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A computational neurodegenerative disease progression score: Method and results

with the Alzheimer's disease neuroimaging initiative cohort

Bruno M. Jedynaka,b,⁎, Andrew Langc, Bo Liua, Elyse Katzd, Yanwei Zhangd, Bradley T. Wymand,

David Raunigd,1, C. Pierre Jedynake, Brian Caffof, Jerry L. Princec

for the Alzheimer's Disease Neuroimaging Initiative2

aDepartment of Applied Math and Statistics, Johns Hopkins University, Baltimore, MD 21218, USA

bCenter for Imaging Science, Johns Hopkins University, Baltimore, MD 21218, USA

cDepartment of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA

dPfizer Inc., Groton, CT 06340, USA

eSelf, Paris, 75011, France

fDepartment of Biostatistics, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD 21205, USA

a b s t r a c ta r t i c l e i n f o

Article history:

Accepted 29 July 2012

Available online 3 August 2012

Keywords:

Neurodegenerative diseases

Alzheimer's disease

Biomarkers

Disease progression score

While neurodegenerative diseases are characterized by steady degeneration over relatively long timelines, it

iswidely believed thattheearlystages are the mostpromising fortherapeuticintervention, before irreversible

neuronal loss occurs. Developing a therapeutic response requires a precise measure of disease progression.

However, since the early stages are for the most part asymptomatic, obtaining accurate measures of disease

progression is difficult. Longitudinal databases of hundreds of subjects observed during several years with

tens of validated biomarkers are becoming available, allowing the use of computational methods. We propose

a widely applicable statistical methodology for creating a disease progression score (DPS), using multiple

biomarkers, for subjects with a neurodegenerative disease. The proposed methodology was evaluated for

Alzheimer's disease (AD) using the publicly available AD Neuroimaging Initiative (ADNI) database, yielding

an Alzheimer's DPS or ADPS score for each subject and each time-point in the database. In addition, a common

description of biomarker changes was produced allowing for an ordering of the biomarkers. The Rey Auditory

Verbal Learning Test delayed recall was found to be the earliest biomarker to become abnormal. The group of

biomarkers comprising the volume of the hippocampus and the protein concentration amyloid beta and Tau

were next in the timeline, and these were followed by three cognitive biomarkers. The proposed methodology

thus has potential to stage individuals according to their state of disease progression relative to a population

and to deduce common behaviors of biomarkers in the disease itself.

© 2012 Elsevier Inc. All rights reserved.

Introduction

Neurodegenerative diseases such as Alzheimer's disease (AD),

Parkinson disease (PD), Huntington disease (HD) and amyotrophic

lateral sclerosis (ALS) involve the loss of structure or function of neu-

rons, including neuronal death (see Martin (2002); Shaw (2005)).

During the earliest stages of these diseases, the progression is slow,

on the time scale of years, (see Sperling et al. (2011) for the case

of AD). It is widely believed that these early stages are the most

promising for therapeutic intervention, before irremediable neuronal

loss occurs.Developing a therapeutic remedyrequiresa precisemeasure

ofdiseaseprogression,i.e.,aquantitywhichwouldbespecifictoapartic-

ulardiseaseandsensitivetosubtlechanges.However,obtainingaccurate

measuresofdiseaseprogressionduringtheearliestphasesofthedisease

is difficult. Indeed, these phases are essentially non-symptomatic and

the clinical tests which characterize the acute phase of the disease are

notsensitiveenoughtoqualifyasameasureofdiseaseprogression.Inre-

sponse, the medical research community has contributed to developing

and validating biomarkers. Biomarkers for neurodegenerative diseases

include protein counts (in the cerebrospinal fluid), blood analysis,

brain imaging, including molecular and MR, genetic analysis and neuro-

psychological tests. Structural imaging biomarkers are unique in that

they allow one to characterize the size, shape, and health of various

brain substructures at the organ level while being noninvasive (see

e.g. Qiu et al. (2008) forAD,Rizk-Jacksonet al. (2011) forHD). Functional

imaging provides a spatially localized image of the physiological

NeuroImage 63 (2012) 1478–1486

⁎ Corresponding author at: Whitehead 208B, Johns Hopkins University, 3400 North

Charles Street, Baltimore, MD, 21218, USA. Fax: +1 410 516 7459.

E-mail address: bruno.jedynak@jhu.edu (B.M. Jedynak).

1Present address: ICON Medical Imaging, Warrington, PA 18976, USA.

2Data used in preparation of this article were obtained from the Alzheimer's Disease

Neuroimaging Initiative (ADNI) database (adni.loni.ucla.edu). As such, the investigators

within the ADNI contributed to the design and implementation of ADNI and/or provided

data but did not participate in analysis or writing of this report. A complete listing of

ADNI investigators can be found at: http://adni.loni.ucla.edu/wp-content/uploads/how_

to_apply/ADNI_Acknowledgement_List.pdf.

1053-8119/$ – see front matter © 2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.neuroimage.2012.07.059

Contents lists available at SciVerse ScienceDirect

NeuroImage

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processes occurring in the brain. See Brooks and Pavese (2011) for

a review of imaging biomarkers in PD and Turner et al. (2011) for

ALS. Due to the complexity of the neurodegenerative diseases and

variabilities within the human population, research efforts have been

pooled in order to create datasets with a large number of subjects,

time-points and biomarkers. The Alzheimer's Disease Neuroimaging

Initiative (ADNI), see http://adni.loni.ucla.edu/, was launched in 2003

by the National Institute on Aging, the National Institute of Biomedical

ImagingandBioengineering,theFood andDrugAdministration,private

pharmaceutical companies and non-profit organizations, as a $60 mil-

lion, 5-year public/private partnership. A related effort is taking place

for PD. The Parkinson Progression Marker Initiative (PPMI), see http://

www.ppmi-info.org/, is a comprehensive observational, international,

multicenter study designed to identify PD progression biomarkers

both to improve understanding of disease etiology and course and to

provide crucial tools to enhance the likelihood of success of PD modify-

ing therapeutic trials. Huntington disease is caused by a mutation in a

single gene, HTT, with full penetrance, making it feasible to identify

presymptomatic individuals who will develop the disease but do

not show yet any clinical symptoms, see Hayden (1981). At least two

largestudies(Predict-HD,see

TrackOn-HD, see http://hdresearch.ucl.ac.uk/current-studies/trackon-

hd/) are underway to identify sensitive biomarkers for HD. Similar ef-

forts are recently taking place for ALS, see Turner et al. (2009); Labbe

(2012).Theavailabilityoflargedatasetsforneurodegenerativediseases

opens new opportunities for computational methods which could have

a strong impact in the study, the development of therapeutics and the

follow-up of patients with neurodegenerative diseases.

We present in this article a generic computational method for

computing a disease progression score (DPS) by combining bio-

markers. ADNI is, as of today, the largest publicly available longitudi-

nal dataset of biomarkers related to a neurodegenerative disease. It is

therefore the dataset which we have chosen to evaluate our method.

Since we will work with the ADNI dataset, we recall some preliminary

information on AD as well as the validated biomarkers for AD in

Section 2. The method for computing a DPS, which is the main contri-

bution of this paper, is presented in Section 3. Results with the ADNI

dataset appear in Section 4 and finally in Section 5, we discuss the

results in the context of ADNI, and their consequence in the study of

AD and other neurodegenerative diseases.

https://www.predict-hd.net/and

Alzheimer's disease

Althoughthispaperdescribesamethodapplicabletoanyneurode-

generative disease, our current evaluation involves the ADNI dataset

and therefore it is informative to use this disease as a framework

for motivating the method. The classical characterization of late-

onset Alzheimer's disease progression is a time-ordered succession

of three stages: normal (N), mild cognitive impairment (MCI), and

AD. Physical measurements of disease progression, i.e., biomarkers,

are used to classify patients into these three stages, but it has been

challenging to reliably define finer stages of the disease. As a result,

staging of the disease remains coarse and the evaluation of therapies

are difficult at the earliest stages when intervention is most likely to

be effective, see Hampel et al. (2008).

Cognitive biomarkers such as the clinical dementia rating sum-

of-boxes (having scores from 0 to 18) and the mini-mental state

exam (having integer scores from 0 to 30) have finer discrete levels,

see Berg et al. (1988); Folstein et al. (1975). But it has been reported

in Mungas and Reed (2000) and Duara et al. (2011) that these mea-

surements have poor dynamic range in the earliest stages of AD. On

the other hand, Mosconi et al. (2007) has shown that the early stages

of AD can be characterized using both imaging and biochemical bio-

markers. Following these observations, Jack et al. (2010) proposed

that there is a single disease progression and that different bio-

markers characterize the disease during different stages. They hy-

pothesized the biomarker changes and disease progression shown

in Fig. 1 (reproduced with permission from Jack et al. (2010)). In

this hypothesized model, the amyloid beta (Aβ42) protein changes

first, followed by changes in the protein Tau, then structural changes

in the brain (gray matter loss), and lastly a deterioration of cognitive

function resulting in dementia. Based on Fig. 1 we expect to find that

no single biomarker has the dynamic range to cover the full spectrum

of the disease. Given the limitations of any single biomarker, there is

likely benefit in developing methods that can combine multiple bio-

markers in a nonlinear fashion in order to represent—using a single

measure—progression throughout the entire disease. This is a key

motivation for the process we report in this paper. An important

byproduct of this effort is a plot similar to that of Fig. 1, but derived

from data using multiple biomarkers which reveal key differences in

the ordering of the biomarker dynamics over the course of disease.

Method

Principles for temporal standardization of multiple biomarkers

The available data are longitudinal measurements of multiple bio‐

markers for hundreds of subjects. Our research first describes and

then evaluates a disease progression score, notated DPS, which stan-

dardizes subject time-lines onto a common temporal scale. The DPS

Fig. 1. This graph represents a conceptualization of the timing of key biomarkers transitions from “Normal” to “Abnormal” as subjects go through the three stages of Alzheimer's

disease: “Cognitively Normal”, “MCI”, and “Dementia.” This plot is reproduced from “Hypothetical model of dynamic biomarkers of the Alzheimer's pathological cascade,” Jack CR Jr,

Knopman DS, Jagust WJ, Shaw LM, Aisen PS, Weiner MW,Petersen RC, Trojanowski JQ., Lancet Neurol. 2010 Jan;9(1):119-28.

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serves as a new (derived) biomarker enabling both disease staging

in single subjects and a data-driven characterization of biomarker

dynamics in the entire population.

The method we use to achieve standardization is based on three

assumptions:

1. All subjects follow a common disease progression but differ in their

age of onset and rate of progression;

2. As the disease progresses, each biomarker changes continuously

and monotonically following a sigmoid shaped curve; and

3. In the longitudinal period over which biomarkers are observed, the

rate of progression of a given subject is constant.

The proposed computation assigns to each subject and each time-

point a score denoted the DPS. Note that all subjects are expected to

undergo the same biological and cognitive changes when they reach

the same DPS.

Statistical model for DPS

The age t of subject i is to be transformed into the DPS sias follows

sit ð Þ ¼ αit þ βi

ð1Þ

upon estimation of the subject dependent parameters αi and βi,

which indicate rate and onset of disease, respectively. A linear trans-

formation is justified when the interval over which longitudinal

observations of subjects occur is short relative to disease duration

(true at present in the ADNI database). This could be generalized to

nonlinear functions in the case of cohorts with longer longitudinal

base. Our objective is to standardize all I subjects by estimating α=

(α1, …, αI) and β=(β1, …, βI). The subject dependent parameters α

and β are deliberately modeled as fixed effects, not random effects,

as the DPS may ultimately be used as a covariate.

The longitudinal dynamic of each biomarker is assumed to be the

same across the population and can be represented as a sigmoidal

function f of DPS s. Sigmoidal functions capture the relative quiescent

states of a biomarker in the early and late parts of the disease progres-

sion while being parsimonious. Using θk=(ak, bk, ck, dk) to represent

the vector of sigmoid function parameters for the k-th biomarker, we

can write the form of the k-th biomarker as

f s;θk

ð Þ ¼ ak1 þ e−bks−ck

ðÞ

??−1þ dk:

ð2Þ

The minimum and maximum values of the sigmoid function are dk

and dk+ak, and the value of s for which the biomarker is the most dy-

namic, having maximum slope akbk/4 corresponding to its inflection

point, is ck. A closely related model is the trilinear model in Brooks

et al. (1993). Caroli et al. (2010) and Sabuncu et al. (2011) noticed

that sigmoids offer a parsimonious parametric model which is

often a better fit than linear models for biomarkers. Sigmoids are

also similar in form to the conceptual evolution of biomarkers

envisioned in Jack et al. (2010) for AD (Fig. 1). Among parametric

models, alternatives include the generalized sigmoid in Richards

(1959) and polynomials of low order.

Databases for neurodegenerative diseases contain measurements

yijkof biomarker k for subject i at visit j. Since there are often irregu-

larities in data collection, we use I to denote the set of triples (i, j, k)

for which measurements are available. Each biomarker observation

can then be written as

yijk¼ f αitijþ βi;θk

??

þ σk?ijk;

i;j;k

ðÞ∈I;

ð3Þ

where tijis the age of subject i at visit j. Observation noise in each

biomarker is modeled for simplicity by the product of ?ijk, which

are independent random variables with zero mean and unit variance.

σkis the standard deviation of biomarker k. The collection of standard

deviations σ=(σ1, …, σK) comprise another unknown that must be

estimated.

The unknowns in this problem are α, β, θ, and σ and the least

squares problem associated with the observation model in (3) is

l α;β;θ;σ

ð Þ ¼

∑

i;j;k

ðÞ∈I

logσkþ

1

2σk

2

yijk−f αitijþ βi;θk

????2

ð4Þ

Parameter fitting

Parameter fitting is performed using alternating least squares

wherein the parameters θ, α, β, and σ are optimized iteratively

starting from the values computed in the previous step. The details

of the fitting algorithm are shown in Alg. 1. Because of the additive

form of (4), optimization over θ is done serially over each of the K bio-

markers. Similarly, optimization over (α, β) is performed serially over

each of the I subjects. Fitting of θ, α, and β requires optimization of con-

tinuouslydifferentiablenonconvexfunctions,whichiscarriedoutusing

the Levenberg–Marquardt algorithm (Lines 4 and 8), see Levenberg

(1944). Ik(line 4) is the number of subjects and visits available for

biomarker k. The denominator in the equation of Line 5 is the number

of degrees of freedom. Because unconstrained optimization can pro-

duce unfeasible parameters, parameters are projected onto the feasible

space after the main loop (Lines 12–16), see (5) below. This does not

change the value of the objective function in (4). Our experiments

presented in Section 4 confirm that successful fitting is accomplished

in 15 iterations for the ADNI dataset; i.e., L=15 on Line 2, standard

optimization stopping criteria can be used otherwise. The parameters

α and β are centered and rescaled in Lines 17–19 in Alg. 1 for

identifiability reasons which are explained in the next section.

Identifiability

The units of DPS are arbitrarily defined, which implies that we

must choose two specific numerical values in order to fully specify

the DPS. This situation is analogous to the selection of a scale for tem-

perature, where the numerical values of the freezing and boiling

points of water determine the scale. Note that calibration is not spe-

cific to the DPS. It is in fact needed for most if not all biomarkers

(see Hughes et al. (1982)). In our experiments with ADNI, we chose

to fix the DPS such that after computation of DPS for the entire pop-

ulation, the computed DPS for all visits of subjects with normal clini-

cal assessment - subjects of type N -had a median (mN) and a median

absolute deviation (σN) which are set respectively to zero and one.

This is accomplished in Lines 17–19 in Alg. 1.

Algorithm 1. Algorithm for fitting of the parameters

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Note that (3) is invariant with respect to the following two trans-

formations, for two constants γ1≠0 and γ2:

ak;bk;ck;dk;αi;βi;σk

ðÞ

↦ ak;γ1bk;γ−1

↦ ak;bk;γ2þ ck;dk;αi;γ2þ βi;σk

1ck;dk;γ−1

1αi;γ−1

1βi;σk

?

ð

?

Þ

Note also that the sigmoid function verifies

f t;−a1;−b1;c1;d1þ a1

ð Þ ¼ f t;a1;b1;c1;d1

ðÞð5Þ

In order to build an identifiable model, we define the restricted

parameter set

ϱ¼

ρ¼ a;b;α;β;σ

ðÞ;I−1X

I

i¼1

αi¼ α0;I−1X

I

i¼1

βi¼ β0;bk>0;ak≠0 for all k∈I

()

for some α0≠0 and β0. Necessary conditions on the available data I

for guaranteeing the identifiability of the parameters are as follows:

1. For each biomarker, there is at least one subject i with αi≠0 and

with at least 4 distinct time-points in I.

2. For each subject, there is at least one biomarker which is available

at 2 time points in I

A proof is provided in the Appendix A. In practice, a sufficient

number of data points per parameter are needed in order to obtain

tight estimators. Examining first the case with no missing data, the

number of equations in (3) is IJK. The number of parameters is

2I+5K, counting two parameters per subject, and five per bio-

markers: four for the sigmoid and one for the standard deviation. In

applications where I is large compared to K, the number of data points

per parameter is close to JK/2. Note that longitudinal data (J>1) is

critical for such modeling. However, a small number J of time-points

together with a small number K of biomarkers is acceptable. The sub-

set of ADNI that we used in our results has numerous missing data

points. Nevertheless, the identifiability conditions are met. The tight-

ness of the estimators of the biomarker parameters is measured using

bootstrapping as reported in the Results section.

The ADNI dataset

Data used in the preparation of this article were obtained from

the ADNI database (adni.loni.ucla.edu). The ADNI was launched in

2003 by the National Institute on Aging (NIA), the National Institute

of Biomedical Imaging and Bioengineering (NIBIB), the Food and

Drug Administration (FDA), private pharmaceutical companies and

non-profit organizations, as a $60 million, 5- year public-private part-

nership. The primary goal of ADNI has been to test whether serial

magnetic resonance imaging (MRI), positron emission tomography

(PET), other biological markers, and clinical and neuropsychological

assessment can be combined to measure the progression of mild

cognitive impairment (MCI) and early Alzheimer's disease (AD). De-

termination of sensitive and specific markers of very early AD pro-

gression is intended to aid researchers and clinicians to develop

new treatments and monitor their effectiveness, as well as lessen

the time and cost of clinical trials.

The Principal Investigator of this initiative is Michael W. Weiner,

MD, VA Medical Center and University of California – San Francisco.

ADNI is the result of efforts of many co-investigators from a broad

range of academic institutions and private corporations, and subjects

have been recruited from over 50 sites across the U.S. and Canada. The

initial goal of ADNI was to recruit 800 adults, ages 55 to 90, to partic-

ipate in the research, approximately 200 cognitively normal older

individuals to be followed for 3 years, 400 people with MCI to be

followed for 3 years and 200 people with early AD to be followed

for 2 years. For up-to-date information, see www.adni-info.org.

The ADNI, ADNI GO, and ADNI 2 biomarker datasets were

downloaded from the ADNI server (http://adni.loni.ucla.edu/) on

November 24, 2011. The following seven biomarkers were selected

for use based on their relevance in assessing the progression of AD.

HIPPOis the sum of the two lateral hippocampal volumes (Freesurfer

version 4.4.0 for longitudinal data http://surfer.nmr.mgh.harvard.

edu) normalized by dividing by the intracranial volume. ADAS is the

Alzheimer's Disease Assessment Scale-cognitive subscale. MMSE is

the Mini-Mental State Examination score. TAU and ABETA (our abbre-

viation for Aβ42) are protein levels measured from the cerebrospinal

fluid. CDRSB is the Clinical Dementia Rating Sum of Boxes score and

RAVLT30 is the Rey Auditory Verbal Learning Test, 30 minute recall.

A detailed description of the ADNI population, protocols and bio-

markers is provided at http://adni.loni.ucla.edu/. Of the seven bio-

markers, only ADAS and RAVLT30 were available at the time of

download from the ADNI 2/GO dataset. The protocol for these bio-

markers is the same in ADNI, ADNI 2, and ADNI GO. All visits without

dateinformationwereremoved. Subjectsnothaving at leasttwomea-

surementsforat least oneoftheseven biomarkers werealsoremoved.

Finally, subjects not having at least two measurements of the HIPPO

biomarker were removed. The total number of subjects remaining

was 687, where 389 were male, 275 were female, and 23 had un-

knowngender.Thetotalnumberofvisitswas3658,andtheclinicaldi-

agnoses at these visits were 1103 N, 1513 MCI, and 1010 AD. There is

an average of 26.92 (sd=5.52) and a minimum of 11 data points

available per subject for estimating the parameters of the model.

Results

DPS computed for ADNI subjects

The Alzheimer's DPS (ADPS) was computed for all subject visits in

the combined ADNI, ADNI 2, and ADNI GO datasets (with minimal

exclusions as was described in Section 5). Seven biomarkers—HIPPO,

MMSE, TAU, ABETA, CDRSB, RAVLT30, and ADAS—were used together

in the computation in order to compute an ADPS score for each visit

of each subject (Fig. 2). The initial values (Line 1 of Alg. 1) are

obtained as follows: firstly, we set α(0)≡1 and β(0)≡0; secondly, the

sigmoids are replaced by linear functions. The main loop (line 2), is

then executed 15 times. In this case, the optimization problems in

lines 4 and 8 are least squares problems which are solved exactly.

At the end of this initialization step, α(0)and β(0)are set to the corre-

sponding values obtained and the sigmoids are initialized using the

linear fits. The running time of the Algorithm 1, which was coded in

Matlab, was 125 seconds using an Intel Core i7 Q820 running at

1.73 GHz (quadcore). In Fig. 2, overall, N subjects (black) have the

smallest ADPS, MCI subjects (red) have moderate ADPS, and AD sub-

jects (green) have the largest ADPS. Lower ADPS scores are therefore

consistent with the normal population and higher ADPS scores are in-

dicative of increased presence of dementia. Those subjects whose

clinical status changes from MCI to AD (blue) are found mostly be-

tween the red and green colors.

The estimated sigmoidal behaviors of each biomarker were also

computed as part of the normalization process (gray curves on each

plot in Fig. 2). It is observed that individual subject trajectories

fall near these curves and have similar slopes in most cases. This is

expected due to the nature of the optimization criterion used to de-

fine ADPS. However, since ADPS is computed as a joint optimization

considering all seven biomarkers, some data falls fairly far from the

estimated characteristic biomarker curves.

We used bootstrapping via Monte Carlo resampling to quantify

the variance of the estimated parameters. We drew 100 resamples

of the observed dataset by random sampling (with replacement)

from the original collection of subjects, and then recomputed the

ADPS for the entire population. Bootstrap replicates of the estimated

biomarker sigmoids are shown in Fig. 3 and 90% confidence intervals

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B.M. Jedynak et al. / NeuroImage 63 (2012) 1478–1486

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for the parameter ck, i.e. the inflection point of each sigmoid, are

presented in Fig 5(b).

The empirical variance of the residuals ?ijkin (3) is the compo-

nent of the variance which is unexplained by the model. It accounts

for about 38% of the total variance. Hence the model explains 62%

(±1.37%) of the total variance (i.e., 62%=100%−38%.), the stan-

dard deviation (sd) of 1.37% being computed using the bootstrap

samples. If instead of the ADPS, ADAS or MMSE was used as a disease

progression score, fitting sigmoid curves as previously described,

the percentage of explained variance would be respectively 49.4%

(±1.4%) and 46% (±1.4%). The percentage of explained variance

is larger with the ADPS than with the ADAS (p-valueb0.01) or the

MMSE (p-valueb0.01); p-values being obtained using the bootstrap

replicates in both cases.

HIPPO

0.002 0.003 0.004 0.005 0.006 0.007

MMSE

0

5

10

15

20

25

30

TAU (pg/ml)

0

100

200

300

400

Progression of ADNI biomarkers

as function of the Alzheimer's

Disease Progression Score (ADPS)

Normal−Normal

MCI−MCI

AD−AD

MCI−AD

MCI−N

sigmoids

ABETA (pg/ml)

−10 −5 0

ADPS

51015 20

50

100

150

200

250

300

350

CDRSB

−10 −505101520

0

5

10

15

20

RAVLT30

−10 −50510 15 20

0

5

10

15

ADAS

−10 −505 101520

0

10

20

30

40

50

60

70

ADPSADPS ADPS

Fig. 2. The values of seven biomarkers, measured at all visits of all ADNI subjects, are plotted on the normalized ADPS. Each connected polyline represents the consecutive visits of a

single subject, and each line segment is colored according to the subject's clinical diagnoses between visits (see legend). The gray curves are the sigmoid functions representing the

fitted behavior of each biomarker in the normalized space.

HIPPO

0.0030 0.0035 0.0040 0.0045 0.0050

MMSE

−5

0

5

10

15

20

25

30

TAU (pg/ml)

50

100

150

200

250

100 Bootstrap replicates

of the estimated biomarker

sigmoids

ABETA (pg/ml)

−10 −50

ADPS

5101520

140

160

180

200

220

240

CDRSB

−10 −505 10 1520

0

5

10

15

20

RAVLT30

−10 −505101520

0

5

10

15

20

25

ADAS

−10 −505101520

0

50

100

150

200

250

ADPSADPS ADPS

Fig. 3. Bootstrapping yields different biomarker sigmoids with each random substitution. These plots give all the computed sigmoids over the entire bootstrapping exercise. Tight

agreement overall is observed.

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B.M. Jedynak et al. / NeuroImage 63 (2012) 1478–1486

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Relation between ADPS and rate of progression

The rate of progression αiof each subject i is also computed as part of

theADPSparameterfittingalgorithm.Weplottedtherateofprogression

of each subject against their ADPS at baseline to see whether a relation-

ship might exist (Fig. 4). A clear trend of increasing rate of ADPS as a

function of ADPS is observed. The third column of Table 1 provides the

mean rate of change of ADPS in unit of years for each status. AD subjects

progress faster on average than MCI subjects. MCI subjects progress

faster on average than N subjects. Observed during 3 years, an MCI sub-

jectwouldprogressonaverageat0.76ADPSperyear.Thecorresponding

ADPS would then increase by 0.76×3=2.28 units. In our model, the

ADPS of each subject is a linear function of age, or equivalently the rate

of change of ADPS is constant over the time a subject is observed. Retro-

spectively, it is therefore a reasonable approximation for N and MCI

subjects. It might be too simple a model for AD subjects. It is important

to recall that these observations are made in light of the optimization

criterion of ADPS, which uses the commonality of biomarker trends as

a basis for determining rate. Thus, an increasing rate of ADPS truly

means that subjects are progressing through degrading biomarkers at

a faster rate.

Biomarker dynamics

The sigmoidal functions representing common behavior of bio-

marker dynamics of the entire ADNI population can be compared by

scaling (and inverting if necessary) each of them independently to

range from −1 (Normal) to +1 (Abnormal). Plotted as a function

of the normalized ADPS (Fig. 5(a)), these scaled sigmoidal functions

provide a plot similar to the conceptual plot in Jack et al. (2010)

(Fig. 1). Our plot is data driven, of course, representing what the

entire ADNI dataset predicts under our model assumptions. Its sig-

moidal functions also provide information about the time of initial

biomarker change (represented by the heels of the sigmoidal func-

tions), the time of maximum biomarker change (represented by the

inflection point of the sigmoidal functions), and the rate of biomarker

change over the course of its activation (represented by the slopes of

the sigmoidal functions).

In addition to their interpretation as the time of maximum bio-

marker change, the inflection points also could represent a threshold

between normal and abnormal. Therefore, we use them as an indicator

−50

ADPS

510

−6

−4

−2

0

2

4

6

Rate (ADPS per year)

Fig. 4. Rate of the ADPS as function of the ADPS for baseline visits. Black: Normal subjects.

Red: MCI subjects. Green: AD subjects.

Table 1

Mean value (standard deviation) of ADPS and rate of change of ADPS for N,MCI and AD

subjects in ADNI at baseline.

ADPS: Mean (sd)Rate of change of ADPS: Mean (sd)

N

MCI

AD

−0.03 (1.48)

2.85 (1.98)

6.49 (1.61)

−0.08 (0.81)

0.76 (1.11)

1.46 (1.38)

b

a

Fig. 5. (a) Estimated biomarker dynamics as a function of the normalized ADPS. Estimation of the normalized ADPS for all ADNI subjects was carried out, and common biomarker

dynamics represented by sigmoidal functions were simultaneously fitted as part of the ADPS normalization algorithm. Each sigmoidal function was scaled and flipped in order to fit

on a scale going from -1 representing “Normal” to 1 representing “Abnormal”. The positions of vertical lines representing progression from Normal to MCI and MCI to AD were fitted

as optimal separating thresholds between the clinical diagnoses provided in the ADNI database. (b) 90% confidence intervals for the inflection point of each biomarker.

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B.M. Jedynak et al. / NeuroImage 63 (2012) 1478–1486

Page 7

of biomarker timing in the disease process. We recomputed the inflec-

tion point of the normalized biomarker sigmoids for each bootstrap

sample and plotted 90% confidence intervals (Fig 5(b)). Furthermore,

counting pairwise ordering within the bootstrap samples, we find that

RAVLT30 precedes all other 6 other biomarkers (p-valueb0.01) and

HIPPO, ABETA and TAU precede MMSE and ADAS (p-valueb0.02).

Relation between ADPS and clinical status

Conditional probability densities of ADPS given the clinical status

of each subject were computed using Gaussian kernel density estima-

tion (Fig. 5(a)). Since N subjects tend to have a smaller ADPS than

MCI subjects who in turn tend to have a smaller ADPS than AD sub-

jects, this plot confirms that ADPS provides a scale that correlates

strongly with clinical classification of disease. The mean and standard

deviation of the baseline ADPS for N, MCI and AD subjects in ADNI is

provided in Table 1, column 2. The means are well separated from

each other. There is overlap in the baseline ADPS value between N

and MCI and also between MCI and AD, but essentially not between

N and AD. It is worth restating the clinical diagnosis is not used in

computing the ADPS except to determine its units.

Discussion

We combine multiple biomarkers to provide a neurodegenerative

disease progression. In contrast, in the case of AD, Brooks et al. (1993);

Stern et al. (1994); Ashford et al. (1995); Mitnitski et al. (1999) and

others use MMSE or ADAS as measure of disease progression. In Yang

et al. (2011a), the authors synchronize subjects onto a time-line

constructed using ADAS scores. The core assumption is that the rate of

change of ADAS is linear with respect to the ADAS score, resulting in

an exponential model of disease progression. In Walhovd et al. (2010);

Hinrichs et al. (2011), multiple biomarkers are combined to diagnose

AD.InFonteijnetal.(2011)theprogressionofADisdividedintodiscrete

events based on the atrophy of different structures in the brain provid-

ing a probabilistic framework for estimating the global progression of

AD as well as for estimating the position of a single subject's measure-

ments. Longitudinal measurements are not used. In Ververidis et al.

(2010), a Bayesian classifier selects the set of biomarkers which are

most informative for classifying the current state of the disease.

Time-series models are used to predict the future state of the disease.

Yang et al. (2011b) use independent component analysis and support

vector machines to classify subjects into N versus MCI or AD. Our statis-

tical model is related to so-called single index models (see Hardle et al.

(1993); Carroll et al. (1997) and the references therein). However,

our models differ from these, as we assume parsimonious parametric

forms for the index function and allow for multivariate outcomes.

Our modelingtechnique applied tothe ADNI hasprovided confirma-

tion of existing results: Jack et al. (2011) binarized each biomarker into

either normal or abnormal using a threshold or cut point. Cut points

were determined for each biomarker at autopsy and with an indepen-

dent cohort. When using these cut point to determine the ADPS at

which a biomarker changes from normal to abnormal, we find that

ABETAprecedesbothHIPPOandTAUwhichisconsistentwiththeresults

in Jack et al. (2011). We have also obtained surprising results. The fact

that the inflection of RAVLT30 precedes that of all other biomarkers,

andin particular thatof ABETA is surprising, compared to Fig. 1, but con-

sistentwithsomepredictions.Jichaand Carr(2010) refertothestudyin

Bennett et al. (2006) stating, “Retrospective analysis of their neuropsy-

chological test performance demonstrated significant differences in

onlydelayedrecalltasksbetweensubjectswithpathologicalADautopsy

findings and those with normal autopsy findings, suggesting that mem-

ory decline may be present, albeit subtly, in persons with (preclinical)

AD before sufficient cognitive decline to warrant the diagnosis of either

MCI or dementia.”Also, Dubois et al. (2007) advocate that the presence

of an early and significant episodic memory impairment should consti-

tute one of the core diagnostic criteria for AD.

Conclusion

Wereportamultiplebiomarker,data-drivenapproachtoassesstime-

dependent changes of biomarkers in neurodegenerative disease and to

localizesubjectsonascaleofdiseaseprogression,theDPS,overtheentire

range of progression. The statistical model is shown to be identifiable

and bootstrap replicates show that the parameters are estimated tightly

incaseoftheADNIdataset.TheDPSintegratesinformationfrommultiple

biomarkers into a single composite biomarker. Using this approach the

conceptual plot of Jack et al. (2010) can be recreated using the ADNI

data. The sequence of biomarkers obtained by comparing the inflection

point of each biomarker is similar to that in Jack et al. (2010) with an

exception: the RAVLT30 becomes dynamic before all other biomarkers.

The DPS provides a continuous measure of progression over the whole

course of disease, and it could therefore be used to stage individuals for

prognosisandtoevaluatetheeffectsofnoveldrugsatallstagesofthedis-

ease. The method is generic and is applicable to all neurodegenerative

diseases pending availability of the data.

Acknowledgments

Personnel costs for this research were partially supported by a

grant from Pfizer Inc. Other support came from grants numbered

P41EB015909 and R01EB012547 from the National Institute of Bio-

medical Imaging and Bioengineering as well as from an Ossoff Scholar

Award. Data collection and sharing for this project was funded by the

Alzheimer's Disease Neuroimaging Initiative (ADNI) (National Insti-

tutes of Health grant U01 AG024904). ADNI is funded by the National

Institute on Aging, the National Institute of Biomedical Imaging and

Bioengineering, and through generous contributions from the follow-

ing: Abbott; Alzheimer's Association; Alzheimer's Drug Discovery

Foundation; Amorfix Life Sciences Ltd.; AstraZeneca; Bayer HealthCare;

BioClinica, Inc.; Biogen Idec Inc.; Bristol-Myers Squibb Company; Eisai

Inc.; Elan Pharmaceuticals Inc.; Eli Lilly and Company; F. Hoffmann-La

Roche Ltd and its affiliated company Genentech, Inc.; GE Healthcare;

Innogenetics, N.V.; IXICO Ltd.; Janssen Alzheimer Immunotherapy

Research & Development, LLC.; Johnson & Johnson Pharmaceutical

Research & Development LLC.; Medpace, Inc.; Merck & Co., Inc.; Meso

Scale Diagnostics, LLC.; Novartis Pharmaceuticals Corporation; Pfizer

Inc.; Servier; Synarc Inc.; and Takeda Pharmaceutical Company. The

Canadian Institutes of Health Research is providing funds to support

ADNI clinical sites in Canada. Private sector contributions are facilitated

by the Foundation for the National Institutes of Health (www.fnih.org).

The grantee organization is the Northern California Institute for Re-

search and Education, and the study is coordinated by the Alzheimer's

Disease Cooperative Study at the University of California, San Diego.

ADNI data are disseminated by the Laboratory for Neuro Imaging

at the University of California, Los Angeles. This research was also

supported by NIH grants P30 AG010129 and K01 AG030514. The first

author would alsolike to thank PatrickSlama for his insightful remarks.

Appendix A. Proof of Identifiability

Theorem 1. The model {Pρ;ρ∈ϱ} is identifiable as long as the following

2 conditions are verified:

1. For each biomarker, there is at least one subject i with αi≠0 and with

at least 4 distinct time-points at which this biomarker is available.

2. For each subject, there is at least one biomarker which is available at 2

time points.

The proof uses the invertibility of a multivalued function closely

related to f. This property is deferred to lemma 1.

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B.M. Jedynak et al. / NeuroImage 63 (2012) 1478–1486

Page 8

Proof of Theorem 1. Let us assume that the model is not identifiable.

Then there exists 2 sets of parameters in ϱ, ρ=(a, b, c, d, α, β, σ) and

ρ′=(a′, b′, c′, d′, α′, β′, σ′) which differed by at least 1 component,

while verifying Pρ=Pρ′. Equivalently,

f αitijþ βi;ak;bk;ck;dk

??

¼ f α′itijþ β′i;a′k;b′k;c′k;d′k

??

ðA:1Þ

for all i;j;k

We proceed in steps until we verify that necessarily ρ=ρ′. Since

σk¼ σ′k, for all k=1…K, we concentrate on the other parameters.

For each k, let i be a subject such that αi>0 and for which biomarker

k is observed at four different time points ti1, ti2, ti3, ti4. Notate uik=

bkαi, vik=bk(βi−ck),u′ik¼ b′kα′iand v′ik¼ b′kβ′i−c′k

arguments of f and using (A.1),

ðÞ∈I and σk¼ σ′kfor all k

ðÞ. Rearanging the

f tij;ak;uik;−uik

−1vik;dk

??

¼ f tij;a′k;u′ik;−u′ik

−1v′ik;d′k

??

for j=1…4. Note that since ai≠0 and bk≠0, uik≠0 andu′ik≠0. Now,

using Lemma 1, ak¼ a′k, dk¼ d′k, uik¼ u′ik, uik−1vik¼ u′ik

ming up over i and dividing by I in bkαi¼ b′kα′i, we obtain bkα0¼

b′kα0, and since α0≠0, bk¼ b′k. Since bk≠0, it follows that αi¼ α′i

and uik¼ u′ik. Replacing in vik¼ v′ikand summing up over i and divid-

ing by I, we obtain thatck¼ c′k. We have then obtained that for all bio-

markers, ak¼ a′k, bk¼ b′k, ck¼ c′k, dk¼ d′kand σk¼ σ′k. Now, for each

subject i, there is at least one biomarker k for which two time-points

ti1and ti2are available. Replacing in (A.1),

−1v′ik. Sum-

f αitijþ βi;ak;bk;ck;dk

??

¼ f α′

itijþ β′

i;ak;bk;ck;dk

??

ð7Þ

for j=1, …, 2. Since ak≠0 and bk≠0, t→f(t;ak, bk, ck, dk) is invertible

which, together with (7), implies thatαi¼ α′

the proof.

iandβi¼ β′

iconcluding

Lemma 1. The vector values function R4→R4for fixed x1bx2bx3bx4:

defined by

a;b;c;d

ðÞ→ f x1;a;b;c;d

ðÞ;f x2;a;b;c;d

ðÞ;f x3;a;b;c;d

ðÞ;f x4;a;b;c;d

ðÞðÞ

with a≠0, b>0 is invertible.

Proof of Lemma 1. We verify that the Jacobian determinant of this

function is nonzero, which is enough to prove invertibility using the

inverse function theorem of multivariate calculus. Let c′=ebc

f x;a;b;c′;d

ðÞ ¼

a

1 þ c′e−bxþ d

It is equivalent to show the Jacobian determinant of

a;b;c′;d

ðÞ→ f x1;a;b;c′;d

ðÞ;f x2;a;b;c′;d

ðÞ;f x3;a;b;c′;d

ðÞ;f x4;a;b;c′;d

ðÞðÞ

is non zero.

The ith row of the Jacobian matrix is:

1 þ c′e−bxi

??−2

1 þ e−bxi;ac′xie−bxi;−ae−bxi;1 þ 2c′e−bxiþ c′2e−2bxi

hi

Column linear transformation will not change the singularity of

the Jacobian matrix. After some linear transformations, the ith row is:

1 þ c′e−bxi

??−2

1;

xie−bxi;

e−bxi;

e−2bxi

hi

Suppose the Jacobian matrix is singular, i.e. there exists (not all

zero) coefficients k, l, m, n such that

k þ lxie−bxiþ me−bxiþ ne−2bxi¼ 0;i ¼ 1;…;4

then the function

g x ð Þ ¼ k þ lxe−bxþ me−bxþ ne−2bx

must have four real roots. Differentiating twice,

2b2ne−bx−lb

would need to have 2 real roots. Since it is not the case, the Jacobian

matrix is invertible, which concludes the proof.

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