An implicit solver for 1D arterial network models

In this study the one dimensional blood flow equations are solved using a newly proposed enhanced trapezoidal rule method ETM, which is an extension to the simplified trapezoidal rule method STM. At vessel junctions the conservation of mass and conservation of total pressure are held as system constraints using Lagrange multipliers that can be physically interpreted as external flow rates. The ETM scheme is compared with published arterial network benchmark problems and a dam break problem. Strengths of the ETM scheme include being simple to implement, intuitive connection to lumped parameter models, and no restrictive stability criteria such as the CFL number. The ETM scheme does not require the use of characteristics at vessel junctions, or for inlet and outlet boundary conditions. The ETM forms an implicit system of equations which requires only one global solve per time step for pressure, followed by flow rate update on the elemental system of equations, thus no iterations are required per time step. Consistent results are found for all benchmark cases and for a 56 vessel arterial network problem it gives very satisfactory solutions at a spatial and time discretisation that results in a maximum CFL of 3, taking 4.44 seconds per cardiac cycle. By increasing the time step and element size to produce a maximum CFL number of 15 the method takes only 0.39 seconds per cardiac cycle with only a small compromise on accuracy

A Computational Pipeline to Investigate Longitudinal Blood Flow Changes in the Circle of Willis of Patients with Stable and Growing Aneurysms

Changes in cerebral blood flow are often associated with the initiation and development of different life-threatening medical conditions including aneurysm rupture and ischemic stroke. Nevertheless, it is not fully clear how haemodynamic changes in time across the Circle of Willis (CoW) are related with intracranial aneurysm (IA) growth. In this work, we introduced a novel reduced-order modelling strategy for the systematic quantification of longitudinal blood flow changes across the whole CoW in patients with stable and unstable/growing aneurysm. Magnetic Resonance Angiography (MRA) images were converted into one-dimensional (1-D) vessel networks through a semi-automated procedure, with a level of geometric reconstruction accuracy controlled by user-dependent parameters. The proposed pipeline was used to systematically analyse longitudinal haemodynamic changes in seven different clinical cases. Our preliminary simulation results indicate that growing aneurysms are not necessarily associated with significant changes in mean flow over time. A concise sensitivity analysis also shed light on which modelling aspects need to be further characterized in order to have reliable patient-specific predictions. This study poses the basis for investigating how time-dependent changes in the vasculature affect the haemodynamics across the whole CoW in patients with stable and growing aneurysms

Data-driven modelling of the FRC network for studying the fluid flow in the conduit system

The human immune system is characterized by enormous cellular and anatomical complexity. Lymph nodes are key centers of immune reactivity, organized into distinct structural and functional modules including the T-cell zone, fibroblastic reticular cell (FRC) network and the conduit system. A thorough understanding of the modular organization is a prerequisite for lymphoid organ tissue-engineering. Due to the biological complexity of lymphoid organs, the development of mathematical models capable of elaborating the lymph node architecture and functional organization, has remained a major challenge in computational biology. Here, we present a computational method to model the geometry of the FRC network and fluid flow in the conduit system. It differs from the blood vascular network image-based reconstruction approaches as it develops the parameterized geometric model using the real statistics of the node degree and the edge length distributions. The FRC network model is then used to analyze the fluid flow through the underlying conduit system. A first observation is that the pressure gradient is approximately linear, which suggests homogeneity of the network. Furthermore, calculated permeability values View the MathML source show the generated network is isotropic, while investigating random variations of pipe radii (with a given mean and standard deviation) shows a significant effect on the permeability. This framework can now be further explored to systematically correlate fundamental characteristics of the FRC conduit system to more global material properties such as permeability

Formulation of Generalized Mass Transfer Correlations for Blood Oxygenator Design

This paper numerically investigates non-Newtonian blood flow with oxygen and carbon dioxide transport across and along an array of uniformly square and staggered arranged fibres at various porosity (e) levels, focussing on a low Reynolds number regime (Re < 10). The objective is to establish suitable mass transfer correlations, expressed in the form of Sherwood number (Sh = f (e,Re,Sc)), that identifies the link from local mass transfer investigations to full-device analyses. The development of a concentration field is initially investigated and expressions are established covering the range from a typical deoxygenated condition up to a full oxygenated condition. An important step is identified where a cut-off point in those expressions is required to avoid any under- or over-estimation on the Sherwood number. Geometrical features of a typical commercial blood oxygenator is adopted and results in general show that a balance in pressure drop, shear stress and mass transfer is required to avoid potential blood trauma or clotting formation. Different definitions of mass transfer correlations are found for oxygen/carbon dioxide, parallel/transverse flow and square/staggered configurations, respectively. From this set of correlations, it is found that transverse flow has better gas transfer than parallel flow which is consistent with reported literature. The mass transfer dependency on fibre configuration is observed to be pronounced at low porosity. This approach provides an initial platform when one is looking to improve the mass transfer performance in a blood oxygenator without the need to conduct any numerical simulations or experiments

Computational investigation of the Laplace law in compression therapy

This study aims to use computational methods for elucidating the effect of limb shape on subgarment and subcutaneous pressures, stresses and strains. A framework was built that generates computational models from 3D arm scans using a depth sensing camera. Finite Element Analysis (FEA) was performed on the scans taken from 23 lymphoedema patients. Subgarment pressures were calculated based on local curvature for each patient and showed a large variability of pressure across each arm. Across the cohort an average maximum subgarment pressure of 5100 Pa was found as opposed to an intended garment pressure of 2500 Pa. Subcutaneous results show that stresses/strains in the adipose tissues more closely follow the subgarment pressures than in the stiffer skin tissues. Another novel finding was that a negative axial gradient in subgarment pressure (from wrist to elbow) consistently led to positive axial gradients for the Von Mises stresses in the adipose tissues; a phenomenon caused by a combination of arm shape and the stiffness ratio between skin and adipose tissues. In conclusion, this work fills a knowledge gap in compression therapy in clinical practice and can inform garment design or lead to optimal treatment strategies

Investigation of Shape with Patients Suffering from Unilateral Lymphoedema

This study investigates the use of a 3D depth sensing camera for analysing the shape of lymphoedematous arms, and seeks to identify suitable metrics for monitoring lymphoedema clinically. A fast, simple protocol was developed for scanning upper limb lymphoedema, after which a robust data pre- and post-processing framework was built that consistently and quickly identifies arm shape and volume. The framework was then tested on 24 patients with mild unilateral lymphoedema, who were also assessed using tape measurements. The scanning protocol developed led to scanning times of about 20–30 s. Shape related metrics such as circumference and circularity were used to distinguish between affected and healthy arms (p ≤ 0.05). Swelling maps were also derived to identify the distribution of oedema on arms. Topology and shape could be used to monitor or even diagnose lymphoedema using the provided framework. Such metrics provide more detailed information to a lymphoedema specialist than solely volume. Although tested on a small cohort, these results show promise for further research into better diagnostics of lymphoedema and for future adoption of the proposed methods across lymphoedema clinics

A comparative study of fractional step method in its quasi-implicit, semi-implicit and fully-explicit forms for incompressible flows

The present review describes and analyses a class of finite element fractional step methodsfor solving the incompressible Navier-Stokes equations. Our objective is not to reproduce the extensivecontributions on the subject, but to report on our long-term experience with and provide a unified overviewof a particular approach: the characteristic based split method. Three procedures, the semi-implicit, quasi-implicit and fully-explicit, are studied and compared. This work provides a thorough assessment of theaccuracy and efficiency of these schemes, both for a first and second order pressure split. In transientproblems, the quasi-implicit form significantly outperforms the fully-explicit approach. The second order(pressure) fractional step method displays significant convergence and accuracy benefits when the quasi-implicit projection method is employed. The fully-explicit method, utilising artificial compressibility and apseudo time stepping procedure, requires no second order fractional split to achieve second order or higheraccuracy. While the fully-explicit form is efficient for steady state problems, due to its ability to handle localtime stepping, the quasi-implicit is the best choice for transient flow calculations with time independent boundary conditions. The semi-implicit form, with its stability restrictions, is the least favoured of all the three forms for incompressible flow calculations

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