Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures

This work was motivated by the problems of analysing detailed 3D models of vascular districts with complex anatomy. It suggests an approach to prescribing realistic boundary conditions to use in order to obtain information on local as well as global haemodynamics. A method was developed which simultaneously solves Navier-Stokes equations for local information and a non-linear system of ordinary differential equations for global information. This is based on the principle that an anatomically detailed 3D model of a cardiovascular district can be achieved by using the finite element method. In turn the finite element method requires a specific boundary condition set. The approach outlined in this work is to include the system of ordinary differential equations in the boundary condition set. Such a multiscale approach was first applied to two controls: (i) a 3D model of a straight tube in a simple hydraulic network and (ii) a 3D model of a straight coronary vessel in a lumped-parameter model of the cardiovascular system. The results obtained are very close to the solutions available for the pipe geometry. This paper also presents preliminary results from the application of the methodology to a particular haemodynamic problem: namely the fluid dynamics of a systemic-to-pulmonary shunt in paediatric cardiac surgery.     

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.

     

A new approach to blood flow simulation in vascular networks

A proper analysis of blood flow is contingent upon accurate modelling of the branching pattern and vascular geometry of the network of interest. It is challenging to reconstruct the entire vascular network of any organ experimentally, in particular the pulmonary vasculature, because of its very high number of vessels, complexity of the branching pattern and poor accessibility in vivo. The objective of our research is to develop an innovative approach for the reconstruction of the full pulmonary vascular tree from available morphometric data. Our method consists of the use of morphometric data on those parts of the pulmonary vascular tree that are too small to reconstruct by medical imaging methods. This method is a three-step technique that reconstructs the entire pulmonary arterial tree down to the capillary bed. Vessels greater than 2 mm are reconstructed from direct volume and surface analysis using contrast-enhanced computed tomography. Vessels smaller than 2 mm are reconstructed from available morphometric and distensibility data and rearranged by applying Murray's laws. Implementation of morphometric data to reconstruct the branching pattern and applying Murray's laws to every vessel bifurcation simultaneously leads to an accurate vascular tree reconstruction. The reconstruction algorithm generates full arterial tree topography down to the ?rst capillary bifurcation. Geometry of each order of the vascular tree is generated separately to minimize the construction and simulation time. The node-to-node connectivity along with the diameter and length of every vessel segment is established and order numbers, according to the diameter-de?ned Strahler system, are assigned. In conclusion, the present model provides a morphological foundation for future analysis of blood flow in the pulmonary circulation     

Automated Quantification and Integrative Analysis of 2D and 3D Mitochondrial Shape and Network Properties

Mitochondrial morphology and function are coupled in healthy cells, during pathological conditions and (adaptation to) endogenous and exogenous stress. In this sense mitochondrial shape can range from small globular compartments to complex filamentous networks, even within the same cell. Understanding how mitochondrial morphological changes (i.e. “mitochondrial dynamics”) are linked to cellular (patho) physiology is currently the subject of intense study and requires detailed quantitative information. During the last decade, various computational approaches have been developed for automated 2-dimensional (2D) analysis of mitochondrial morphology and number in microscopy images. Although these strategies are well suited for analysis of adhering cells with a flat morphology they are not applicable for thicker cells, which require a three-dimensional (3D) image acquisition and analysis procedure. Here we developed and validated an automated image analysis algorithm allowing simultaneous 3D quantification of mitochondrial morphology and network properties in human endothelial cells (HUVECs). Cells expressing a mitochondria-targeted green fluorescence protein (mitoGFP) were visualized by 3D confocal microscopy and mitochondrial morphology was quantified using both the established 2D method and the new 3D strategy. We demonstrate that both analyses can be used to characterize and discriminate between various mitochondrial morphologies and network properties. However, the results from 2D and 3D analysis were not equivalent when filamentous mitochondria in normal HUVECs were compared with circular/spherical mitochondria in metabolically stressed HUVECs treated with rotenone (ROT). 2D quantification suggested that metabolic stress induced mitochondrial fragmentation and loss of biomass. In contrast, 3D analysis revealed that the mitochondrial network structure was dissolved without affecting the amount and size of the organelles. Thus, our results demonstrate that 3D imaging and quantification are crucial for proper understanding of mitochondrial shape and topology in non-flat cells. In summary, we here present an integrative method for unbiased 3D quantification of mitochondrial shape and network properties in mammalian cells.     

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

     

Fractal dimension of the middle meningeal vessels: variation and evolution in Homo erectus, Neanderthals, and modern humans

The middle meningeal vascular network leaves its traces on the endocranial surface because of the tight relationship between neurocranial development and brain growth. Analysing the endocast of fossil specimens, it is therefore possible to describe the morphology of these structures, leading inferences on the cerebral physiology and metabolism in extinct human groups. In this paper, general features of the meningeal vascular traces are described for specimens included in the Homo erectus, Homo neanderthalensis, and Homo sapiens hypodigms. The complexity of the arterial network is quantified by its fractal dimension, calculated through the box-counting method. Modern humans show significant differences from the other two taxa because of the anterior vascular dominance and the larger fractal dimension. Neither the fractal dimension nor the anterior development are merely associated with cranial size increase. Considering the differences between Neanderthals and modern humans, these results may be interpreted in terms of phylogeny, cerebral functions, or cranial structural network.     

Early changes in vascular dynamics in relation to twin-twin transfusion syndrome.

A clearer understanding of the early determinants of normal and abnormal vascular development is pivotal in order to identify those at increased risk of later vascular disease, and perhaps to prevent it by early intervention. Measurement of pulse wave velocity(PWV) has been used in the postnatal evaluation of the monochorionic(MC) twins. They are genetically identical and those with twin-twin transfusion syndrome(TTTS) provide an ideal natural model in whom to study the influence of differing haemodynamic stresses on the developing vascular tree. We investigated firstly whether surviving twin pairs with TTTS have altered arterial distensibility in childhood by comparing PWV in the radial arteries of surviving MC twin pairs with TTTS and in two control groups, one cohort of MC twins without TTTS and another dichorionic group (DC) Secondly, we tested a cohort of TTTS twin pair survivors treated with laser photocoagulation. The co-twin pairs in the group managed palliatively with amnioreduction showed increased PWV in the donor and reduced PWV in the recipient twins. This was neither seen in the laser-treated, nor in the control groups. Our studies suggest that a period of haemodynamic imbalance gives rise to changes in a muscular conduit artery that persist at least into infancy and it seems that by correcting the abnormal haemodynamics relatively soon after the disease process had begun, the alterations in elasticity are prevented. These studies are the first to demonstrate fetal programming of the vascular bed in humans, and prevention or reversal of this programming by an intervention in mid-gestation.     

Nanoparticle transport and delivery in a heterogeneous pulmonary vasculature

Quantitative understanding of nanoparticles delivery in a complex vascular networks is very challenging because it involves interplay of transport, hydrodynamic force, and multivalent interactions across different scales. Heterogeneous pulmonary network includes up to 16 generations of vessels in its arterial tree. Modeling the complete pulmonary vascular system in 3D is computationally unrealistic. To save computational cost, a model reconstructed from MRI scanned images is cut into an arbitrary pathway consisting of the upper 4-generations. The remaining generations are represented by an artificially rebuilt pathway. Physiological data such as branch information and connectivity matrix are used for geometry reconstruction. A lumped model is used to model the flow resistance of the branches that are cut off from the truncated pathway. Moreover, since the nanoparticle binding process is stochastic in nature, a binding probability function is used to simplify the carrier attachment and detachment processes. The stitched realistic and artificial geometries coupled with the lumped model at the unresolved outlets are used to resolve the flow field within the truncated arterial tree. Then, the biodistribution of 200 nm, 700 nm and 2 µm particles at different vessel generations is studied. At the end, 0.2–0.5% nanocarrier deposition is predicted during one time passage of drug carriers through pulmonary vascular tree. Our truncated approach enabled us to efficiently model hemodynamics and accordingly particle distribution in a complex 3D vasculature providing a simple, yet efficient predictive tool to study drug delivery at organ level.     

Quantification of the non-planarity of the human carotid bifurcation

The carotid bifurcation has been a region of particular interest due to its predilection for clinically significant atherosclerosis. It has been shown that the vessel geometry is a major determinant of the local haemodynamic properties which are believed to be associated with the location of atherosclerotic lesions. Current knowledge of the geometry of the carotid bifurcation is insufficient and restricted to basic geometric parameters. To provide some means of quantifying the degree of complexity of the 3D shape of the bifurcation, we made an initial attempt by evaluating the non-planarity of an arterial bifurcation based upon the singular value decomposition theorem.In this paper we present our results obtained on the right carotid bifurcations of six normal subjects, each of whom was scanned twice using the 2D time-of-flight MR sequence. The acquired 2D cross sectional images were processed by using our in-house software which comprises 2D segmentation, 3D reconstruction and smoothing. The centroids of each transverse slices were determined and used as input data for the non-planarity analysis. Our results using the singular value decomposition method have demonstrated discernible differences in non-planarity among individuals. Comparisons with the planarity definition proposed by other investigators suggest that the singular value decomposition method offers more information about the linearity and planarity of the bifurcation. However, it is also realised that a single measure of non-planarity can never fully characterise a bifurcation owing to the great variety of geometries.     

Automated reconstruction of three-dimensional neuronal morphology from laser scanning microscopy images

Experimental and theoretical studies demonstrate that both global dendritic branching topology and fine spine geometry are crucial determinants of neuronal function, its plasticity and pathology. Importantly, simulation studies indicate that the interaction between local and global morphologic properties is pivotal in determining dendritic information processing and the induction of synapse-specific plasticity. The ability to reconstruct and quantify dendritic processes at high resolution is therefore an essential prerequisite to understanding the structural determinants of neuronal function. Existing methods of digitizing 3D neuronal structure use interactive manual computer tracing from 2D microscopy images. This method is time-consuming, subjective and lacks precision. In particular, fine details of dendritic varicosities, continuous dendritic taper, and spine morphology cannot be captured by these systems. We describe a technique for automated reconstruction of 3D neuronal morphology from multiple stacks of tiled confocal and multiphoton laser scanning microscopy (CLSM and MPLSM) images. The system is capable of representing both global and local structural variations, including gross dendritic branching topology, dendritic varicosities, and fine spine morphology with sufficient resolution for accurate 3D morphometric analyses and realistic biophysical compartment modeling. Our system provides a much needed tool for automated digitization and reconstruction of 3D neuronal morphology that reliably captures detail on spatial scales spanning several orders of magnitude, that avoids the subjective errors that arise during manual tracing with existing digitization systems, and that runs on a standard desktop workstation.     

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.     

Accuracy and reproducibility of CFD predicted wall shear stress using 3D ultrasound images

Computational fluid dynamics (CFD) flow simulation techniques have the potential to enhance our understanding of how haemodynamic factors are involved in atherosclerosis. Recently, 3D ultrasound has emerged as an alternative to other 3D imaging techniques, such as magnetic resonance angiography (MRA). The method can be used to generate realistic vascular geometry suitable for CFD simulations. In order to assess accuracy and reproducibility of the procedure from image acquisition to reconstruction to CFD simulation, a human carotid artery bifurcation phantom was scanned three times using 3D ultrasound. The geometry was reconstructed and flow simulations were carried out on the three sets as well as on a model generated using computer aided design (CAD) from the geometric information given by the manufacturer. It was found that the three reconstructed sets showed good reproducibility as well as satisfactory quantitative agreement with the CAD model. Analyzing two selected locations probably representing the 'worst cases,' accuracy comparing ultrasound and CAD reconstructed models was estimated to be between 7.2% and 7.7% of the maximum instantaneous WSS and reproducibility comparing the three scans to be between 8.2% and 10.7% of their average maximum.     

Review of Zero-D and 1-D Models of Blood Flow in the Cardiovascular System

Background  

Zero-dimensional (lumped parameter) and one dimensional models, based on simplified representations of the components of the cardiovascular system, can contribute strongly to our understanding of circulatory physiology. Zero-D models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst one-D (distributed parameter) models add the facility to represent efficiently the effects of pulse wave transmission in the arterial network at greatly reduced computational expense compared to higher dimensional computational fluid dynamics studies. There is extensive literature on both types of models.     

Blood flow multiscale phenomena

The cardiovascular disease is one of most frequent cause deaths in modern society. The objective of this work is analyse the effect of dynamic vascular geometry (curvature, torsion, bifurcation) and pulsatile blood nature on secondary flow, wall shear stress and platelet deposition. The problem was examined as multi-scale physical phenomena using perturbation analysis and numerical modelling. The secondary flow determined as influence pulsatile pressure, vascular tube time-dependent bending and torsion on the main axial flow. Bifurcation and branching phenomena are analysed experimentally through, blood-like fluid pulsatile flow across elastic rubber-like Y-model model. The problem complex geometry near branching in platelet deposit modelling is resolved numerically as Falker-Skan flow.     

Computational simulation of flow-induced arterial remodeling of the pancreaticoduodenal arcade associated with celiac artery stenosis

Arterial remodeling of the pancreaticoduodenal arcade, which enables collateral flow to the liver, spleen, and stomach, is a well-recognized clinical sign of celiac artery (CA) stenosis. However, the hemodynamic changes due to remodeling are poorly understood, despite their importance in surgical procedures such as pancreaticoduodenectomy. In this study, a framework to simulate remodeling of the arterial network following pathological flow alterations was developed and applied to investigate the hemodynamic characteristics of patients with CA stenosis. A one-dimensional–zero-dimensional cardiovascular model was used for blood flow simulation. After introducing CA stenosis into the normal network, arterial remodeling was simulated by iteratively changing the diameter of each artery until time-averaged wall shear stress reached its value under normal conditions. A representative case was simulated to validate the present framework, followed by simulation cases to investigate the impact of stenosis severity on remodeling outcome. A markedly dilated arcade was observed whose diameter agreed well with the corresponding values measured in subjects with CA stenosis, confirming the ability of the framework to predict arterial remodeling. A series of simulations clarified how the geometry and hemodynamics after remodeling change with stenosis severity. In particular, the arterial remodeling and resulting blood flow redistribution were found to maintain adequate organ blood supply regardless of stenosis severity. Furthermore, it was suggested that flow conditions in patients with CA stenosis could be estimated from geometric factors, namely, stenosis severity and arcade diameter, which can be preoperatively and non-invasively measured using diagnostic medical images.     

Lumped Parameter and Feedback Control Models of the Auto-regulatory Response in the Circle of Willis

The Circle of Willis (CoW) 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. A 1-dimensional model of the CoW has been created to simulate a series of possible clinical scenarios such as occlusions in afferent arteries, absent or string-like circulus vessels, or arterial infarctions. The model captures cerebral haemodynamic auto-regulation by using a proportional-integral-derivative (PID) controller to modify efferent resistances and maintain optimal efferent flowrates for a given circle geometry and afferent blood pressure. Results match limited clinical data and results obtained in prior studies to within 6%. In addition, a set of boundary conditions and geometry is presented for which the auto-regulated system cannot provide the necessary efferent flowrates and perfusion, representing a condition with increased risk of stroke and highlighting the importance of modelling the haemodynamics of the CoW. The system model created is computationally simple so it can be used to identify at-risk cerebral arterial geometries and conditions prior to surgery or other clinical procedures.     

Compound Ex Vivo and In Silico Method for Hemodynamic Analysis of Stented Arteries

Hemodynamic factors such as low wall shear stress have been shown to influence endothelial healing and atherogenesis in stent-free vessels. However, in stented vessels, a reliable quantitative analysis of such relations has not been possible due to the lack of a suitable method for the accurate acquisition of blood flow. The objective of this work was to develop a method for the precise reconstruction of hemodynamics and quantification of wall shear stress in stented vessels. We have developed such a method that can be applied to vessels stented in or ex vivo and processed ex vivo. Here we stented the coronary arteries of ex vivo porcine hearts, performed vascular corrosion casting, acquired the vessel geometry using micro-computed tomography and reconstructed blood flow and shear stress using computational fluid dynamics. The method yields accurate local flow information through anatomic fidelity, capturing in detail the stent geometry, arterial tissue prolapse, radial and axial arterial deformation as well as strut malapposition. This novel compound method may serve as a unique tool for spatially resolved analysis of the relationship between hemodynamic factors and vascular biology. It can further be employed to optimize stent design and stenting strategies.