly included in the study. Therefore, the findings in the study should be interpreted with cautions when comes to the biological and clinical significance.     

Three-dimensional hemodynamics analysis of the circle of Willis in the patient-specific nonintegral arterial structures

The hemodynamic alteration in the cerebral circulation caused by the geometric variations in the cerebral circulation arterial network of the circle of Wills (CoW) can lead to fatal ischemic attacks in the brain. The geometric variations due to impairment in the arterial network result in incomplete cerebral arterial structure of CoW and inadequate blood supply to the brain. Therefore, it is of great importance to understand the hemodynamics of the CoW, for efficiently and precisely evaluating the status of blood supply to the brain. In this paper, three-dimensional computational fluid dynamics of the main CoW vasculature coupled with zero-dimensional lumped parameter model boundary condition for the CoW outflow boundaries is developed for analysis of the blood flow distribution in the incomplete CoW cerebral arterial structures. The geometric models in our study cover the arterial segments from the aorta to the cerebral arteries, which can allow us to take into account the innate patient-specific resistance of the arterial trees. Numerical simulations of the governing fluid mechanics are performed to determine the CoW arterial structural hemodynamics, for illustrating the redistribution of the blood flow in CoW due to the structural variations. We have evaluated our coupling methodology in five patient-specific cases that were diagnosed with the absence of efferent vessels or impairment in the connective arteries in their CoWs. The velocity profiles calculated by our approach in the segments of the patient-specific arterial structures are found to be very close to the Doppler ultrasound measurements. The accuracy and consistency of our hemodynamic results have been improved (to \(16.1 \pm 18.5\) %) compared to that of the pure-resistance boundary conditions (of 43.5 \(\pm \) 28 %). Based on our grouping of the five cases according to the occurrence of unilateral occlusion in vertebral arteries, the inter-comparison has shown that (i) the flow reduction in posterior cerebral arteries is the consequence of the unilateral vertebral arterial occlusion, and (ii) the flow rate in the anterior cerebral arteries is correlated with the posterior structural variations. This study shows that our coupling approach is capable of providing comprehensive information of the hemodynamic alterations in the pathological CoW arterial structures. The information generated by our methodology can enable evaluation of both the functional and structural status of the clinically significant symptoms, for assisting the treatment decision-making.     

Tissue-fluid interface analysis using biphasic finite element method

Numerical studies on fluid-structure interaction have primarily relied on decoupling the solid and fluid sub-domains with the interactions treated as external boundary conditions on the individual sub-domains. The finite element applications for the fluid-structure interactions can be divided into iterative algorithms and sequential algorithms. In this paper, a new computational methodology for the analysis of tissue-fluid interaction problems is presented. The whole computational domain is treated as a single biphasic continuum, and the same space and time discretisation is carried out for the sub-domains using a penalty-based finite element model. This procedure does not require the explicit modelling of additional boundary conditions or interface elements. The developed biphasic interface finite element model is used in analysing blood flow through normal and stenotic arteries. The increase in fluid flow velocity when passing through a stenosed artery and the drop in pressure at the region are captured using this method.     

Status of the "dead point" of blood flow in anastomoses of superficial cerebral arteries as affected by an acute increase of blood pressure

Changes in pial arteries diameter and the condition of blood flow "dead point" in arterial anastomoses were established using the brain window during an acute increase of mean arterial pressure (MAP) induced by intravenous injection of norepinephrine (NE) with microcineangiography and the analysis of films and frames on a montage table and TAS ("Leitz"). During an acute increase of MAP the movement of blood flow "dead point" in anastomoses and the expansion of plasma segments occurred much more frequently than in normotension. The stabilization of blood flow "dead point" was observed at high constant MAP. Pronounced dilation of both pial arteries and veins first occurred in anastomoses, then spread to arterial branches. It is assumed that high vulnerability of the brain vessels of the borderline zones is due to breakthrough in autoregulation of cerebral blood flow on its upper limit and depends on the repeatedly changing directions of the blood flow and its moving "dead point", as the peripheral resistance of arterial anastomoses-forming branches under these circumstances changes in an irregular manner.     

Towards enabling a cardiovascular digital twin for human systemic circulation using inverse analysis

An exponential rise in patient data provides an excellent opportunity to improve the existing health care infrastructure. In the present work, a method to enable cardiovascular digital twin is proposed using inverse analysis. Conventionally, accurate analytical solutions for inverse analysis in linear problems have been proposed and used. However, these methods fail or are not efficient for nonlinear systems, such as blood flow in the cardiovascular system (systemic circulation) that involves high degree of nonlinearity. To address this, a methodology for inverse analysis using recurrent neural network for the cardiovascular system is proposed in this work, using a virtual patient database. Blood pressure waveforms in various vessels of the body are inversely calculated with the help of long short-term memory (LSTM) cells by inputting pressure waveforms from three non-invasively accessible blood vessels (carotid, femoral and brachial arteries). The inverse analysis system built this way is applied to the detection of abdominal aortic aneurysm (AAA) and its severity using neural networks.

     

Effects of transients on pulsatile flow in arteries

Analytical solutions to the model problem of unsteady Newtonian fluid flow in straight, elastic-walled vessels can provide: theoretical insights into the flow of blood in arteries; a theoretical basis for clinical measurements in diagnoses of arterial flow rates; and guidance for boundary conditions in numerical simulations of flow in finite computational domains. However, while Womersley's analyses of blood flow assume solution forms that treat the flow as periodic and continuously unsteady, many flow variables in the smaller arteries are not continuously unsteady at all. They are characterized more accurately as rapid transient motions followed by a period of recovery to a stationary state, repeated in successive cycles. These flows are not continually unsteady ones described by Womersley's solutions but unsteady flows restarted from rest in each cycle, characterized as initial-boundary value problems. In this paper, we compare the Womersley and initial-boundary value solutions for model transients that stop then restart, explain these previously unreported limitations of Womersley's solutions, and demonstrate how the initial-boundary value solutions provide excellent agreement with measurements of blood flow in the anterior tibial and popliteal arteries of patients. Some consequences of these findings for understanding and interpreting measurements of blood flow, and for prescribing boundary conditions in computer simulations of arterial blood flow are discussed.     

Investigation of hemodynamics in the development of dissecting aneurysm within patient-specific dissecting aneurismal aortas using computational fluid dynamics (CFD) simulations

Aortic dissecting aneurysm is one of the most catastrophic cardiovascular emergencies that carries high mortality. It was pointed out from clinical observations that the aneurysm development is likely to be related to the hemodynamics condition of the dissected aorta. In order to gain more insight on the formation and progression of dissecting aneurysm, hemodynamic parameters including flow pattern, velocity distribution, aortic wall pressure and shear stress, which are difficult to measure in vivo, are evaluated using numerical simulations. Pulsatile blood flow in patient-specific dissecting aneurismal aortas before and after the formation of lumenal aneurysm (pre-aneurysm and post-aneurysm) is investigated by computational fluid dynamics (CFD) simulations. Realistic time-dependent boundary conditions are prescribed at various arteries of the complete aorta models. This study suggests the helical development of false lumen around true lumen may be related to the helical nature of hemodynamic flow in aorta. Narrowing of the aorta is responsible for the massive recirculation in the poststenosis region in the lumenal aneurysm development. High pressure difference of 0.21 kPa between true and false lumens in the pre-aneurismal aorta infers the possible lumenal aneurysm site in the descending aorta. It is also found that relatively high time-averaged wall shear stress (in the range of 4-8 kPa) may be associated with tear initiation and propagation. CFD modeling assists in medical planning by providing blood flow patterns, wall pressure and wall shear stress. This helps to understand various phenomena in the development of dissecting aneurysm.     

Repeated unit cell (RUC) approach for pure bending analysis of coronary stents

Flexibility is one of the key properties of coronary stents. The objective of this paper is to characterize the bending behaviour of stents through finite element analysis with repeated unit cell (RUC) models. General periodic boundary conditions for the RUC under the pure bending condition are formulated. It is found that the proposed RUC approach can provide accurate numerical results of bending behaviour of stents with much less computational costs. Bending stiffness, post-yield bending behaviour and the relationship between moment and bending curvature are investigated for Palmaz-Schatz stents and stents with the V- and S-shaped links. It is found that the effect of link geometry on the bending behaviour of stent is significant. The behaviour of stents subjected to cyclic bending is also investigated.     

Theoretical model of blood flow autoregulation: roles of myogenic, shear-dependent, and metabolic responses

The autoregulation of blood flow, the maintenance of almost constant blood flow in the face of variations in arterial pressure, is characteristic of many tissue types. Here, contributions to the autoregulation of pressure-dependent, shear stress-dependent, and metabolic vasoactive responses are analyzed using a theoretical model. Seven segments, connected in series, represent classes of vessels: arteries, large arterioles, small arterioles, capillaries, small venules, large venules, and veins. The large and small arterioles respond actively to local changes in pressure and wall shear stress and to the downstream metabolic state communicated via conducted responses. All other segments are considered fixed resistances. The myogenic, shear-dependent, and metabolic responses of the arteriolar segments are represented by a theoretical model based on experimental data from isolated vessels. To assess autoregulation, the predicted flow at an arterial pressure of 130 mmHg is compared with that at 80 mmHg. If the degree of vascular smooth muscle activation is held constant at 0.5, there is a fivefold increase in blood flow. When myogenic variation of tone is included, flow increases by a factor of 1.66 over the same pressure range, indicating weak autoregulation. The inclusion of both myogenic and shear-dependent responses results in an increase in flow by a factor of 2.43. A further addition of the metabolic response produces strong autoregulation with flow increasing by a factor of 1.18 and gives results consistent with experimental observation. The model results indicate that the combined effects of myogenic and metabolic regulation overcome the vasodilatory effect of the shear response and lead to the autoregulation of blood flow.     

Repeated unit cell (RUC) approach for pure bending analysis of coronary stents

Flexibility is one of the key properties of coronary stents. The objective of this paper is to characterize the bending behaviour of stents through finite element analysis with repeated unit cell (RUC) models. General periodic boundary conditions for the RUC under the pure bending condition are formulated. It is found that the proposed RUC approach can provide accurate numerical results of bending behaviour of stents with much less computational costs. Bending stiffness, post-yield bending behaviour and the relationship between moment and bending curvature are investigated for Palmaz–Schatz stents and stents with the V- and S-shaped links. It is found that the effect of link geometry on the bending behaviour of stent is significant. The behaviour of stents subjected to cyclic bending is also investigated.     

Influence of moderate normoxic hypercapnia on the autoregulation of the cerebral circulation in the unanesthetized rabbit]

In the unanesthetized rabbit autoregulation of cerebral blood flow was evaluated by continuous recording of local cerebral blood flow during progressive hypotension induced by exsanguination. Under hypercapnia induced by CO2, 8 per cent in air, autoregulation was not suppressed but an increase of the threshold under which autoregulation disappears was noted.     

Tissue-growth-based synthetic tree generation and perfusion simulation

Biological tissues receive oxygen and nutrients from blood vessels by developing an indispensable supply and demand relationship with the blood vessels. We implemented a synthetic tree generation algorithm by considering the interactions between the tissues and blood vessels. We first segment major arteries using medical image data and synthetic trees are generated originating from these segmented arteries. They grow into extensive networks of small vessels to fill the supplied tissues and satisfy the metabolic demand of them. Further, the algorithm is optimized to be executed in parallel without affecting the generated tree volumes. The generated vascular trees are used to simulate blood perfusion in the tissues by performing multiscale blood flow simulations. One-dimensional blood flow equations were used to solve for blood flow and pressure in the generated vascular trees and Darcy flow equations were solved for blood perfusion in the tissues using a porous model assumption. Both equations are coupled at terminal segments explicitly. The proposed methods were applied to idealized models with different tree resolutions and metabolic demands for validation. The methods demonstrated that realistic synthetic trees were generated with significantly less computational expense compared to that of a constrained constructive optimization method. The methods were then applied to cerebrovascular arteries supplying a human brain and coronary arteries supplying the left and right ventricles to demonstrate the capabilities of the proposed methods. The proposed methods can be utilized to quantify tissue perfusion and predict areas prone to ischemia in patient-specific geometries.

     

Autoregulation of Brain Circulation in Severe Arterial Hypertension

Cerebral blood flow was studied by the arteriovenous oxygen difference method in patients with severe hypertension and in normotensive controls. The blood pressure was lowered to study the lower limit of autoregulation (the pressure below which cerebral blood flow decreases) and the pressure limit of brain hypoxia. Both limits were shifted upwards in the hypertensive patients, probably as a consequence of hypertrophy of the arteriolar walls. These findings have practical implications for antihypertensive therapy.When the blood pressure was raised some patients showed an upper limit of autoregulation beyond which an increase of cerebral blood flow above the resting value was seen without clinical symptoms. No evidence of vasospasm was found in any patient at high blood pressure. These observations may be of importance for the understanding of the pathogenesis of hypertensive encephalopathy.     

Enhanced myogenic tone in cerebral arteries from a rabbit model of subarachnoid hemorrhage

Cerebral artery vasospasm is a major cause of death and disability in patients experiencing subarachnoid hemorrhage (SAH). Currently, little is known regarding the impact of SAH on small diameter (100-200 microm) cerebral arteries, which play an important role in the autoregulation of cerebral blood flow. With the use of a rabbit SAH model and in vitro video microscopy, cerebral artery diameter was measured in response to elevations in intravascular pressure. Cerebral arteries from SAH animals constricted more (approximately twofold) to pressure within the physiological range of 60-100 mmHg compared with control or sham-operated animals. Pressure-induced constriction (myogenic tone) was also enhanced in arteries from control animals organ cultured in the presence of oxyhemoglobin, an effect independent of the vascular endothelium or nitric oxide synthesis. Finally, arteries from both control and SAH animals dilated as intravascular pressure was elevated above 140 mmHg. This study provides evidence for a role of oxyhemoglobin in impaired autoregulation (i.e., enhanced myogenic tone) in small diameter cerebral arteries during SAH. Furthermore, therapeutic strategies that improve clinical outcome in SAH patients (e.g., supraphysiological intravascular pressure) are effective in dilating small diameter cerebral arteries isolated from SAH animals.     

One-dimensional and three-dimensional models of cerebrovascular flow

The Circle of Willis is a ring-like structure of blood vessels found beneath the hypothalamus at the base of the brain. Its main function is to distribute oxygen-rich arterial blood to the cerebral mass. One-dimensional (1D) and three-dimensional (3D) computational fluid dynamics (CFD) models of the Circle of Willis have been created to provide a simulation tool which can potentially be used to identify at-risk cerebral arterial geometries and conditions and replicate clinical scenarios, such as occlusions in afferent arteries and absent circulus vessels. Both models capture cerebral haemodynamic autoregulation using a proportional-integral (PI) controller to modify efferent artery resistances to maintain optimal efferent flow rates for a given circle geometry and afferent blood pressure. The models can be used to identify at-risk cerebral arterial geometries and conditions prior to surgery or other clinical procedures. The 1D model is particularly relevant in this instance, with its fast solution time suitable for real-time clinical decisions. Results show the excellent correlation between models for the transient efferent flux profile. The assumption of strictly Poiseuille flow in the 1D model allows more flow through the geometrically extreme communicating arteries than the 3D model. This discrepancy was overcome by increasing the resistance to flow in the anterior communicating artery in the 1D model to better match the resistance seen in the 3D results.