An integrated geometric modelling framework for patient-specific computational haemodynamic study on wide-ranged vascular network

Patient-specific haemodynamic computations have been used as an effective tool in researches on cardiovascular disease associated with haemodynamics such as atherosclerosis and aneurysm. Recent development of computer resource has enabled 3D haemodynamic computations in wide-spread arterial network but there are still difficulties in modelling vascular geometry because of noise and limited resolution in medical images. In this paper, an integrated framework to model an arterial network tree for patient-specific computational haemodynamic study is developed. With this framework, 3D vascular geometry reconstruction of an arterial network and quantification of its geometric feature are aimed. The combination of 3D haemodynamic computation and vascular morphology quantification helps better understand the relationship between vascular morphology and haemodynamic force behind 'geometric risk factor' for cardiovascular diseases. The proposed method is applied to an intracranial arterial network to demonstrate its accuracy and effectiveness. The results are compared with the marching-cubes (MC) method. The comparison shows that the present modelling method can reconstruct a wide-ranged vascular network anatomically more accurate than the MC method, particularly in peripheral circulation where the image resolution is low in comparison to the vessel diameter, because of the recognition of an arterial network connectivity based on its centreline.     

Fluid–structure interaction in a fully coupled three-dimensional mitral–atrium–pulmonary model

This paper aims to investigate detailed mechanical interactions between the pulmonary haemodynamics and left heart function in pathophysiological situations (e.g. atrial fibrillation and acute mitral regurgitation). This is achieved by developing a complex computational framework for a coupled pulmonary circulation, left atrium and mitral valve model. The left atrium and mitral valve are modelled with physiologically realistic three-dimensional geometries, fibre-reinforced hyperelastic materials and fluid–structure interaction, and the pulmonary vessels are modelled as one-dimensional network ended with structured trees, with specified vessel geometries and wall material properties. This new coupled model reveals some interesting results which could be of diagnostic values. For example, the wave propagation through the pulmonary vasculature can lead to different arrival times for the second systolic flow wave (S2 wave) among the pulmonary veins, forming vortex rings inside the left atrium. In the case of acute mitral regurgitation, the left atrium experiences an increased energy dissipation and pressure elevation. The pulmonary veins can experience increased wave intensities, reversal flow during systole and increased early-diastolic flow wave (D wave), which in turn causes an additional flow wave across the mitral valve (L wave), as well as a reversal flow at the left atrial appendage orifice. In the case of atrial fibrillation, we show that the loss of active contraction is associated with a slower flow inside the left atrial appendage and disappearances of the late-diastole atrial reversal wave (AR wave) and the first systolic wave (S1 wave) in pulmonary veins. The haemodynamic changes along the pulmonary vessel trees on different scales from microscopic vessels to the main pulmonary artery can all be captured in this model. The work promises a potential in quantifying disease progression and medical treatments of various pulmonary diseases such as the pulmonary hypertension due to a left heart dysfunction.

     

A mathematical model of maternal vascular growth and remodeling and changes in maternal hemodynamics in uncomplicated pregnancy

The maternal vasculature undergoes tremendous growth and remodeling (G&R) that enables a?>?15-fold increase in blood flow through the uterine vasculature from conception to term. Hemodynamic metrics (e.g., uterine artery pulsatility index, UA-PI) are useful for the prognosis of pregnancy complications; however, improved characterization of the maternal hemodynamics is necessary to improve prognosis. The goal of this paper is to develop a mathematical framework to characterize maternal vascular G&R and hemodynamics in uncomplicated human pregnancies. A validated 1D model of the human vascular tree from the literature was adapted and inlet blood flow waveforms at the ascending aorta at 4 week increments from 0 to 40 weeks of gestation were prescribed. Peripheral resistances of each terminal vessel were adjusted to achieve target flow rates and mean arterial pressure at each gestational age. Vessel growth was governed by wall shear stress (and axial lengthening in uterine vessels), and changes in vessel distensibility were related to vessel growth. Uterine artery velocity waveforms generated from this model closely resembled ultrasound results from the literature. The literature UA-PI values changed significantly across gestation, increasing in the first month of gestation, then dramatically decreasing from 4 to 20 weeks. Our results captured well the time-course of vessel geometry, material properties, and UA-PI. This 1D fluid-G&R model captured the salient hemodynamic features across a broad range of clinical reports and across gestation for uncomplicated human pregnancy. While results capture available data well, this study highlights significant gaps in available data required to better understand vascular remodeling in pregnancy.

     

Skeletal muscle blood bed in growing rats

Morphometric study of vascular bifurcations and capillaries in biceps femoris artery basin was performed in 2-week old, 1-month old, and adult Wistar rats. Proceeding from the results, the blood tree was digitally reconstructed, and the main haemodynamic parameters in digital simulation were evaluated in this vascular bed: the blood flow velocity, wall shear stress, drop of perfusion pressure, and resistance. The comparison with a similar study of cerebral vascular bed revealed peculiarities of the vascular bed formation and intraorgan haemodynamics in these organs.     

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

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

     

Effects of Externally Applied Pressure on the Haemodynamics of the Lower Limb

The effects of externally applied pressure of 5-150 mm Hg on the haemodynamics of the leg of dog and man were investigated. The criteria used for the assessments included femoral arterial and venous blood flow as well as vascular hydraulic conductance. The results indicated that external pressure of 5 mm Hg results in a very small non-significant increase in the femoral arterial and venous flow. Higher external pressure of 15 mm Hg or more significantly reduces the femoral arterial and venous flows as well as the vascular conductance. It therefore seems that compression produced by bandaging in horizontal supine subjects has little or no haemodynamic value and may prove to be harmful unless carefully controlled.     

Investigation of material modeling in fluid–structure interaction analysis of an idealized three-layered abdominal aorta: aneurysm initiation and fully developed aneurysms

Different material models for an idealized three-layered abdominal aorta are compared using computational techniques to study aneurysm initiation and fully developed aneurysms. The computational model includes fluid–structure interaction (FSI) between the blood vessel and the blood. In order to model aneurysm initiation, the medial region was degenerated to mimic the medial loss occurring in the inception of an aneurysm. Various cases are considered in order to understand their effects on the initiation of an abdominal aortic aneurysm. The layers of the blood vessel were modeled using either linear elastic materials or Mooney–Rivlin (otherwise known as hyperelastic) type materials. The degenerated medial region was also modeled in either linear elastic or hyperelastic-type materials and assumed to be in the shape of an arc with a thin width or a circular ring with different widths. The blood viscosity effect was also considered in the initiation mechanism. In addition, dynamic analysis of the blood vessel was performed without interaction with the blood flow by applying time-dependent pressure inside the lumen in a three-layered abdominal aorta. The stresses, strains, and displacements were compared for a healthy aorta, an initiated aneurysm and a fully developed aneurysm. The study shows that the material modeling of the vessel has a sizable effect on aneurysm initiation and fully developed aneurysms. Different material modeling of degeneration regions also affects the stress–strain response of aneurysm initiation. Additionally, the structural analysis without considering FSI (called noFSI) overestimates the peak von Mises stress by 52% at the interfaces of the layers.     

Effects of stent stiffness on local haemodynamics with particular reference to wave reflections

The placement of a rigid stent within an elastic vessel produces wave reflection sites at the entrance to and exit from the stent. The net haemodynamic effects of these reflections depend critically on the degree of stiffness of the stent and on its length and position within the diseased vessel, variables that have been found to affect the clinical performance of a stent. Here these effects are examined analytically, using a segmented tube model. The results indicate that the presence of the stent within the larger diseased vessel has the effect of producing higher pressure at the vessel entrance than that at exit. This pressure difference, when superimposed on the underlying pressure distribution within the vessel, has the net effect of actually aiding rather than impeding the flow, but the extent of this depends on the length and position of the stent. A short stent placed near the entrance of the diseased vessel may be favoured clinically for producing the least perturbation in the underlying haemodynamics and thus reducing the chance of restenosis, while a long stent placed near the exit may be favoured for producing a positive pressure difference and thus aiding the flow.     

A problem in the mathematical biophysics of blood circulation: II. Relation between pressure and flow of a viscous fluid in an elastic distensible tube

In an elastic distensible tube, like a blood vessel, the radius is determined by the equality of the hydrostatic pressure and the elastic forces. If a viscous fluid flows through such a tube, there is a pressure drop along the line of flow. This results in a variation of the radius of the tube along the axis. An approximate expression, valid within a limited range of values, is derived for the radius of the tube as a function of the distance along the axis. Another approximate expression is derived for the relation between pressure drop and total flow in such a case. For sufficiently high rates of flow the pressure drop does not vary linearly with the flow, as in the usual poiseuille's law, but more rapidly.     

An optimization principle for vascular radius including the effects of smooth muscle tone.

An optimization principle is proposed for the regulation of vascular morphology. This principle, which extends Murray's law, is based on the hypothesis that blood vessel diameter is controlled by a mechanism that minimizes the total energy required to drive the blood flow, to maintain the blood supply, and to support smooth muscle tone. A theoretical analysis reveals that the proposed principle predicts that the optimum shear stress on the vessel wall due to blood flow increases with blood pressure. This result agrees qualitatively with published findings that the fluid shear stress in veins is significantly smaller than it is in arteries.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Computer simulation of blood flow in the human arm

This paper considers a finite element method to characterize blood flow in the human arm arteries. A set of different pressure waveforms, which represent normal and diseased heart pulses, is used for the proximal boundary conditions, and a modified Windkessel model is used for the distal arterial boundary conditions. A comparison of the distal pressure and flow waveforms, for each different proximal pressure, is made to determine whether such waveforms are significantly altered from normal waveforms. The results show that the distal pressure and/or flow waveforms in certain cases are sufficiently different to be possibly used as a diagnostic indicator of an abnormal heart condition. Also considered is the effect of stenosis, change of compliance, and dilatation of the distal beds on the pressure and flow waveforms. A stenosis which has an area reduction of greater than approximately 75% is found to significantly alter both the distal pressure and flow waveforms. Changes in arterial compliance, however, do not strongly influence the waveforms. Dilatation of distal vascular beds is simulated by reducing the lumped resistance of these beds, and this reduction increases mean flow and decreases mean distal pressure, but has little effect on the basic shape of either the pressure or flow waveform.     

Mathematical Modelling of a Brain Tumour Initiation and Early Development: A Coupled Model of Glioblastoma Growth,Pre-Existing Vessel Co-Option,Angiogenesis and Blood Perfusion

We propose a coupled mathematical modelling system to investigate glioblastoma growth in response to dynamic changes in chemical and haemodynamic microenvironments caused by pre-existing vessel co-option, remodelling, collapse and angiogenesis. A typical tree-like architecture network with different orders for vessel diameter is designed to model pre-existing vasculature in host tissue. The chemical substances including oxygen, vascular endothelial growth factor, extra-cellular matrix and matrix degradation enzymes are calculated based on the haemodynamic environment which is obtained by coupled modelling of intravascular blood flow with interstitial fluid flow. The haemodynamic changes, including vessel diameter and permeability, are introduced to reflect a series of pathological characteristics of abnormal tumour vessels including vessel dilation, leakage, angiogenesis, regression and collapse. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. The simulation focuses on the avascular phase of tumour development and stops at an early phase of angiogenesis. The model is able to demonstrate the main features of glioblastoma growth in this phase such as the formation of pseudopalisades, cell migration along the host vessels, the pre-existing vasculature co-option, angiogenesis and remodelling. The model also enables us to examine the influence of initial conditions and local environment on the early phase of glioblastoma growth.     

Three-dimensional flows in a hyperelastic vessel under external pressure

We study the collapsible behaviour of a vessel conveying viscous flows subject to external pressure, a scenario that could occur in many physiological applications. The vessel is modelled as a three-dimensional cylindrical tube of nonlinear hyperelastic material. To solve the fully coupled fluid–structure interaction, we have developed a novel approach based on the Arbitrary Lagrangian–Eulerian (ALE) method and the frontal solver. The method of rotating spines is used to enable an automatic mesh adaptation. The numerical code is verified extensively with published results and those obtained using the commercial packages in simpler cases, e.g. ANSYS for the structure with the prescribed flow, and FLUENT for the fluid flow with prescribed structure deformation. We examine three different hyperelastic material models for the tube for the first time in this context and show that at the small strain, all three material models give similar results. However, for the large strain, results differ depending on the material model used. We further study the behaviour of the tube under a mode-3 buckling and reveal its complex flow patterns under various external pressures. To understand these flow patterns, we show how energy dissipation is associated with the boundary layers created at the narrowest collapsed section of the tube, and how the transverse flow forms a virtual sink to feed a strong axial jet. We found that the energy dissipation associated with the recirculation does not coincide with the flow separation zone itself, but overlaps with the streamlines that divide the three recirculation zones. Finally, we examine the bifurcation diagrams for both mode-3 and mode-2 collapses and reveal that multiple solutions exist for a range of the Reynolds number. Our work is a step towards modelling more realistic physiological flows in collapsible arteries and veins.     

A data-driven model to study utero-ovarian blood flow physiology during pregnancy

In this paper, we describe a mathematical model of the cardiovascular system in human pregnancy. An automated, closed-loop 1D–0D modelling framework was developed, and we demonstrate its efficacy in (1) reproducing measured multi-variate cardiovascular variables (pulse pressure, total peripheral resistance and cardiac output) and (2) providing automated estimates of variables that have not been measured (uterine arterial and venous blood flow, pulse wave velocity, pulsatility index). This is the first model capable of estimating volumetric blood flow to the uterus via the utero-ovarian communicating arteries. It is also the first model capable of capturing wave propagation phenomena in the utero-ovarian circulation, which are important for the accurate estimation of arterial stiffness in contemporary obstetric practice. The model will provide a basis for future studies aiming to elucidate the physiological mechanisms underlying the dynamic properties (changing shapes) of vascular flow waveforms that are observed with advancing gestation. This in turn will facilitate the development of methods for the earlier detection of pathologies that have an influence on vascular structure and behaviour.

     

Numerical simulation of blood and interstitial flow through a solid tumor

A theoretical framework is presented for describing blood flow through the irregular vasculature of a solid tumor. The tumor capillary bed is modeled as a capillary tree of bifurcating segments whose geometrical construction involves deterministic and random parameters. Blood flow along the individual capillaries accounts for plasma leakage through the capillary walls due to the transmural pressure according to Sterling’s law. The extravasation flow into the interstitium is described by Darcy’s law for a biological porous medium. The pressure field developing in the interstitium is computed by solving Laplace’s equation subject to derived boundary conditions at the capillary vessel walls. Given the arterial, venous, and tumor surface pressures, the problem is formulated as a coupled system of integral and differential equations arising from the interstitium and capillary flow transport equations. Numerical discretization yields a system of linear algebraic equations for the interstitial and capillary segment pressures whose solution is found by iterative methods. Results of numerical computations document the effect of the interstitial hydraulic and vascular permeability on the fractional plasma leakage. Given the material properties, the fractional leakage reaches a maximum at a particular grade of the bifurcating vascular tree.     

Differential regional haemodynamic effects of the non-peptide angiotensin II antagonist, DuP 753, in water-replete and water-deprived Brattleboro rats

Regional haemodynamic effects of DuP 753 were assessed in conscious unrestrained Brattleboro (i.e. vasopressin-deficient) rats, chronically instrumented with miniaturised pulsed Doppler flow probes and intravascular catheters. Responses to DuP 753 were assessed in water-replete animals and in animals following 14h water deprivation to render cardiovascular status dependent on the renin angiotensin system (RAS). In water-replete animals DuP 753 had little effect on mean arterial blood pressure, but there were dose-dependent tachycardias and increases in renal blood flow, associated with vasodilatations. There were also increases in mesenteric blood flow and vascular conductance, but these were not clearly dose-related; DuP 753 had no significant effects on hindquarters haemodynamics. In water-deprived animals DuP 753 caused dose-dependent hypotension, tachycardia, and renal and mesenteric vasodilatations; hindquarters vasodilatation was not dose-dependent. In both conditions, in the presence of DuP 753 (10 mg/kg), captopril had little additional effect. DuP 753 appears to be an extremely effective tool for assessing the role of the RAS in cardiovascular regulation.     

Mathematical modeling of the response of a resistive vessel to pressure

The response of small arterial vessels to internal pressure makes an essential contribution to autoregulation in the vascular bed. It is believed that free cytosolic Ca²⁺ concentration plays a pivotal role in the regulation of smooth muscle contractility and hence of the vascular lumen. A simple mathematical model of blood flow in a resistive vessel is suggested. The model is based on the experimental data obtained for cerebral arteries, but may be used for any other resistive vessel. The model not only describes the regulation of the vascular lumen by transmural pressure but also shows realistic behavior of the vessel radius and cytosolic [Ca²⁺] at different rates of pressure change. Possible variations in the radius along the vessel due to the Bayliss effect are considered.     

Numerical analysis of dynamic temperature in response to different levels of reactive hyperaemia in a three-dimensional image-based hand model

Vascular reactivity (VR) is considered as an effective index to predict the risk of cardiovascular events. A cost-effective alternative technique used to evaluate VR called digital thermal monitoring (DTM) is based on the response of finger temperature to vessel occlusion and reperfusion. In this work, a simulation has been developed to investigate hand temperature in response to vessel occlusion and perfusion. The simulation consists of image-based mesh generation and finite element analysis of blood flow and heat transfer in tissues. In order to reconstruct a real geometric model of human hand, a computer programme including automatic image processing for sequential MR data and mesh generation based on the transfinite interpolation method is developed. In the finite element analysis part, blood flow perfused in solid tissues is considered as fluid phase through porous media. Heat transfer in tissues is described by Pennes bioheat equation and blood perfusion rate is obtained from Darcy velocities. Capillary pressure, blood perfusion and temperature distribution of hand are obtained. The results reveal that fingertip temperature is strongly dependent on larger arterial pressure. This simulation is of potential to quantify the indices used for evaluating the VR in DTM test if it is integrated with the haemodynamic model of blood circulation in upper limb.