# Glucodensities: a new representation of glucose profiles using
distributional data analysis

Marcos Matabuena1,2,∗ Alexander Petersen3 Juan C.Vidal2 and Francisco Gude1
1Unidad de Epidemiología Clínica  Hospital Cínico Universitario de Santiago de
Compostela  Spain
2CiTIUS (Centro Singular de Investigación en Tecnoloxías Intelixentes)
Universidade de Santiago de Compostela  Spain
3Department of Statistics and Applied Probability  University of California
Santa Barbara
<EMAIL_ADDRESS>

###### Abstract

Biosensor data has the potential ability to improve disease control and
detection. However, the analysis of these data under free-living conditions is
not feasible with current statistical techniques. To address this challenge,
we introduce a new functional representation of biosensor data, termed the
glucodensity, together with a data analysis framework based on distances
between them. The new data analysis procedure is illustrated through an
application in diabetes with continuous-time glucose monitoring (CGM) data. In
this domain, we show marked improvement with respect to state of the art
analysis methods. In particular, our findings demonstrate that i) the
glucodensity possesses an extraordinary clinical sensitivity to capture the
typical biomarkers used in the standard clinical practice in diabetes, ii)
previous biomarkers cannot accurately predict glucodensity, so that the latter
is a richer source of information, and iii) the glucodensity is a natural
generalization of the time in range metric, this being the gold standard in
the handling of CGM data. Furthermore, the new method overcomes many of the
drawbacks of time in range metrics, and provides deeper insight into assessing
glucose metabolism.

##  1 Introduction

The steadily increasing availability and prominence of biosensor data have
given rise to new methodological challenges for their statistical analysis. A
primary feature of these data is that the monitored individuals are in free-
living conditions, making a direct analysis of the recorded time series
between groups of patients problematic if not infeasible. A clear example of
such data is found in the study of diabetes, where continuous glucose
monitoring (CGM) is increasingly used. The elevation of glucose is distinct
between individuals and is influenced by factors such as mealtimes, diet
composition, or physical exercise (Ewings et al., 2015). Consequently, an
exciting topic of debate is how to exploit the enormous wealth of information
recorded by CGM to draw more reliable conclusions about the glucose
homeostasis rather than the cursory summary measures such as fasting plasma
glucose (FPG) or glycated hemoglobin (A1c) (Zaccardi and Khunti, 2018).

Since $2010$, the American Diabetes Association (ADA) has included measurement
of A1c levels to both diagnosis and diabetes control Association et al.
(2018). A1c levels reflect underlying glucose levels over the preceding $2–3$
months, testing is convenient because blood samples can be obtained at any
time of day, overnight fasting is not required, and A1c within‐patient
reproducibility is superior to that of fasting plasma glucose and oral glucose
tolerance tests (OGTTs) Selvin et al. (2007). However, recent articles have
provided evidence for the need to go beyond A1c and use new measures for
glycemic control (Group, 2018; Bergenstal, 2015) in order to capture more
diverse aspects of the temporally evolving glucose levels beyond the average,
for example, glucose variability and time in range metrics. The time in metric
range measures the proportion of time an individual’s glucose levels is
maintained in different target zones. In the case of diabetes, these can
include ranges corresponding to hypoglycemia and hyperglycemia. In an
innovative article, Beck et al. (2019) validated the time in range metric,
showing that it is a good predictor of long-term microvascular complications
despite just measuring glucose values seven times per day. Lu et al. (2018)
reached similar conclusions but using CGM technology only for $24$ hours in
each patient. At the same time, it is well-known that two patients may have
the same glycosylated hemoglobin and a completely different glycemic profile
(Beck et al., 2017). These new approaches and findings have lead clinical
specialists to consider that continuous glucose measurement during long
monitoring periods can lead to more accurate results in research and clinical
practice than in standard methods (Hirsch et al., 2019). In fact, since
$2012$, the European Medicine Agency (for Medicinal Products for Human Use et
al., 2012) recommends the use of CGM to validate the effect of drugs for
treatment or prevention of diabetes mellitus.

Traditionally, CGM was designed for the risk management in real-time for type
$1$ diabetes and the control of glucose values with insulin pumps (Kovatchev
et al., 2009; Feig et al., 2017; DiMeglio et al., 2018). Notwithstanding, more
recent applications of CGM have been more general. They involve, for example,
screening patients, optimizing diet, epidemiological studies, assessing
patient prognosis, and supporting treatment prescription, and have even been
used in healthy populations (Freeman and Lyons, 2008; Hall et al., 2018). In
addition to the increasing utility of CGM data, the technology is gradually
becoming cheaper, and new devices capable of measuring glucose in a non-
invasive way, for example, with glasses (Nichols et al., 2013), are quickly
emerging. All of these advances are facilitating the adoption of CGM in
standard clinical practice.

In $2012$, a panel of experts discussed how to represent CGM data in an “easy
to view format” (Bergenstal et al., 2013). They also analyzed the convenience
of using glycemic variability measures and other summary measures such as time
in range to extract the recorded information from CGM. In $2019$, ADA
established an updated version of clinical standards to use and define target
zones with time in range metrics (Battelino et al., 2019). In a more recent
review about the CGM metric, they establish time in range as a gold standard
measure (Nguyen et al., ).

Motivated by the problem of analyzing data gathered via CGM more precisely,
while still leveraging the advantages possessed by time in range metrics, we
propose an approach based on the construction of a functional profile of
glucose values for each subject. Conceptually, the approach is a natural
extension of time in range metrics in which the ranges shrink and size and
increase in number, so that new profile effectively measures the proportion of
time each patient spends at each specific glucose concentration rather than a
coarsely defined range. As a result of this, the new functional profile, which
we refer to as a glucodensity, automatically and simultaneously captures all
parameters arising from individual glucose distributions. Figure 1 illustrates
a set of constructed glucodensities that represent the data objects for which
we will propose the use of a tailored set of statistical methods.

Mathematically, glucodensities constitute functional distributional data since
each glucodensity represents a distribution of glucose concentrations. As
such, these complex and constrained curves cannot be directly analyzed with
the usual techniques. To overcome this, we introduce a framework for the
analysis of glucodensities by compiling suitable methods that are based on the
calculation of glucodensities distances. We also reveal the superior clinical
capacity of our representation compared to classical measures of diabetes.
Finally, we demonstrate that our representation has a higher sensitivity than
the standard time in range metric to explain the glycemic differences between
patients in various settings, including regression analysis. A new shiny
interface to use the methods outlined in this paper is available at
https://tec.citius.usc.es/diabetes.

Figure 1: Example of a set of glucodensities estimated from a random sample of the AEGIS population-based study Biomarker | Clinical significance
---|---
A1c |  Gold standard marker
in diabetes diagnosis and control |
HOMA-IR |  Measurements to quantify insulin
resistance and $\beta$-cell function |
CONGA |
MODD |  Summary indices of
glucose variability |
MAGE |
Table 1: Clinical importance of biomarkers used in the statistical analysis

###  1.1 Outline

The structure of this paper is as follows. First, we briefly describe the
AEGIS study, and the methods used. We then formally introduce the concept of
glucodensity, the estimation methods, and some essential statistical
background to understand the statistical procedures introduced in the paper.
Subsequently, we explain the regression models used in the validation of the
representation. Afterward, we show the results that demonstrate the
superiority of glucodensity over glucose representations of state-art. Then,
we illustrate the use with real data of the glucodensities methodology in two-
sample testing and cluster analysis. Finally, we discuss the clinical
implications of these results, their limitations, and the new perspectives of
the glucodensities method in medicine and device technology.

##  2 Sample and Procedures

####  2.0.1 Study design

A subset of the subjects in the A Estrada Glycation and Inflammation Study
(AEGIS; trial $NCT01796184$ at www.clinicaltrials.gov) provided the sample for
the present work. In the latter cross-sectional study, an age-stratified
random sample of the population (aged $\geq 18)$ was drawn from Spain’s
National Health System Registry. A detailed description has been published
elsewhere (Gude et al., 2017). For a one year beginning in March, subjects
were periodically examined at their primary care centre where they $(i)$
completed an interviewer-administered structured questionnaire; $(ii)$
provided a lifestyle description; $(iii)$ were subjected to biochemical
measurements; and $(iv)$ were prepared for CGM (lasting $6$ days). The
subjects who made up the present sample were the $581$ ($361$ women, $220$
men) who completed at least $2$ days of monitoring, out of an original $622$
persons who consented to undergo a $6$-day period of CGM. Another $41$
original subjects were withdrawn from the study due to non-compliance with
protocol demands (n = $4$) or difficulties in handling the device (n = $37$).
The characteristics of the participants are shown on the Table 2.

| Men $(n=220)$ | Women $(n=361)$
---|---|---
Age, years | $47.8\pm 14.8$ | $48.2\pm 14.5$
A1c, % | $5.6\pm 0.9$ | $5.5\pm 0.7$
FPG $mg/dL$ | $97\pm 23$ | $91\pm 21$
HOMA-IR $mg/dL.\mu UI/m$ | $3.97\pm 5.56$ | $2.74\pm 2.47$
BMI $kg/m^{2}$ | $28.9\pm 4.7$ | $27.7\pm 5.3$
CONGA $mg/dL$ | $0.88\pm 0.40$ | $0.86\pm 0.36$
MAGE $mg/dL$ | $33.6\pm 22.3$ | $31.2\pm 14.6$
MODD | $0.84\pm 0.58$ | $0.77\pm 0.33$
Table 2: Characteristics of AEGIS study participants by sex. Mean and standard
deviation are shown. $BMI$ \- body mass index; $FPG$ \- fasting plasma
glucose; $A1c$ \- glycated haemoglobin; $HOMA-IR$ \- homeostasis model
assessment-insulin resistance; $CONGA$ \- glycemic variability in terms of
continuous overall net glycemic action; $MODD$ \- mean of daily differences;
$MAGE$ \- mean amplitude of glycemic excursions.

####  2.0.2 Ethical approval and informed consent

The present study was reviewed and approved by the Clinical Research Ethics
Committee from Galicia, Spain (CEIC$2012$-$025$). Written informed consent was
obtained from each participant in the study, which conformed to the current
Helsinki Declaration.

####  2.0.3 Laboratory determinations

Glucose was determined in plasma samples from fasting participants by the
glucose oxidase peroxidase method. A1c was determined by high-performance
liquid chromatography in a Menarini Diagnostics HA-$8160$ analyser; all A1c
values were converted to DCCT-aligned values (Hoelzel et al., 2004). Insulin
resistance was estimated using the homeostasis model assessment method (HOMA-
IR) as the fasting concentration of plasma insulin ($\mu$ units/mL) $\times$
plasma glucose (mg/dL)/ $405$ (Matthews et al., 1985).

####  2.0.4 Glycaemic variability

Glycaemic variability was measured in terms of continuous overall net glycemic
action (CONGA) (McDonnell et al., 2005), the mean amplitude of glycaemic
excursions (MAGE) (Service et al., 1970), and the mean of the daily
differences (MODD) (Molnar et al., 1972) in glucose concentration.

####  2.0.5 CGM Procedures

At the start of each monitoring period, a research nurse inserted a sensor
(Enlite™, Medtronic, Inc, Northridge, CA, USA) subcutaneously into the
subject’s abdomen, and instructed him/her in the use of the iPro™ CGM device
(Medtronic, Inc, Northridge, CA, USA). The sensor continuously measures the
interstitial glucose level $40-400$ (range mg/dL) of the subcutaneous tissue,
recording values every $5$ min. Participants were also provided with a
conventional OneTouchR VerioR Pro glucometer (LifeScan, Milpitas, CA, USA) as
well as compatible lancets and test strips for calibrating the CGM. All
subjects were asked to make at least three capillary blood glucose
measurements (usually before main meals). These readings were taken without
checking the current CGM reading. On the seventh day the sensor was removed
and the data downloaded and stored for further analysis. If the number of
data-acquisition “skips” per day totalled more than $2$ h, the entire day’s
data were discarded.

###  2.1 Time-in-range metric

The time in the range metric was calculated with two different methods. In the
first, through the CGM records of the AEGIS study, we estimate the deciles of
CGM records with normoglycemic patients and use as cut-offs the deciles (Table
3). In the second, we use cut-off points established by the ADA in the $2019$
Medical guideline (Battelino et al., 2019) (Table 4).

Range $1$ | $<85$
---|---
Range $2$ | $85-90$
Range $3$ | $91-94$
Range $4$ | $95-98$
Range $5$ | $99-101$
Range $6$ | $102-105$
Range $7$ | $106-109$
Range $8$ | $110-115$
Range $9$ | $116-124$
Range $10$ | $>125$
Table 3: Cut-offs for metric time in range using own estimations throught normoglucemic individuals of AEGIS study Range $1$ | $<54$
---|---
Range $2$ | $54-69$
Range $3$ | $70-180$
Range $4$ | $181-250$
Range $5$ | $>250$
Table 4: Cut-offs for metric time in range following ADA guidelines Battelino
et al. (2019)

####  2.1.1 Statistical procedure

The density functions for each individual was estimated with non-parametric
Nadaraya-Watson procedure. For this purpose, we used a Gaussian kernel and
rule of thumb as a smoothing parameter. In addition, we estimate quantile
representation for $2$-Wasserstein methods using the empirical distribution.

The following three regression models were used: i) The non-parametric kernel
functional regression model through $2$-Wasserstein distance with the
glucodensity as predictor (Ferraty and Vieu, 2006); ii) A global
$2$-Wasserstein regression model where the glucodensity is response (Petersen
and Müller, 2019); and iii) $k$-nearest neighbor algorithm in the case of time
in range metrics with $k=10$ neighbors.

In the case of time in range metrics, we applied the isometric log-ratio (ilr)
transformation for compositional data prior to fitting the model. To avoid
problems with zeros, a fixed positive constant was added to each each range,
which were then normalized to add to $1$.

All analyses were carried out using R software. Functional data analysis was
performed using the fda.usc package (Febrero-Bande and de la Fuente, 2012),
which is freely available at https://cran.r-project.org/, and our own
implementations of the ANOVA test of Dubey and Müller (2019) or Frèchet
regression in Petersen and Müller (2019) using the $2$-Wasserstein distance.
The glucodensities and their quantile representation were estimated using the
R basis functions.

##  3 Definition and Estimation of the Glucodensity

For patient $i$, denote the gathered glucose monitoring data by pairs
$(t_{ij},X_{ij})$, $j=1,\ldots,m_{i},$ where the $t_{ij}$ represent recording
times that are typically equally spaced across the observation interval, and
$X_{ij}$ is the glucose level at time $t_{ij}\in[0,T_{i}].$ Note that the
number of records $m_{i}$, the spacing between them, and the overall
observation length $T_{i}$ can vary by patient. One can think of these data as
discrete observations of a continuous latent process $Y_{i}(t),$ with
$X_{ij}=Y_{i}(t_{ij}).$ The glucodensity for this patient is defined in terms
of this latent process as $f_{i}(x)=F_{i}^{\prime}(x),$ where

$F_{i}(x)=\frac{1}{T_{i}}\int_{0}^{T_{i}}\mathbf{1}\left(Y_{i}(t)\leq
x\right)\mathrm{d}t\quad\text{for }\,\quad\inf_{t\in[0,T_{i}]}Y_{i}(t)\leq
x\leq\sup_{t\in[0,T_{i}]}Y_{i}(t)$

is the proportion of the observation interval in which the glucose levels
remain below $x.$ Since $F_{i}$ are increasing from 0 to 1, the data to be
modeled are a set of probability density functions $f_{i},$ $i=1,\ldots,n.$

Of course, neither $F_{i}$ nor the glucodensity $f_{i}$ is observed in
practice, but one can construct an approximation through a density estimate
$\tilde{f}_{i}(\cdot)$ obtained from the observed sample. In this case of CGM
data, the glucodensities may have different support and shape. Therefore, we
suggest using a non-parametric approach to estimate each density function. For
example, using a kernel-type estimator, we have

$\tilde{f}_{i}(x)=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}K_{h_{i}}(x-X_{ij}),$

where $h_{i}>0$ is the smoothing parameter and
$K_{h_{i}}(s)=\frac{1}{h_{i}}K(\frac{s}{h_{i}})$. The choice of $K$ does not
have a big impact on the efficiency of the estimator, but the value of $h_{i}$
is crucial.

Several alternatives for selecting the smoothing parameter have been proposed
in the literature, including cross-validation, minimizing the estimated mean
integrated squared error (MISE), or a “rule of thumb” derived from the
assumption that the density is Gaussian. In this last case, the choice can be
explicitly written as $\tilde{h}_{i}=1.06\tilde{\sigma_{i}}m_{i}^{-1/5}$,
where $\tilde{\sigma_{i}}$ is the sample standard deviation of the $X_{ij}$.
For more details, see Silverman (1986). Other approaches for the density
function estimation include the use orthogonal series (e.g., Fourier or
Wavelet) expansions, splines, or smoothing of histograms. For further details
the reader is referred to Antoniadis (1997); Izenman (1991); Müller and
Petersen (2014).

###  3.1 Distance-based Descriptive Statistics

Let $[a,b]$ be an interval of the real line, which may be unbounded, and
suppose that each glucodensity $f_{i}$ has support contained in $[a,b]$. From
a statistical point of view, the sample $f_{1},\ldots,f_{n}$ may be modeled
and analyzed using methods of functional data analysis (Ramsay et al., 2005;
Wang et al., 2016). However, since the $f_{i}$ must be positive and satisfy
$\int_{a}^{b}f_{i}(x)dx=1,$ classical methods have in recent years been
adapted to account for the nonlinear, distributional structure of density
samples (Petersen and Müller, 2016; Hron et al., 2016). The general approach
is to define a metric or distance between densities that, in turn, leads to
descriptive statistics that respect the unique density properties. For
example, define the data space of glucodensities as
$A:=\\{f:[a,b]\to\mathbb{R}^{+}:\int_{a}^{b}f(x)dx=1\hskip
5.69046pt\text{and}\hskip 5.69046pt\int_{a}^{b}x^{2}f(x)dx<\infty\\}$. Given
two arbitrary glucodensities $f,g\in A$, the $2$-Wasserstein distance
(Villani, 2008) between $f$ and $g$ is

$d_{W^{2}}(f,g)=\sqrt{\int_{a}^{b}(F^{-1}(x)-G^{-1}(x))^{2}}dx,$ (1)

where $F$ and $G$ are the cumulative distribution functions (cdfs) of the
density functions $f$ and $g$.

The $2$-Wasserstein distance is a natural distance to measure the similarity
between density functions through its representation in the space of the
quantile (inverse cdf) functions and it has already been successfully applied
in biological problems. Furthermore, it has computational and modeling
advantages compared to the usual $L^{2}[a,b]$ metric when glucodensities have
different support within $[a,b]$. Finally, it has a physical interpretation in
the theory of optimal transport.

As glucodensities are distributional data, the subsequent application of the
usual techniques for functional data, such as estimation of mean, covariance,
and regression models, may lead to misleading results. Hence, we have chosen
to use models based on the $2$-Wasserstein distance, although other choices
are possible. As a starting point, based on the notion of distance we can
generalise the mean and variance of a random variable that takes values in an
abstract space with metric structure (Fréchet, 1948). As we will see, similar
adaptations can be developed for regression, hypothesis testing, or to perform
cluster analysis. Given a distance $d:A\times A\to\mathbb{R}^{+},$ of which
$d_{W^{2}}$ is one example, and a random variable $f$ defined on $A$, the
_Fréchet mean_ of $f$ is

$\mu_{f}=\operatorname*{arg\,min}_{g\in A}E(d^{2}(f,g)).$

The _Fréchet variance_ of $Z$ is then

$\sigma_{f}^{2}=E(d^{2}(f,\mu_{f})).$

If the choice of distance is the Wasserstein metric $d_{W^{2}},$ these are
given the names of Wasserstein mean and variance, respectively. In the
following subsections we will extend these concepts of Frèchet to statistical
methodologies of regression, clustering, and hypothesis testing based on the
notion of distance.

##  4 Regression models with glucodensities

####  4.0.1 Non-parametric regression model with Glucodensity as the Predictor

Let $f$ be a functional random variable taking values in $(A,d_{W^{2}})$ and
$Y$ a random variable that take values in the real line. We assume the
following regression relationship between $f$ and $Y$, which represent the
predictor and response variables, respectively:

$Y=g(f)+\epsilon$ (2)

where $g:A\to\mathbb{R}$ is an unknown smooth function, and the random error
$\epsilon$ satisfies $E(\epsilon)=0$.

Given a sample $\\{(f_{i},Y_{i})\in A\times\mathbb{R}\\}_{i=1}^{n}$, most non-
parametric estimators $\tilde{g}(\cdot)$ have the form of a weighted average
of the responses

$\tilde{g}(x)=\sum_{i=1}^{n}w_{ni}(x)Y_{i}.$ (3)

In general, the weights $w_{ni}(x)$ depend on the distance between each
$f_{i}$ and $x$, with larger distances receiving lower weights, and satisfy
$\sum_{i=1}^{n}w_{ni}(x)=1$ (Ferraty and Vieu, 2006). A typical choice would
be the Nadaraya–Watson weights

$w_{ni}(x)=\frac{K(\frac{d(x,f_{i})}{h})}{\sum_{i=1}^{n}(K(\frac{d(x,f_{j})}{h}))},$
(4)

where $h$ is a smoothing parameter and $K:\mathbb{R}\to\mathbb{R}$ is a known
univariate probability density function called the kernel. For more details
about this procedure see Ferraty and Vieu (2006). As an alternative for the
above method, we can use the kernel methods in Reproductive Kernel Hilbert
Spaces (RKHS) (Preda, 2007; Szabó et al., 2016).

###  4.1 Regression model with Glucodensity as the Response

In the case of the regression models with a density function as response, the
literature is not very extensive to the current date (Nerini and Ghattas,
2007; Han et al., 2019; Petersen and Müller, 2019; Capitaine et al., 2019;
Talska et al., 2018). In this article we use the model proposed in Petersen
and Müller (2019) which allows us to incorporate the desired metric
$d_{W^{2}}$ and is a direct generalization of classical linear regression. The
primary rationale for our use of this model is that, unlike the other
approaches mentioned above, there is a methodology developed to performance
inferential procedures such as confidence bands and hypothesis testing in
order to establish the significance of the input variables in the model
Petersen et al. (2019).

Let $f$ be a random variable (e.g. a glucodensity) that take values in the
space of $(A,d_{W^{2}})$ defined above. Consider a random vector
$U\subset\mathbb{R}^{d}$ that contains the set of predictors. Our interest is
in the Frèchet regression function, or function of conditional Frèchet means,

$\overline{f}(u):=\operatorname*{arg\,min}_{g\in
A}E(d^{2}_{W^{2}}(f,g)|U=u),\quad u\in\mathbb{R}^{d}$ (5)

Petersen and Müller (2019) imposes a particular model for $\overline{f}$ that,
in direct analogy to classical linear regression, takes the form of a weighted
Frèchet mean:

$\overline{f}(u)=\operatorname*{arg\,min}_{g\in
A}E(s(U,u)d^{2}_{W^{2}}(f,g)),u\in\mathbb{R}^{d}.$ (6)

Here, the weight function is

$s(U,u)=1+(U-\mu)^{T}\Sigma^{-1}(u-\mu),\hskip
5.69046pt\mu=E(U),\Sigma=\mathrm{Cov}(U),$ (7)

and $\Sigma$ is assumed to be positive definite.

Given a sample $(U_{i},f_{i}),$ $i=1,\ldots,n,$ of independent pairs each
distributed as $(U,f),$ one can proceed to estimate $\overline{f}(u)$ for any
desired input $u.$ Due to the intimate connection between the Wasserstein
metric and quantile functions as in (1), for most inferential procedures it is
sufficient to estimate the conditional Wasserstein mean quantile function
$\overline{Q}(u)$ corresponding to $\overline{f}(u).$ Let $D$ be the set of
quantile functions, $Q_{i}$ the quantile function corresponding to the random
density $f_{i},$ and define empirical weights
$s_{in}(u)=1+(U_{i}-\overline{U})^{T}\hat{\Sigma}^{-1}(u-\overline{U}),$ where
$\overline{U}$ and $\tilde{\Sigma}$ are the sample mean and variance of the
$U_{i},$ respectively. The natural estimator under $d_{W_{2}}$ is the weighted
empirical mean quantile function

$\tilde{\overline{Q}}(u)=\operatorname*{arg\,min}_{Q\in
D}\sum_{i=1}^{n}s_{in}(x)\lVert Q-Q_{i}\rVert^{2},$ (8)

where $\lVert\cdot\rVert$ denotes the $L^{2}[0,1]$ norm on $D$.

A straightforward algorithm for computing $\tilde{\overline{Q}}(u)$ is shown
in Supplementary Material of original reference Petersen et al. (2019). In
addition, two algorithms are given to estimate the confidence bands at a given
significance level $\alpha$ for both the quantile functional parameter
$\overline{Q}(u)$ and the density parameter $\overline{f}(u)$.

##  5 Results

###  5.1 Clinical validation of the glucodensity

To validate the glucodensity representation, we use the database from the
AEGIS study (Gude et al., 2017). The database contains the continuous glucose
monitoring data between $2$-$6$ days of $581$ patients from a random sample of
a general population. A detailed description of the data is introduced in
Section 2 together with characteristics of patients in Table 2. To develop the
validation task, we use two different regression models: i) a non-parametric
regression model where the unique predictor is glucodensity, and ii) a linear
regression model where the response is a glucodensity. Further details on the
regression models used can be found in the Section 3. The first model was used
to predict glycated hemoglobin (A1c) (Kilpatrick, 2000), homeostatic model
assessment (HOMA-IR) Ausk et al. (2010), and the following measures of
glycemic variability Service (2013); Monnier et al. (2008); Gude et al.
(2017): continuous overall net glycemic action (CONGA), mean amplitude of
glycemic excursions (MAGE) and mean of daily differences (MODD), through
glucodensity representation. In contrast, the second was used to predict the
glucodensity with the five variables above. Figure 1 gives a visualization of
the sample of glucodensities used in these models. Biological significance in
variables under consideration is described in Table 1.

####  5.1.1 Prediction of biomarkers using the glucodensity

The aim of the first set of regression analyses is to demonstrate that the
glucodensity is sufficiently rich in its information content to recover the
aforementioned biomarkers with high precision. To quantify this precision, we
estimated the $R^{2}$ after fitting a non-parametric model for each biomarker
as the outcome variable, using the glucodensity as the sole predictor (i.e.
independent variable). The $R^{2}$ estimates for A1c, HOMA-IR, MAGE, MODD,
CONGA were $0.79$, $0.79$, $0.92$, $0.86$, and $0.92$ respectively. To
supplement the results, Figure 2 shows the predicted values against the
observed values, where the outstanding predictive capacity of the glucodensity
can be seen independently of high or low response values.

Figure 2: Real values vs Estimated values when glucodensity is predictor

####  5.1.2 Prediction of the glucodensity using biomarkers

In the second regression analysis with the glucodensity as the outcome
variable, we aim to show that the previous measurements commonly used in the
clinical practice are not capable of capturing the glucodensity with high
accuracy. This fact is not completely surprising because, as noted by some
authors (Zaccardi and Khunti, 2018), the information provided by a CGM is more
precise than that contained in summary measures. To accomplish this, we
computed a suitable version of $R^{2}$ for this task after fitting a
regression model where the response is a glucodensity, and the previous
variables are the predictors. In this case, the $R^{2}$ estimated was $0.74$.
As predicted, compared to the previous section’s results, we were not able to
accurately capture the complex nature of glucodensities, even while using the
combined predictive power of several commonly used summary measures. Moreover,
in some cases, the differences in prediction can be significant (see Figure
3).

Figure 3: Residuals in quantile space prediction glucodensities

###  5.2 Comparison of time in range metrics with glucodensities

To illustrate the higher clinical sensitivity of glucodensities compared to
time in range metrics, we compared the ability of each representation to
predict A1c, HOMA-IR, and glycemic variability metrics MODD, MAGE, and CONGA,
using the data from AEGIS study. The predictive capacity of the glucodensity
representation was illustrated above, and this section gives the corresponding
results for time in range metrics, where these were calculated according to
two sets of cutoffs. In the first, the deciles of the normoglycemic
individuals from the AEGIS study were used, while in the second those proposed
by the ADA were used. Tables 3 and 4 in Section 2 show the exact cutoff values
for both cases. Since the time in range metrics constitute a sample of
standard compositional data, the isometric log-ratio (ilr) transformation was
employed in combination with a $k$-nearest neighbor algorithm as a regression
model for predicting the scalar variables. Methodological details about this
statistical procedure can be found in Section 2.

####  5.2.1 Prediction of A1c, HOMA-IR and glycemic variability measures using
time in range metrics

Figure 4 compares the real and estimated values of the previous five variables
under the two time in range metrics under consideration with. Table 5 provides
the estimates of $R^{2}$ for each variable and metric.

| A1c | HOMA-IR | CONGA | MAGE | MODD
---|---|---|---|---|---
Normoglycemic cut-off | $0.63$ | $0.22$ | $0.68$ | $0.65$ | $0.65$
ADA cut-off | $0.61$ | $0.08$ | $0.73$ | $0.69$ | $0.60$
Table 5: $R^{2}$ estimated with time in range metrics under consideration

The predictive capacity is significantly worse than that attained by the
glucodensity methodology. The superiority of the glucodensity is particularly
noteworthy in the case of the HOMA-IR variable, where the association is quite
weak for time in range metrics. Even for the other variables where the values
of $R^{2}$ are moderate, the larger residuals seen in diabetes patients with
more severe alterations of glucose metabolism indicate that time in range
metrics are particularly poorly suited for such patients. Interestingly, we do
not observe substantial or consistent differences between the two time in
range metrics used, as deciles perform better than ADA criteria for two of the
variables, while in other instances the ordering was reversed.

Figure 4: Real values vs. Estimated values when time in range metric is the
predictor. Blue, time in range metric with cut-offs calculated with
normoglycemic patients of AEGIS database. Red, time in range metric using of
cut-offs suggested by ADA.

##  6 Hyphotesis and clustering analysis with glucodensities

###  6.1 Analysis of variance with glucodensities

As a special case of regression, suppose we have a sample $f_{1}$, $\ldots$
$f_{n}$ of glucodensities defined on $(A,d_{W})$ belonging to $k$ different
groups $G_{1},G_{2},\cdots,G_{k}$ that partition $\\{1,\ldots,n\\}$ and are of
size $n_{j}$ $(j=1,\cdots,k)$, so that $\sum_{j=1}^{k}n_{j}=n$. If the goal is
to simply test whether the Wasserstein means are equal for each group,
Petersen et al. (2019) developed testing procedures based on model (6) for
this purpose. An advantage of this model is its flexibility, which allows for
multiple factor layouts as well as tests for interactions. However, the
theoretical properties of these tests require a type of equal variance
assumption that may be restrictive for some data sets.

More generally, one may wish to test the null hypothesis that the population
distributions of the $k$ groups share common Wasserstein means and variances,
against the alternative that at least one of the groups has a different
population distribution compared to the others in terms of either its
Wasserstein mean or variance. In this scenario, Dubey and Müller (2019)
investigated a test statistic based on the group proportions
$\lambda_{j,n}=n_{j}n^{-1}$, the groupwise sample Wasserstein means
$\tilde{\mu}_{j}=\operatorname*{arg\,min}_{g\in A}\sum_{i\in
G_{j}}d_{W^{2}}^{2}(f_{i},g)$ and variances
$\tilde{V}_{j}=n_{j}^{-1}\sum_{i\in
G_{j}}d_{W^{2}}^{2}(f_{i},\tilde{\mu}_{j}),$ the pooled Wasserstein mean
$\hat{\mu}_{p}=\operatorname*{arg\,min}_{g\in A}\sum_{j=1}^{k}\sum_{i\in
G_{j}}d^{2}_{W^{2}}(f_{i},g)$ and variance
$\tilde{V}_{p}=n^{-1}\sum_{j=1}^{k}\sum_{i\in
G_{j}}d^{2}_{W^{2}}(f_{i},\tilde{\mu}_{p}),$ and finally the quantities

$\tilde{\sigma}_{j}^{2}=\frac{1}{n_{j}}\sum_{i\in
G_{j}}d^{4}_{W^{2}}(f_{i},\hat{\mu_{j}})-\left\\{\frac{1}{n_{j}}\sum_{i\in
G_{j}}d^{2}_{W^{2}}(f_{i},\hat{\mu_{j}})\right\\}^{2}$

as estimates of the variance of $\hat{V}_{j}.$ Then, with

$F_{n}=\tilde{V}_{p}-\sum_{j=1}^{k}\lambda_{j,n}\tilde{V}_{j},\quad
R_{n}=\sum_{j<l}\frac{\lambda_{j,n},\lambda_{l,n}}{\tilde{\sigma}^{2}_{l}\tilde{\sigma}^{2}_{j}}(\tilde{V}_{j}-\tilde{V}_{l}),$

the proposed test statistic is

$T_{n}=\frac{nR_{n}}{\sum_{j=1}^{k}\lambda_{j,n}\tilde{\sigma}^{-2}_{j}}+\frac{nF^{2}_{n}}{\sum_{j=1}^{k}\lambda_{j,n}\tilde{\sigma}^{2}_{j}}.$
(9)

Dubey and Müller (2019) demonstrated that the corresponding test is
distribution-free, in that the limiting distribution of $T_{n}$ does not
depend on the underlying distribution under some assumptions. In practice, it
was also demonstrated that it can be useful to calibrate the test under the
null hypothesis via a simple empirical bootstrap over the preceding
statistics. For more details we refer the reader to the supplementary material
of the original reference.

###  6.2 Energy distance methods with glucodensities

The energy distance is a statistical distance between two distribution
functions proposed in 1984 by Gábor J. Székely (Szekely and Rizzo (2017)).
This distance is inspired by the concept of gravitational energy between two
bodies, and has experienced a rise in appeal for modern statistical
applications due to its applicability to data of a complex nature such as
functions, graphs or objects that live in negative type space.

Consider independent random variables $Y,Y^{\prime}\sim F$ and
$Z,Z^{\prime}\sim G$ that are defined on a (semi)metric space $(\Omega,\rho)$
of negative type, where $\rho:V\times V\to\mathbb{R}$ is the semi-metric.
Though the notation in this section is quite general, in particular we have in
mind the case $(\Omega,\rho)=(A,d_{W^{2}})$ corresponding to glucodensities.
The energy distance associated with $\rho$ between the distribution $F$ and
$G$ is

$\epsilon_{\rho}(F,G)=2E(\rho(Y,Z))-E(\rho(Y,Y^{\prime}))-E(\rho(Z,Z^{\prime})).$

Given random samples $Y_{1},\dots,Y_{n}\overset{\mathrm{iid}}{\sim}F$ and
$Z_{1},\dots,Z_{m}\overset{\mathrm{iid}}{\sim}G$, the sample energy distance
is

$\tilde{\epsilon}_{\rho}(F,G)=2\frac{1}{nm}\sum_{i=1}^{n}\sum_{j=1}^{m}\rho(Y_{i},Z_{j})-\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{i=1}^{n}\rho(Y_{i},Y_{j})-\frac{1}{m^{2}}\sum_{i=1}^{m}\sum_{i=1}^{m}\rho(Z_{i},Z_{j}).$

The asymptotic distribution of the above statistic for a null hypothesis
$(H_{0}:F=G)$ as well as for the alternative $(H_{a}:F\neq G)$ is dependent on
the chosen semimetric $\rho$. Besides, its expression is difficult to
calculate and to implement in practice. Hence, when using the energy distance
based methods, the distribution under the null hypothesis is usually
calibrated with a permutation method. Alternatives to calibrate the
distribution under the null hypothesis include the wild or a weighted
boostrap, as described in Leucht and Neumann (2013); Jiménez-Gamero et al.
(2019). The energy distance can also be extended to handle samples from more
than two populations. Given $k$ independent samples
$Y_{j1},\ldots,Y_{jn_{j}}\overset{\mathrm{iid}}{\sim}F_{j}$, $j=1,\ldots,k,$
the energy distance statistic is

$\tilde{\epsilon}_{\rho}(F_{1},\ldots,F_{k})\sum_{1\leq j<l\leq
k}\frac{n_{j}n_{l}}{2n}[2g_{jl}-g_{jj}-g_{ll}],\,\,g_{jl}=\frac{1}{n_{j}n_{l}}\sum_{i=1}^{n_{j}}\sum_{i^{\prime}=1}^{n_{l}}\rho(Y_{ji},Y_{li^{\prime}}),$
(10)

where $n=n_{1}+\cdots,n_{k}.$

We now explain how this statistic can be adapted to perform clustering.
Consider random pairs $(Y_{i},I_{i}),$ $i=1,\ldots,n,$ where $Y_{i}$ is
observed and takes values in $(\Omega,\rho),$ while $I_{i}\in\\{1,\ldots,k\\}$
is an unobserved label of cluster membership. The task is to recover the true
clusters $C_{j}^{*}=\\{i:I_{i}=j\\},$ $j=1,\ldots,k.$ Let $C_{1},\ldots,C_{k}$
be a generic partition of $\\{1,\ldots,n\\}$, and denote the size of each
cluster by $|C_{j}|$. Then a clustering may be chosen by optimizing the
statistic

$S_{\rho}(C_{1},\ldots,C_{k})=\sum_{1\leq j<l\leq
k}\frac{n_{j}n_{l}}{2n}[2\tilde{g}_{jl}-\tilde{g}_{jj}-\tilde{g}_{ll}],\,\,\tilde{g}_{jl}=\frac{1}{|C_{j}||C_{l}|}\sum_{(i,i^{\prime})\in
C_{j}\times C_{l}}\rho(Y_{i},Y_{i}^{\prime})$ (11)

over all possible clusters $C_{j}.$ At first view, this seems computationally
intractable due to the appearance of distances between the elements of each
cluster. However, defining

$W_{\rho}(C_{1},\ldots,C_{k})=\sum_{j=1}^{k}\frac{|C_{j}|}{2}\tilde{g}_{jj},\\\
$ (12)

it can be proven that $S_{\rho}+W_{\rho}$ is constant. This implies that
maximizing $S_{\rho}$ is equivalent to minimizing $W_{\rho}$.

In Franca et al. (2020), the authors show the equivalence between the previous
optimization problem with the clustering procedure kernel $k$-means. The
latter relationship allows the solving of kernel Kgroup clustering procedure
through the popular heuristics algorithms as Hartigan and Lloyd that allow
finding the optimal solution with the k-means algorithm.

###  6.3  Example of hypothesis testing and clustering analysis with
glucodensity methodology

Below, we illustrate the methodology of glucodensities in hypothesis testing
and cluster analysis with the $2$-Wasserstein distance. We use the ANOVA test
(Dubey and Müller, 2019) and the $k$-groups algorithm Franca et al. (2020).

####  6.3.1 Hypothesis testing

An interesting question to address in an epidemiological study is whether
there are differences between men and women in the glycemic profile. The ANOVA
test is an important instrument to establish whether there are statistically
significant differences in mean and variance with glucodensities where there
are two or more patient groups. After applying this method with AEGIS data,
the test yields a p-value equal to $0.10$. Therefore, there is no
statistically significant difference between men and women at the significance
level of $5$ percent.

Figure 5 shows the glucodensity samples for each gender using their quantile
representations. The pointwise means of these quantile functions constitute
the quantile function of the sample Wasserstein mean glucodensites. These,
together with pointwise standard deviation curves, are also shown in Figure 5.
On average, the groups are quite similar. However, certain discrepancies are
observed between both groups in terms of their variance, although not large
enough for the test to show statistical significant differences.

Figure 5: (Left two panels) Glucodensities for men and women of the AEGIS
study, plotted as quantile functions; (Third panel) 2-Wasserstein mean
quantile functions for each group; (Fourth Panel) Cross-sectional standard
deviation curves for quantile functions in each group.

####  6.3.2 Clustering analysis

Cluster analysis is an essential tool for identifying subgroups of patients
with similar characteristics. As an example, with the diabetes patients’ data
from the AEGIS study, we perform a cluster analysis using three clusters. To
establish when a patient has diabetes, we use the doctor’s previous diagnostic
criteria, or if individuals currently have their glucose values measured with
A1c and FPG in the ranges established by the ADA to be classified in that
category.

Figure 6 contains the results of applying the cluster analysis in diabetes
patients. The algorithm has identified three differentiated groups of
patients. The first group is patients with normal glucose values, probably
because they are on medication, and the diagnosis of diabetes was made in the
past. The second group are patients with slightly altered diabetes metabolism.
Finally, the last group is patients with severely altered glucose values, and
as can be seen in the glucodensities, their glucose is continuously
fluctuating. The two-dimensional graphical representation of the density
function of A1c and FPG helps to validate these findings.

Figure 6: Clustering analysis of diabetes patients in AEGIS study

##  7 Discussion

The primary contribution of this article is to propose a new representation of
CGM data called glucodensity. We have validated this representation from a
clinical point of view, proving that it is more accurate than time in range
metrics.

###  7.1 Diabetes etiology and biological components to capture in a
mathematical represention

Diabetes encompasses a heterogeneous group of impaired glucose metabolism,
such as the frequent presence of hyperglycemias or hypoglycemias Association
et al. (2018). Anomalous glucose fluctuations are another essential trait of
dysglycemic regulation Monnier and Colette (2011); Monnier et al. (2008). The
use of glycemic control measures that go beyond the average glucose values
such as A1c and also capture i) the impact of time spent at each glucose
concentration on the glucose deregulation process, ii) the oscillations of
glucose associated with cellular damage Monnier and Colette (2011), is crucial
in the management of patients with diabetes as in the assessment of glucose
metabolism with a high degree of precision.

###  7.2 Clinical validation of glucodensity

Our proposal accurately captures the components of diabetes mentioned above.
Using clinical data, we evaluated the clinical sensitivity against established
biomarkers in diabetes. We found a high association between A1c, HOMA-IR,
CONGA, MODD, MAGE, and glucodensity. In the case of the HOMA-IR variable, the
predictive ability does not seem excellent, although, to the best of our
knowledge, no known marker shows a predictive ability against that variable.
However, our model can provide consistent values in moderate and large HOMA-IR
values. While the fit for the variable A1c was not perfect, we must consider
that the time scale for the A1c and the glucodensities were quite different.
A1c is a measure that reflects the average glucose over $2-4$ months while
monitoring patients for less than $1$ week to compute the glucodensity. Our
$R^{2}$ of $0.79$ is better than the average glucose recorded by the
monitoring period ($R^{2}=0.61$), which indicates that an individual’s glucose
distributional values may give extra information to the long-term glucose
averages.

In the prediction of glucodensity from A1c, HOMA-IR, and glycemic variability
measures, the estimated $R^{2}$ shows a moderate relationship between those
variables. However, we are introducing the essential variables of the glucose
deregulation process. A possible explanation of this is that the use of the
summary measures commonly used in diabetes can hardly capture an individual’s
glycemic profile. Glucose metabolism is very complex and highly dependent on
the patient’s conditions. For example, the cellular mechanisms are different
in type I and type II diabetes. In the former, there is an inhibition of
$\beta$-cell function and consequent non-insulin production, while insulin
secretion is reduced in the latter (Taylor, 2013). In this context, the
introduction of the concept of glucodensity provides greater clinical accuracy
to the possible decisions derived from such representation compared to
traditional methods because we utilize the entire distribution of glucose
concentrations of an individual over time.

###  7.3 Time in range metrics vs. Glucodensity

While time in range metrics may also achieve the previous aim, they do so to a
clearly lesser extent than the glucodensity. Our proposal can capture the
differences between individuals in each glucose concentration.
Notwithstanding, time in range only measures glucose differences along
intervals with the subsequent loss of information. Also, time in range metrics
are substantially limited since the target zones must be defined previously,
and these may also depend on the study population or the aim of the analysis.

Empirical results demonstrate the advantages of our proposal out of the
theoretical framework. The ability of glucodensity to predict A1c, HOMA-IR,
and the CONGA, MAGE, and MODD variability measures is surprisingly high, much
higher than that achieved with the range metric despite using two different
target zones: the deciles of normoglycemic patients glucose values and the
target zones prescribed by the ADA.

The estimated $R^{2}$ between glucodensities and A1c is similar than that
reported by other authors between A1c and average glucose values Nathan et al.
(2007). However, in this study, patients are monitored only for $2-6$ days and
not for weeks. Two possible factors we must consider in the analysis of the
results. First, there are both diabetic and non-diabetic patients in our
database, and, second, the glucodensity captures A1c better because it
represents the entire distribution of glucose concentration values, while
glycation rates are known to increase with glucose concentrations Singh et al.
(2001). In particular, the estimated $R^{2}$ between A1c and the mean glucose
in our database is only $0.61$.

###  7.4 Statistical considerations

From a statistical standpoint, glucodensities are a special constrained type
of functional data known as distributional data; therefore, it is not possible
to directly use the usual statistical techniques. In this paper, we have
proposed a framework for the analysis of these distributional data based on
distances with existing techniques for hypothesis testing, cluster analysis,
and regression models. However, further methodological development is
necessary, as it can be the case of mixed models or causal inference methods
where there is no available methodology.

###  7.5 Limitations

A potential limitation of our representation is that it ignores the order of
events. Instead, it analyzes only the distribution of glucose values. However,
the event sequence may not be a critical component in diabetes modeling. The
main factor of microvascular and macrovascular complications is chronic
hyperglycemia Cryer (2014); Škrha et al. (2016), and this is captured with
high accuracy by our models. Moreover, an essential aspect of managing
diabetes patients is hypoglycemia control, and our proposal also captures
this. Finally, the third component of dysglycemia Monnier et al. (2008),
glucose variability, can accurately predict by our representations, at least,
through metrics CONGA, MAGE, and MODD.

The sample size used may also be a limitation from a statistical point of
view. Nevertheless, in the field of diabetes, the AEGIS study is the world’s
largest databases and, unlike other studies, is composed of randomly selected
individuals from a general population and non-participants Zeevi et al.
(2015). Finally, for study validation, perhaps the most reliable way of
validating the new representation is in terms of the patients’ long-term
prognosis. However, to the best of our knowledge, no study with a reasonable
sample size has this information from the intensive use of CGM technology.
Moreover, we have established the clinical validity from variables that do
have a clear and established relationship with the prognosis and prevalence of
diabetes as evidenced in the current literature in the field.

###  7.6 Potential Applications

Adopting the concept of glucodensity in clinical practice and biomedical
research could be very promising in the following ways.

  * •

To have a simple and more accurate representation of the glycaemic profile of
an individual. This representation is especially useful in the management of
diabetic patients and to assess the effects of an intervention.

  * •

To establish if there are statistically significant differences between
patients subjected to different interventions, for example, in a clinical
trial.

  * •

To identify different subtypes of patients based on their glycaemic condition
and other variables. Cluster analysis of glucodensities can create new patient
subtypes based on the risk of diabetes or other complications. Furthermore, it
allows us to describe the etiology of the disease better by creating groups of
subjects whose glucose profiles and other clinical characteristics are
similar.

  * •

To establish the prognosis or risk of a patient or to analyze the relationship
of an individual’s glycaemic profile with different clinical variables in
epidemiological studies.

  * •

To predict changes in the glycaemic profile based on the characteristics of
the individuals and the intervention performed. For example: how does the
glucodensity vary according to the diet?

  * •

To recommend the most advantageous treatments for a patient. Following the
previous idea, a causal inference model could be fitted where the response is
glucodensity, for example, to establish which diet is the most beneficial for
the individual to achieve a suitable glucose levels.

###  7.7 Future work

We introduce glucodensities methodology with CGM data. However, our
methodology is also valid for data from other biosensors such as
accelerometers to measure physical activity levels. In this domain, the time
in range metric is one of the most used representations, and perhaps the
adoption of our approach can lead to better results Dumuid et al. (2018,
2020). The adoption of new methodology with other biosensors may be an
essential research issue to be addressed in the future.

In the diabetes field, it will be necessary to evaluate the predictive
capacity of the glucodensity in the long-term prognosis of patients. In
addition, it would be interesting to assess, in more extended monitoring
periods, the reproducibility between days and weeks with the representation
constructed. One way to accomplish this is to compute the intraclass
correlation coefficient (ICC) using, for example, the methodology proposed
recently in Xu et al.  and based on distances between functions.

## Acknowledgements

We thanks Russell Lyons for his discussions on the use of energy-distance
based methods with glucodensities.

This work has received financial support from Carlos III Health Institute,
Grant/Award Number: PI16/01395; Ministry of Economy and Competitiveness
(SPAIN) European Regional Development Fund (FEDER); the Axencia Galega de
Innovación, Consellería de Economía, Emprego e Industria, Xunta de Galicia,
Spain, Grant/Award Number: GPC IN607B 2018/01; National Science Foundation,
Grant/Award Number: DMS-1811888; the Spanish Ministry of Economy and
Competitiveness Grant/Award Number: TIN2015-73566-JIN and TIN2017-84796-C21-R.

## Competing Interests

The authors declare no competing interests.

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