Mathematical Analysis of Ultrafast Ultrasound Imaging

This paper provides a mathematical analysis of ultrafast ultrasound imaging. This newly emerging modality for biomedical imaging uses plane waves instead of focused waves in order to achieve very high frame rates. We derive the point spread function of the system in the Born approximation for wave propagation and study its properties. We consider dynamic data for blood flow imaging, and introduce a suitable random model for blood cells. We show that a singular value decomposition method can successfully remove the clutter signal by using the different spatial coherence of tissue and blood signals, thereby providing high-resolution images of blood vessels, even in cases when the clutter and blood speeds are comparable in magnitude. Several numerical simulations are presented to illustrate and validate the approach.Comment: 25 pages, 13 figure

Numerical Investigation of Physical Parameters in Cardiac Vessels as a New Medical Support Science for Complex Blood Flow Characteristics

تقترح هذه الدراسة نهجًا رياضيًا وتجربة عددية لحل بسيط لتدفق الدم القلبي إلى الأوعية الدموية للقلب. تمت دراسة نموذج رياضي لتدفق الدم البشري عبر الفروع الشريانية وحسابه باستخدام معادلة نافييه-ستوكس التفاضلية الجزئية مع تحليل العناصر المحدودة (FEA). علاوة على ذلك ، يتم تطبيق FEA على التدفق الثابت للسوائل اللزجة ثنائية الأبعاد من خلال أشكال هندسية مختلفة. تتحدد صلاحية الطريقة الحسابية بمقارنة التجارب العددية مع نتائج تحليل الوظائف المختلفة. أظهر التحليل العددي أن أعلى سرعة لتدفق الدم تبلغ 1.22 سم / ثانية حدثت في مركز الوعاء الذي يميل إلى أن يكون رقائقيًا ويتأثر بعامل لزوجة منخفض قدره 0.0015 باسكال. بالإضافة إلى ذلك ، تحدث الدورة الدموية في جميع الأوعية الدموية بسبب ارتفاع الضغط في القلب ويقل الضغط عندما يعود من الأوعية الدموية بنفس المعايير. أخيرًا ، عندما تكون اللزوجة عالية ، تميل المقادير القصوى لتدفق الدم نحو جدار الوعاء الدموي بنفس سرعة التدرج ونصف قطره تقريبًا.This study proposes a mathematical approach and numerical experiment for a simple solution of cardiac blood flow to the heart's blood vessels. A mathematical model of human blood flow through arterial branches was studied and calculated using the Navier-Stokes partial differential equation with finite element analysis (FEA) approach. Furthermore, FEA is applied to the steady flow of two-dimensional viscous liquids through different geometries. The validity of the computational method is determined by comparing numerical experiments with the results of the analysis of different functions. Numerical analysis showed that the highest blood flow velocity of 1.22 cm/s occurred in the center of the vessel which tends to be laminar and is influenced by a low viscosity factor of 0.0015 Pa.s. In addition, circulation throughout the blood vessels occurs due to high pressure in the heart and the pressure becomes lower when it returns from the blood vessels at the same parameters. Finally, when the viscosity is high, the extreme magnitudes of blood flow tend toward the vessel wall at approximately the same velocity and radius of the gradient

CFD Multiphase Modeling of Blood Cells Segregation in Flow through Microtubes

Cardiovascular diseases, commonly referred as Heart Diseases, involve heart and blood vessels associated to the cardiovascular system. So called blood wetted medical devices are widely used in treatment of heart diseases as they help to provide better blood flow to patients. However, when blood is flowing through medical devices, it can be damaged due to lack of compatibility with surrounding non-biological walls of pipes, connectors and containers, thermal and osmotic effects, or most prominently due to excessive shear stresses on blood cells by medical devices. Though laboratory tests are vital for design improvements, they have proven to be costly, time intensive and ethically controversial. On the other hand, Computational Fluid Dynamics is a promising and inexpensive tool for simulating blood flow. The aim of this project is to improve and validate existing numerical model of blood cells segregation in flow through microtube. An improved numerical model of blood cells segregation is of interest for further evaluation of blood damage for design purposes of medical devices. The proposed model is based on Granular Kinetic Theory and represents a continuation of previous work by Mendygarin et al. [4] by sensitive analysis of red blood cells Sauter diameter according to local flow conditions

Numerical methods and applications for reduced models of blood flow

The human cardiovascular system is a vastly complex collection of interacting components, including vessels, organ systems, valves, regulatory mechanisms, microcirculations, remodeling tissue, and electrophysiological signals. Experimental, mathematical, and computational research efforts have explored various hemodynamic questions; the scope of this literature is a testament to the intricate nature of cardiovascular physiology. In this work, we focus on computational modeling of blood flow in the major vessels of the human body. We consider theoretical questions related to the numerical approximation of reduced models for blood flow, posed as nonlinear hyperbolic systems in one space dimension. Further, we apply this modeling framework to abnormal physiologies resulting from surgical intervention in patients with congenital heart defects. This thesis contains three main parts: (i) a discussion of the implementation and analysis for numerical discretizations of reduced models for blood flow, (ii) an investigation of solutions to different classes of models in the realm of smooth and discontinuous solutions, and (iii) an application of these models within a multiscale framework for simulating flow in patients with hypoplastic left heart syndrome. The two numerical discretizations studied in this thesis are a characteristics-based method for approximating the Riemann-invariants of reduced blood flow models, and a discontinuous Galerkin scheme for approximating solutions to the reduced models directly. A priori error estimates are derived in particular cases for both methods. Further, two classes of hyperbolic systems for blood flow, namely the mass-momentum and the mass-velocity formulations, are systematically compared with each numerical method and physiologically relevant networks of vessels and boundary conditions. Lastly, closed loop vessel network models of various Fontan physiologies are constructed. Arterial and venous trees are built from networks of one-dimensional vessels while the heart, valves, vessel junctions, and organ beds are modeled by systems of algebraic and ordinary differential equations

Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation

A novel multiscale mathematical and computational model of the pulmonary circulation is presented and used to analyse both arterial and venous pressure and flow. This work is a major advance over previous studies by Olufsen et al. (Ann Biomed Eng 28:1281–1299, 2012) which only considered the arterial circulation. For the first three generations of vessels within the pulmonary circulation, geometry is specified from patient-specific measurements obtained using magnetic resonance imaging (MRI). Blood flow and pressure in the larger arteries and veins are predicted using a nonlinear, cross-sectional-area-averaged system of equations for a Newtonian fluid in an elastic tube. Inflow into the main pulmonary artery is obtained from MRI measurements, while pressure entering the left atrium from the main pulmonary vein is kept constant at the normal mean value of 2 mmHg. Each terminal vessel in the network of ‘large’ arteries is connected to its corresponding terminal vein via a network of vessels representing the vascular bed of smaller arteries and veins. We develop and implement an algorithm to calculate the admittance of each vascular bed, using bifurcating structured trees and recursion. The structured-tree models take into account the geometry and material properties of the ‘smaller’ arteries and veins of radii ≥ 50 μ m. We study the effects on flow and pressure associated with three classes of pulmonary hypertension expressed via stiffening of larger and smaller vessels, and vascular rarefaction. The results of simulating these pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment for circulatory diseases within the lung

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

One-Dimensional Navier-Stokes Finite Element Flow Model

This technical report documents the theoretical, computational, and practical aspects of the one-dimensional Navier-Stokes finite element flow model. The document is particularly useful to those who are interested in implementing, validating and utilizing this relatively-simple and widely-used model.Comment: 46 pages, 1 tabl

Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods

One-dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ greatly in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we systematically compare several reduced models of blood flow for physiologically relevant vessel parameters, network topology, and boundary data. The models are discretized by a class of Runge-Kutta discontinuous Galerkin methods