Mathematical analysis of blood flow model through channels with flexible walls

A simplified mathematical model of blood flow through flexible arteries is developed and analyzed. The resulting system of non-linear, non-homogeneous PDE\u27s is analyzed numerically using the Richtmyer Lax-Wendroff method. Numerical and theoretical results show excellent agreement suggesting that in physiologically relevant situations shocks only develop outside the domain of interest. These results suggest that when the model assumptions are satisfied the model provides sufficient regularity to yield a physically reasonable representation of flow through a flexible artery. We conclude with a discussion of future directions for this model

Non-Newtonian Rheology in Blood Circulation

Blood is a complex suspension that demonstrates several non-Newtonian rheological characteristics such as deformation-rate dependency, viscoelasticity and yield stress. In this paper we outline some issues related to the non-Newtonian effects in blood circulation system and present modeling approaches based mostly on the past work in this field.Comment: 26 pages, 5 figures, 2 table

Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement

We present a new model and a novel loosely coupled partitioned numerical scheme modeling fluid-structure interaction (FSI) in blood flow allowing non-zero longitudinal displacement. Arterial walls are modeled by a {linearly viscoelastic, cylindrical Koiter shell model capturing both radial and longitudinal displacement}. Fluid flow is modeled by the Navier-Stokes equations for an incompressible, viscous fluid. The two are fully coupled via kinematic and dynamic coupling conditions. Our numerical scheme is based on a new modified Lie operator splitting that decouples the fluid and structure sub-problems in a way that leads to a loosely coupled scheme which is {unconditionally} stable. This was achieved by a clever use of the kinematic coupling condition at the fluid and structure sub-problems, leading to an implicit coupling between the fluid and structure velocities. The proposed scheme is a modification of the recently introduced "kinematically coupled scheme" for which the newly proposed modified Lie splitting significantly increases the accuracy. The performance and accuracy of the scheme were studied on a couple of instructive examples including a comparison with a monolithic scheme. It was shown that the accuracy of our scheme was comparable to that of the monolithic scheme, while our scheme retains all the main advantages of partitioned schemes, such as modularity, simple implementation, and low computational costs

Arterial Pulse Waveform under the watch of Left Ventricular Ejection time: A physiological outlook

The behavior of arterial pulse waves was studied in connection with time interval at different phases of propagation. The essence of the study was to have a clue about the incidence of the time of pulse wave propagation on cardio-vascular parameters. Model analysis shows that arterial waveforms behave like solitons. It was seen, from the soliton solution of the arterial pulse waveform, that time interval between the phases of propagation, which corresponds with left ventricular ejection time (LVET), could supply some information about apparent pathogenesis. Keywords: pressure; waveform; soliton; incompressible; mathematical; physiology

Mathematical model of the cerebral circulation and distribution of cerebrospinal fluid

Shifts in cerebral fluid are known to be important in a number of diseases, and in conditions of microgravity such as space travel. In this work we develop a fluid mechanical model from firstprinciples incorporating key features of the flow of both blood and cerebrospinal fluid (CSF) in the intracranial and spinal spaces. For the cerebral blood vessels, we model the arteries and veins as symmetric bifurcating trees with constant geometrical scaling factors between generations, assume one-dimensional flow in each vessel and account for elastic effects via a pressure-area relationship, and we assume the capillaries have a constant resistance. We treat the vessel walls as porous media to find the transmural flux of plasma. We assume flow between the other compartments to be proportional to the pressure difference; additionally, the flow to the outer-dural space is assumed to be one-way. The set of ordinary differential equations for the evolution of the fluid pressures and volumes of each compartment can be solved numerically. Additional features include autoregulation, which we model by ensuring constant pressure at the microcirculation, meaning the resulting model must be solved iteratively. Also, we can model the effect of postural changes by including hydrostatic effects in the spinal column. The results are in accordance with physiological measurements and indicate that the pressure in the vasculature is highly sensitive to changes in vessel geometry, which also affects the transmural flux, whilst ventricular and spinal subarachnoid spaces are sensitive to compliances. We investigate transitions from supine to standing and upside down positions and also the effect of the external pressure surrounding the outer-dural spinal compartment. The model is computationally inexpensive and can be used as a platform for further analysis of cerebrovascular behaviour.Open Acces

The 'Sphere': A Dedicated Bifurcation Aneurysm Flow-Diverter Device.

We present flow-based results from the early stage design cycle, based on computational modeling, of a prototype flow-diverter device, known as the 'Sphere', intended to treat bifurcation aneurysms of the cerebral vasculature. The device is available in a range of diameters and geometries and is constructed from a single loop of NITINOL(®) wire. The 'Sphere' reduces aneurysm inflow by means of a high-density, patterned, elliptical surface that partially occludes the aneurysm neck. The device is secured in the healthy parent vessel by two armatures in the shape of open loops, resulting in negligible disruption of parent or daughter vessel flow. The device is virtually deployed in six anatomically accurate bifurcation aneurysms: three located at the Basilar tip and three located at the terminus bifurcation of the Internal Carotid artery (at the meeting of the middle cerebral and anterior cerebral arteries). Both steady state and transient flow simulations reveal that the device presents with a range of aneurysm inflow reductions, with mean flow reductions falling in the range of 30.6-71.8% across the different geometries. A significant difference is noted between steady state and transient simulations in one geometry, where a zone of flow recirculation is not captured in the steady state simulation. Across all six aneurysms, the device reduces the WSS magnitude within the aneurysm sac, resulting in a hemodynamic environment closer to that of a healthy vessel. We conclude from extensive CFD analysis that the 'Sphere' device offers very significant levels of flow reduction in a number of anatomically accurate aneurysm sizes and locations, with many advantages compared to current clinical cylindrical flow-diverter designs. Analysis of the device's mechanical properties and deployability will follow in future publications

Simulation of haemodynamic flow in head and neck chemotherapy

In recent years, intra arterial chemotherapy has become an important component in head and neck cancer treatment. However, therapy success can vary significantly and consistent treatment guidelines are missing. The purpose of this study was to create a computer simulation of the chemical agent injection in the head and neck arteries to investigate the distribution and concentration of the chemical. This is of great interest for medical scientists and vital for prognosis. Realistic three dimensional patient specific geometry was created from image scan data. Engineering principles such as conservation of mass and momentum, turbulence models and a multiphase model were applied in a computational fluid dynamics (CFD) software. At first, a steady cause and effect study with various turbulence and material models was made without the chemical component. It was discovered, that a non-\Newtonian material model is compulsory for blood. The shear stress transport k-w turbulence model is appropriate for the whole velocity range and of superior robustness. These conclusions were used in the following two-component transient simulation. Pulsatile blood flow, turbulence, the chemical agent injection via a catheter and the mixture between blood and the chemical were considered. The principal conclusion was; the modelled catheter position right before the common carotid artery bifurcation produced an ineffective cisplatin distribution consistent throughout all the arteries. Due to high wall shear stress and turbulence at the inner bifurcation wall, serious complications during the treatment could occur, for instance heamolysis or acute vascular endothelial changes. To the best of the author\u27s knowledge a novel \CFD based approach was introduced, suitable for the optimization and the formulation of treatment guidelines in intra arterial injection chemotherapy. After a required validation, this model can be modified to investigate the influence of various catheter positions and dose rate schemes in future simulations

Mathematical Modeling and Numerical Simulation of Atherosclerosis Based on a Novel Surgeon’s View

This paper deals with the mathematical modeling of atherosclerosis based on a novel hypothesis proposed by a surgeon, Prof. Dr. Axel Haverich (Circulation 135(3):205–207, 2017). Atherosclerosis is referred as the thickening of the artery walls. Currently, there are two schools of thoughts for explaining the root of such phenomenon: thickening due to substance deposition and thickening as a result of inflammatory overgrowth. The hypothesis favored here is the second paradigm stating that the atherosclerosis is nothing else than the inflammatory response of of the wall tissues as a result of disruption in wall nourishment. It is known that a network of capillaries called vasa vasorum (VV) accounts for the nourishment of the wall in addition to the natural diffusion of nutrient from the blood passing through the lumen. Disruption of nutrient flow to the wall tissues may take place due to the occlusion of vasa vasorums with viruses, bacteria and very fine dust particles such as air pollutants referred to as PM 2.5. They can enter the body through the respiratory system at the first place and then reach the circulatory system. Hence in the new hypothesis, the root of atherosclerotic vessel is perceived as the malfunction of microvessels that nourish the vessel. A large number of clinical observation support this hypothesis. Recently and highly related to this work, and after the COVID-19 pandemic, one of the most prevalent disease in the lungs are attributed to the atherosclerotic pulmonary arteries, see Boyle and Haverich (Eur J Cardio Thorac Surg 58(6):1109–1110, 2020). In this work, a general framework is developed based on a multiphysics mathematical model to capture the wall deformation, nutrient availability and the inflammatory response. For the mechanical response an anisotropic constitutive relation is invoked in order to account for the presence of collagen fibers in the artery wall. A diffusion–reaction equation governs the transport of the nutrient within the wall. The inflammation (overgrowth) is described using a phase-field type equation with a double well potential which captures a sharp interface between two regions of the tissues, namely the healthy and the overgrowing part. The kinematics of the growth is treated by classical multiplicative decomposition of the gradient deformation. The inflammation is represented by means of a phase-field variable. A novel driving mechanism for the phase field is proposed for modeling the progression of the pathology. The model is 3D and fully based on the continuum description of the problem. The numerical implementation is carried out using FEM. Predictions of the model are compared with the clinical observations. The versatility and applicability of the model and the numerical tool allow