# Modeling Reactive Hyperemia to better understand and assess Microvascular
Function: a review of techniques

Alberto Coccarelli1 Michael D. Nelson2

(1 Zienkiewicz Centre for Computational Engineering, Faculty of Science and
Engineering, Swansea University, UK
2Department of Kinesiology, University of Texas at Arlington, USA
corresponding author<EMAIL_ADDRESS>
###### Abstract

Reactive hyperemia is a well-established technique for the non-invasive
evaluation of the peripheral 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
remain 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 estimation 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 Circulation, 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 tomography (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 on the development of safe and cost-effective alternative methods for
measuring microvascular reactivity [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).

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.

Fundamentally, the approach measures microvascular 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 velocity/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 weaknesses [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 predictions 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 resulting 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 correct 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 evaluated by employing detailed three-dimensional (3D)
fluid-structure interaction models [19] or simplified 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 vasculature 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:

$\begin{split}\frac{\partial A}{\partial t}+\frac{\partial(Au)}{\partial
z}=0,\\\ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial
z}+\frac{1}{\rho}\frac{\partial P}{\partial z}+\frac{\mu
u}{\rho}\frac{8\pi}{A}=0,\end{split}$ (1)

where $t$ is time, $\mu$ is the fluid dynamic viscosity, $\rho$ is 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 ($D$=$\sqrt{4A/\pi}$)

$P=P_{ext}+\beta(\sqrt{A}-\sqrt{A_{0}}),$ (2)

where $P_{ext}$ is the external pressure from the surrounding tissue, $\beta$
is a parameter representing the wall elasticity and $A_{0}$ is 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 structural (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 the central part of the tube, RBCs
travel faster than the surrounding plasma (referred as Fahraeus effect) and
the shorter RBC’s transit time with respect to the total blood flow has a
profound effect on oxygen transport and exchange with tissues. The discharge
hematocrit ($H_{D}$), which is the ratio between RBCs volume flux and total
blood volume flux along the tube, appears higher than the tube hematocrit
($H_{T}$), which is the volume occupied by RBCs over the tube volume at a
certain time instant (see Figure 2).

Figure 2: Hematocrit in blood vessel. The discharge hematocrit, being the
volume fraction of RBCs in blood delivered through the vessel by the flow, is
obtained as ratio the vessel volumetric rate of RBCs and volumetric blood flow
($H_{D}$=$\dot{V}_{RBC}/\dot{V}_{blood}$). The tube hematocrit is given as
ration between the volume occupied by RBCs and total blood volume in the
vessel ($H_{T}$=$n_{RBC}{V}_{RBC}/{V}_{blood}$, where $n_{RBC}$ is the number
of RBCs in the vessel).

The mean velocity of RBCs ($u_{RBC}$) can be estimated if the ratio between
$H_{T}$ and $H_{D}$ is known (since ${H_{T}}/{H_{D}}$=$u/{u_{RBC}}$), for
which Pries et al. [36] identified the following correlation:

$\frac{H_{T}}{{H}_{D}}=H_{D}+(1-H_{D})(1+1.7\text{exp}(-0.35D)-0.6\text{exp}(-0.01D)).$
(3)

The latter expression was then modified by Pries et al. [37] to account for
the reduction in diameter due to the endothelial surface layer. In tubes with
diameters below 300 $\mu$m, the viscosity of blood reduces with decreasing
diameter. This phenomenon, discovered by Fahraeus and Lindquist [38] (referred
as Fahraeus-Lindquist effect) appears to be associated with the tendency of
RBCs to migrate away from the vessel wall [35]. The resulting blood viscosity
can be evaluated as a product between the plasma viscosity and the relative
apparent viscosity ($\mu_{rel}$), with the latter accounting for the effect of
the hematocrit. Pries and co-workers defined a set of correlations which
relate apparent viscosity with discharge hematocrit, and diameter of
microvessel for in-vitro [39] and in-vivo conditions [36, 40]. In the latter
case, the relative apparent viscosity is defined as:

$\mu_{rel}=[1+(\mu_{45})\frac{(1-H_{D})^{C_{\mu}}-1}{(1-0.45)^{C_{\mu}}-1}(\frac{D}{D-1.1})^{2}](\frac{D}{D-1.1})^{2},$
(4)

in which $\mu_{45}$ and $C_{\mu}$ are experimentally derived correlations
dependent on the vessel diameter. The presence of cell-free layers within the
microvessels is also responsible for the phase separation effect, which
consists in a non-proportional partition of RBCs and plasma flows at diverging
bifurcations. The experimentally-derived correlations by Pries et al. [41, 37]
represent the reference approach for including this phenomenon within vascular
networks. An alternative RBCs phase separation model was subsequently
introduced by Gould and Linninger [42], although its benefits over the one by
Pries et al. [41, 37] have been debated [43].

Opposite to the continuum approach, blood flow across microvessels can also be
described via motion of multiple discrete particles [44, 45] and the effect
associated to the particulate nature can be assessed for different channel
sizes [46]. To transit across the smallest capillaries, the RBC membrane
undergoes significant deformation, with the intensity of applied forces on the
capillary lumen playing a significant mechanobiological role [47]. We expect
that the surge in blood flow during reactive hyperemia substantially changes
the RBC-capillary lumen interaction. Variation in RBC number and velocity
across the capillary bed can be experimentally analyzed by using intravital
video recordings [48] but the resulting pressure field remains uncertain.
Discrete multi-scale RBC modeling approaches [49, 50, 51] can be used for
reproducing the mechanical and rheological properties of the cell and their
interaction with plasma. Casquero et al. [52] used a NURBS-based fluid-
structure interaction framework for assessing the deformation of RBCs, which
were modeled as compound capsules (or solid membrane) where the nucleus is
treated as bulky deformable solid. These multidimensional models allow the
direct assessment of the physics underlying the interaction between RBCs and
microvessel wall but they require efficient parallel algorithms [53, 54] for
being scalable to realistic and physiologically relevant problems.

###  2.3 Lumped circulation models

Lumped or zero-dimensional (0D) modeling represents an approach in which the
system’s quantities are not (directly) space-dependent but depend only on
time. Their governing equations can be directly derived from formulations of
high-fidelity/space-dependent models (such as full 3D heart chambers, arteries
and veins) and the space-dependent characteristics are encapsulated through a
set of parameters which are generally optimized using experimental data [55].
0D models may be employed alone for describing the full cardiovascular system
or parts of it and can be represented through electrical analogy. Lumped
models can also be used in combination with higher-fidelity counterparts to
represent those vascular territories where a low level of space details is
required.

Compartmental lumped models [56, 57, 58] split the vasculature into different
flow compartments, each of them characterized by specific parameters
representing vascular features such as flow resistance and compliance. In this
case, the full haemodynamic profiles across the vasculature can be obtained by
simply computing a set of (time-dependent) ordinary differential equations.
This approach involves very low computational costs and allows an easy
integration and/or comparison with quantities averaged over space obtained
from some experimental imaging techniques (such as doppler ultrasound or NIRS-
DCS). Therefore, due to its simplicity, 0D modeling should be considered as
first-step for modeling blood flow variables during reactive hyperemia. On the
other hand, lumped modeling alone cannot adequately capture the wave-
propagation phenomena underlying the reactive hyperemia response and therefore
does not provide a suitable platform for a full mechanistic understanding of
the different space-dependent interactions between vascular territories.
Nevertheless we re-iterate that, for various questions related to reactive
hyperemia, lumped modeling should be considered for the description of the
whole system’s dynamics, with space-dependent models introduced as ‘sub-
models’ for further refinement if necessary.

###  2.4 Vascular network definition

The topology of the fluid network can either be reconstructed from images [59,
60, 48] or generated by using branching (either deterministic or stochastic)
algorithms [61, 62]. From CT or MRI scans it is possible to obtain, after
segmentation, networks of large blood vessels [59] but diameters smaller than
500 $\mu$m are generally below the standard angiography spatial resolution. On
the other hand, microvascular networks can be digitally reconstructed, down to
the capillary scale, by leveraging recent developments in high-resolution
microscopy [60]. To facilitate the integration of experimental datasets in
network models, methods were developed for classifying microvessels without
requiring any flow information [60] or estimating flow rate with unknown
boundary conditions [63]. The microvascular arrangement and consequent levels
of blood perfusion strongly depend on local morphological and functional
properties of the tissue such as O2 consumption rate, level of venous drainage
and presence of anastomosis [64]. Skeletal muscles are perfused by bundles of
parallel capillaries which transport blood from a common terminal arteriole to
the same post-capillary venule. Mendelson et al. [48] used intravital video
microscopy to show that capillary modules may present substantial structural
differences in terms of topology, as well as functional heterogeneities in
RBCs haemodynamics within the same module and across modules. This study
proposed a new paradigm for capillary flow organization which is pivotal for
explaining the complex flow regulation occurring during skeletal muscle
contraction. Capillary modules are indeed interconnected to form continuous
columns, defined as capillary fascicle, that align with the structure of the
muscle fascicle [48]. Blood flow through the arterial-venous vasculature
(arteries $\rightarrow$ arterioles $\rightarrow$ capillaries $\rightarrow$
venules$\rightarrow$ veins) is complex, dictated by varying vessel wall
morphology and geometry. Due to this, multi-scale network models can be used
to accurately capture the blood flow transition across the mesoscale (large to
microvessel) [61, 62].

To compute haemodynamic variables along a portion of the vasculature, it is
necessary to define the boundary conditions at the network extremities to
represent the remaining part of the cardiovascular system that is not
explicitly modelled in space (see for instance [65]). For computing blood flow
in large arteries during reactive hyperemia, the inlet boundary condition
could be imposed either at the heart level or at a (proximal) brachial artery
site (as long as the inlet is not beyond the occlusion site). However, the
latter case is practically less feasible than the former as it requires the
prescription of both flow and pressure (which are difficult to measure
simultaneously) for not omitting the potential contribution of backwards
reflections. The left ventricle (or the proximal aortic node) can therefore be
used as the inflow prescription point [24, 66, 20, 67], with its complex
pumping function simplified using a lumped model, which is able to capture the
essential features of inflow data.

To close the fluid network, it is always necessary to define appropriate
outflow conditions, which represent the haemodynamics characterizing the
downstream systems. Venous hydrodynamic conditions (i.e., position of the
limb) may play a significant role in the experimental settings [18], as they
directly affect the downstream pressure level. Therefore. inclusion of a part
of the venous system into the computer model may be useful for reflecting
changes in the functional state of its components. Others have also proposed
using a closed loop flow model, in which the arterial network system is
coupled with a venous counterpart at the heart and capillary levels [68, 25].
While this holistic approach incorporates additional features, like the venous
network, translation of this model to a patient-specific context is hindered
by the large numbers of parameters which need to be identified/measured. Due
to their simplicity and adaptability, lumped Windkessel models represent the
most popular approach for modeling vascular territories beyond the main system
of interest [25]. These lumped models can be combined in series to form
complex interconnected compartments, allowing to establish a link between
their functional states and the outflow conditions. Alternatively, more
complex methods can be used to represent the downstream circulation, including
the structured (or fractal) tree approach [69], its derived versions [70, 71],
and the porous-media based approach [72, 73]. While these models encapsulate
more details of the downstream network, they come with higher computational
costs.

##  3 Coordinated regulation

Tissue ischemia generates hyperpolarization of the smallest microvessels
(capillaries mainly) which is then conducted upstream through EC gap junctions
to relax the arteriolar vasculature and favour a more consistent/systemic
increase in blood flow. In the time period between tissue ischemia generation
and the full recovery of the system, intrinsic pressure and flow control
mechanisms simultaneously work together to restore homeostasis; the entirety
of which is extremely complex and difficult to quantify [74, 75].

###  3.1 Generation of the hyperpolarizing signal

Tissue metabolism involves the production and consumption of different
substances, some of which have been traditionally associated with hypoxia
(i.e., adenosine, K+, CO2, lactate and H+). In reactive hyperemia, the
mismatch between oxygen demand and supply is caused by a sudden decrease in
oxygenated blood flow and not by a change in tissue metabolic activity, which
instead drives functional hyperemia. The review by Rosenberry and Nelson [12]
reports an exhaustive description of the cellular pathways involved from the
stimulus generation to vessel dilation. Accordingly, the stimulus originating
the hyperpolarization is yet to be defined but most likely involves a
combination of K+, ATP, bradykin, H2O2 and epoxyeicosatrienoic acids. These
molecules cause vessel hyperpolarization by activating different vascular
channels including inwardly rectifying potassium channels and Na+/K+ pumps
[15]. Recent findings [76] on the capillary EC function showed that membrane
hyperpolarization may trigger a complex cascade of Ca2+ signaling pathways
which include cross-talk between intracellular Ca2+ stores and transient
receptor potential channels. In a hypoxic environment, RBCs may also
contribute to microvascular dilation by releasing substances into the
bloodstream such as ATP [77, 78, 79, 80, 81, 82].

In the computer model, the originating stimulus can be represented either as a
single variable in a set of kinetic equations describing oxygen/tissue
signaling or simply as perturbation of the concentration of species in the
extracellular space. In the reference model by Moshkforoush et al. [83] the
effect of cerebral metabolic activity on capillary ECs was represented as an
increase in extracellular $K^{+}$ concentration, which in turn affects
transmembrane K+ channels and membrane potential. Hyperpolarization of
capillaries can be modeled by employing electrochemical models [84, 85, 83,
86] describing the dynamics of intracellular species (such as Ca2+, K+ and
Na+) and membrane potential in stimulated ECs. These models are considered as
‘space-homogeneous‘ (or lumped/0D) because their variables represent
quantities averaged over the cell volume and therefore only dependent on time.
In the seminal work by Wiesner et al. [84] the effect of an exogenous agent
(thrombin) was simulated by combining the EC Ca2+ dynamics with kinetics
models representing the time evolution of ligands involved in the receptor-
activation pathway. Modeling studies that quantitatively characterize the
function of a component of this orchestra, such as [86], are therefore
instrumental for deciphering the emergent cell dynamics features. The
capabilities of space-homogeneous models can be further extended by adding
transport equations for describing diffusion of species [87] or models
representing cytosolic micro-domains [88] for assessing the interaction
between sub-cellular components. Space-homogenous models allow an easy
extension to cluster of cells for computing the resulting intra-cellular
signaling. On the other hand, this would be hard to achieve in case a multi-
dimensional approach for cell dynamics was used. Therefore, cellular space-
homogeneous models represent the most trivial option as bottom level of a
multi-scale vessel framework. Furthermore, O2-induced release of ATP from RBCs
across capillary networks can also be accounted for in the model by following
the work by Goldman et al. [89, 90]. However, a recent theoretical study [91]
suggested that, with respect to other O2-driven regulatory pathways, the role
of vasoactive molecules release by RBCs is limited.

###  3.2 Conducted vasodilation

Ascending vasodilation allows to export the local capillary hyperpolarization
to a widespread fraction of the microvasculature. The endothelium serves as
the predominant cellular pathway for signal conduction as the
hyperpolarization is transmitted from cell to cell along the vessel wall
through gap junctions [1]. According to Welsh et al., the conduction of the
signal is enabled by a defined pattern of charge movement along the vascular
wall which is the result of the interplay between tissue structure, gap
junction resistivity and ion channel activity [92]. Despite the vessel axis
being the main direction of propagation, the hyperpolarizing signal is also
transmitted into surrounding SMCs through myoendothelial junctions to promote
vascular relaxation [93]. Endothelial Ca2+-activated K+ (SKCa/IKCa) channels
seem to play a key role in tuning electrical conduction along microvessels by
modulating signal dissipation through changes in transmembrane resistance
[94]. Experiments in rat cremaster arterioles indicate that vasoactive
conduction is favoured by circulation of autocrine and paracrine mediators ATP
and K+, which is increased under hyperemic conditions [95].

Early efforts in modeling electrical communication across vascular branching
include the works by Diep et al. [96], in which vascular intercellular gap
junctions in the skeletal muscle were represented by ohmic resistors. For
investigating the signal transduction involved in conducted vaso-reactivity,
Kapela et al. integrated their previous modeling efforts [85, 97] into a
multicellular model of a rat mesenteric arteriole which comprises a detailed
description of ECs-SMCs units, coupled via non-selective gap junctions [98].
This framework was used in a subsequent study [99] for estimating from
experimental measurements the electrical resistance associated to cell
membrane and gap junctions. More recently, the same group presented an
experimentally-validated computational framework for reproducing the
propagation of the hyperpolarizing signal across a cerebral microvascular
network [83]. This provides some key information regarding how the activity of
different ion channels can impact on the ascending signal and its re-
generation. However, this framework neither quantifies the dynamic changes in
microvascular compliance nor indicates whether the metabolic demand is
satisfied but it represents a solid modeling base to start from.
Alternatively, Arciero et al. [100] developed a multi-compartmental framework
based on a previous autoregulation wall mechanics model to represent the
adjustments in diameter across a microvascular network induced by vaso-
conducted metabolic response. In this work the stimulation considered was the
hypoxic-induced ATP release by RBCs and the hemodynamic model was coupled with
Krogh-type models for representing the mass exchange with tissue.

###  3.3 Vessel dilation

The conducted changes in membrane potential spread from capillaries to
arterioles. Indeed, in rabbit skeletal muscle [101, 102] and studies on brain
function [103] capillaries are able to control local blood through pericytes
contraction. In the study by Horn et al. [104], a feed artery occlusion caused
a heterogeneous flow distribution in capillary beds with the presence of
capillary no-reflow, and this appeared as result of the complex interaction
between myogenic and metabolic mechanisms occurring at the arteriolar level.
In simulated capillary modules, RBCs flow appeared to be independent from the
network resistance [105]. This indicates that RBCs flow across capillaries is
mainly regulated at control sites located at-pre and post-capillary levels.
Overall, the role of single capillaries on skeletal muscle local flow
regulation still needs to be fully elucidate but its extent seems minor with
respect to the upstream microcirculation [106]. Experimental evidence
indicated that also venules and veins are endowed with a contractile
apparatus, and therefore they may play a role in the pressure re-distribution
occurring during and after the arterial occlusion [107, 108, 109]. However,
their high blood volume capacity suggests a minor role of these vessels
especially in the first part of haemodynamic response after occlusion release.

The level of dilation of each microvessel depends on the magnitude of the
hyperpolarization signal that has reached its location and by the ion channels
expression levels of its wall constituents [110]. A change in the wall
membrane potential promotes the increase in vessel diameter in potentially two
ways. First, as direct effect, hyperpolarization of the SMC causes a decrease
in intracellular Ca2+ which ultimately diminishes the activity of myosin light
chain kinase with reduction of cross-bridges formation. A second effect on the
contractile machinery may be mediated through ECs’ Ca2+ dynamics. Indeed, in
these cells, the decrease in membrane potential causes an increase in
intracellular Ca2+ which promotes pathways the formation of autocoids such as
NO which ultimately affect the cross-bridges by phosphorylating the myosin
light chain phosphatase. Both pathways, by reducing the fraction of attached
actin and myosin filaments, leads to distension of the SMC and consequent
vessel dilation. However, Crecelius et al. [15] showed that, in the forearm,
there is no combined role of NO and PGs in peak RH. Therefore, EC Ca2+
dynamics appears to be crucial for the conduction of the hyperpolarizing
signal along the endothelium and to the adjacent SMCs while its paracrine
function is less relevant. In skeletal muscle resistance arteries, Ca2+
sensitization pathways are also known to play a role in the myogenic response
[111], but these seem to be present only above a certain blood pressure level
($\sim$ 60 mmHg) [112].

To accurately model the change in vessel diameter due to membrane
hyperpolarization a mechanistic multi-scale model is required. The contractile
machinery model needs to translate the biochemical information from SMC Ca2+
signaling into tissue deformation. For this purpose space-homogenous models
for describing cellular signaling seems the most viable choice. The SMC Ca2+
dynamics model developed by Kapela et al. [113], based on 26 partial
differential equations, provides a comprehensive description of the
interaction between several ionic channels upon different agonist stimulation.
Pioneering work by Yang et al. [114, 115] bridges the multi-scale components
of SMC contractility. This framework, devised for computing the myogenic
response in rat cerebrovascular arteries, includes a description for the
cross-bridges kinetics and corresponding cell length variation, which
establishes a mechanistic link between intracellular Ca2+ variation to
structural vessel deformation. Similarly, Layton’s group developed other
reference models [116, 117] able to accurately mimick the myogenic response in
renal afferent arterioles. On the contrary, Carlson and Secomb [118] presented
a minimalistic vessel wall mechanics model that, without including any
cellular/sub-cellular component, provides a good agreement for both passive
and active tension against myograph experimental data from single isolated
microvessels. This type of approach is extremely attractive due to its
simplicity and limited computational cost but a dynamic interplay with ECs
network carrying the hyperpolarizing signal remains challenging to establish.
The multi-scale models following the approach by Murtada et al. [119, 120, 27,
28] provide a rigorous way to encapsulate the information regarding the actin-
myosin interaction into a tissue/continuum mechanics level by using an active
strain energy function. This class of models represents the best option for
coupling the resulting wall deformation and stress with the blood flow within
the vessel.

###  3.4 Regulated networks and clinical data integration

To date, there is a limited number of models able to describe the coordinated
haemodynamic response to arterial occlusion through a multi-scale vascular
system. The work by Yamazaki and Kamiyama [121] is one of the earliest attempt
to simulate flow-mediated dilation in a conduit vessel by including multiple
scales (from cell to vessel) into one modeling framework. Jin et al. [122],
extended this work by including a realistic blood flow network model. This
model closely approximates experimental recordings made at the conduit artery
level but its representation of the downstream microvascular dynamics remains
limited. Some studies [123, 124, 125] have also assessed the impact of
microvascular diameter changes on blood rheology and tissue perfusion across
different vasculatures. It is however important to note that, despite such
models allowing quantification of flow characteristics upon an imposed
variations in microvessel resistance, there is no direct link with a realistic
coordinated regulatory response.

Due to their intrinsic simplicity, compartmental lumped models represent the
standard approach for evaluating the blood flow dynamics across different
levels of the micro-vasculature, without directly describing propagation
phenomena. For each compartment, vascular changes due to regulation can be
represented by varying (in time) its lumped parameters (such as resistor and
capacitor, see Figure 3) according to a more or less complex internal variable
evolution equations [126, 127, 128, 129, 130].

Figure 3: Example of regulated blood flow compartmental model with potential
clinical data integration. $P_{ai}$: inlet arterial pressure; $Q_{a}$:
arterial flow; $R_{a}$: arterial resistance; $C_{a}$: arterial compliance;
$Q_{ma}$: micro-arterial/arteriolar flow; $R_{ma}$: micro-arterial/arteriolar
resistance; $C_{ma}$: micro-arterial/arteriolar compliance; $Q_{c}$: capillary
flow; $R_{c}$: capillary resistance; $P_{vi}$: inlet venous pressure;
$P_{ext}$: external pressure from surrounding tissue. $R_{ma}$ and $C_{ma}$
are modulated to reflect vascular changes due to regulatory mechanisms (green
arrows).

de Mul et al. [126, 131] were among the first to describe the fluid-dynamics
occurring during post-arterial occlusion reactive hyperemia by using
compartmental modeling, through which vascular territories are split into
arterial and capillary systems. In [126], measurements from Laser-Doppler
perfusion monitoring were used for estimating two time constants
characterizing the dynamics, which differed across different types of patients
(healthy vs unhealthy). Likewise, Vo et al. [127] modelled reactive hyperemia
response to venous occlusion and by using NIRS data. Solovyev et al. [128]
defined an hybrid model, combining a lumped model for blood flow with an agent
based model of skin injury, to assess the role of blood flow on the
development of pressure ulcers in spinal-cord injury patients. Recently, a
multi-compartmental model incorporating oxygen transport, tissue metabolism,
and vascular regulation mechanisms was proposed by Chen and Wright [129] to
characterize and interpret MRI-derived microvascular reactive hyperemia (by
arterial spin labeling) in calf muscles of human volunteers. Through all these
modeling approaches it is generally possible to evaluate quantities that can
be directly associated with in-vivo experimental measurements carried out in
human patients. However, this type of approach requires the condensation of
most regulatory mechanisms within few model parameters, which might not have a
clear physical meaning and that in most of the times need to be estimated by
experimental fitting. At the same time, some of these compacted model
parameters can be used for a straightforward discrimination of different types
of cardiovascular profiles. As such, given the complexity of reactive
hyperemia, this class of models is likely most appropriate for capturing the
interaction between vascular compartments and their contribution to the
regulated haemodynamic response.

##  4 Discussion and concluding remarks

There is a pressing need to enhance our understanding on microvascular
function and its regulation. Advances in this direction will ultimately lead
to the development of new diagnostic, progress monitoring and therapeutical
approaches for diseases characterized by (micro-)vascular dysfunction. Through
this work, we sought to provide a guide for modeling reactive hyperemia, which
represents a standard method for assessing microvascular reactivity. Based on
the current knowledge on reactive hyperemia, we reported a series of models
that can be used for describing physiological phenomena that occurs at
different space and time scales, spanning from cellular ionic currents to
systemic circulation flow dynamics. Several issues hinder the accurate
modeling of reactive hyperemia. First and foremost, from an experimental point
of view, isolating single vascular control mechanisms and vascular compartment
contribution to the resulting haemodynamics is technically very challenging if
not impossible. On the other hand, it is possible to measure the combined
response at locations across different experimental conditions, in order to
best inform the computer model. The use of multiple simultaneous recordings of
blood flow via Doppler ultrasound as well as RBCs flux via NIRS-DCS may also
be helpful in the identification of model parameters and quantification of
their uncertainties [132].

There is need to define, test and use suitable modeling approaches that,
despite the complexity of the system analysed, are capable of capturing the
relevant interactions between vascular compartments. Furthermore, there is
need for computational models that can efficiently deal with different scales
in space and time as this aspect is essential for the successful translation
of the computer model into clinical practice. Connections between components
of the hierarchical multi-scale models play a crucial role. Further
investigation and validation is needed before modeling may inform clinical
decision making. To this end, high-fidelity models that leverage large
existing datasets augmented by machine learning/data-driven approaches may be
needed.

## Declaration of competing interest

The authors declare that there is no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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