# Different impact of heart rate variability in the deep cerebral and central
hemodynamics at rest: an _in silico_ investigation

Stefania Scarsoglio 1,∗, and Luca Ridolfi 2

###### Abstract

Background: Heart rate variability (HRV), defined as the variability between
consecutive heartbeats, is a surrogate measure of cardiac vagal tone. It is
widely accepted that a decreased HRV is associated to several risk factors and
cardiovascular diseases. However, a possible association between HRV and
altered cerebral hemodynamics is still debated, suffering from HRV short-term
measures and the paucity of high-resolution deep cerebral data. We propose a
computational approach to evaluate the deep cerebral and central hemodynamics
subject to physiological alterations of HRV in an ideal young healthy patient
at rest.

Methods: The cardiovascular-cerebral model is composed by electrical
components able to reproduce the response of the different cardiovascular
regions and their features. The model was validated over more than thirty
studies and recently exploited to understand the hemodynamic mechanisms
between cardiac arrythmia and cognitive deficit. Three configurations
(baseline, increased HRV, and decreased HRV) are built based on the standard
deviation (SDNN) of RR beats. For each configuration, 5000 RR beats are
simulated to investigate the occurrence of extreme values, alteration of the
regular hemodynamics pattern, and variation of mean perfusion/pressure levels.

Results: In the cerebral circulation, our results show that HRV has overall a
stronger impact on pressure than flow rate mean values but similarly alters
pressure and flow rate in terms of extreme events. By comparing reduced and
increased HRV, this latter induces a higher probability of altered mean and
extreme values, and is therefore more detrimental at distal cerebral level. On
the contrary, at central level a decreased HRV induces a higher cardiac effort
without improving the mechano-contractile performance, thus overall reducing
the heart efficiency.

Conclusions: Present results suggest that: (i) the increase of HRV _per se_
does not seem to be sufficient to trigger a better cerebral hemodynamic
response; (ii) by accounting for both central and cerebral circulations, the
optimal HRV configuration is found at baseline. Given the relation inversely
linking HRV and HR, the presence of this optimal condition can contribute to
explain why the mean HR of the general population settles around the baseline
value (70 bpm).

##  1 Keywords:

heart rate variability, cardiovascular modeling, cerebral circulation,
computational hemodynamics, time-series analysis

##  2 Introduction

Heart rate variability (HRV), defined as the variability between successive RR
heartbeats, has grown as _hot topic_ , with a simple ”Heart rate variability”
topic search currently listing more than 44,000 results on Web of Science. The
great interest elicited within the scientific community ranges from
psychophysiology [1] to exercise [2], cardiovascular risk factors [3], and
chronic fatigue syndrome [4]. This increasing and wide interest is also due to
the fact that HRV measurements are easy to perform, quite reliable and non-
invasive. In particular, HRV can be considered a surrogate measure of the
cardiac vagal tone [1], although there are controversial aspects in using HRV
to determine cardiac vagal tone indep endently from other factors, such as
age, physical condition, and the presence of cardiac pathologies [5]. In this
paper, we refer to the parasympathetic activity within cardiac regulation as
cardiac vagal tone. HRV parameters in time and frequency domain able to
reflect cardiac vagal tone [1, 6, 7] show that an increased cardiac vagal tone
is associated to a higher HRV, while a reduction of the cardiac vagal tone
response is linked to a HRV reduction [8, 9]. Moreover, it has recently being
recognized that HRV (as measured by means of different metrics, such as SDNN,
RMSSD, pvRSA, and HF), inversely correlates with the heart rate (HR), and this
relation should be taken into account when dealing with increased/reduced HRV
[10, 11, 12, 5]. The inverse relation between HR and HRV can be explained
considering that a faster HR reduces the interval between successive beats and
the opportunity for the RR beating to vary, lowering the HRV. On the contrary,
a slower HR increases the cardiac interval and enhances the chance for RR to
vary, raising HRV [11].

From an overall cardiovascular point of view, a HRV increase in the
physiological range seems beneficial at rest, while a reduced HRV is
symptomatic of a stress action, can be predictor of cardiovascular risk
factors and is correlated to higher morbidity and mortality [3, 13, 4]. Thus,
the clinical interest has been mainly focused on the prognostic significance
of HRV related to the risk factors of cardiovascular pathologies. There is
substantial evidence that a decreased HRV enhances several risk factors and is
associated to cardiovascular diseases, such as left ventricle hypertrophy
[14], sudden cardiac death [15], and myocardial infarction [16]. On the
contrary, lower risk factor profiles are associated with increased HRV [17].
The HRV scenario is opposite under physical effort: for increasing moderate-
to-vigorous exercise intensity HRV decreases [18, 2]. During exercise a HRV
decrease is beneficial and forced by an increased HR, however there is an
immediate post-exercise recovery of HRV [2] and exercise training can increase
resting HRV [19].

Although the overall HRV impact on the central hemodynamics is quite clear,
especially related to cardiac disease conditions, its influence on the
cerebral hemodynamics is barely known so far. In fact, definitive association
between HRV and cerebral circulation is still missing and mainly involve
pathologic conditions, such as chronic fatigue syndrome [4, 20] and stroke
[21]. Attention has recently grown about the possible association between
increased HRV and improved cognitive performance. Results are conflicting, as
some studies have demonstrated the association [22, 23, 24], while some others
observed that a reduced HRV does not contribute to cognitive impairment or
even dementia [25, 26, 27, 28]. To understand this controversy it should be
kept in mind that, among many influencing factors such as the sample and the
objectives of the study, most of the literature is based on short term
measures, which have lower prognostic value than 24-hours HRV [6, 29, 11].

Though a possible association is widely debated, the underlying hemodynamic
mechanisms are so far only hypothesized and mostly unclear [24]. It is
presently unknown whether HRV is able to influence the normal pressure and
flow rate pattern at distal-capillary level, to alter the mean perfusion and
pressure levels, and to enhance the occurrence of extreme values. Detailed
information are lacking since current non-invasive techniques, such as
transcranial doppler ultrasonography, are not able to offer high-resolution
data in terms of pressure and flow rate beyond the circle of Willis.
Therefore, in the deep cerebral circulation hemodynamic measurements are
unreliable or almost absent. Thus, it can be of great significance
understanding and predicting from a computational point of view how deep
cerebral hemodynamics is affected by HRV at rest. Despite the intrinsic
limits, computational hemodynamics and cardiovascular modeling are becoming
increasingly important to isolate specific mechanisms and investigate
processes where clinical data are not yet feasible, accurate, or easily
measurable.

We propose an _in-silico_ study where, through a validated combined
cardiovascular-cerebral model, we evaluated the deep cerebral hemodynamics
subject to physiological alterations of HRV in an ideal young healthy patient
at rest, thereby comparing the cerebral to central hemodynamics responses to
HRV changes. With deep cerebral circulation we refer to the microcirculation
beyond the circle of Willis (i.e., distal and capillary-venous circulation),
while the set of parameters defined through cardiac variables and central
arterial pressure characterizes the central hemodynamics. The complete lumped-
parameter model is composed by a suitable combination of electrical
counterparts (resistances, compliances/elastances, inertances), accounting for
the arterial and venous circuits of both systemic and pulmonary circulations,
an active representation of the four cardiac chambers, an accurate valve
motion description, and a short-term baroreceptor mechanism. The cerebral
circulation is divided into three main regions: large arteries, distal
arterial circulation, and capillary/venous circulation. Cerebrovascular
control mechanisms of autoregulation and $CO_{2}$ reactivity are taken into
account. The present computational approach has been validated and recently
exploited in a wide area of applications, such as the hemodynamic response to
exercise in atrial fibrillation [30], the linking mechanisms between cardiac
arrythmia and cognitive impairment [31, 32, 33, 34], and the cardiovascular
deconditioning emerging in altered gravity conditions [35].

We considered three HRV configurations (baseline, increased HRV, decreased
HRV), each of them analyzed over 5000 RR beats, so that our results are not
affected by transient behavior and are statistically significant and stable.
HRV variations were assessed through SDNN, which is the standard deviation of
all RR beats and allowed us to define in a straightforward and univocal way
the RR stochastic beating extraction. The model is able to provide the whole
pressure and flow rate time-series from proximal to distal circulation as well
as beat-to-beat values. The focus was on the possible occurrence of extreme
values, such as hypertensive events, alteration of the regular hemodynamics
pattern, and variation of mean perfusion/pressure levels. All these aspects
are to date mostly unexplored, but extremely useful to understand the role of
HRV on the cerebral circulation and how, in turn, the hemodynamic alteration
can impact the cognitive sphere. The present work can offer precious insights
by: (i) isolating the net basic mechanisms induced by HRV changes into the
cerebral circulation in healthy resting conditions, and (ii) quantifying how
differently HRV acts on the cerebral hemodynamics with respect to the central
hemodynamics, thus fostering necessary future clinical measurements within
this topic.

##  3 Materials and Methods

###  3.1 HRV configurations

We considered three HRV configurations: baseline condition (reported in the
following sections with blue color), increased HRV (in red color) and
decreased HRV (in black color). To assess HRV variations we adopted the
simplest HRV metric, i.e., SDNN - which is the standard deviation of all RR
beats. SDNN belongs to the time-domain analyses, which are computationally
simpler and easier to apply than frequency-domain analyses [2]. Since we
focused on resting configurations lasting hours, we chose SDNN because it is
the gold standard for long-term measurements [6, 11]. Moreover, the SDNN
adoption allows us to define in a straightforward and univocal way the RR
beating extraction. In terms of SDNN, HRV was found to be dependent on the
circadian day/night cycle [11] \- e.g., 93.06 [ms] day, 121.31 [ms] night,
101.71 [ms] 24-hours [36] \- and to decrease with age in adulthood [37]. In
general, SDNN recording was lower in resting supine condition (65 [ms] [38];
49 [ms] [29]) than in non-resting or upright position (e.g., 93.06 [ms] [36];
127 [ms] [39]). It was recently observed that HRV inversely correlates with HR
through an exponential function [10, 40, 11, 37, 12], and this relation should
be taken into account when dealing with increased/reduced HRV.

To consider SDNN for a healthy young person in resting supine condition during
day (awake) and account for the interplay between SDNN e HR, we adopted the
exponential relation obtained by de Geus et al. [12] in leisure baseline
condition:

$\textmd{SDNN}=309.4\textmd{e}^{-0.021\textmd{HR}}$ (1)

The baseline configuration was set at HR=70 bpm and SDNN=71.14 [ms] was
obtained from the above relation. Recalling that the beating period is
RR=60/HR [s], we define the coefficient of variation as $c_{v}$=SDNN/RR. The
resulting $c_{v}$ is 0.08 which well represents normal sinus rhythm daily
values in resting supine condition (e.g., $c_{v}$=0.05 [41]; $c_{v}$=0.07
[38]; $c_{v}$=0.06 [42]). Moreover, $c_{v}$=0.08 as baseline value is close to
the one adopted by our group ($c_{v}$=0.07) to simulate resting supine sinus
rhythm [43]. Increased and decreased HRV configurations were achieved by
changing SDNN by +20% and -20%, respectively. This threshold represents a
significant variation within a range of physiological values. The
corresponding HRs were individuated again using Eq. (1) as proposed by [12].
Fig. 1a reports the SDNN(HR) curve in resting condition introduced by [12]
with the values here chosen and below a summarizing table of the three cases.

For each configuration, we evaluated 5000 RR beats to have statistically
significant and stationary results. RR beatings were extracted from an _in
silico_ pink-correlated Gaussian distribution, which well reproduces the
typical beating features of normal sinus rhythm recorded in vivo [44, 41, 45,
43], having mean and standard deviation values identified as above. In the
range SDNN $\in$ [30, 100] ms, we also evaluated the RMSSD (i.e., the root
mean square of successive RR interval differences) for the RR series extracted
with the adopted time-correlation features and composed by 108 beats. A high
linear correlation value ($R^{2}=0.99$) was found between RMSSD and SDNN, with
fitting law RMSSD=0.65 SDNN + 0.54, as reported in Fig. 1b. Although RMSSD and
SDNN don’t have the same physiological origin and only RMSSD can be assumed to
fully reflect cardiac vagal activity [1, 6, 7], SDNN is a reliable proxy of
RMSSD for the considered RR time series. Bottom panels of Fig. 1 display the
RR series (panel c) and the corresponding probability density functions (PDFs)
for the three HRV cases here investigated (panel d).

###  3.2 Cardiovascular and cerebral circulation modeling

Following RR extraction, the cardiovascular-cerebral model was run to obtain
the hemodynamic cerebral signals. The complete lumped model is composed by
electrical components able to reproduce the response of the different (cardiac
and vascular) regions and their features, in terms of resistance $R$
(diffusive effects), inertance $L$ (inertial effects), and compliance $C$ or
elastance $E$ (elasticity/contractility effects). The cardiovascular dynamics
includes the arterial and venous circuits of both systemic and pulmonary
circulations, an active representation of the four cardiac chambers, and an
accurate valve motion description. A short-term baroreceptor mechanism is also
modeled, accounting for the inotropic effect of both ventricles, as well as
the control of the systemic vasculature (peripheral arterial resistances,
unstressed volume of the venous system, and venous compliance). The
chronotropic effects due to the heart rate regulation are intrinsically taken
into account by the RR extraction. The cerebral circulation is divided into
three principal regions: large arteries, distal arterial circulation, and
capillary/venous circulation. Cerebrovascular control mechanisms of
autoregulation and $CO_{2}$ reactivity are taken into account. The model is
expressed in terms of pressures, $P$, flow rates, $Q$, volumes, $V$, and valve
opening angles, $\theta$.

The cardiovascular model has been proposed and validated over more than thirty
studies to check the consistency of the hemodynamic response during AF:
extensive details of this evaluation are reported in [43]. Then, the model has
been exploited to study the impact of atrial fibrillation on the
cardiovascular system [46, 47, 48, 30]. The complete cardiovascular-cerebral
model has been used to understand the hemodynamic mechanisms between atrial
fibrillation and cognitive deficit [31, 32, 33, 34]. A full description of the
governing equations and model parameters is given in the Supplementary
Material.

We here focused - in terms of pressure $P$ and flow rate $Q$ \- on the left
internal carotid artery-middle cerebral artery (ICA-MCA) pathway, which
already turned out to be representative of the hemodynamics from proximal to
distal cerebral districts [31, 32, 33, 34]. The left ICA-MCA path starts at
the internal carotid level ($P_{a}$: systemic arterial pressure;
$Q_{ICA,left}$: left internal carotid flow rate), goes through the middle
cerebral artery ($P_{MCA,left}$: left middle cerebral artery pressure;
$Q_{MCA,left}$: left middle cerebral artery flow rate), includes the middle
distal regions ($P_{dm,left}$: left middle distal pressure; $Q_{dm,left}$:
left middle distal flow rate), and ends with capillary-venous districts
($P_{c}$: cerebral capillary pressure; $Q_{pv}$: proximal venous flow rate). A
schematic representation of the systemic-pulmonary circulation and ICA-MCA
pathway is reported in Figure 2, together with examples of left ventricle PV
loops, aortic flow rate ($Q_{ao}$), systemic arterial pressure ($P_{a}$), and
cerebral capillary pressure ($P_{c}$) time-series for different HRV
configurations.

###  3.3 Variable definition and data analysis

We recall the definition of mechano-energetic and oxygen consumption indexes,
which will be used to describe the central hemodynamics. End-systolic left
ventricular volume, $V_{lves}$ [ml], is the left ventricle volume at the
closure of the aortic valve, while end-diastolic left ventricular volume,
$V_{lved}$ [ml], corresponds to the closure of the mitral valve. Stroke volume
is defined as $SV=(V_{lved}-V_{lves})$ [ml], ejection fraction is
$EF=V/V_{lved}\cdot 100$ [%]. Cardiac output is $CO=SV\cdot HR$ [l/min],
stroke work per minute, $SW/min$ [J/min], is measured as the area within the
left ventricle pressure-volume loop per beat. Oxygen consumption is evaluated
through three estimates [49]: (i) the rate pressure product,
$RPP=P_{a,syst}\cdot HR$ [mmHg/min], where $P_{a,syst}$ is the aortic systolic
pressure; (ii) the tension time index per minute,
$TTI/min=\overline{P}_{lv}\cdot RR\cdot HR$ [mmHg s/min], where the symbol
$\overline{f}$ indicates the mean value of the generic hemodynamic variable
$f$ averaged over a RR beat (in this case $f=P_{lv}$); and (iii) the pressure
volume area per minute, $PVA/min=(PE+SW)\cdot HR$, where
$PE=[P_{lves}\cdot(V_{lves}-V_{lv,un})/2-P_{lved}\cdot(V_{lved}-V_{lv,un})/4]$
is the elastic potential energy ($V_{lv,un}$ = 5 ml is the unstressed left
ventricle volume), $SW$ is the stroke work, $P_{lves}$ and $P_{lved}$ are the
end-systolic and end-diastolic left ventricle pressures. The left ventricular
efficiency $LVE$ is defined by the ratio $SW/PVA$.

The cerebral hemodynamics is assessed by means of two main approaches: (i)
analysis of the continuous time-series, where the signal is continuous and
defined by all the temporal instants of the whole time-series, and (ii) beat-
to-beat analysis, where the signal was discretized and one-per-beat data were
obtained. In so doing, beat-to-beat signals are composed by 5000 values,
corresponding to the 5000 RR beats simulated.

For the analysis of the continuous time-series, extremely high or low cerebral
hemodynamic values were evaluated through the percentile analysis of the
hemodynamic signals. We adapted the definition used for studying atrial
fibrillation [31, 34], to assess the possibility of extreme events related to
HRV changes. For the increased and decreased HRV cases, we estimated to which
percentile the baseline (10th and 90th) reference thresholds correspond, by
quantifying the probability of assuming rare values. Top panels of Fig. 3 (a,
b) provide a representative graphical representation of the percentile
analysis for $P_{dm,left}$. The 10th and 90th percentiles individuated in
baseline correspond to the 2nd and 85th percentiles in the decreased HRV case
(panel a), while the 10th and 90th baseline percentiles correspond to the 22nd
and 94th percentiles in the increased HRV case (panel b). In so doing, we
quantify whether HRV changes are able to make extreme values more frequently
reached with respect to the baseline case (e.g., in Fig. 3b the 10th baseline
percentile corresponds to the 22nd percentile for increased HRV).

For the beat-to-beat analysis, the $i-th$ element of the discretized series
may contain the average value ($\overline{Q}$ and $\overline{P}$), the maximum
($Q^{max}$ and $P^{max}$) and minimum ($Q^{min}$ and $P^{min}$) values, as
well as the pulse value ($Q^{pv}$ and $P^{pv}$, defined as the difference
between maximum and minimum values) of the related haemodynamic variables
computed over the $i-th$ beat. In Fig. 3d examples of $\overline{P}$,
$P^{max}$, $P^{min}$, and $P^{pv}$ are represented for $P_{dm,left}$.

The average values per beat, $\overline{P}$ and $\overline{Q}$, were still
exploited to enrich the beat-to-beat analysis, by accounting for the
persistence of extreme values over the whole beat [31, 34], and not only the
occurrence of instantaneous peak values. We introduce hypoperfusions (or
hypotensive events), which take place when the mean flow rate per beat
$\overline{Q}$ (or mean pressure per beat $\overline{P}$) is below the 10th
percentile referred to the whole baseline signal. On the contrary,
hyperperfusions (or hypertensive events) emerge when $\overline{Q}$ (or
$\overline{P}$) is above the 90th percentile of the baseline condition. Note
that, by definition, extreme events cannot emerge in the baseline
configuration. Bottom panels of Fig. 3 (c, d) show examples of hypertensive
and hypotensive events for $P_{dm,left}$.

##  4 Results

###  4.1 Continuous $P$ and $Q$ time-series analysis

Table 1 reports the main statistics (mean $\mu$, standard deviation $\sigma$,
and coefficient of variation $c_{v}$) of the continuous time-series of the
hemodynamic variables along the ICA-MCA pathway. Figure 4 shows probability
density functions (PDFs) of the whole hemodynamic time-series for the three
configurations. Considering the continuous hemodynamic signals, HRV influenced
much more pressure than flow rate series. In fact, mean values were overall
maintained (see Table 1) and PDFs revealed self-similar features (see Fig. 4)
for flow rates $Q$, while HRV significantly affected mean values and PDFs of
pressures $P$ (only $P_{c}$ at the end of the pathway regained a self-similar
shape for the three configurations). At a given district, HRV impacted
$\sigma$ relative variations with respect to baseline similarly for both
pressure and flow rate, while $c_{v}$ relative variations induced by HRV with
respect to baseline were more evident towards the distal-capillary
circulation.

It is then useful to evaluate how extreme values of the continuous time-series
$P(t)$ and $Q(t)$ distribute as HRV changes, by considering the percentile
analysis (as represented in Fig. 3a-b). By assessing to which percentile the
baseline reference thresholds correspond in the configurations with altered
HRV, we can quantify whether HRV is able to modify the probability of reaching
extremely high/low values (see Fig. 5). Distal-capillary regions were the most
affected by HRV: the 10th percentile for baseline $P_{dm,left}$ corresponds to
over the 20th percentile for increased HRV, while the 90th percentile for
baseline $P_{c}$ corresponds to about the 82th percentile for the reduced HRV
configuration (see top panels of Fig. 5). It is worth noting that a similar
scenario was found for flow rates as well. The increased HRV enhanced extreme
values of the continuous time-series towards the venous circulation: the 10th
percentile corresponds to the 17th, while the 90th to the 82th (see bottom
panels of Fig. 5).

###  4.2 Beat-to-beat analysis

The scenario of higher impact of HRV on cerebral pressures than flow rates was
also confirmed in the beat-to-beat analysis. Table 2 displays mean and
standard deviation values for the minimum ($P^{min}$ and $Q^{min}$), average
($\overline{P}$ and $\overline{Q}$), maximum ($P^{max}$ and $Q^{max}$), and
pulse ($P^{pv}$ and $Q^{pv}$) values per beat of pressures $P$ and flow rates
$Q$ as computed over 5000 RR beats. PDFs of $\overline{P}$ and $\overline{Q}$
along the ICA-MCA pathway are depicted in Fig. 6. The altered HRV did not
substantially modify $\overline{Q}$ values, with relative variations of
increased and decreased HRV $<$0.1% with respect to baseline, see the
$\overline{Q}$ column in Table 2 and the corresponding PDFs (see Fig. 6, right
panels). Relative variations of minimum ($Q^{min}$) and maximum ($Q^{max}$)
values per beat were within 8%. $\overline{P}$ values as well as the
corresponding PDFs were instead much more affected by HRV: only at the
capillary level, similar $\overline{P}_{c}$ values were recovered for the
three HRV configurations (with relative variations $<$0.1%, see bottom left
panel of Fig. 6 and the mean value column of Table 2), while in all the other
regions relative $\overline{P}$ variations were within 3%. Relative variations
of maximum ($P^{max}$) and minimum ($P^{min}$) values per beat did not exceed
5% with respect to the baseline configuration (see Table 2).

The different impact of HRV change on pressure and flow rate can be explained
noting that cerebral control mechanisms act on the continuous flow rate values
at the distal level only, by modifying pial arterial-arteriolar compliances
and resistances [50]. Thanks to the autoregulation effects, mean values of the
continuous flow rate time-series (Table 1) and beat-to-beat $\overline{Q}$
recordings (Table 2) were preserved unvaried in the three configurations,
while the same was not true for cerebral pressure. Despite cerebral
autoregulation mechanisms are concentrated in the distal district solely,
their effects involve both upstream and downstream regions. In fact, the mean
flow rate is guaranteed as constant for the three configurations throughout
the ICA-MCA pathway (i.e., from $Q_{ICA,left}$ to $Q_{pv}$, see $Q$ mean
values in Table 1 and $\overline{Q}$ in Table 2).

Based on what emerged so far, we expect a higher occurrence of extreme events
in the beat-averaged pressure, $\overline{P}$, than beat-averaged flow rate,
$\overline{Q}$. Table 3 reports the total number of rare one-beats
hypotensive, hypertensive, hypoperfusion, and hyperperfusion events. We recall
that these events, by definition, cannot occur in the baseline configuration,
which is taken to set the reference thresholds (10th and 90th percentiles).
For the other configurations, one of these events happens at a specific
district if the mean value per beat for $\overline{P}$ or $\overline{Q}$
reaches extremely high (above the 90th percentile of the baseline
configuration) or low (below the 10th percentile of the baseline
configuration) values. In particular, out of 5000 RR beats we counted 42
hypotensive events at distal level for the increased HRV configuration, and 69
hypertensive events for reduced HRV. No hypoperfusions were found, while other
more occasional events, such as 9 distal hyperperfusion events, 6 capillary
hypertensive events, and 4 distal hypertensive events, were induced by the
higher HRV configuration. The reason of the low occurrence of extreme
perfusive events can be understood by observing that the widening of the PDFs
for the continuous flow rate time-series caused by increased HRV (see the red
curves in right panels of Fig. 4) is related to instantaneous peak values
rather than enduring events in time. Thus, increased HRV was not able to
trigger significant extreme perfusive events over a whole beat. In fact, PDFs
of $\overline{Q}$ values were quite coincident for the three configurations
(right panels of Fig. 6), leading to very sporadic hyperperfusion events and
no hypoperfusion events at all. It should be also recalled that HRV variations
were taken in a physiological range ($\pm$20% of the baseline value),
therefore it is reasonable not expecting a large number of extreme events
based on averaged-beat variables.

##  5 Discussion

The present study aims at computationally investigating the role of HRV on the
deep cerebral and central hemodynamics. SDNN has been adopted as a proxy
measure for HRV and, although it does not have the same physiological origin
reflecting cardiac vagal activity as RMSSD, it is highly correlated with the
latter and more suited to the proposed computational approach. HRV seems to
have overall a stronger impact on $\overline{P}$ than $\overline{Q}$ values,
due to autoregulation effects acting to maintain an adequate average cerebral
perfusion. However, in terms of extreme (minimum and maximum) values and
percentile distributions, HRV similarly altered both pressure and flow rate.
By comparing reduced and increased HRV, this latter induced a higher
probability of altered mean values and extreme values (the only exception was
the $P_{dm,left}$ case with reduced HRV and hypertensive events), and is
therefore more detrimental at distal cerebral level.

As a further confirmation of the worsening linked to a higher HRV, it is
observed that starting with an increased HRV, the variability of hemodynamic
variables increased from proximal to distal regions along the left ICA-MCA
pathway. On the contrary, starting with a reduced HRV, the hemodynamic
variability decreased towards the deep cerebral circulation. This behavior is
evident for the continuous signals in terms of $c_{v}$ ratio (as reported in
Fig. 7), as well as at beat-to-beat level, through relative variations of
pulse values, $P^{pv}$ and $Q^{pv}$ (see Fig. 8). It should be noted that the
relative variations of $P^{pv}$ and $Q^{pv}$ showed this increasing/decreasing
trend, although $P^{min}$, $P^{max}$, $Q^{min}$, and $Q^{max}$ did not
singularly present, as mentioned before, a similar behavior: relative
variations of $P^{min}$ and $P^{max}$ ($Q^{min}$ and $Q^{max}$) with respect
to baseline did not exceed 5% (8%) along the ICA-MCA pathway (please refer to
Table 2, columns with minimum and maximum values). Moreover, as displayed by
Figures 7 and 8, the behavior was analogously found for both pressure and flow
rate.

Even though all results fell within a range of physiological values, increased
HRV seems more deleterious than reduced HRV along the proximal-to-distal
cerebral pathway. An unsafe increased pulsatility of the hemodynamic signals
was found especially at capillary-venous levels, and can promote extremely
high/low pressure and flow rate values.

This result can be explained recalling what observed for the cerebral
hemodynamics during atrial fibrillation [32]. The intrinsic structural latency
revealed by the cerebral microcirculation and combined with a sequence of in-
series irregular RR beats produced an alteration of the cerebral circulation.
Indeed, due to the delay related to mechanical and structural properties, the
inertia of the system increased towards the microvasculature.

Atrial fibrillation represents a succession of disturbances in terms of RR
beating, which are then transmitted from the carotid level to the deep
cerebral circulation. Each of these perturbations singularly produced an
alteration of the cerebral circulation. The continuous sequence of transient
perturbations at the carotid entrance represented by the fibrillated beating
did not allow the system to recover the physiological state before another
disturbance arrives. When this disturbance chain spread throughout the
cerebral vessel network, the distal and capillary regions remained altered for
longer. The behavior is similar to a system of springs in series and parallel,
which is externally forced at one end: the stiffness of each spring combines
with the others and the oscillation survives even if the external perturbation
is ceased. It is important to note that this mechanism occurred regardless of
whether the beat in atrial fibrillation is accelerated or not. In fact, the
comparison between NSR and atrial fibrillation [32] was proposed at the same
mean heart rate, highlighting the role of RR variability in promoting the
observed hemodynamic alteration.

With due proportions, the basic mechanism detected in atrial fibrillation
seems to be similar in the case of increased HRV. Clearly, fibrillated
outcomes were much more exacerbated and pathological since variability was
much higher, and the RR beating was uncorrelated too. However, also here the
component of increased variability seems to plausibly combine with the
intrinsic latency of the system, causing the observed altered scenario.

The hemodynamic framework is inverted if we focus at central level, in terms
of cardiac contractility, efficiency end energetic indexes (see Table 4). The
decreased HRV configuration showed an increase of oxygen consumption indexes
(+16% $RPP$, +6% $TTI/min$, +11% $PVA/min$) and cardiac work (+9% $EW/min$),
in front of a reduction of the ejection fraction (-5%) and a limited increase
of cardiac output (+6%). The reduction of the left ventricle efficiency $LVE$,
though slight (-2%), confirms that a higher cardiac effort did not translate
into a better mechanical and contractile cardiac performance. Moreover,
variations of mean central flow rate (in terms of $CO$) did not entail average
changes of the flow rate in the deep cerebral circulation (see cerebral mean
flow rates in Tables 1 and 2), due to the autoregulation mechanisms at the
distal cerebral level. In other words, the greater central energy expenditure
did not produce any gain in terms of mean flow rate in the cerebral
microcirculation.

To extend the discussion, we considered a configuration of heart failure as
representative of a very different condition from the healthy state and often
present as a co-morbidity in the elderly population. With respect to the
baseline heart failure condition, we analyzed two HRV configurations also
here, increased (+20 % SDNN) and decreased (-20 % SDNN) HRV. Details of the
heart failure configurations are provided in the Supplementary Material. For
the deep cerebral hemodynamics, the percentile analysis and the PDFs of the
continuous hemodynamic time-series in the heart failure case are reported and
compared with the healthy case in Fig. 9. Although the values achieved are
different from the case healthy (see the blue thick and thin curves), the HRV
variations with respect to the baseline are quite close to the healthy case
(see the red/black thick and thin curves), especially in the distal regions.
Table 5 shows the values of the central hemodynamic parameters in the heart
failure case. Compared to the healthy case, the values achieved are very
different and show a rather compromised hemodynamic framework (e.g., at
baseline $CO$ = 3.55 l/min, $EF$ = 16.41 %, $LVE$ = 0.31). However, the HRV
variations compared to baseline are in agreement with the healthy case
(decreased HRV: +12% $RPP$, +3% $TTI/min$, +10% $PVA/min$, +4% $EW/min$, -8%
$EF$, +3% $CO$, -5% $LVE$).

For both healthy and heart failure cases, at central level there is a
qualitative agreement with what has been widely observed in literature, that
is an increased HRV is beneficial for the cardiac performance and efficiency.
At cerebral level, where instead results are still debated, an increase of HRV
only - regardless any other variation, compensatory mechanism or pathological
condition - seems to be associated to a worse hemodynamic response. A
plausible interpretation of these findings with respect to the possible
association between HRV and cognitive performance observed in some literature
studies is that we only considered the net contribution of HRV alterations,
without any complementary and further variation of the sympathetic system. In
fact, changes of the sympathetic activity are hardly quantifiable and out of
the objectives of the present study. According to the neurovisceral
integration model vagally-mediated HRV metrics, such as RMSSD and HF, are
expected to be linked to cognitive performance, but not SDNN [51]. Based on
our outcomes the increase of HRV, measured by means of SDNN, does not seem to
be sufficient _per se_ to trigger a better cerebral hemodynamic response.

###  5.1 Limitations

The present computational approach has some limiting aspects related to the
modeling hypotheses. We considered the same young healthy resting subject as
forced through three different HRV configurations. In particular, the three
configurations only differ by the entrance inputs, the beating sequence RR,
while the remaining hemodynamic framework is set as in baseline condition,
regardless of other compensatory mechanisms that may occur or altered
sympathetic activity. Cerebral autoregulation and autonomic regulation are not
fully coupled, but this interplay is present in one-way direction only:
through the baroreceptor mechanisms acting on the systemic arterial pressure,
autonomic regulation influences the cerebral autoregulation, but there are no
feedbacks from cerebral blood flow to autonomic control. Moreover, coronary
circulation was not included into the model. In the end, we evaluated a
monitoring temporal window of about one hour (5000 RR beats), thus no long-
term remodelling effects related to increased or decreased HRV were taken into
account.

##  6 Conclusions

In conclusion, by recalling the relation inversely linking SDNN and HR [10,
11, 12], at deep cerebral level a higher SDNN is worse than a higher HR. On
the contrary, at central level a higher HR is more negatively impacting than a
higher SDNN. From an overall point of view which contemporarily accounts for
both central and deep cerebral circulations, present results suggested that
the optimal HRV configuration is found at baseline. In fact, an increased HRV
(lower HR) is detrimental for the cerebral circulation causing possible
hypertensive events and extreme pressure and flow rate values, while a
decreased HRV (higher HR) induces a higher cardiac effort without improving
the mechano-contractile performance, thus overall reducing the heart
efficiency. The presence of this optimal condition is the main finding and can
contribute to explain why the mean HR of the general population settles around
70 bpm: the baseline value is a good compromise able to guarantee an adequate
hemodynamic response for both central and deep cerebral regions, preserving
from extreme cerebral hemodynamic events and at the same time maintaining
satisfactory cardiac efficiency.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

All authors conceived and designed the experiments. SS performed the
experiments. All authors analyzed the data, contributed
reagents/materials/analysis tools. SS wrote the paper. All authors reviewed
and approved the final version of the manuscript.

## Data Availability Statement

The datasets generated for this study are available on request to the
corresponding author.

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## Figure captions

Figure 1: Top panels (a)-(b): Relation SDNN(HR) and RMSSD(SDNN). Middle panel:
table with the chosen values of the three configurations. Bottom panels
(c)-(d): RR series and probability density functions of the three
configurations. Red: increased HRV; blue: baseline; black: decreased HRV.
Figure 2: Sketch of the cardiovascular-cerebral model for the three HRV
configurations (red: increased HRV, blue: baseline, black: decreased HRV).
(left) Central and systemic-pulmonary circulation with representative time-
series of the aortic flow rate ($Q_{ao}$) and left ventricle PV loops (5
seconds length). (right) Cerebral circulation with the three main regions
(large arteries, distal arteries, capillary-venous circulation), together with
examples of the systemic arterial pressure, $P_{a}$ (bottom), and cerebral
capillary pressure, $P_{c}$ (top), time-series. Figure 3: (a)-(b) Examples of
percentile evaluation for $P_{dm,left}$ ($p(P_{dm,left})$ is the probability
density function). 10% and 90% blue dashed lines individuate the 10th and 90th
percentiles in the baseline configuration (panels a-b, blue areas), while they
correspond to the 2nd and 85th percentiles in the decreased HRV configuration
(panel a, black areas), and to the 22nd and 94th percentiles in the increased
HRV configuration (panel b, red areas). (c)-(d) $P_{dm,left}$ examples of
hypertensive event lasting 2 beats in the decreased HRV configuration (average
pressure per beat, $\overline{P}$, is represented by grey horizontal lines and
hypertensive events are marked with bold grey lines) and hypotensive event in
the increased HRV configuration ($\overline{P}$ is represented by red
horizontal lines and hypothensive events are marked with bold red lines). The
10th and 90th percentile baseline thresholds are displayed through the dashed
blue horizontal lines, while full circles indicate the heartbeat extremes. In
panel (d) examples of $P^{max}$ ($\triangle$), $P^{min}$ ($\Box$), and
$P^{pv}$ ($\leftrightarrow$) values per beat are evidenced for $P_{dm,left}$.
Figure 4: Probability density functions of the continuous hemodynamic time-
series along the ICA-MCA pathway. Left panels: pressures. Right panels: flow
rates. Black: decreased HRV; blue: baseline; red: increased HRV. Figure 5:
Percentile values in increased (red) and decreased (black) HRV to which the
10th (left panels) and 90th (right panels) percentiles in baseline
configuration correspond. Pressure (top panels) and flow rate (bottom panels)
along the ICA-MCA pathway. Figure 6: Probability density functions of pressure
and flow rate values averaged per beat, $\overline{P}$ and $\overline{Q}$,
along the ICA-MCA pathway. Left panels: pressures $\overline{P}_{a}$,
$\overline{P}_{MCA,left}$, $\overline{P}_{dm,left}$, $\overline{P}_{c}$. Right
panels: flow rates $\overline{Q}_{ICA,left}$, $\overline{Q}_{MCA,left}$,
$\overline{Q}_{dm,left}$, $\overline{Q}_{pv}$. Black: decreased HRV; blue:
baseline; red: increased HRV. Figure 7: Ratio between the coefficient of
variation $c_{v}$ in increased/decreased (red/black) HRV cases and the
coefficient of variation in baseline condition. Ratios for RR time-series and
pressure/flow rate continuous time-series along the ICA-MCA pathaway are
shown. Figure 8: Relative variations [%] of pulse values per beat with respect
to baseline: $(P^{pv}–P^{pv}_{baseline})/P^{pv}_{baseline}\cdot 100$ and
$(Q^{pv}–Q^{pv}_{baseline})/Q^{pv}_{baseline}\cdot 100$. Pulse values,
$P^{pv}$ and $Q^{pv}$, are evaluated at each site of the ICA-MCA pathway.
Black: decreased HRV/baseline. Red: increased HRV/baseline.

Figure 9: Heart failure (thick) vs healthy (thin) percentile analyses.
Percentile values in increased (red) and decreased (black) HRV to which the
10th and 90th percentiles in baseline configuration correspond. Pressure (top
panel) and flow rate (bottom panel) along the ICA-MCA pathway. Inset panels
represent the PDFs of the continuous hemodynamic time-series. Table 1: Basic
statistics (mean: $\mu$, standard deviation: $\sigma$, coefficient of
variation: $c_{v}$) of the continuous time-series along the ICA-MCA pathway:
$P_{a}$, $Q_{ICA,left}$, $P_{MCA,left}$, $Q_{MCA,left}$, $P_{dm,left}$,
$Q_{dm,left}$, $P_{c}$, and $Q_{pv}$.
Configuration | Pressure P [mmHg] | Flow rate Q [ml/s]
---|---|---
| $\mu$ | $\sigma$ | $c_{v}$ | $\mu$ | $\sigma$ | $c_{v}$
| _$P_{a}$_ | _$Q_{ICA,left}$_
Decreased HRV (-20%) | 99.30 | 13.40 | 0.13 | 4.74 | 1.68 | 0.35
Baseline | 96.91 | 14.42 | 0.15 | 4.73 | 1.83 | 0.39
Increased HRV (+20%) | 94.41 | 15.40 | 0.16 | 4.72 | 1.99 | 0.42
| _$P_{MCA,left}$_ | _$Q_{MCA,left}$_
Decreased HRV (-20%) | 96.60 | 12.51 | 0.13 | 3.75 | 1.22 | 0.33
Baseline | 94.22 | 13.44 | 0.14 | 3.74 | 1.35 | 0.36
Increased HRV (+20%) | 91.72 | 14.32 | 0.16 | 3.72 | 1.48 | 0.40
| _$P_{dm,left}$_ | _$Q_{dm,left}$_
Decreased HRV (-20%) | 58.11 | 2.79 | 0.05 | 3.75 | 0.43 | 0.11
Baseline | 56.91 | 3.38 | 0.06 | 3.74 | 0.51 | 0.14
Increased HRV (+20%) | 55.66 | 3.97 | 0.07 | 3.72 | 0.60 | 0.16
| _$P_{c}$_ | _$Q_{pv}$_
Decreased HRV (-20%) | 25.02 | 1.85 | 0.07 | 12.49 | 1.52 | 0.12
Baseline | 25.00 | 2.33 | 0.09 | 12.46 | 1.87 | 0.15
Increased HRV (+20%) | 24.97 | 2.85 | 0.11 | 12.43 | 2.24 | 0.18
Table 2: Mean and standard deviation (in brackets) values for the minimum
($P^{min}$ and $Q^{min}$), average ($\overline{P}$ and $\overline{Q}$),
maximum ($P^{max}$ and $Q^{max}$), and pulse ($P^{pv}$ and $Q^{pv}$) values
per beat of $P$ and $Q$ along the ICA-MCA pathway: $P_{a}$, $Q_{ICA,left}$,
$P_{MCA,left}$, $Q_{MCA,left}$, $P_{dm,left}$, $Q_{dm,left}$, $P_{c}$, and
$Q_{pv}$.
Configuration | Min value | Mean value | Max value | Pulse value
---|---|---|---|---
| $P_{a}^{min}$ | $\overline{P}_{a}$ | $P_{a}^{max}$ | $P_{a}^{pv}$
Decreased HRV (-20%) | 78.43 (2.65) | 99.42 (1.86) | 121.26 (1.36) | 42.82 (1.56)
Baseline | 74.52 (3.02) | 97.06 (2.19) | 120.45 (1.64) | 45.93 (1.72)
Increased HRV (+20%) | 70.63 (3.31) | 94.59 (2.47) | 119.54 (1.82) | 48.91 (1.89)
| $P_{MCA,left}^{min}$ | $\overline{P}_{MCA,left}$ | $P_{MCA,left}^{max}$ | $P_{MCA,left}^{pv}$
Decreased HRV (-20%) | 77.08 (2.58) | 96.71 (1.83) | 117.11 (1.36) | 40.03 (1.44)
Baseline | 73.27 (2.94) | 94.36 (2.16) | 116.16 (1.63) | 42.89 (1.58)
Increased HRV (+20%) | 69.48 (3.24) | 91.90 (2.43) | 115.11 (1.81) | 45.63 (1.73)
| $P_{dm,left}^{min}$ | $\overline{P}_{dm,left}$ | $P_{dm,left}^{max}$ | $P_{dm,left}^{pv}$
Decreased HRV (-20%) | 53.72 (1.56) | 58.16 (1.30) | 61.04 (1.25) | 7.33 (0.82)
Baseline | 51.59 (1.84) | 56.97 (1.52) | 60.54 (1.50) | 8.95 (1.02)
Increased HRV (+20%) | 49.34 (2.05) | 55.73 (1.65) | 60.05 (1.61) | 10.71 (1.21)
| $P_{c}^{min}$ | $\overline{P}_{c}$ | $P_{c}^{max}$ | $P_{c}^{pv}$
Decreased HRV (-20%) | 22.05 (0.74) | 25.03 (0.66) | 27.12 (0.78) | 5.07 (0.64)
Baseline | 21.29 (0.90) | 25.01 (0.77) | 27.70 (0.93) | 6.41 (0.81)
Increased HRV (+20%) | 20.45 (0.99) | 24.98 (0.83) | 28.37 (1.07) | 7.92 (0.99)
| $Q_{ICA,left}^{min}$ | $\overline{Q}_{ICA,left}$ | $Q_{ICA,left}^{max}$ | $Q_{ICA,left}^{pv}$
Decreased HRV (-20%) | 2.32 (0.16) | 4.75 (0.14) | 7.30 (0.24) | 4.99 (0.23)
Baseline | 2.14 (0.15) | 4.74 (0.17) | 7.53 (0.28) | 5.40 (0.29)
Increased HRV (+20%) | 1.98 (0.15) | 4.73 (0.18) | 7.80 (0.33) | 5.81 (0.34)
| $Q_{MCA,left}^{min}$ | $\overline{Q}_{MCA,left}$ | $Q_{MCA,left}^{max}$ | $Q_{MCA,left}^{pv}$
Decreased HRV (-20%) | 1.96 (0.13) | 3.75 (0.13) | 5.76 (0.21) | 3.81 (0.20)
Baseline | 1.81 (0.14) | 3.74 (0.15) | 5.96 (0.26) | 4.16 (0.25)
Increased HRV (+20%) | 1.66 (0.14) | 3.73 (0.16) | 6.17 (0.29) | 4.51 (0.29)
| $Q_{dm,left}^{min}$ | $\overline{Q}_{dm,left}$ | $Q_{dm,left}^{max}$ | $Q_{dm,left}^{pv}$
Decreased HRV (-20%) | 3.06 (0.17) | 3.75 (0.15) | 4.23 (0.18) | 1.17 (0.13)
Baseline | 2.91 (0.19) | 3.74 (0.18) | 4.33 (0.22) | 1.42 (0.16)
Increased HRV (+20%) | 2.75 (0.20) | 3.73 (0.19) | 4.44 (0.24) | 1.68 (0.19)
| $Q_{pv}^{min}$ | $\overline{Q}_{pv}$ | $Q_{pv}^{max}$ | $Q_{pv}^{pv}$
Decreased HRV (-20%) | 9.98 (0.52) | 12.49 (0.43) | 14.26 (0.53) | 4.28 (0.46)
Baseline | 9.41 (0.62) | 12.47 (0.48) | 14.70 (0.60) | 5.29 (0.56)
Increased HRV (+20%) | 8.82 (0.66) | 12.44 (0.51) | 15.19 (0.68) | 6.37 (0.66)
Table 3: Total number of one-beat extreme events (out of 5000 RR beats) along
the ICA-MCA pathway for the two decreased and increased HRV configurations.
Configuration | Pressure $\overline{P}$ | Flow rate $\overline{Q}$
---|---|---
| $\overline{P}_{a}$ | $\overline{P}_{MCA,left}$ | $\overline{P}_{dm,left}$ | $\overline{P}_{c}$ | $\overline{Q}_{ICA,left}$ | $\overline{Q}_{MCA,left}$ | $\overline{Q}_{dm,left}$ | $\overline{Q}_{pv}$
| Hypotensive events | Hypoperfusion events
Decreased HRV (-20%) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Increased HRV (+20%) | 0 | 0 | 42 | 0 | 0 | 0 | 0 | 0
| Hypertensive events | Hyperperfusion events
Decreased HRV (-20%) | 0 | 0 | 69 | 0 | 0 | 0 | 0 | 0
Increased HRV (+20%) | 0 | 0 | 4 | 6 | 0 | 0 | 9 | 0
Table 4: Cardiac parameters and oxygen consumption indexes at central level:
mean (std). $V_{lved}$: end-diastolic left ventricular volume, $V_{lves}$:
end-systolic left ventricular volume, $SV$: stroke volume, $EF$: ejection
fraction, $CO$: cardiac output, $SW/min$: stroke work per minute, $RPP$: rate
pressure product, $TTI/min$: tension time index per minute, $PVA/min$:
pressure volume area per minute, $LVE$: left ventricle efficiency.
Variable | Decreased HRV (-20%) | Baseline | Increased HRV (+20%)
---|---|---|---
$V_{lved}$ [ml] | 126.99 (1.71) | 130.87 (1.79) | 134.43 (1.95)
$V_{lves}$ [ml] | 58.80 (1.16) | 56.73 (1.30) | 54.78 (1.38)
$SV$ [ml] | 68.18 (2.77) | 74.14 (2.97) | 79.65 (3.22)
$EF$ [%] | 53.67 (1.49) | 56.63 (1.53) | 59.23 (1.58)
$CO$ [l/min] | 5.52 (0.30) | 5.21 (0.32) | 4.91 (0.31)
$SW/min$ [J/min] | 75.44 (4.84) | 69.46 (5.01) | 63.72 (4.98)
$RPP$ [mmHg/min] | 9837.29 (835.45) | 8494.19 (786.38) | 7393.16 (733.68)
$TTI/min$ [mmHgs/min] | 2775.53 (100.87) | 2622.53 (102.08) | 2488.01 (101.07)
$PVA/min$ [J/min] | 101.85 (7.15) | 91.98 (7.26) | 82.93 (7.13)
$LVE$ | 0.74 (0.01) | 0.76 (0.01) | 0.77 (0.01)
Table 5: Heart failure. Cardiac parameters and oxygen consumption indexes at
central level: mean (std). $V_{lved}$: end-diastolic left ventricular volume,
$V_{lves}$: end-systolic left ventricular volume, $SV$: stroke volume, $EF$:
ejection fraction, $CO$: cardiac output, $SW/min$: stroke work per minute,
$RPP$: rate pressure product, $TTI/min$: tension time index per minute,
$PVA/min$: pressure volume area per minute, $LVE$: left ventricle efficiency.
Variable | Decreased HRV (-20%) | Baseline | Increased HRV (+20%)
---|---|---|---
$V_{lved}$ [ml] | 251.07 (1.16) | 253.01 (1.11) | 254.51 (1.01)
$V_{lves}$ [ml] | 213.19 (0.81) | 211.47 (1.17) | 209.43 (1.56)
$SV$ [ml] | 37.88 (1.87) | 41.54 (2.19) | 45.08 (2.47)
$EF$ [%] | 15.08 (0.68) | 16.41 (0.80) | 17.71 (0.90)
$CO$ [l/min] | 3.64 (0.17) | 3.55 (0.17) | 3.46 (0.18)
$SW/min$ [J/min] | 39.22 (1.95) | 37.55 (1.97) | 35.98 (1.99)
$RPP$ [mmHg/min] | 10243.26 (699.45) | 9116.87 (691.62) | 8189.49 (672.01)
$TTI/min$ [mmHgs/min] | 2709.65 (55.37) | 2621.87 (58.59) | 2543.39 (61.17)
$PVA/min$ [J/min] | 132.13 (7.64) | 120.59 (7.81) | 110.73 (7.71)
$LVE$ | 0.30 (0.01) | 0.31 (0.01) | 0.33 (0.01)


# Why Do Students Drop Out? University Dropout Prediction and Associated
Factor Analysis Using Machine Learning Techniques

Sean Kim∗1,2, Eliot Yoo∗1,3, and Samuel Kim1

1 IF Research Lab, La Palma, CA, USA
2 Oxford Academy, Cypress, CA, USA
3 Cypress High School, Cypress, CA, USA
{seankim.hahjean<EMAIL_ADDRESS>
<EMAIL_ADDRESS>∗ These authors contributed equally. The names are
listed in alphabetical order.

###### Abstract

Graduation and dropout rates have always been a serious consideration for
educational institutions and students. High dropout rates negatively impact
both the lives of individual students and institutions. To address this
problem, this study examined university dropout prediction using academic,
demographic, socioeconomic, and macroeconomic data types. Additionally, we
performed associated factor analysis to analyze which type of data would be
most influential on the performance of machine learning models in predicting
graduation and dropout status. These features were used to train four binary
classifiers to determine if students would graduate or drop out. The overall
performance of the classifiers in predicting dropout status had an average
ROC-AUC score of 0.935. The data type most influential to the model
performance was found to be academic data, with the average ROC-AUC score
dropping from 0.935 to 0.811 when excluding all academic-related features from
the data set. Preliminary results indicate that a correlation does exist
between data types and dropout status.

##  I INTRODUCTION

Student dropout represents a significant and complex issue within higher
education [1]. Not only does it interfere with educational development, but
dropping out of college also has consequences for individual students and
institutions. At the individual level, there is a waste of personal resources,
time, and money and a loss in employment prospects and earnings potential [2].
On the other hand, college graduates tend to have significantly higher
personal income and happiness and lower depression and stress levels [3]. At
the institutional level, universities are negatively impacted educationally
and economically [2]. Due to the various negative repercussions of university
dropout, improving student retention would benefit individual students and
educational institutions. Predicting which students are likely to drop out or
graduate accurately would help alleviate dropout rates in higher education. By
identifying at-risk students most likely to be at risk of dropping out, more
resources can be focused on them, which may lead to improved retention and
graduation rates.

Studies have examined a complex array of unique factors that include more than
just academic information to examine dropout in higher education, including
socio-demographic, economic, and mental factors [2, 3, 4, 5]. Incorporating a
wide array of relevant factors is vital because non-academic variables have
been indicated to be involved with school dropout [6]. Common predictors
examining students’ background info include demographics, socioeconomic
status, prior academic experience, performance, and student-related records
such as age, gender, and status [7].

Various studies have studied the effects of specific predictors for university
dropout but generally have little consensus on what specific factors are
important [4, 6, 8, 9]. According to [5], a wide range of different entry
qualifications are associated with greater chances of student dropout. There
was no specific pattern of demographic variables particularly important to
drop out. On the other hand, the effects of socioeconomic, psychological, and
educational variables needed to be better investigated further.

Our study aims to examine the impact that demographic, socioeconomic,
academic, and macroeconomic data have on predicting student dropout in
university with different machine-learning techniques.

##  II Procedure

###  II-A Data Source

We utilized a data set collected in another study related to dropout
prediction, which consists of records related to students with 17 different
undergraduate degrees from 2008 to 2019. The data set was created using data
from the Polytechnic Institute of Portalegre as well as data from the General
Directorate of Higher Education and the Contemporary Portugal Database. The
information includes demographic data, socioeconomic data, macroeconomic data,
data at the time of student enrollment, and academic data at the end of
semesters. There are 4424 total records with 35 attributes. See [10] for
further information regarding the specifics of the data set.

###  II-B Exploratory Data Analysis

Of the 4424 total records of students, 1421 had the status of dropout, 2209
had the status of graduate, and 794 were enrolled, as shown in Figure 1. We
implemented a binary classification task by focusing on the 2209 graduates and
1421 dropouts for 3630 total records in our study, removing the 794 enrolled
students from our data due to ambiguous status. The final class distribution
of our data is shown in Figure 2. 2381 students were females, while 1249
students were male. Figure 3 depicts the distribution of dropouts and
graduates across the two genders, which reveals that females tended to be more
academically successful than males. Figure 4 and Figure 5 show the
distribution of dropouts and graduates across marital status, daytime or
evening attendance, displacement status, educational special needs, debtor
status, whether tuition fees were up to date, scholarship holders, and
international student status. Features related to socioeconomic status, such
as late tuition fees and debtor status, seemed to correlate highly with
dropout status. Dropout rates were highest for students with late tuition
fees, with approximately 94 percent of such students dropping out. 76 percent
of debtors were dropouts. On the other hand, 86 percent of scholarship holders
became graduates, which may be due to a higher desire to succeed and confirms
existing findings about the relationship between scholarships and student
success [11]. Concerning marital status, single students appeared more likely
to graduate, while legally separated students were likelier to drop out.

Correlation heat maps were created to analyze relationships between different
features. Figure 6 graphs all academic, demographic, socioeconomic, and
macroeconomic features. There appears to be some correlation between certain
groups such as “Nationality” and “International,” “Mother’s occupation” and
“Father’s occupation,” and “Curricular units 1st sem (enrolled)” and
“Curricular units 2nd sem (enrolled)” which is consistent with previous
findings [10].

Figure 1: Class distribution of original data set. Figure 2: Class
distribution after the ‘Enrolled’ class was dropped and represents the final
class distributions used in the study.  Figure 3: Distribution of dropouts and
graduates by gender. Figure 4: Distribution of dropouts and graduates by
marital status, daytime/evening attendance, displaced status, and educational
special needs status.  Figure 5: Distribution of dropouts and graduates by
debtor status, tuition fee status, scholarship holder status, and
international student status.  Figure 6: Correlation Heat maps for all
features.

##  III Experiments

###  III-A Methodology

Each subject had labels stating their status in their school: graduated,
dropped out, or currently enrolled. As currently enrolled students in the data
set could not be used to train the machine learning models to determine
whether a student would graduate or drop out, they were cut out from the
primary data set, leaving 3630 subjects (removed 794 subjects). The remaining
students were then split randomly into testing and training data sets (the
testing set is 20%), with their labels being graduation status. Using this
data, four different machine learning classifiers were evaluated on their
performance on the binary classification task of student dropout prediction.

A secondary test was run to find the type of data most influential on the
results of student dropout prediction. The four types of data were
demographic, socioeconomic, macroeconomic, and academic data. Demographic data
included marital status, nationality, displaced, gender, age at enrollment,
and international. Socioeconomic data included father’s qualification,
mother’s qualification, father’s occupation, mother’s occupation, educational
special needs, debtor, tuition fees up to date, and scholarship holder.
Macroeconomic data included the unemployment rate, inflation rate, and GDP.
The remaining 17 features were grouped as academic data.

We looked at the impact on performance after excluding one data group from the
data set to find the most influential data type. While training the machine
learning classifiers for dropout prediction, one of the four data types was
excluded (For example, only using demographic, socioeconomic, and
macroeconomic for one trial, then using only demographic, socioeconomic, and
academic data for the next, and so on). This was repeated a total of four
times, with a different group excluded each time. Thus, the performance of
machine learning classifiers was evaluated on four distinct data sets, each
one missing either demographic, socioeconomic, macroeconomic, or academic
data. The performance of machine learning classifiers when using all four data
types was used as a baseline.

Multiple classifiers were trained on the data set to find correlations between
the multiple features and their target. Using numerous classifiers allows us
to validate the results and ensure a higher accuracy in results. The machine
learning classifiers that were used include the Random Forest, Decision Tree
(max depth = 5), Support Vector (random_state= 42, kernel= ’linear’), and
k-nearest Neighbors (n_neighbors = 20) with their specific parameters.

###  III-B Results

Receiver Operating Characteristic (ROC) Curves were graphed, and ROC-AUC (Area
Under Curve) scores were used to evaluate the performance of each model. The
four classifiers had ROC-AUC scores of 0.911 or higher (see Table I). Figure 7
shows the ROC curves for each classifier and its performance compared to each
other. The best-performing classifier was the Random Forest classifier, which
had a 0.955 ROC-AUC score. The classifier with the second best result was the
Support Vector Classifier, with a 0.953 ROC-AUC in its performance to predict
the students’ outcomes. The K-Nearest Neighbors Classifier had a ROC-AUC score
of 0.92, and the Decision Tree Classifier had a 0.911 ROC-AUC score. These
show a relatively good performance in predicting the students’ chances of
graduating or dropping out. We can assume a positive relationship between the
features of the subjects mentioned earlier and their performance in school.
The numerous classifiers help confirm this as they had similar accurate
performances that show these results were not all random.

Figure 7: The ROC curve graph for the performance of all the classifiers with
the whole data set.

We found that the academic data was the most influential to the results as the
average ROC-AUC score dropped 0.124 (see Table II for all AUC scores for each
classifier), lowering the scores significantly more than the other four tests.
This is also evident in Figure 8, as the ROC curves are considerably flatter
for academic data than the other tests. These results are reasonable as the
academic data type accounted for 17 of the 34 total features, which is
considerably greater than the other 3 data types and is a limitation of this
study. The second most influential data type was the socioeconomic data, with
its classifiers’ average ROC-AUC score dropping 0.013, followed by the
demographic data, with the average ROC-AUC score dropping 0.001, and the least
influential features was macroeconomic data, with the average ROC-AUC score of
0.935, not dropping from the overall test. Thus, our results indicate that
while academic data significantly impacted the performance of the machine
learning classifiers, macroeconomic data had little to no impact.

| SVC | DT | RF | KNN | Average | STDV
---|---|---|---|---|---|---
ROC-AUC | 0.953 | 0.911 | 0.955 | 0.92 | 0.935 | 0.023
TABLE I: The AUC scores for all of the classifiers, including their average
and standard deviation

(a) ROC Curve with Academic Data Excluded

(b) ROC Curve with Demographic Data Excluded

(c) ROC Curve with Macroeconomic Data Excluded

(d) ROC Curve with Socioeconomic Data Excluded

Figure 8: ROC curve graphs for each of the tests, with each excluding one different category. | Excl. Academic Data | Excl. Demographic Data | Excl. Macroeconomic Data | Excl. Socioeconomic Data
---|---|---|---|---
SVC | 0.821 | 0.950 | 0.952 | 0.939
DT | 0.798 | 0.912 | 0.904 | 0.894
RF | 0.823 | 0.953 | 0.954 | 0.945
KNN | 0.80 | 0.92 | 0.93 | 0.91
Average | 0.811 | 0.934 | 0.935 | 0.922
STDV | 0.013 | 0.208 | 0.023 | 0.024
TABLE II: AUC scores for each of the four tests including their averages and
standard deviations

###  III-C Feature Importance

Calculating feature importance is a crucial aspect of machine learning tasks
as it allows for a clear understanding of what features may be contributing
most to the final decision made by models. Through feature importance, future
work can gain insight into types of features that may be more suitable for
predicting dropout and graduation in higher academia. Feature importance was
found for the best-performing classifier from our experiments, the Random
Forest Classifier. The top three features considered most important were:
“Curricular units 1st sem (approved)”, “Curricular units 2nd sem (approved)”,
and “Curricular units 2nd sem (grade)”.

###  III-D Limitations

A limitation of this study was the imbalanced number of features in each data
category. For example, academic data contained 17 features, while
macroeconomic data only included 3 features. The imbalanced features may have
affected the study results and should be considered in future work.

##  IV Conclusion

This study aimed to explore the connections between a student’s academic,
demographic, socioeconomic, and macroeconomic data and their chances of
graduating or dropping out and analyzing the impact of each data type on the
results. Various machine learning classifiers were used to tackle the binary
classification task of dropout prediction. Of the classifiers trained, Random
Forest had the best performance, and academic data was found to have the
highest impact on performance for the task of dropout prediction. This
research can be applied in future studies predicting dropout for students to
train more accurate machine learning models and obtain relevant and impactful
data for this task. Dropout prediction can help identify at-risk students at
an early stage, which can help determine where to divert additional resources
to reduce dropout rates.

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