Modeling Reactive Hyperemia to better understand and assess Microvascular Function a review of techniques Alberto Coccarelli1Michael D. Nelson2

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Modeling Reactive Hyperemia to better understand and assess

Microvascular Function: a review of techniques

Alberto Coccarelli1Michael D. Nelson2

1Zienkiewicz Centre for Computational Engineering, Faculty of Science and

Engineering, Swansea University, UK

2Department of Kinesiology, University of Texas at Arlington, USA

corresponding author: alberto.coccarelli@swansea.ac.uk

Abstract

Reactive hyperemia is a well-established technique for the non-invasive evaluation of the periph-

eral microcirculatory function, measured as the magnitude of limb re-perfusion after a brief period

of ischemia. Despite widespread adoption by researchers and clinicians alike, many uncertainties re-

main surrounding interpretation, compounded by patient-speciﬁc confounding factors (such as blood

pressure or the metabolic rate of the ischemic limb). Mathematical modeling can accelerate our

understanding of the physiology underlying the reactive hyperemia response and guide in the esti-

mation of quantities which are diﬃcult to measure experimentally. In this work, we aim to provide a

comprehensive guide for mathematical modeling techniques that can be used for describing the key

phenomena involved in the reactive hyperemia response, alongside their limitations and advantages.

The reported methodologies can be used for investigating speciﬁc reactive hyperemia aspects alone,

or can be combined into a computational framework to be used in (pre-)clinical settings.

Keywords: Reactive Hyperemia, Microvascular Function, Non-invasive Testing, Peripheral Circu-

lation, Computational Haemodynamics, Multi-scale modeling

1 Introduction

The microcirculation is the essential ‘end point’ of the cardiovascular system, consisting of a vast

network of microvessels perfusing the body’s organ tissues, whose main function is delivering oxygen

and nutrients and removing waste products. At the local level, the microcirculation continuously

regulates the levels of blood ﬂow and pressure across its network to satisfy metabolic demands, to

redistribute hydraulic loads and to promote inﬂammatory processes. To this end, a sophisticated

hierarchical control system (intrinsic, metabolic and neurohormonal) operates by regulating the

diameter of the microvessels, causing vaso-dilation or vaso-constriction, which in turn modulates

their hydrodynamic resistance. When a signiﬁcant increase in tissue metabolic demand occurs, a

large fraction of the microcirculation needs to be recruited for coordinating the vaso-dilation. To

achieve this, the vaso-dilation arising from the lower microcirculation can ‘ascend’ into feed arteries

by means of an electrical signal conducted along the endothelial cells (ECs), hyper-polarizing vascular

smooth muscle cells (SMCs) causing relaxation [1].

Since microcirculation represents the ‘mesoscale’ functionally bridging the systemic circulation

to perfused tissues, microvascular dysfunction (MVD) ultimately compromises oxygen delivery to

the end organ(s) [2]. Indeed, it is within the microcirculation that the earliest signs of several

cardiovascular diseases manifest themselves. For example, the Fireﬁghters and Their Endothelium

(FATE) study [3] showed that microvascular health represents a powerful independent predictor of

cardiovascular events in primary prevention. Thus, assessment of microvascular function may not

only have utility for providing novel insights into the pathophysiology of the patient, but also presents

an important opportunity for early disease detection and risk stratiﬁcation [4].

Microvascular function can be evaluated invasively at the level of the end-organ using vaso-active

agents combined with pressure/ﬂow wires and non-invasively using advanced imaging techniques such

as computed tomography (CT), magnetic resonance imaging (MRI) and positron emission tomog-

raphy (PET) [5, 6]. Invasive approaches can indeed selectively partition macro- from microvascular

function but are not suitable nor feasible in all individuals. While non-invasive CT, PET and MRI

address many of these concerns, exposing participants to ionizing radiation, conﬁned spaces, and/or

strong magnetic ﬁelds also limits widespread adoption. As such, there has been considerable focus

arXiv:2210.10901v1 [q-bio.TO] 19 Oct 2022

on the development of safe and cost-eﬀective alternative methods for measuring microvascular reac-

tivity [7]. Accordingly, much attention has been placed on the peripheral vasculature, as it is easily

accessible and reasonably reﬂects vascular function in other end-organs of interest [8, 9, 10]. Although

not yet established in clinical practice, methods such as reactive hyperemia provide prognostically-

signiﬁcant indicators, capable of diﬀerentiating clinical phenotypes [11, 12]. In its simplest form,

reactive hyperemia represents the magnitude of limb reperfusion following a brief period of ischemia

induced by arterial occlusion (see Figure 1). Fundamentally, the approach measures microvascular

Figure 1: Microvascular conditions before, during and after occlusion. The size of the arrow indicates

the magnitude of blood ﬂow. The vasodilation in resistance arteries may have a smaller entity than in

arterioles due to the limited distance that can be reached by the ascending hyperpolarizing signal.

vasoreactivity to metabolic substances produced in response to tissue ischemia. Here we consider

reactive hyperemia to be limited only to the initial increase in ﬂow that immediately follows the

restoration of ﬂow. Indeed, the blood ﬂow response after this initial period is aﬀected by regulatory

mechanisms that fall outside of pure ‘microvascular function’. Multiple methods exist for evaluating

reactive hyperemia, including limb distension by venous occlusion plethysmography, blood veloc-

ity/ﬂow via Doppler ultrasound of an upstream conduit vessel, kinetic changes in tissue oxygenation

by near-infrared spectroscopy (NIRS) [13, 14, 15, 16, 17], and tissue perfusion by (near-infrared)

diﬀuse correlation spectroscopy (NIRS-DCS) [18]; each of which with its own strengths and weak-

nesses [12]. As detailed in this review article, the factors governing reactive hyperemia are multi-

factorial and diﬃcult to quantify. Thus, the best measurement approach is one that accounts for the

greatest number of such complexities.

Computer modeling can be used as a complementary tool for hypothesis testing, making predic-

tions and quantifying the dynamics underlying vascular regulation that cannot be directly observed

during in vivo experimentation. Here, we report on various modeling techniques used to describe

reactive hyperemia dynamics across the circulation, and describe the eﬀects of various assumptions

on hierarchical blood ﬂow control system. The review concludes with an integration of knowledge,

relating the theoretical modeling with existing data collected by our group and others (by way of

peripheral ischemia-reperfusion), identifying key opportunities for future discovery.

2 Haemodynamics

Occlusion of a conduit blood vessel, like the brachial artery, has a direct negative impact on the re-

sulting pressure and ﬂow distribution downstream of the occlusion. The reduction in pressure quickly

propagates from the occlusion site down to the level of the exchange vessels, ultimately impairing

oxygen delivery. The ensuing tissue ischemia results in microvascular vasodilation (intended to cor-

rect the error signal), such that when the occlusion is reversed, the ensuing blood ﬂow is markedly

elevated (i.e. reactive hyperemia).

2.1 Flow in (large) compliant vessels

The exchange of forces between blood and vascular wall and its resulting displacement can be eval-

uated by employing detailed three-dimensional (3D) ﬂuid-structure interaction models [19] or sim-

pliﬁed one-dimensional (1D) blood ﬂow modeling approaches which consider ﬁeld variations only

along the main ﬂow (axial) direction [20]. The latter methodology is less accurate (especially around

localized anatomical details), but has many competitive computational advantages and can be easily

adopted for vessel networks [21]. This makes 1D blood ﬂow modeling the best option for describing

the ﬂuid mechanics in reactive hyperemia, which involves propagating phenomena along the vascu-

lature with time scale of many seconds. Furthermore, through 1D blood ﬂow modeling it is also

possible to perform wave intensity analysis which allows the quantiﬁcation of pressure waveforms

travelling forward to the microcirculation and backward to the heart [20]. In arteries (and arterioles)

the blood behaviour can be generally approximated as homogeneous and Newtonian since the size of

the red blood cells carried by the plasma is considerably smaller (10 times) than the vessel diameter.

Furthermore, ﬂow is also generally considered incompressible and laminar, with a Poiseuille velocity

proﬁle. In 1D blood ﬂow modeling, each compliant vessel can be treated as axisymmetric, with

blood velocity (u), pressure (P) and ﬂow described as continuous variables along its axial direction

z. These quantities are averaged over the cross-sectional area (A) and their variation along the radial

direction is considered negligible. The Navier-Stokes equations for 1D blood ﬂow in compliant vessels

can be expressed in terms of cross-sectional area and velocity averaged over the cross-section:

∂A

∂t +∂(Au)

∂z = 0,

∂u

∂t +u∂u

∂z +1

∂P

∂z +µu

8π

A= 0,

(1)

where tis time, µis the ﬂuid dynamic viscosity, ρis the ﬂuid density, while Q=Au is the (volumetric)

ﬂow rate. It is worth noting that (1) can also be written in terms of ﬂow rate and pressure [22] or

cross-sectional area and ﬂow rate [23]. The mechanics underlying the vascular wall deformation

appears to be complex, mainly due to vessel visco-elastic properties and the capacity to produce

active tone for diameter regulation. To describe the interaction between blood and vessel wall,

diﬀerent approaches can be used [24, 25, 26]. In the simplest case, the ﬂuid pressure is related to the

cross-section of the vessel via a linear function with respect to the luminal diameter (D=p4A/π)

P=Pext +β(√A−√A0),(2)

where Pext is the external pressure from the surrounding tissue, βis a parameter representing the

wall elasticity and A0is the unstressed cross-section area. However, modeling reactive hyperemia

requires a vessel wall model able to describe the hyperpolarization-induced dilation and then the

resulting (compliant) structural response to hyperemic ﬂow. Given such complexity, wall mechanics

models derived from conservation laws retaining mechanobiological features [27, 28] are preferable

over tube laws which are purely phenomenological; indeed the latter, for this speciﬁc application,

would require the introduction of several non-physical parameters. It is also noted that, if desired,

wall viscoelasticity can be integrated into the blood pressure-wall deformation law by using more

complex constitutive models [29, 30, 31] with a consequent decrease in computational eﬃciency. The

haemodynamic features (viscosity and density) and the geometric (diameter and length) and struc-

tural (stiﬀness) wall properties can be made vessel speciﬁc and can reﬂect diﬀerent physiological and

pathological conditions such as ageing or hypertension [20, 32]. The blood ﬂow variables described

by system (1) and (2) can be computed by employing a broad variety of numerical schemes including

ﬁnite diﬀerences, ﬁnite elements and ﬁnite volumes and can be extended to large vessel networks by

imposing mass and momentum conservation at the interface between vessels [33, 20, 34, 22].

2.2 Biphasic ﬂow in microvessels

During reactive hyperemia, ﬂow in the microvessels is altered from its physiological range due ﬁrst

to the sudden pressure reduction induced by the upstream occlusion and then by arteriolar luminal

expansion consequent to the wall relaxation driven by ischemic tissues. Due to its morphology and

function, the downstream vasculature constitutes the site of major blood pressure drop along the

cardiovascular system. The work by Secomb [35] provides a detailed characterization of the ﬂow

through microcirculatory networks. In these microvessels Reynolds number is <1, and therefore the

blood ﬂow can be described with a good level of accuracy as incompressible Stokes ﬂuid, for which the

convective component is neglected. Blood ﬂow can still be described by (1) but most assume a rigid

microvessel wall, which implies considering only the momentum conservation equation for relating

blood ﬂow and pressure. Furthermore, the biphasic nature of ﬂow requires a modeling framework

that accounts for the main rheological properties of its components. Since they are concentrated in

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ModelingReactiveHyperemiatobetterunderstandandassessMicrovascularFunction:areviewoftechniquesAlbertoCoccarelli1MichaelD.Nelson21ZienkiewiczCentreforComputationalEngineering,FacultyofScienceandEngineering,SwanseaUniversity,UK2DepartmentofKinesiology,UniversityofTexasatArlington,USAcorrespondingauthor...

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Modeling Reactive Hyperemia to better understand and assess Microvascular Function a review of techniques Alberto Coccarelli1Michael D. Nelson2

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