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- Abstract
- 1. Introduction
- 2. Alzheimer’s disease
- 3. Method
- 4. Results
- 5. Discussion
- 6. Conclusion
- References

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Neuroimage. Author manuscript; available in PMC 2013 November 15.

Published in final edited form as:

Published online 2012 August 3. doi: 10.1016/j.neuroimage.2012.07.059

PMCID: PMC3472161

NIHMSID: NIHMS399496

Bruno M. Jedynak,^{a,}^{b,}^{*} Andrew Lang,^{c} Bo Liu,^{a} Elyse Katz,^{d} Yanwei Zhang,^{d} Bradley T. Wyman,^{d} David Raunig,^{d,}^{**} C. Pierre Jedynak,^{e} Brian Caffo,^{f} and Jerry L. Prince^{c}, for the Alzheimer’s Disease Neuroimaging Initiative^{g}

The publisher's final edited version of this article is available at Neuroimage

See other articles in PMC that cite the published article.

While neurodegenerative diseases are characterized by steady degeneration over relatively long timelines, it is widely believed that the early stages are the most promising for therapeutic intervention, 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.

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 neurons, 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 remedy requires a precise measure of disease progression, i.e., a quantity which would be specific to a particular disease and sensitive to subtle changes. However, obtaining accurate measures of disease progression during the earliest phases of the disease is difficult. Indeed, these phases are essentially non-symptomatic and the clinical tests which characterize the acute phase of the disease are not sensitive enough to qualify as a measure of disease progression. In response, 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 neuropsychological 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) for AD, Rizk-Jackson et al. (2011) for HD). Functional imaging provides a spatially localized image of the physiological 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 Imaging and Bioengineering, the Food and Drug Administration, private pharmaceutical companies and non-profit organizations, as a $60 million, 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 modifying 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 yet show yet any clinical symptoms, see Hayden (1981). At least two large studies (Predict-HD, see https://www.predict-hd.net/ and TrackOn-HD, see http://hdresearch.ucl.ac.uk/current-studies/trackon-hd/) are underway to identify sensitive biomarkers for HD. Similar efforts are recently taking place for ALS, see Turner et al. (2009); Labbe (2012). The availability of large datasets for neurodegenerative diseases 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 biomarkers. ADNI is, as of today, the largest publicly available longitudinal 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 contribution 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 of neurodegenerative diseases.

Although this paper describes a method applicable to any neurodegenerative 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 measurements 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 biomarkers. Following these observations, Jack et al. (2010) proposed that there is a single disease progression and that different biomarkers characterize the disease during different stages. They hypothesized 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 (A*β*_{42}) 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 biomarkers 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.

The available data are longitudinal measurements of multiple biomarkers for hundreds of subjects. Our research first describes and then evaluates a disease progression score, notated DPS, which standardizes subject time-lines onto a common temporal scale. The DPS 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:

- All subjects follow a common disease progression but differ in their age of onset and rate of progression;
- As the disease progresses, each biomarker changes continuously and monotonically following a sigmoid shaped curve; and
- 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.

The age *t* of subject *i* is to be transformed into the DPS *s _{i}* as follows

(1)

upon estimation of the subject dependent parameters *α _{i}* and

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 progression while being parsimonious. Using *θ _{k}* = (

(2)

The minimum and maximum values of the sigmoid function are *d _{k}* and

Databases for neurodegenerative diseases contain measurements *y _{ijk}* of biomarker

(3)

where *t _{ij}* is the age of subject

The unknowns in this problem are *α, β, θ*, and *σ* and the least squares problem associated with the observation model in (3) is

(4)

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* biomarkers. Similarly, optimization over (*α, β*) is performed serially over each of the *I* subjects. Fitting of *θ, α*, and *β* requires optimization of continuously differentiable nonconvex functions, which is carried out using the Levenberg-Marquardt algorithm (Lines 4 and 8), see Levenberg (1944). *I _{k}* (line 4) is the number of subjects and visits available for biomarker

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 temperature, where the numerical values of the freezing and boiling points of water determine the scale. Note that calibration is not specific 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 population, the computed DPS for all visits of subjects with normal clinical assessment had a median (*m _{N}*) and a median absolute deviation (

Algorithm for the fitting of the parameters

- Inititialize
*α*^{(0)},*β*^{(0)} **for***l*= 1 to*L***do****for***k*= 1 to*K***do****end for****for***i*= 1 to*I***do****end for***α*^{(0)}=*α*^{(1)},*β*^{(0)}=*β*^{(1)}**end for****for***k*= 1 to*K***do****if***b*< 0_{k}**then****end if****end for****for***i*= 1 to*I***do****end for**

Note that (3) is invariant with respect to the following 2 transformations, for 2 constant *γ*_{1} ≠ 0 and *γ*_{2}:

Note also that the sigmoid function verifies

(5)

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

for some *α*_{0} ≠ 0 and *β*_{0}. Necessary conditions on the available data *I* for guaranteing the identifiability of the parameters are as follows:

- For each biomarker, there is at least 1 subject
*i*with*α*≠ 0 and with at least 4 distinct time-points in_{i}*I*. - For each subject, there is at least 1 biomarker which is available at 2 time points in
*I*

A proof is provided in the appendix. 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 2*I* + 5*K*, counting two parameters per subject, and five per biomarkers: four for the sigmoid and one for the sd. 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 subset of ADNI presented in section 4 has numerous missing data. Nevertheless, the identifiability conditions are met. The tightness of the estimators of the biomarker parameters is measured using bootstrap and reported in section 4.1.

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 partnership. 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 Alzheimers disease (AD). Determination of sensitive and specific markers of very early AD progression 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 participate 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. *HIPPO* is 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 intra-cranial volume. *ADAS* is the Alzheimer’s Disease Assessment Scale-cognitive subscale. *MMSE* is the Mini-Mental State Examination score. *TAU* and *ABETA* (our abbreviation for A*β*_{42}) 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 biomarkers is provided at http://adni.loni.ucla.edu/. Of the seven biomarkers, only *ADAS* and *RAVLT30* were available at the time of download from the ADNI 2/GO dataset. The protocol for these biomarkers is the same in ADNI, ADNI 2, and ADNI GO. All visits without date information were removed. Subjects not having at least two measurements for at least one of the seven biomarkers were also removed. 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 unknown gender. The total number of visits was 3658, and the clinical diagnoses 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.

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 3.5). 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 i7Q820 running at 1.73 GHz (quadcore). Overall, N subjects (black) have the smallest ADPS, MCI subjects (red) have moderate ADPS, and AD subjects (green) have the largest ADPS. Lower ADPS scores are therefore consistent with the normal population and higher ADPS scores are indicative of increased presence of dementia. Those subjects whose clinical status changes from MCI to AD (blue) are found mostly between the red and green colors.

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 **...**

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 define 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 for the parameter *c _{k}*, i.e the inflection point of each sigmoid, are presented in Fig 5(b).

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.

(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 **...**

The empirical variance of the residuals *ε _{ijk}* in (3) is the component 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 standard 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-value < 0.01) or the MMSE (p-value <0.01); p-values being obtained using the bootstrap replicates in both cases.

The rate of progression *α _{i}* of each subject

Rate of the ADPS as function of the ADPS for baseline visits. **Black:** Normal subjects. **Red:** MCI subjects **Green:** AD subjects.

The sigmoidal functions representing common behavior of biomarker 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 sigmoidal functions also provide information about the time of initial biomarker change (represented by the heels of the sigmoidal functions), 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 biomarker change, the inflection points also could represent a threshold between normal and abnormal. Therefore, we use them as an indicator of biomarker timing in the disease process. We recomputed the inflection 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-value < 0.01) and *HIPPO, ABETA* and *TAU* precede *MMSE* and *ADAS* (p-value < 0.02).

Conditional probability densities of ADPS given the clinical status of each subject were computed using Gaussian kernel density estimation (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 subjects, 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.

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. In Fonteijn et al. (2011) the progression of AD is divided into discrete events based on the atrophy of different structures in the brain providing a probabilistic framework for estimating the global progression of AD as well as for estimating the position of a single subject’s measurements. 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 statistical 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 modeling technique applied to the ADNI has provided confirmation 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 independent cohort. When using these cut point to determine the ADPS at which a biomarker changes from normal to abnormal, we find that *ABETA* precedes both *HIPPO* and *TAU* which is consistent with the results in Jack et al. (2011). We have also obtained surprising results. The fact that the inflection of *RAVLT30* precedes that of all other biomarkers, and in particular that of *ABETA* is surprising, compared to Fig. 1, but consistent with some predictions. Jicha and Carr (2010) refer to the study in Bennett et al. (2006) stating, “Retrospective analysis of their neuropsychological test performance demonstrated significant differences in only delayed recall tasks between subjects with pathological AD autopsy findings and those with normal autopsy findings, suggesting that memory 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 constitute one of the core diagnostic criteria for AD.

We report a multiple biomarker, data-driven approach to assess time-dependent changes of biomarkers in neurodegenerative disease and to localize subjects on a scale of disease progression, the DPS, over the entire range of progression. The statistical model is shown to be identifiable and bootstrap replicates show that the parameters are estimated tightly in case of the ADNI dataset. The DPS integrates information from multiple 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 prognosis and to evaluate the effects of novel drugs at all stages of the disease. The method is generic and is applicable to all neurodegenerative diseases pending availability of the data.

- A computational neurodegenerative disease progression score (DPS) is proposed
- The DPS combines measurements from multiple biomarkers
- Validation with the Alzheimer’s Disease Neuroimaging Initiative (ADNI) cohort
- An Alzheimer’s DPS (ADPS) is computed for each subject and time-point in ADNI
- Evidence for a common Alzheimer’s disease progression within ADNI subjects

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 Biomedical 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 Institutes 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 following: Abbott; Alzheimers Association; Alzheimers 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 Research 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 also like to thank Patrick Slama for his insightful remarks.

The model {*P _{ρ}*;

*For each biomarker, there is at least 1 subject i with α*≠ 0_{i}*and with at least 4 distinct time-points at which this biomarker is available*.- For each subject, there is at least 1 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.

Let us assume that the model is not identifiable. Then there exists 2 sets of parameters in *, ρ* = (*a, b, c, d, α, β, σ*) and *ρ′* = (*a′, b′, c′, d′, α′, β′, σ′*) which differe by at least 1 component, while verifying *P _{ρ}* =

(A.1)

for all (*i, j, k*) *I* and
for all *k*

We proceed in steps until we verify that necessarily *ρ* = *ρ′*. Since
, for all *k* = 1 … *K*, we concentrate on the other parameters. For each *k*, let *i* be a subject such that *α _{i}* > 0 and for which biomarker

for *j* = 1 … 4. Note that since *a _{i}* ≠ 0 and

(A.2)

for *j* = 1, …, 2. Since *a _{k}* ≠ 0 and

*The vector values function R*^{4} → *R*^{4}
*for fixed x*_{1} < *x*_{2} < *x*_{3} < *x*_{4}: *defined by*

*with a* ≠ 0, *b* > 0 *is invertible.*

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′* = *e ^{bc}*

It is equivalent to show the Jacobian determinant of

is non zero.

The *i*th row of the Jacobian matrix is:

Column linear transformation will not change the singularity of the Jacobian matrix. After some linear transformations, the *i*th row is:

Suppose the Jacobian matrix is singular, i.e. there exists (not all zero) coefficients *k, l, m, n* such that

then the function

must have four real roots. Differentiating twice,

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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