Mathematical methods for modeling the microcirculation

The microcirculation plays a major role in maintaining homeostasis in the body. Alterations or dysfunctions of the microcirculation can lead to several types of serious diseases. It is not surprising, then, that the microcirculation has been an object of intense theoretical and experimental study over the past few decades. Mathematical approaches offer a valuable method for quantifying the relationships between various mechanical, hemodynamic, and regulatory factors of the microcirculation and the pathophysiology of numerous diseases. This work provides an overview of several mathematical models that describe and investigate the many different aspects of the microcirculation, including geometry of the vascular bed, blood flow in the vascular networks, solute transport and delivery to the surrounding tissue, and vessel wall mechanics under passive and active stimuli. Representing relevant phenomena across multiple spatial scales remains a major challenge in modeling the microcirculation. Nevertheless, the depth and breadth of mathematical modeling with applications in the microcirculation is demonstrated in this work. A special emphasis is placed on models of the retinal circulation, including models that predict the influence of ocular hemodynamic alterations with the progression of ocular diseases such as glaucoma

Modeling and simulation of blood circulation and perfusion

Numeriske simuleringer har hatt vesentlig betydning for vår forståelse av perfusjon og blodsirkulasjon, og simuleringer gir også viktig innsikt under utviklingen av medisinske anvendelser. Teknologiske fremskritt har muliggjort bruken av mer realistiske modeller, ikke bare i form av mer kompleks fysikk, men også ved at en kan studere sirkulasjonen i hele organer. Disse kjennetegnene er ofte av interesse da fysiologiske egenskaper er forskjellige på tvers av romlige størrelsesordener. Denne avhandlingen fokuserer på modellering og simulering av blodstrøm, sporstofftransport og perfusjon i organvev. De fysiske prosessene er uttrykt i en flerskala strømningsmodell der segmenterte arterier og vener danner en nettverksmodell for vaskulær strømning, og som er knyttet til en mikrosirkulasjonsmodell. Den ikkeobserverbare vaskulaturen beskrevet av modellen simuleres både med en kontinuerlig og en diskretisert tilnærming. Vi presenterer et flerskala rammeverk for å studere blodsirkulasjon. Det nytenkende aspektet ved rammeverket består i å kombinere en eksisterende hybrid strømningsmodell for flerskala sirkulasjon med vaskulærfremkalte ikke-lineariteter som har opphav i bl.a. veggelastisiteten og kurvaturen til blodkarene. Anvendelsen av en passende betingelse fra lineær algebra gjør at vi effektivt kan løse det tilknyttede ikke-lineære systemet ved bruk av en iterativ Newtons metode, og det relativt enkle rammeverket beskriver slik blodsirkulasjon i et komplekst fysisk domene med en lav beregningsmessig kostnad. Modellene og deres tilhørende implementeringer presenteres i artiklene som utgjør Del II i avhandlingen. Her foreslår vi et rammeverk for å generere digitale fantomer for avbildning av perfusjon, og ved å evaluere kinetikkmodeller for sporstoff demonstrerer vi de betydelige verdiene som finnes i etterbehandling av medisinske data. I tillegg undersøker vi optimale vaskulære nettverk som avslører en kompleks gjensidig avhengighet mellom geometrien til det vaskulære nettverket, kapillærene og organene. Resultatene fra denne avhandlingen bidrar til en bedre forståelse av blodperfusjonsmodeller i vev og potensialet til disse, samt potensialet som vitenskapelig databehandling har i medisinske anvendelser utover perfusjonsavbildning.Numerical simulations have become essential for understanding blood circulation and perfusion, as well as providing important insights for medical applications. More realistic models have become possible with technological advances, not only in the form of more complex physics, but also in the flow detail of an entire organ circulation. These characteristics are frequently of interest because blood vessels at different spatial scales have different physiological properties. This thesis focuses on the modeling and simulations of blood flow, tracer transport, and perfusion in an organ tissue. The physical processes are expressed in a multiscale flow model with segmented arteries and veins forming a vascular network flow model that is connected to a microcirculation model. The unobservable vasculature, including small vessels and capillaries, represented by the connection model, is simulated by using a continuum and discrete approach. A multiscale framework for solving blood circulation is presented. The novelty of this framework comes from combining an existing hybrid flow model for a multiscale circulation with vasculature-induced nonlinearities such as vessel wall elasticity and vessel curvature. By using an appropriate linear algebra precondition, the corresponding nonlinear system can be efficiently solved by using an iterative Newton method. This allows us to formulate more realistic blood circulation in a complex physical domain by employing a relatively simple framework with a low computational cost. The models and their implementation are presented in the papers that constitute Part II of this thesis. In the paper section, we propose a framework to generate a digital phantom for perfusion imaging. Moreover, we evaluate tracer kinetic models demonstrating the significant value of post-processing of medical data. We also investigate optimal vascular networks revealing a complex interdependence between the geometry of the vascular network, the capillary bed and organ shape. The results of this thesis contribute to a better understanding of blood perfusion models in tissue and their potential, as well as the potential of scientific computing, for medical applications not limited to perfusion imaging.Doktorgradsavhandlin

A Multi-Scale CFD Analysis of Patient-Specific Geometries to Tailor LVAD Cannula Implantation Under Pulsatile Flow Conditions: an investigation aimed at reducing stroke incidence in LVADs

A Left Ventricular Assist Device (LVAD) is a mechanical pump that provides temporary circulatory support when used as bridge-to-transplantation and relieves workload demand placed on a failing heart allowing for myocardia recovery when used as destination therapy. Stroke is the most devastating complication after ventricular assist device (VAD) implantation, with an incidence of 14-47% over 3-6 months. This complication due to thrombus formation and subsequent transport through the vasculature to cerebral vessels continues to limit the widespread implementation of VAD therapy. Patient-specific computational fluid dynamics (CFD) analysis may elucidate ways to reduce this risk. We employed a multi-scale model of the aortic circulation in order to examine the effects on flow conditions resulting from varying the VAD cannula implantation location and angle of incidence of the anastomosis to the ascending aorta based on a patient-specific geometry obtained from CT scans. The multi-scale computation consists of a 0D lumped parameter model (LPM) of the circulation modeled via a 50 degree of freedom (DOF) electrical circuit analogy that includes an LVAD model coupled to a 3D computational fluid dynamics model of the circulation. An in-house adaptive Runge-Kutta method is utilized to solve the 50 DOF LPM, and the Starccm+ CFD code is utilized to solve the flowfield. This 0D-3D coupling for the flow is accomplished iteratively with the 0D LPM providing the pulsatile boundary conditions that drive the 3D CFD time-accurate computations of the flowfield. Investigated angle configurations include cannula implantations at 30°, 60° and 90° to the right lateral wall of the ascending aorta. We also considered placements of the VAD cannula along the ascending aorta in which distances of the VAD anastomosis is varied relative to the take-off of the innominate artery. We implemented a mixed Eulerian-Lagrangian particle-tracking scheme to quantify the number of stroke-inducing particles reaching cerebral vessel outlets and included flow visualization through streamlines to identify regions of strong vorticity and flow stagnation, which can promote thrombus formation. Thrombi were modeled as spheres with perfectly elastic interactions numerically released randomly in time and space at cannula inlet plane. Based on clinical observation of the range of thrombus sizes encountered in such cases, particle diameters of 2.5mm and 3.5mm were investigated in our numerical computations. Pulsatile flow results for aforementioned angles suggest that a 90° cannula implementation causes flow impingement on the left lateral aortic wall and appears to be highly thrombogenic due to large momentum losses and zones of large re-circulation and that shallow and intermediate cannula angles promote more regular flow carrying particles towards the lower body potentially reducing stroke risk. Indications from this pulsatile numerical study suggest that up to a 50% reduction in stroke rate can be achieve with tailoring of cannula implantation. Results are consistent with significant reduction in stroke incidence achieved by tailoring cannula implantation as reported in previous steady flow computations carried out by our group. As such, results of this study suggest that a simple surgical maneuver in the process of VAD implantation may significantly improve patient life

Advancements in blood rheology and hemodynamics simulation with a brief history

Blood rheology is a complex field of study that investigates blood flow behavior, vital for understanding its role in physiological and pathological conditions. This article delves into various rheological models that describe blood behavior, ranging from Generalized Newtonian models to more sophisticated thixotropic and elastoviscoplastic models. One such model, the Horner-Armstrong-Wagner-Beris (HAWB) model, offers valuable insights into the dynamic interplay of reversible and irreversible phenomena in blood flow. Recent advancements, such as the mHAWB framework, provide enhanced accuracy and versatility in modeling blood rheology, holding great potential for diagnostic and therapeutic applications. Moreover, microscopic and mesoscopic simulations have paved the way for deeper insights into blood behavior, bridging the gap between theory and experiment. Multiscale models offer a promising approach to capturing the complexities of blood rheology at various length scales. Finally, we explore the clinical implications of blood rheology, including its significance in conditions like polycythemia, neonatal respiratory distress, and circulatory inadequacy. By understanding blood rheology comprehensively, we can advance our knowledge of complex blood flow dynamics and its potential applications in healthcare

Multiscale modelling of metabolism and transport phenomena in living tissues

The range of applications of mathematical modelling in biosciences has recently expanded to encompass problems posed by biomedicine and sport sciences. Topics of interest are for instance the prediction of the response of an athlete to exercise, the pharmacokinetics of a chemical compound, or the detection of illicit drugs. In this work, we consider some of these problems, related to metabolism, circulation and mass transport in tissues. First, we address the quantitative analysis of the biochemical reactions that are responsible of energy production in muscle cells. These reactions are strictly dependent on chemical exchanges between blood and tissues, by several physiological auto-regulation mechanisms. For this reason, we consider coupled problems in which the reaction phenomena are influenced by transport in blood. In particular, the problem of local blood perfusion and supply of substrates to tissues is studied in detail. The processes underlying the interaction between metabolism and circulation feature a multiscale nature: for instance, although metabolism takes place in cells, it modifies the hemodynamics of peripheral (capillaries) and central (heart) circulation. Therefore, we will set up a hierarchy of models, corresponding to these different scales. At first, we adopt an integrative approach, based on a compartmental model of the whole-body response to exercise, or more generally to variations in skeletal muscle metabolism. This model is the higher level of the hierarchy, describing the interactions between organs. Then, we increase the level of detail and focus on isolated tissues and vessels, considering more accurate one-dimensional models for blood flow and mass transport, as well as coupled 1D-3D models of tissue perfusion. In the latter models, the microvascular matrix is represented as a three-dimensional homogeneous medium, where larger vessels are described as 1D networks: circulation, transport and reaction of biochemical species are modelled at both the scales. The models considered in this work may provide a multi-scale analysis of metabolic processes, such as those induced by exercise, that often begin at cellular level, progressively propagate up through the hierarchy of scales, until adaptation of the whole body is reached. Examples of simulations, dealing with exercise protocols or clinical study cases, are provided to support the range of applications

Global energy minimisation of arterial trees with application to embolic stroke

Computer generation of optimal arterial trees has previously been limited to the production of locally optimal configurations. The application of a global optimisation algorithm allows for the generation of vasculatures with consistent structure. Comparison of this structure to that of in-vivo vasculatures allows the determination of to what extent the vascular structure is the result of energy minimisation. In this thesis an algorithm capable of generation globally optimal vascular trees in geometries derived from medical imaging is developed. We begin by outlining a small set of constraints which capture physiological principles guiding the organisation of arterial trees. The constraints are then used to produce an algorithm capable of finding the minimal energy configuration of a given arterial tree. The algorithm is used to produce both coronary and cerebral vasculature, and the latter is generated in geometries segmented from MRI data of a human brain. The trees are compared both morphologically and structurally to those found in-vivo. The morphological comparisons for the coronary vasculature show excellent agreement with experiment. The positions of the larger coronary arteries in the generated trees agree extremely well with experiment, suggesting that structure of the coronary vasculature is the result of energy minimisation. The generated cerebral vasculature approximates the vascular territories of the major cerebral arteries, however the morphological comparisons show that the structure of the cerebral arteries is likely not the result of energy minimisation. The cerebral vasculatures is used to extend a statistical model of embolic stroke to include the effects of branching asymmetry, and an analytic approximation to the statistical model of embolic stroke is developed and validated. It is found that branching asymmetry produces an overall reduction in the level of blockage occuring during an embolic event

Methods and Algorithms for Cardiovascular Hemodynamics with Applications to Noninvasive Monitoring of Proximal Blood Pressure and Cardiac Output Using Pulse Transit Time

Advanced health monitoring and diagnostics technology are essential to reduce the unrivaled number of human fatalities due to cardiovascular diseases (CVDs). Traditionally, gold standard CVD diagnosis involves direct measurements of the aortic blood pressure (central BP) and flow by cardiac catheterization, which can lead to certain complications. Understanding the inner-workings of the cardiovascular system through patient-specific cardiovascular modeling can provide new means to CVD diagnosis and relating treatment. BP and flow waves propagate back and forth from heart to the peripheral sites, while carrying information about the properties of the arterial network. Their speed of propagation, magnitude and shape are directly related to the properties of blood and arterial vasculature. Obtaining functional and anatomical information about the arteries through clinical measurements and medical imaging, the digital twin of the arterial network of interest can be generated. The latter enables prediction of BP and flow waveforms along this network. Point of care devices (POCDs) can now conduct in-home measurements of cardiovascular signals, such as electrocardiogram (ECG), photoplethysmogram (PPG), ballistocardiogram (BCG) and even direct measurements of the pulse transit time (PTT). This vital information provides new opportunities for designing accurate patient-specific computational models eliminating, in many cases, the need for invasive measurements. One of the main efforts in this area is the development of noninvasive cuffless BP measurement using patient’s PTT. Commonly, BP prediction is carried out with regression models assuming direct or indirect relationships between BP and PTT. However, accounting for the nonlinear FSI mechanics of the arteries and the cardiac output is indispensable. In this work, a monotonicity-preserving quasi-1D FSI modeling platform is developed, capable of capturing the hyper-viscoelastic vessel wall deformation and nonlinear blood flow dynamics in arbitrary arterial networks. Special attention has been dedicated to the correct modeling of discontinuities, such as mechanical properties mismatch associated with the stent insertion, and the intertwining dynamics of multiscale 3D and 1D models when simulating the arterial network with an aneurysm. The developed platform, titled Cardiovascular Flow ANalysis (CardioFAN), is validated against well-known numerical, in vitro and in vivo arterial network measurements showing average prediction errors of 5.2%, 2.8% and 1.6% for blood flow, lumen cross-sectional area, and BP, respectively. CardioFAN evaluates the local PTT, which enables patient-specific calibration and its application to input signal reconstruction. The calibration is performed based on BP, stroke volume and PTT measured by POCDs. The calibrated model is then used in conjunction with noninvasively measured peripheral BP and PTT to inversely restore the cardiac output, proximal BP and aortic deformation in human subjects. The reconstructed results show average RMSEs of 1.4% for systolic and 4.6% for diastolic BPs, as well as 8.4% for cardiac output. This work is the first successful attempt in implementation of deterministic cardiovascular models as add-ons to wearable and smart POCD results, enabling continuous noninvasive monitoring of cardiovascular health to facilitate CVD diagnosis