⭐3: Uncovering the heterogeneity and temporal complexity of neurodegenerative diseases with Subtype and Stage Inference (SuStain)

Young, A.L., Marinescu, R.V., Oxtoby, N.P. et al.
Blog post written by: Lisa Schmierer

Introduction

Every 3.2 seconds one person worldwide is diagnosed with dementia. [1] With up to 70%, Alzheimer's Disease (AD) covers most of the cases, followed by Vascular (10-20%) and Frontotemporal Dementia (FTD) (10%). [2] Although so many people suffer from dementia, the illness is still uncurable. One reason is, that neurodegenerative disorders such as FTD and AD are highly individual: Biomarkers, e.g. brain volume or protein measurements, as well as the symptoms, vary between patients. Researchers observed multiple subtypes for most neurodegenerative diseases (ND), each of which follows a distinct progression pattern. The condition of a single patient, therefore, does not only depend on the disease stage, the underlying subtype matters too. Previous studies only focused on one part of the problem: Either researchers try to differentiate different disease subtypes (subtype modeling) or explain how a disease progresses (disease progression modeling).

Disease Progression Modeling

Disease progression models make two core assumptions. Firstly, the relevant biomarkers need to be monotonic. Once a biomarker has worsened, it must not get better again, and vice versa. The second assumption, however, is a bigger problem when modeling neurodegenerative diseases: Disease progression models assume that the symptoms develop similarly for all patients, i.e., they can only derive a single disease progression pattern. Since multiple subtypes exist for diseases such as AD and FTD and each subtype has a distinct progression pattern, disease progression models cannot explain the whole disease's complexity.

Fig. 1: Disease progression models reconstruct a single progression pattern from the underlying data.

Subtype Modeling

The case is different for subtype modeling. Those cluster algorithms can differentiate multiple subtypes. However, these models struggle if the data contains measurements from multiple disease stages. If this is the case, clusters can comprise different subtypes at a single stage or multiple subtypes at multiple stages – the result becomes meaningless. This approach too fails to explain the differences between multiple subtypes and multiple disease stages.

Fig. 2: Subtype models only work if the underlying data contains samples from a single stage.
Otherwise, multiple subtypes and/or stages might be mixed up within one cluster.

SuStaIn – Stage and Subtype Inference

To predict not only disease subtype but also disease stage, Alexandra L Young et al. bring together subtype, and disease progression modeling in one approach: The SuStaIn-algorithm, short for Stage and Subtype inference. The unsupervised machine-learning technique deduces how many subtypes are present in the dataset and determines how the disease progresses for each of these subtypes. This allows predicting how likely a patient is to belong to each subtype and stage later on. In their tests, even patients at an early stage strongly correlated with a single disease subtype. The authors claim that this might especially leverage precision medicine.

Fig. 3: SuStaIn takes data as input that contains multiple stages and subtypes.
After the fitting, it outputs multiple subtypes and the corresponding progression pattern.

Mathematical Model

The core of SuStaIn is the underlying mathematical model, the so-called linear z-score model. In their approach, the authors built on the event-based model presented by Fonteijn HM, et al. [4] The shared core idea of both models is to describe disease progression as a certain order $\begin{array}{l}S\end{array}$ of events $\begin{array}{l}E\end{array}$ . The event $\begin{array}{l}E_i\end{array}$ indicates, that biomarker $\begin{array}{l}i, i=1,…,I\end{array}$ changed. The goal is to find the sequence of events that makes it most likely to observe the collected data $\begin{array}{l}X\end{array}$ , i.e. to find the maximum likelihood of $\begin{array}{l}P(X|S)\end{array}$ .

In contrast to the event-based model that only deals with discrete events, the z-score model can handle continuous events that happen over time. This expansion is necessary since most biomarkers for neurodegenerative disease do not progress discretely. For brain volumes, for example, there is not only a normal and abnormal brain size: It is important to quantify how abnormal a certain brain volume is compared to a healthy control group to determine whether the disease is at an early or at a later stage.

To collect the dataset $\begin{array}{l}\{X = \{x_{i,j} | i=1…I, j=1…J\}\end{array}$ , each biomarker $\begin{array}{l}i\end{array}$ was measured for each subject $\begin{array}{l}j\end{array}$ . Hence, the measurement $\begin{array}{l}x_{i,j}\end{array}$ corresponds to the $\begin{array}{l}i^{th}\end{array}$ biomarker in the $\begin{array}{l}j^{th}\end{array}$ subject. Each biomarker is associated with a set of z-scores. The value of a z-score describes how many standard deviations a measurement is away from the mean. In other words, the z-score quantifies how abnormal a certain value, in this case, a certain biomarker, is.

Since z-scores could become arbitrarily large, the model defines a maximum z-score, $\begin{array}{l}z_{i, max}\end{array}$ , for each biomarker. In total, we can describe the absolute number of biomarkers as $\begin{array}{l}N = \sum_{i=0}^I R_i\end{array}$ . The total number of biomarkers $\begin{array}{l}N\end{array}$ is directly linked to the number of events: The event $\begin{array}{l}E_{i,z}\end{array}$ encodes that biomarker $\begin{array}{l}i\end{array}$ increases to the (next) z-score $\begin{array}{l}z_{ir} = z_{i1}…z_{iR_i}\end{array}$ . Hence, $\begin{array}{l}N\end{array}$ does not only describe the total number of z-scores but also the number of events. Assuming that in one stage only one event occurs, this is also equal to the number of disease stages.

As mentioned above, the algorithm’s goal is to order the $\begin{array}{l}N\end{array}$ events in a sequence, $\begin{array}{l}S = E_{z_{1}1}.....E_{z,i},\end{array}$ which makes it most likely to observe the collected data. To describe at which stage, i.e. at which position in the sequence, an event occurs, the authors use the subscript $\begin{array}{l}k\end{array}$ . A stage $\begin{array}{l}k\end{array}$ lasts from $\begin{array}{l}t = \frac{k}{(N+1)}\end{array}$ to $\begin{array}{l}t = \frac{k+1}{N+1}\end{array}$ . Within this time, a biomarker $\begin{array}{l}i\end{array}$ increases to the next z-score. Hence, the beginning of a stage means, that the biomarker $\begin{array}{l}i\end{array}$ reached the next z-score $\begin{array}{l}z_i\end{array}$ , i.e. event $\begin{array}{l}E_{zi}\end{array}$ happens. To refer to this time point we use $\begin{array}{l}t_{E_{zi}}\end{array}$

The model assumes that during each stage, a biomarker progresses linearly. Over time, this progression is modeled with a piecewise linear function $\begin{array}{l}g_i(t)\end{array}$ that is specific for each biomarker $\begin{array}{l}i\end{array}$ :

$\begin{array}{l}g(t)= \begin{cases} \frac{z_1}{t_{E_{z_1}}}t, \quad 0 < t \leq t_{E_{z_1}} \\ z_1 + \frac{z_2-z_1}{t_{E_{z_2}}-t_{E_{z_1}}}\big(t-t_{E_{z_1}}\big), \quad t_{E_{z_1}}< t< t_{E_{z_2}} \\ \vdots \\ z_{R-1}+\frac{z_R-z_{R-1}}{t_{E_{z_R}}-t_{E_{z_{R-1}}}}\big(t-t_{E_{z_{R-1}}}\big), \quad t_{E_{z_{R-1}}}< t< t_{E_{z_R}} \\ z_R+\frac{z_{max}-z_R}{1-t_{E_{z_{R}}}}\big(t-t_{E_{z_R}}\big), \quad t_{E_{z_R}}< t\leq1 \end{cases}\end{array}$

For each case statement of the function $\begin{array}{l}g_i(t)\end{array}$ , the formulation is retrieved as illustrated below. It adds up the previous z-score with the slope times the amount of time that passed since the last event.

Fig. 4: Illustration of how the function $\begin{array}{l}g_i(t)\end{array}$ is retrieved.

With the definitions above, it is possible to now describe the model likelihood for the linear z-score model:

$\begin{array}{l}\displaystyle P(X|S)= \prod_{j=1}^J \bigg[\sum_{k=0}^N\bigg(\int_{t=\frac{k}{N+1}}^{t=\frac{k+1}{N+1}}\bigg(P(t)\prod_{i=1}^IP\big(x_{ij}|t\big)\bigg)\partial t\bigg)\bigg]\end{array}$

To obtain the probability that corresponds to the whole dataset, we need to multiply over all observed subjects $\begin{array}{l}j\end{array}$ . Furthermore, we have to use the law of total probability to account for the hidden variable $\begin{array}{l}k\end{array}$ , the stage or position we consider in our sequence. Since the biomarkers progress continuously, it is necessary to integrate over the duration of stage $\begin{array}{l}k\end{array}$ to obtain the probability mass. The most inner product accounts for the fact, that we do consider the value of all biomarkers $\begin{array}{l}i\end{array}$ at a time point $\begin{array}{l}t\end{array}$ . We model the probability to measure the biomarker $\begin{array}{l}i\end{array}$ in subject $\begin{array}{l}j\end{array}$ at time point $\begin{array}{l}t, \; P(x_{ij}|t)\end{array}$ with a normal probability distribution function. As a mean, we pick the value of the piecewise linear function of the corresponding biomarker $\begin{array}{l}i\end{array}$ . This is the value that we would expect if the underlying sequence is correct. The authors picked a uniform distribution to describe the prior on the disease time $\begin{array}{l}P(t)\end{array}$ .

So far, the linear z-score model only considers one sequence. However, the SuStaIn algorithm should consider multiple sequences, one for each subtype $\begin{array}{l}c, \: c=1…C\end{array}$ . Therefore, the authors formulate a mixture model of the linear z-score models described above. $\begin{array}{l}\mathrm{f_c}\end{array}$ refers to the proportion of samples assigned to cluster $\begin{array}{l}c\end{array}$ . $\begin{array}{l}M\end{array}$ is the final SuStaIn model.

$\begin{array}{l}\displaystyle P\big(X|M\big)= \sum_{c=1}^Cf_cP\big(X|S_c\big)\end{array}$

Results

For their experiments, the authors trained the algorithm with regional brain volumes from MRI data for two different diseases: AD and FTD. To train SuStaIn for AD, the authors used cross-sectional data from the Alzheimer’s-Neuroimaging-Initiative-Database (ADNI). [5] In the case of FTD, the authors retrieved the MRI data from the Genetic Frontotemporal Dementia Initiative (GENFI).[6]

Frontotemporal Dementia (FTD)

There are three known gene mutations that cause FTD: GRN, MAPT, and C9orf72. Without knowing the present gene mutation in the subject, SuStaIn recovered subtypes, that match the underlying gene mutation. As seen in figure five, the subtype “Asym. Frontal” contains mostly GRN- and the “Temporal” subtype mostly MAPT-mutation carriers. For the C9orf72 mutation, the algorithm recovers two different subtypes. This indicates that there are two different anatomical phenotypes and progression patterns among C9orf72-mutation carriers.

Fig. 5: Part a) shows the progression pattern SuStaIn reconstructed based on MRT data for four different subtypes derived for FTD.
Part b) shows the distribution of the underlying gene mutation across the subtypes. Source: [8]

Another interesting aspect to consider is how reliably SuStaIn can assign unseen subjects to stage and subtype. In figure six, the authors differentiate between subtype (Fig. 6a) and stage (Fig. 6c) assignment. If one sample represented by a dot is in the corner of a triangle, SuStaIn is 100% confident, that this sample belongs to the corresponding subtype. For most of the subjects affected by the disease, the corresponding marker is close to one of the corners. Most unaffected samples however are in the triangle center. This means, that SuStaIn is undecided and each subtype is equally likely, which is reasonable for unaffected patients. Assigning the subjects to stages is equally successful. Unaffected subjects are mostly allocated to early stages. For people with FTD the peak is at stage 20.

For the FTD task, SuStaIn clearly outperforms Subtype models that do not consider disease progression. The proposed algorithm achieved a balanced accuracy of 95% whereas the subtypes-only model only classified 86% of the samples correctly. Unfortunately, the authors do not compare SuStaIn to disease progression models.

Fig. 6: SuStaIn assigns people not affected by FTD to early stages (c) and is less sure which subtype to assign them (a). Source: [8]

Alzheimer's Disease

The ADNI database differentiates between three main disease labels: Cognitively normal (CN), mild cognitively impair (MCI), and Alzheimer's Disease (AD). While subjects assigned to CN do not show any symptoms, subjects labeled MCI show noticeable cognitive impairments. 10 to 20% of MCI patients that are older than 64 years develop dementia within one year after their MCI diagnosis. [7]

The authors performed experiments for the SuStaIn model trained on AD data similar to those for FD. The algorithm derives three subtypes from 3T-weighted MRI scans. (Fig. 7). For most AD patients SuStaIn is confident about the underlying subtype. For people that do not show any signs of dementia, SuStaIn is again undecided for most of the cases. (Fig. 8)

Regarding the stage assignment, SuStaIn predicts early stages for most CN-patients. The number of CN samples rapidly declines for later stages.

Fig. 7: SuStaIn recovers three subtypes from the T3-weighted MRI images: The Typical, Cortical, and Subcortical Subtype. Source: [8]

Fig. 8: The authors showed that SuStaIn can successfully assign patients to subtypes and stages. For most AD patients the model is confident about the subtype (b).
Considering the stage assignment, healthy people are mostly assigned to early stages (d). Source: [8]

For the AD version, the authors also examined whether they could use SuStain to predict how likely a patient with MCI will progress to AD. To do so, they calculated the hazards ratio for multiple variables listed in table one. The ratio describes how much the risk of progressing from MCI to AD changes if the corresponding variable increases by one unit. They found that the risk to convert to AD is 1.57 higher for patients with cortical subtype compared to subcortical. For patients with the typical subtype, it is again 1.57 higher compared to the subcortical.

	SuStaIn subtype	SuStaIn stage	Age	Sex	Education	APOE4
S–C–T	1.57	1.13	0.98	0.98	0.93	1.82

Table 1: Hazards ratio to assess the risk that MCI will progress to AD dependend on different variables.

Discussion

Although the authors demonstrate SuStaIn only for neurodegenerative diseases, the method is applicable to any progressive diseases, e.g. cancer. Since SuStaIn relies on MRI data, it can assign subtypes in-vivo, which is crucial for diagnosing in clinical applications.

A further strength is, that the method is unsupervised: SuStaIn only relies on the provided data and needs no information or hypothesis about the underlying disease. This enables the model to find subtypes beyond current medical assumptions. In the case for FTD for example, it did not only recover three subtypes based on the three different mutations that cause FTD. It also found that there are two different subtypes for the same underlying genetic mutation for which the disease progresses differently.

One downside is, that the model assumes that the biomarker variance is independent. Especially for neurodegenerative diseases this is not the case since different parts of the brain influence each other and are, therefore, not independent. However, the authors found that SuStaIn is robust to biomarker co-variance. A more critical aspect is, that there is no way to be sure, that the subtypes SuStaIn found come from distinct trajectories. It could also be, that the subtypes are extremes of a spectrum of disease progression patterns.

Apart from the proposed method itself, the paper lacks clear explanations. The authors do not go into depth explaining the z-score model and describe the fitting algorithm poorly. Lastly, they do not explain certain design decisions, e.g. why they chose to model the disease progression with a piecewise linear function instead of a polynomial. Even when reading through the reviewers’ comments there was much confusion about the paper's key elements. Although the authors tried to address these issues, I personally feel that they still did not succeed in explaining the technical aspects in an easily understandable way.

List of Abbreviations

AD Alzheimer's Disease

FTD Frontotemporal Dementia

ND Neurodegenerative Diseases

SuStaIn Subtype and Stage Inference

CN Cognitively Normal

MCI Mild Cognitive Impair

ADNI Alzheimer's Neuroimaging Initiative

GENFI Genetic Frontotemporal Dementia Initiative

References

[1] https://www.alzint.org/about/dementia-facts-figures/dementia-statistics/ (accessed 15.01.2023)

[2] https://qbi.uq.edu.au/brain/dementia/types-dementia (accessed 15.01.2023)

[3] https://www.bundesgesundheitsministerium.de/themen/pflege/online-ratgeber-demenz/krankheitsbild-und-verlauf.html (accessed 15.01.2023)

[4] Fonteijn HM, et al. An event-based model for disease progression and its application in familial Alzheimer’s disease and Huntington’s disease. Neuroimage. 2012;60:1880–1889.

[5] https://adni.loni.usc.edu/about/ (accessed 15.01.2023)

[6] https://www.genfi.org (accessed 16.01.2023)

[7] https://www.nia.nih.gov/health/what-mild-cognitive-impairment (accessed 15.01.2023)

[8] Young, A.L., Marinescu, R.V., Oxtoby, N.P. et al. Uncovering the heterogeneity and temporal complexity of neurodegenerative diseases with Subtype and Stage Inference.Nat Commun 9, 4273 (2018).

[9] Bekris LM, Yu C-E, Bird TD, Tsuang DW. Review Article: Genetics of Alzheimer Disease. Journal of Geriatric Psychiatry and Neurology. 2010;23(4):213-227

[10] Jennifer L. Whitwell, Scott A. Przybelski, Stephen D. Weigand, Robert J. Ivnik, Prashanthi Vemuri, Jeffrey L. Gunter, Matthew L. Senjem, Maria M. Shiung, Bradley F. Boeve, David S. Knopman, Joseph E. Parisi, Dennis W. Dickson, Ronald C. Petersen, Clifford R. Jack, Jr, Keith A. Josephs, Distinct anatomical subtypes of the behavioural variant of frontotemporal dementia: a cluster analysis study, Brain, Volume 132, Issue 11, November 2009, Pages 2932–2946

[11] Annie M. Racine, Rebecca L. Koscik, Sara E. Berman, Christopher R. Nicholas, Lindsay R. Clark, Ozioma C. Okonkwo, Howard A. Rowley, Sanjay Asthana, Barbara B. Bendlin, Kaj Blennow, Henrik Zetterberg, Carey E. Gleason, Cynthia M. Carlsson, Sterling C. Johnson, Biomarker clusters are differentially associated with longitudinal cognitive decline in late midlife, Brain, Volume 139, Issue 8, August 2016, Pages 2261–2274

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