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Introduction

Neurodegenerative disorders, such as frontotemporal dementia (FTD) and Alzheimer’s disease (AD), are biologically heterogeneous, producing high variance in in vivo disease biomarkers, such as volumetric measurements from imaging, protein measurements from lumbar puncture or behavioural measurements from psychometrics, which reduces their utility in disease studies and management. Key contributors to this heterogeneity are that individuals belong to a range of disease subtypes (giving rise to phenotypic heterogeneity) and are at different stages of a dynamic disease process (producing temporal heterogeneity). Previous studies aiming to explain biomarker variance typically focus on a single aspect of this heterogeneity: phenotypic heterogeneity at a coarse, typically late, disease stage or temporal heterogeneity in a broad population. However, the inability to disentangle the range of subtypes from the development and progression of each over time limits the biological insight these techniques can provide, as well as their utility for patient stratification. Constructing a comprehensive picture separating phenotypic and temporal heterogeneity, i.e. identifying distinct subtypes and characterising the development and progression of each, remains a major current challenge. However, such a picture would provide insights into underlying disease mechanisms, and enable accurate fine-grained patient stratification and prognostication, facilitating precision medicine in clinical trials and healthcare.

Previous studies of neurodegenerative disease heterogeneity have focussed on either temporal heterogeneity (i.e. subjects appear different at different disease stages) or phenotypic heterogeneity (i.e. distinct groups of subjects appear different even at the same disease stage), but rarely both. We refer to these two approaches as stages-only models, which account for temporal heterogeneity but not phenotypic heterogeneity, and subtypes-only models, which account for phenotypic heterogeneity but not temporal heterogeneity. Stages-only models arise for example from regression against disease stage5,6, and data-driven disease progression modelling7,8,9,10,11,12,13,14,15. Although such models have enabled deeper understanding of the temporal progression of a range of conditions, the inherent assumption that all individuals have a single phenotype, i.e. follow approximately the same trajectory, is a key limitation. At best, this limits the biological insight and the accuracy of stratification they can provide, but potentially could also lead to erroneous conclusions. Subtypes-only models use, for example, clustering (e.g. refs. 16,17,18,19,20,21,22,23) to identify distinct groups, or group individuals using information independent of the model, such as genetics (e.g. ref. 24) or post-mortem examination (e.g. refs. 25,26,27,28) for models based on in vivo imaging. With typical subtypes-only models, the limitation is the inherent assumption that all subjects are at a common disease stage so that the cohort has no temporal heterogeneity. This requires a priori staging and selection of individuals, which is typically crude in practice leaving models that are not specific to subtype differences. Models of both disease subtype and stage heterogeneity have been constructed previously for the small proportion of neurodegenerative diseases that are inherited on an autosomal-dominant basis. For example, Rohrer et al.29 investigate temporal heterogeneity within genetic groups by regressing imaging markers against an estimated age of onset (from family history). However, such studies lack the ability to identify within-genotype phenotypes, and the temporal resolution of the recovered genotype progression patterns is limited by inaccuracy of the a priori staging.

This paper presents Subtype and Stage Inference (SuStaIn): a computational technique that disentangles temporal and phenotypic heterogeneity to identify population subgroups with common patterns of disease progression. We demonstrate SuStaIn using structural magnetic resonance imaging (MRI) data sets from cohorts of genetic FTD and AD patients. In each case, SuStaIn provides a data-driven taxonomy (set of subtypes and stages), as well as detailed pictures of the progression of neurodegeneration within each of the data-driven subgroups. From the genetic FTD data set, SuStaIn identifies subtypes from imaging alone that map closely onto the genotypes and reconstructs patterns of neurodegeneration that reflect analysis of the individual genetic groups. This provides a validation of SuStaIn’s ability to identify subgroups with distinct temporal progression patterns, as the different genotypes are known to have distinct patterns of neurodegeneration visible as brain atrophy in MRI29. However, SuStaIn further uncovers two distinct within-genotype phenotypes for carriers of a mutation in the C9orf72 gene, while finding the MAPT and GRN mutation groups are more homogeneous. In AD, SuStaIn identifies three distinct subtypes and reconstructs their previously unseen temporal progression. In both neurodegenerative diseases, we demonstrate strong assignment of individuals to the SuStaIn subtypes, which is in contrast to subtypes-only models in the literature (e.g. ref. 23). Even at very early stages, at least a proportion of individuals show strong alignment with particular subtypes, which highlights the potential utility in precision medicine. In AD, we show that SuStaIn subtype and stage enhance the ability to predict conversion between diagnostic categories substantially beyond subtypes-only or stages-only models.

Results

Subtype and stage inference

Figure 1 provides a conceptual overview of the SuStaIn modelling technique. SuStaIn is an unsupervised machine-learning technique that identifies population subgroups with common patterns of disease progression. SuStaIn builds on and combines ideas from clustering (e.g. refs. 16,17,18,19,20,21,22,23) and data-driven disease progression modelling (e.g. refs. 7,8,9,10,12). The combination uniquely enables SuStaIn to group individuals with common phenotypes across the range of disease stages. It determines the number of subtypes that the available data can support, reconstructs the trajectory of stages within each subtype, and assigns a probability of each subtype and stage to each subject. These features provide insights into the underlying disease biology and a mechanism for in vivo fine-grained stratification at early disease stages.

Synthetic data

A simulation study (see Supplementary Methods, Supplementary Results, Supplementary Discussion and Supplementary Figures 112) verifies the ability of the SuStaIn algorithm to recover predefined subtypes and their progression patterns from heterogeneous data sets with comparable numbers of subjects, biomarkers and clusters (subtypes) to those used in this study.

Subtype progression patterns

We demonstrate SuStaIn in two neurodegenerative diseases, genetic FTD and sporadic AD, using cross-sectional regional brain volumes from MRI data in the GENetic Frontotemporal dementia Initiative (GENFI) and the Alzheimer’s Disease Neuroimaging Initiative (ADNI). GENFI investigates biomarker changes in carriers of mutations in GRN, MAPT and C9orf72 genes, which cause FTD. GRN and MAPT mutations are known to be associated with distinct phenotypes, whereas C9orf72 is a heterogeneous group30. Here, GENFI serves as a test data set with a partially known ground truth for validation, as we expect SuStaIn to identify genetic groups as distinct phenotypic subtypes. However, it further supports investigation of the phenotypic and temporal heterogeneity within genotypes. Specifically, we ran SuStaIn on the combined data set from all 172 mutation carriers in GENFI (Fig. 2a), without genotypes, and compared the resulting subtype assignments and progression patterns with (a) participant’s genotype labels (Fig. 2b), and (b) subtype progression patterns obtained from each genotype separately (Supplementary Figure 13; 76 GRN carriers, 63 C9orf72 carriers, 33 MAPT carriers). Next, we used SuStaIn to identify sporadic AD subtypes from ADNI (793 subjects, including 524 with mild cognitive impairment (MCI) or AD) and characterise their progression from early to late disease stages (Fig. 3). We tested consistency of the SuStaIn subtypes in a largely independent data set—ADNI 1.5T MRI (576 subjects, including 396 with MCI or AD) scans (Fig. 4) rather than the main 3T data set used for Fig. 3. In each disease, cross-validation tests the reproducibility of the subtypes and estimated progression patterns (Supplementary Figure 14).

SuStaIn reveals within-genotype phenotypes in FTD

Figure 2 shows that SuStaIn successfully identifies the progression patterns of the different genetic groups in GENFI, without prior knowledge of genotype, and further suggests that phenotypic heterogeneity of the C9orf72 group results from two neuroanatomical subtypes. Figure 2a shows the four subtypes that SuStaIn finds from the full set of all mutation carriers in GENFI. We refer to them as the asymmetric frontal lobe subtype, temporal lobe subtype, frontotemporal lobe subtype and subcortical subtype. Figure 2b reveals that GRN mutation carriers are the main contributors to the asymmetric frontal lobe subtype, MAPT mutation carriers are the main contributors to the temporal lobe subtype, and C9orf72 mutation carriers are the main contributors to both the frontotemporal lobe subtype and the subcortical subtype. This suggests that there are two distinct subtypes in the C9orf72 group. Application of SuStaIn to each genetic group separately supports this finding by demonstrating that the GRN mutation carriers are best described as a single asymmetric frontal lobe subtype, the MAPT mutation carriers are best described as a temporal lobe subtype and the C9orf72 mutation carriers are best described as two distinct disease subtypes: a frontotemporal lobe subtype and a subcortical subtype. SuStaIn additionally finds a subsidiary cluster in the MAPT group for which the progression pattern has high uncertainty. This high uncertainty likely prevents the cluster from being detected when applying SuStaIn to all mutation carriers in Fig. 2 as this small number of subjects can be sufficiently modelled by the three alternative subtype progression patterns. Supplementary Figure 13 shows that the subtype progression patterns for each genetic group are in good agreement with those found in the full set of all mutation carriers (Fig. 2a). Supplementary Figure 14A shows that the four subtypes estimated in Fig. 2a are reproducible under cross-validation, with a high average similarity between cross-validation folds of >93% for each subtype. Altogether these results provide strong validation of SuStaIn’s ability to recover distinct subtypes and their progression patterns from a heterogeneous data set, while simultaneously disentangling the heterogeneity of the C9orf72 group into two distinct subtypes.

SuStaIn identifies three subtype progression patterns in AD

Figure 3 shows the temporal progression of the three neuroanatomical subtypes that SuStaIn identifies from ADNI, which we term typical, cortical and subcortical. SuStaIn reveals that for the typical subtype, atrophy starts in the hippocampus and amygdala; for the cortical subtype in the nucleus accumbens, insula and cingulate; and for the subcortical subtype in the pallidum, putamen, nucleus accumbens and caudate. Supplementary Figure 14B shows that these three subtypes are reproducible under cross-validation, giving an average similarity between cross-validation folds of >92% for each subtype.

AD subtypes are reproducible in an independent data set

Figure 4 shows that the three subtypes in Fig. 3 are reproducible in a largely independent data set (<5% subjects in common) consisting of regional brain volumes derived from 1.5T rather than 3T MRI scans. From the 1.5T data, SuStaIn broadly replicates the three major clusters found in the 3T data, again finding a typical, cortical and subcortical subtype. The origin of atrophy for each subtype is in general agreement with the 3T data: atrophy begins in the hippocampus and amygdala for the typical subtype, in the insula and cingulate for the cortical subtype; and in the pallidum, putamen and caudate for the subcortical subtype. The main difference compared to the 3T data is that the nucleus accumbens is not indicated as an early region to atrophy in the 1.5T data for the cortical and subcortical subtypes. SuStaIn additionally identifies a small proportion (4%) of outliers with a parietal subtype in the 1.5T data.

Disease subtyping and staging

We investigated SuStaIn’s capability for reliable stratification in each neurodegenerative disease (Fig. 5) to determine the potential for homogeneous cohort identification. First, we assessed how reliably SuStaIn assigns patients to subtypes (Fig. 5a, b). Specifically, in genetic FTD, we tested the consistency of SuStaIn subtypes with the different genotypes in symptomatic mutation carriers (Table 1), and compared this consistency against models that do not account for temporal heterogeneity (Table 2). Second, we assessed the reliability of the SuStaIn stages in each disease (Fig. 5c, d) by comparison with clinical diagnostic categories. In ADNI, where clinical follow-up information is available, we further examined the ability of SuStaIn subtypes and stages to predict relevant outcomes, by determining whether SuStaIn subtype and/or stage modify the risk of conversion between diagnostic categories (Table 3).

SuStaIn provides utility for patient stratification

Figure 5 illustrates the ability of SuStaIn to provide disease subtyping and staging information for each neurodegenerative disease. Figure 5a shows that the strength of assignment (see Methods: Strength of assignment to subtype) to the SuStaIn subtypes in genetic FTD increases as the diseases progress, with 88% of the symptomatic mutation carriers in GENFI being strongly assigned (i.e. >50% likelihood of a particular subtype). Figure 5b shows that the strength of assignment to the SuStaIn subtypes in AD also increases with disease progression, with a strong assignment of individuals to subtypes in 78% of ADNI participants with an AD diagnosis. The strong assignment of the AD subtypes that SuStaIn achieves by accounting for temporal heterogeneity is in contrast to previous studies23 that model phenotypic but not temporal heterogeneity. Moreover, the strong assignment is seen even at early disease stages (MCI), where many subjects cluster around the vertices of the triangles: 37% of MCI subjects are strongly assigned to a subtype. Figures 5c, d show that the distribution of SuStaIn stages differs between diagnostic groups in both GENFI and ADNI, and provides a good separation of presymptomatic and symptomatic mutation carriers, and cognitively normal (CN) and AD.

SuStaIn subtypes discriminate FTD genotype

Table 1 shows the classification accuracy obtained using the SuStaIn subtypes in Fig. 2 to discriminate the genotype of affected mutation carriers in GENFI. While the use of MRI to identify genotype is not necessary in these subjects, so not clinically relevant, this experiment demonstrates the ability of SuStaIn to identify subtypes in a data set with a known ground truth. The SuStaIn subtypes give a balanced accuracy of 95% for the two-way classification task of distinguishing the homogeneous GRN and MAPT carrier groups. For the more challenging three-way classification task of distinguishing all genotypes in the presence of heterogeneity, the SuStaIn subtypes provide a maximum balanced accuracy of 86%. A high proportion of the homogeneous GRN and MAPT carrier groups are correctly assigned to the asymmetric frontal lobe (93% of affected GRN carriers) and temporal lobe subtype progression patterns (91% of affected MAPT carriers). The heterogeneous C9orf72 carrier group are much more difficult to classify, with a total of 75% of affected C9orf72 carriers being assigned to the frontotemporal lobe and subcortical subtypes. Apart from heterogeneity, the C9orf72 carriers are also more difficult to classify because the frontotemporal lobe and subcortical subtype progression patterns are more similar to the other subtypes; by evaluating the similarity of each pair of subtype progression patterns (see Methods: Similarity between two subtype progression patterns) we find that the asymmetric frontal lobe and temporal lobe subtypes have the most distinct progression patterns of any pair of subtypes; the asymmetric frontal lobe and frontotemporal lobe subtypes have the most similar progression patterns of any pair of subtypes. The precise strategy of assigning subjects to subtype can alter the classification rates somewhat and Supplementary Table 1 examines this effect.

Genotype discrimination out-performs subtypes-only models

Table 2 shows the classification accuracy obtained using a subtypes-only model (Fig. 6), which does not account for temporal heterogeneity, to discriminate the genotype of affected mutation carriers in GENFI. The SuStaIn subtypes out-perform the subtypes-only model. The subtypes-only model gives a balanced accuracy of 92% compared to 95% using SuStaIn for the two-way classification task of distinguishing GRN and MAPT carrier groups; the subtypes-only model gives a maximum balanced accuracy of 69% compared to 86% using SuStaIn for the three-way classification task of distinguishing all genotypes. In the subtypes-only model the majority of misclassifications arise from the earlier stage affected GRN and MAPT carriers being assigned to the mild frontotemporal subtype associated with C9orf72 carriers. See also Supplementary Table 2.

SuStaIn subtypes and stages have predictive utility in AD

Table 3 shows that the SuStaIn subtypes and stages have predictive utility for the risk of conversion between diagnostic categories in ADNI. By fitting a Cox Proportional Hazards model, we found significant effects (t-test) of baseline SuStaIn subtype (p = 2.44 × 10−3) and stage (p = 8.76 × 10−11) on an individual’s risk of conversion from MCI to AD. Of the SuStaIn subtypes, the subcortical subtype is associated with the lowest risk of conversion, while the typical subtype is associated with the highest risk of conversion. Supplementary Table 3 shows that SuStaIn out-performs subtypes-only and stages-only models at estimating the risk of conversion between diagnostic categories in ADNI. By performing likelihood ratio tests comparing SuStaIn to subtypes-only and stages-only we find that SuStaIn provides a significantly better fit (likelihood ratio test) than both subtypes-only (p = 3.96 × 10−5) and stages-only (p = 7.18 × 10−4) models. This shows that both the subtypes and stages estimated by SuStaIn provide additional information for predicting the risk of conversion from MCI to AD.

Discussion

In this study we introduce SuStaIn—a powerful tool for data-driven disease phenotype discovery, providing insights into disease aetiology, and enhanced power for patient stratification in clinical trials and healthcare. Results from the GENFI data set first validate that SuStaIn can successfully recover known distinct progression patterns in genetic FTD corresponding to different genotypes. Moreover, SuStaIn identifies and characterises within-group heterogeneity for carriers of a mutation in the C9orf72 gene as distinct temporal progression patterns in two subtypes. The results demonstrate the utility of SuStaIn for data-driven disease phenotype discovery, and provide biological insight into the C9orf72 mutation. Application of SuStaIn to the 3T ADNI data set recovers three distinct AD subtypes with final stages that reflect post-mortem neuropathological findings. Results from a largely independent AD data set (ADNI 1.5T) corroborate these three subtypes. The disease subtype characterisation SuStaIn provides goes much further than post-mortem neuropathological studies3,28, or other machine-learning techniques23, by characterising the temporal trajectory of each subtype, enabling in vivo stratification of subjects by disease stage as well as disease subtype. We demonstrate the ability of SuStaIn to stratify in vivo by both subtype and stage in genetic FTD and AD. In genetic FTD, we show that the SuStaIn neuroimaging subtypes can distinguish affected carriers belonging to different genetic groups with high classification accuracy. In AD, we demonstrate strong assignment of subjects to the SuStaIn subtypes even at early disease stages (MCI), and that the SuStaIn subtypes and stages have added utility for predicting conversion between clinical diagnoses, beyond more traditional stages-only or subtypes-only models.

Previous studies in genetic FTD have found asymmetric frontotemporoparietal lobe volume loss in GRN carriers, temporal lobe volume loss in MAPT carriers, and widespread symmetric grey matter atrophy and volume loss in the cerebellum in C9orf72 carriers24. The asymmetric frontal lobe subtype and temporal lobe subtype in Fig. 2 show clear similarities with previous studies of regional volume loss in GRN and MAPT mutation carriers respectively. However, SuStaIn provides much greater detail and accuracy by avoiding reliance on crude a priori staging, e.g. via mean familial age of onset. The frontotemporal lobe subtype and subcortical subtype in Fig. 2 both have features previously associated with C9orf72 mutation carriers, but SuStaIn assigns these features to two distinct disease subtypes, and further reveals the temporal progression of each subtype.

Several biological factors may produce the two subtypes observed in C9orf72 mutation carriers, either individually or in combination. Clinically, while there is significant overlap, patients typically present with either a behavioural variant FTD or amyotrophic lateral sclerosis as their main phenotype31, and they can progress at various rates; genetically, the expansion length is variable and there are additional genetic modifiers (e.g. TMEM106B and ATXN2) that alter phenotype32,33,34; and pathologically, most cases have either type A or type B TDP-43 pathology31. While further study is required to determine the biological factors that influence neuroanatomical phenotype, these findings demonstrate the power of SuStaIn in identifying hitherto unrecognised disease subtypes using clinical data, and thus open up the potential to link variations in genetics, pathology and neuroanatomy.

We also find evidence for the presence of a subsidiary group in the MAPT mutation carriers, but numbers are too small to determine whether this group have a distinct progression pattern. Among individuals with significant evidence of MRI atrophy (SuStaIn stage of ≥5), four individuals (two pairs of individuals from the same families) of 13 were identified as belonging to the subsidiary group. Although MAPT mutations have been commonly thought to have a very specific pattern of atrophy affecting the anterior and medial temporal lobes predominantly, one previous paper has shown that there can be a second pattern of atrophy in specific mutations, where the lateral temporal lobes are affected more than the medial regions35. Interestingly, the two pairs of individuals who constitute the subsidiary group in our analysis all have P301L mutations, a mutation that falls into this second alternate atrophy pattern group in ref. 35. None of the nine individuals assigned to the predominant progression pattern in our analysis have P301L mutations, or V337M mutations, the other mutation identified in ref. 35 as having an alternate atrophy pattern. This suggests that SuStaIn may be able to identify particular MAPT mutations that fall into this alternate group, but larger studies will be required to confirm this.

In AD, post-mortem histology3 and retrospectively analysed MRI scans close to the time of death28 observe three distinct patterns of atrophy in late-stage AD patients: one focussed on the temporal lobe that is similar to the late stages of the typical SuStaIn subtype; one affecting predominantly cortical regions cf. late stages of the cortical SuStaIn subtype; and one with stronger subcortical involvement cf. late stages of the subcortical SuStaIn subtype. This gives confidence in the SuStaIn subtypes, which provide much greater information by revealing the progression of each subtype over time, including the earliest sites of regional volume loss. Moreover, and importantly for practical utility, the SuStaIn subtypes can be assigned in vivo using MRI, enabling linkage of late-stage pathological observations with early-stage neurodegeneration.

The three AD subtypes found in the 3T MRI data set are corroborated by the largely independent 1.5T MRI data set. However, some small differences arise between the subtype progression patterns of the three subtypes recovered in each data set. These differences lie predominantly in how early the nucleus accumbens begins to atrophy in the different subtypes: across all three subtypes the nucleus accumbens shows atrophy earlier in the 3T subtypes than the 1.5T subtypes. A possible explanation for this is that the volume of the nucleus accumbens, which is relatively small, can be estimated more accurately using the higher field strength 3T MRI scans than the 1.5T MRI scans, and thus atrophy in the nucleus accumbens can be identified from an earlier stage in the 3T data set compared to the 1.5T data set.

In the 1.5T MRI data set we additionally find a small proportion (4%) of outliers with a parietal subtype. This small subgroup may represent a posterior cortical atrophy phenotype: comparing the Alzheimer’s disease Assessment Scale-cognitive subscale (ADAS-cog) scores between individuals with an AD diagnosis that are assigned to the parietal subgroup (N = 6) and the typical AD subgroup (N = 65), we find that the parietal subgroup have worse performance (Mann–Whitney U test) on certain praxic (Q6. Ideational Praxis, p = 6.1 × 10−3, z = 2.7) and spatially-demanding (Q14. Number Cancellation, p = 4.9 × 10−3, z = 2.8) subtests, but similar performance (Mann–Whitney U test) in memory domains (Q8. Word Recognition, p = 0.81, z = −0.2; Q1. Word Recall, p = 0.48, z = 0.70). Additionally, the parietal subgroup is on average 10.3 years younger (p = 2.8 × 10−3, z = −3.0, Mann–Whitney U test) than the typical AD subgroup.

The temporal spreading patterns for distinct subtypes estimated by SuStaIn offer biological insight. For example, the progression pattern of each subtype provides a view of how neurodegeneration spreads from a distinct origin over the rest of the brain that is uncorrupted by phenotypic heterogeneity. A key advantage of SuStaIn is that it provides a purely data-driven, hypothesis-free, reconstruction of the progression of neurodegenerative disease subtypes. However, these observations also have great potential to inform mechanistic models36,37 of neurodegenerative disease, which explain their temporal progression via various hypothetical mechanisms of disease propagation over brain networks. Current mechanistic models implicitly assume a single-disease progression pattern—an assumption often violated in patient data sets, but much more reasonable if focussed on particular SuStaIn subtypes.

SuStaIn shows strong capabilities for patient stratification in AD, which we are able to validate in genetic FTD where we expect the subtype assignments to correspond to distinct genotypes. SuStaIn provides high classification accuracy for differentiating the different mutation types in genetic FTD, and the AD subtypes are clearly assignable. In genetic FTD, SuStaIn out-performs a subtypes-only model, giving a balanced classification accuracy of 86% for distinguishing genotype compared to 69% for the subtypes-only model. This provides compelling evidence that there is substantial heterogeneity in disease stage within different phenotypes, and that modelling this disease stage heterogeneity is important for better patient stratification. This is further demonstrated in AD, in which SuStaIn’s subtypes and stages substantially out-perform subtypes-only and stages-only models for predicting conversion between diagnostic categories. These early results are highly promising, particularly given that the particular choice of biomarkers used here (coarse regional brain volumes) is not optimised for stratification. In this initial study, we chose to use MRI to maximise the number of subjects with all available measurements, to simplify the interpretation of the results, and to enhance the clinical utility. However, future work will test the added benefit of including a wider range of biomarkers and a more fine-grained set of regional volumes in SuStaIn for patient stratification. For example in AD, incorporation of amyloid and neurofibrillary tangle measures, e.g. from amyloid and tau positron emission tomography (PET) scans, will enable stratification of individuals at the very earliest disease stages.

The previous study of Zhang et al.23 also looked at the assignment of AD subtypes using a subtypes-only model that does not account for temporal heterogeneity in disease stage. In contrast to the study of Zhang et al.23, we observe strong assignment of AD patients to the subtypes (Fig. 5b) emphasising the importance of accounting for heterogeneity in disease stage. This assignability clearly increases with disease progression, with the subtypes being most strongly assigned in clinically diagnosed AD patients. However, even at early stages (MCI), many subjects cluster around the vertices of the triangles showing strong potential for identifying early-stage cohorts representative of each subtype.

The model underlying SuStaIn makes several assumptions to enable the simultaneous estimation of subtypes and their progression. One assumption is that biomarker variance is independent. In reality biomarkers tend to co-vary due to shared biological processes. However, simulation experiments (Supplementary Figure 7) show that the subtype progression patterns recovered by SuStaIn are robust to biomarker covariance. Nevertheless, refinements might come from modelling covariance among strongly dependent biomarkers, e.g. using model selection criteria to identify a minimal set of necessary covariance parameters, and future work will explore this idea.

To enable the modelling of purely cross-sectional data, here we make an assumption of an arbitrary timescale. This formulation can also work with longitudinal data when available, although here we reserve the longitudinal information to validate the clinical utility of SuStaIn to make future predictions at an individual’s first visit. However, extensions to SuStaIn that utilise longitudinal information to further provide a well-defined timescale are an important area for future work.

Here we make a further implicit assumption that the cohort is correctly diagnosed. While the genetic tests in GENFI ensure this, the clinical diagnoses of AD in ADNI are less reliable and the proportion of misdiagnosis, e.g. of depression or other neurological diseases, is non-negligible. Simulation of the effect of misdiagnosis (Supplementary Figure 9) demonstrates that SuStaIn can robustly recover subtype progression patterns under a substantial proportion of outliers, up to 20%. Nevertheless, future adaptions of the SuStaIn model might include a broad outlier class to capture individuals that do not fit any of the main clusters.

One caveat on the findings is that the underlying data may come from a spectrum of disease progression patterns, rather than a set of distinct trajectories as SuStaIn is designed to estimate. Simulations (Supplementary Figure 11) demonstrate that SuStaIn may still recover multiple distinct progression patterns from data generated by a spectrum of progression patterns. In this case the distinct progression patterns identified by SuStaIn still provide useful information about the extrema within the underlying spectrum of progression patterns. Here, however, the alignment with genotypes in genetic FTD and neuropathological observations in AD provide confidence that the distinct subtypes are genuine. Future work will extend SuStaIn to be able to represent spectra of progression patterns (e.g. using Mallow’s models as in refs. 38,39).

We introduce SuStaIn—a tool to disentangle and characterise the temporal and phenotypic heterogeneity of neurodegenerative diseases. We use it to elucidate the temporal and phenotypic heterogeneity of both genetic FTD and AD subtypes with previously unseen detail. We further demonstrate SuStaIn’s potential as a patient stratification tool in AD by showing strong alignment of subjects with specific subtypes even at early disease stages, as well as added power to predict conversion between clinical diagnoses. SuStaIn has the potential to make substantial clinical impact as a tool for precision medicine and is readily applicable to any progressive disease, including other neurodegenerative diseases, respiratory diseases and cancers.

Methods

GENFI data set

We used cross-sectional volumetric MRI data from GENFI (http://www.genfi.org.uk/). Subjects were included from the second data freeze of GENFI, which in total consisted of 365 participants recruited across 13 centres in the United Kingdom, Canada, Italy, The Netherlands, Sweden and Portugal. A total of 313 participants had a usable volumetric T1-weighted MRI scan for analysis (15 participants did not have a scan and the other participants were excluded as the scans were of unsuitable quality due to motion, other imaging artefacts or pathology unlikely to be attributed to FTD). The 313 participants included 141 non-carriers, 123 presymptomatic carriers and 49 symptomatic carriers. Of the 123 presymptomatic mutation carriers there were 62 GRN, 39 C9orf72 and 22 MAPT carriers. Of the 49 symptomatic carriers, there were 14 GRN, 24 C9orf72 and 11 MAPT carriers. The acquisition and post-processing procedures for GENFI have been previously described in ref. 29. Briefly, cortical and subcortical volumes were generated using a multi-atlas segmentation propagation approach40, combining cortical regions of interest to calculate grey matter volumes of the entire cortex, separated into the frontal, temporal, parietal, occipital, cingulate and insula cortices. In addition to regional volumetric measures, we also included a measure of asymmetry, which is calculated as the absolute value of the difference between the volumes of the right and left hemispheres, normalised by the total volume of both hemispheres. This asymmetry measure was log transformed to improve normality. See Supplementary Table 4 for a summary of the biomarkers used in the SuStaIn modelling.

Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (http://adni.loni.usc.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 organisations, as a \$60 million, 5-year public-private partnership. For up-to-date information, see http://www.adni-info.org. Written consent was obtained from all participants, and the study was approved by the Institutional Review Board at each participating institution.

z-scores

We expressed each regional volume measurement as a z-score relative to a control population: in GENFI we used data from all non-carriers, in ADNI we used amyloid-negative CN subjects, defined as those with a CSF Aβ1–42 measurement >192 pg per ml41. This gave us a control population of 48 amyloid-negative CN subjects for the 3T data set, and 56 amyloid-negative CN subjects for the 1.5T data set. We used these control populations to determine whether the effects of age, sex, education or number of APOE4 alleles (ADNI only) were significant, and if so to regress them out. We then normalised each data set relative to its control population, so that the control population had a mean of 0 and standard deviation of 1. Because regional brain volumes decrease over time the z-scores become negative with disease progression, so for simplicity we took the negative value of the z-scores so that the z-scores would increase as the brain volumes became more abnormal.

SuStaIn modelling

We formulate the model underlying SuStaIn as groups of subjects with distinct patterns of biomarker evolution (see Mathematical Model). We refer to a group of subjects with a particular biomarker progression pattern as a subtype. The biomarker evolution of each subtype is described as a linear z-score model in which each biomarker follows a piecewise linear trajectory over a common timeframe. The noise level for each biomarker is assumed constant over the timeframe and is derived from a control population (see Mathematical model). This linear z-score model is based on the event-based model in refs. 7,8,38, but reformulates the events so that they represent the continuous linear accumulation of a biomarker from one z-score to another, rather than an instantaneous switch from a normal to an abnormal level. A key advantage of this formulation is that it can work with purely cross-sectional data because it requires no information about the timescale of change, but instead uses events as control points of piecewise linear segments with arbitrary duration. The model fitting considers increasing number of subtypes C, for which we estimate the proportion of subjects f that belong to each subtype, and the order SC in which biomarkers reach each z-score for each subtype c = 1 … C. We determine the optimal number of subtypes C for a particular data set through ten-fold cross-validation (see Cross-validation).

Mathematical model

The linear z-score model underlying SuStaIn is a continuous generalisation of the original event-based model7,8, which we describe first.

The event-based model in refs. 7,8 describes disease progression as a series of events, where each event corresponds to a biomarker transitioning from a normal to an abnormal level. The occurrence of an event, Ei, for biomarker i = 1 … I, is informed by the measurements xij of biomarker i in subject j, j = 1 … J. The whole data set X = {xij | i = 1 … I, j = 1 … J} is the set of measurements of each biomarker in each subject. The most likely ordering of the events is the sequence S that maximises the data likelihood

$$P\left( {{\bf{X}}|{\bf{S}}} \right) = \mathop {\prod }\limits_{j = 1}^J \left[ {\mathop {\sum }\limits_{k = 0}^I \left( {P(k)\mathop {\prod }\limits_{i = 1}^k P\left( {x_{ij}|E_i} \right)\mathop {\prod }\limits_{i = k + 1}^I P\left( {x_{ij}|\neg E_i} \right)} \right)} \right],$$
(1)

where P(x | Ei) and P(x | ¬Ei) are the likelihoods of measurement x given that biomarker i has or has not become abnormal, respectively. P(k) is the prior likelihood of being at stage k, at which the events E1, ..., Ek have occurred, and the events Ek+1, …, EI have yet to occur. The model uses a uniform prior on the stage, so that P(k) = 1/(I + 1), k = 0 … I, i.e. a priori individuals are equally likely to belong to any stage along the progression pattern. The likelihoods P(x | Ei) and P(x | ¬Ei) are modelled as normal distributions.

The linear z-score model we use in this work reformulates the event-based model in (1) by replacing the instantaneous normal to abnormal events with events that represent the (more biologically plausible) linear accumulation of a biomarker from one z-score to another. The linear z-score model consists of a set of N z-score events Eiz, which correspond to the linear increase of biomarker i = 1 … I to a z-score zir = zi1$$z_{iR_i}$$, i.e. each biomarker is associated with its own set of z-scores, and so N = $$\mathop {\sum }\limits_i {\kern 1pt} R_i$$. Each biomarker also has an associated maximum z-score, zmax, which it accumulates to at the end of stage N. We consider a continuous time axis, t, which we choose to go from t = 0 to t = 1 for simplicity (the scaling is arbitrary). At each disease stage k, which goes from t = $${\textstyle{k \over {N + 1}}}$$ to t = $${\textstyle{{k + 1} \over {N + 1}}}$$, a z-score event Eiz occurs. The biomarkers evolve as time t progresses according to a piecewise linear function gi(t), where

$$g\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{z_1}}{{t_{E_{z_1}}}}t,0 < t \le t_{E_{z_1}}} \\ {z_1 + \frac{{z_2 - z_1}}{{t_{E_{z_2}} - t_{E_{z_1}}}}\left( {t - t_{E_{z_1}}} \right),t_{E_{z_1}} < t \le t_{E_{z_2}}} \\ \vdots \\ {z_{R - 1} + \frac{{z_R - z_{R - 1}}}{{t_{E_{z_R}} - t_{E_{z_{R - 1}}}}}\left( {t - t_{E_{z_{R - 1}}}} \right),t_{E_{z_{R - 1}}} < t \le t_{E_{z_R}}} \\ {z_R + \frac{{z_{max} - z_R}}{{1 - t_{E_{z_R}}}}\left( {t - t_{E_{z_R}}} \right),t_{E_{z_R}} < t \le 1} \end{array}} \right..$$

Thus, the times $$t_{E_{iz}}$$ are determined by the position of the z-score event Eiz in the sequence S, so if event Eiz occurs in position k in the sequence then $$t_{E_{iz}}$$ = $${\textstyle{{k + 1} \over {N + 1}}}$$.

To formulate the model likelihood for the linear z-score model we replace Eq. (1) with

$$P\left( {{\bf{X}}|{\bf{S}}} \right) = \mathop {\prod }\limits_{j = 1}^J \left[ {\mathop {\sum }\limits_{k = 0}^N \left( {\mathop {\int }\nolimits_{t = \frac{k}{{N + 1}}}^{t = \frac{{k + 1}}{{N + 1}}} \left( {P(t)\mathop {\prod }\limits_{i = 1}^I {\kern 1pt} P\left( {x_{ij}|t} \right)} \right)\partial t} \right)} \right],$$
(2)

where,

$$P\left( {x_{ij}|t} \right) = {\mathrm{NormPDF}}\left( {x_{ij},g_i\left( t \right),\sigma _i} \right).$$

NormPDF(x, μ, σ) is the normal probability distribution function, with mean μ and standard deviation σ, evaluated at x. We assume the prior on the disease time is uniform, as in the original event-based model.

The SuStaIn model is a mixture of linear z-score models, hence we have

$$P\left( {{\bf{X}}|{\bf{M}}} \right) = \mathop {\sum }\limits_{c = 1}^C {\kern 1pt} f_c{\kern 1pt} P\left( {{\bf{X}}|{\bf{S}}_c} \right),$$

where C is the number of clusters (subtypes), f is the proportion of subjects assigned to a particular cluster (subtype), and M is the overall SuStaIn model.

Model fitting

Supplementary Figure 15 provides a flowchart detailing the processes involved in the SuStaIn model fitting. Model fitting requires simultaneously optimising subtype membership, subtype trajectory and the posterior distributions of both. In particular, the cost function here depends on the sequence ordering, which to our knowledge standard algorithms do not handle. We therefore derive our own algorithm to fit SuStaIn, based on the well-established methods developed for the event-based model (7,8,42,43), for which we demonstrate convergence and optimality in simulation (see Supplementary Results: Convergence) and in the data sets used here (see Convergence). As shown in the black box in Supplementary Figure 15, the SuStaIn model is fitted hierarchically, with the number of clusters being estimated via model selection criteria obtained from cross-validation. The hierarchical fitting initialises the fitting of each C-cluster (subtype) model from the previous C-1-cluster model, i.e. the clustering problem is solved sequentially from C = 1Cmax (where Cmax is the maximum number of clusters being fitted), initialising each model using the previous model. For the initial cluster (C = 1), we use the single-cluster expectation maximisation (E-M) procedure shown in the green box in Supplementary Figure 15, and described subsequently. We fit subsequent cluster numbers (C > 1) hierarchically by generating C-1 candidate C-cluster models using the split-cluster E-M procedure shown in the blue box in Supplementary Figure 15, and described subsequently. From these C-1 candidate C-cluster models, the model with the highest likelihood is chosen.

The split-cluster E-M procedure shown in the blue box in Supplementary Figure 15 is used to generate each of the C-1 candidate C cluster models. For each of the C-1 clusters, the split-cluster E-M procedure first finds the optimal split of cluster c into two clusters. To find the optimal split of cluster c into two clusters, the data points belonging to cluster c are randomly assigned to two separate clusters. The optimal model parameters for these two data subsets are then obtained using the single-cluster E-M procedure (green box in Supplementary Figure 15). These cluster parameters are used to initialise the fitting of a two-cluster model to the subset of the data belonging to cluster c, using E-M. This two-cluster solution is then used together with the other C-2 clusters to initialise the fitting of the C-cluster model. The C-cluster model is then optimised using E-M, alternating between updating the sequences Sc for each cluster and the fractions fc. This procedure is repeated for 25 different start points (random cluster assignments) to find the maximum likelihood solution (see Convergence).

The single-cluster E-M procedure shown in the green box in Supplementary Figure 15 is used to find the optimal model parameters (the sequence S in which the biomarkers reach each z-score) for a single-cluster. In the single-cluster E-M procedure the sequence S is initialised randomly. This sequence is then optimised using E-M by going through each z-score event E in turn and finding its optimal position in the sequence relative to the other z-score events, i.e. by fixing the order of the subsequence T = S/E and maximising the likelihood of the sequence by changing the position of event e in the subsequence T. The sequence S is updated until convergence. Again the single-cluster sequence S is optimised from 25 different random starting sequences to find the maximum likelihood solution (see Convergence).

Convergence

At several points in the model fitting we perform a greedy optimisation from a number of different starting points and choose the maximum likelihood sequence or set of sequences. The multiple runs safeguard against local minima. However, in fact, we find that the optimisation displays good convergence: runs from all start points typically converge to a solution with likelihood within a 1 × 10−3 % of the maximum likelihood, and within the uncertainty estimated by the uncertainty estimation procedure (see Uncertainty estimation). The convergence of the SuStaIn algorithm and ability to locate the global minimum and correct solution is further demonstrated in simulation using synthetic data in (Supplementary Results: Convergence).

Uncertainty estimation

In addition to estimating the most probable sequence Sc for each subtype, we can determine the relative likelihood of all sequences for each subtype by evaluating the probability of each possible sequence. This gives us an estimate of the uncertainty in the ordering Sc, which we summarise by plotting the probability that each z-score event appears at each position in the sequence for each subtype. We visualise this probability (see Fig. 2 for example) using different colours to indicate the cumulative probability each region has reached a particular z-score: the cumulative probability of a region going from a z-score of 0-sigma to 1-sigma ranges from 0 in white to 1 in red, the cumulative probability of a region going from a z-score of 1-sigma to 2-sigma ranges from 0 in red to 1 in magenta, and the cumulative probability of a region going from a z-score of 2-sigma to 3-sigma ranges from 0 in magenta to 1 in blue. In practise the number of sequences is too large to evaluate all possible sequences so we use Markov Chain Monte Carlo (MCMC) sampling to provide an approximation to this uncertainty, as in7,8. As in refs. 7,8, we take 1,000,000 MCMC samples initialised from the maximum likelihood solution, checking that the MCMC trace shows good mixing properties.

Cross-validation

We use ten-fold cross-validation here for two distinct purposes: (i) to evaluate the optimal number of subtypes and (ii) to evaluate the consistency of the subtype progression patterns. We evaluated the optimal number of subtypes using the Cross-Validation Information Criterion (CVIC)44, i.e. by evaluating the likelihood of each c-subtype model from c = 1 … C on the test data for each fold and choosing the model with the highest out-of-sample likelihood P(X | M), or equivalently the lowest value of the CVIC, across all folds. The CVIC is defined as CVIC = −2 × log(P(X | M)), where P(X | M) is the probability of the data for a particular SuStaIn model, M, i.e. P(X | M) = $$\mathop {\sum }\limits_{c = 1}^C {\kern 1pt} P\left( {{\bf{X}}|{\bf{S}}_c} \right)P\left( {{\bf{S}}_c} \right)$$. In cases where the evidence for a more complex model was not strong (a difference of less than 6 between the CVIC and the minimum CVIC across models, or equivalently a difference of less than 3 between the out-of-sample log-likelihood and the minimum out-of-sample log-likelihood across models), we favoured the less complex model to avoid over-fitting45. To evaluate the consistency of the subtype progression patterns we performed ten-fold cross-validation by dividing the data into ten folds and re-fitting the model to each subset of the data, with one of the folds retained for testing each time. We report the consistency of the models across folds by computing the similarity between the progression patterns of two subtypes (see Similarity between two subtype progression patterns): the model fitted to each fold and the model fitted to the whole data set.

Similarity between two subtype progression patterns

To enable the comparison of subtype progression patterns in data subsets (Supplementary Figure 13B) and across cross-validation folds (CVS in Figs. 24, and Supplementary Figures 13A and 14), we measure the similarity of pairs of subtype progression patterns using the Bhattacharyya coefficient46. We evaluate the Bhattacharyya coefficient between the position of each biomarker event in the two subtype progression patterns, averaged across biomarker events and MCMC samples. The Bhattacharyya coefficient measures the similarity of the distribution of the position of biomarker events in the subtype sequences and ranges from 0 (maximum dissimilarity) to 1 (maximum similarity).

Patient subtyping and staging

We assigned subjects to subtypes and stages predicted by the SuStaIn model (Fig. 5, Tables 1 and 3) by first evaluating the likelihood that they belonged to each subtype (by integrating over disease stage) and choosing the subtype with the highest likelihood, and then evaluating the probability they belonged to each stage of the most probable subtype and choosing the stage with the highest likelihood. When evaluating the likelihood we integrated over the set of MCMC samples to account for the uncertainty in the model parameters, rather than just evaluating the likelihood at the maximum likelihood parameters. This means that a patient’s model stage indicates the average position over the posterior distribution on the sequence given the data.

Strength of assignment to subtype

We evaluated the strength of an individual’s assignment to a particular subtype by comparing the probability that they were at stage ≤2 (i.e. they had no major imaging abnormalities and therefore could not be assigned to a particular subtype) with the probability that they belonged to each SuStaIn subtype (probability for each subtype summed over stages 3+). The strength of the assignment was evaluated as their maximum probability of belonging to one of the subtypes. We considered those with a maximum probability of belonging to a particular subtype of greater than a half as having a strong assignment to a subtype.

Comparison to subtypes-only and stages-only models

We compared our SuStaIn model to a subtypes-only model and a stages-only model. In the subtypes-only model, individuals are clustered together into groups based on the similarity of their biomarker measurements—without accounting for heterogeneity in disease stage. The stages-only model is a disease progression model where all subjects are assumed to be samples of a single common progression pattern—without accounting for heterogeneity in disease subtype. We formulated the subtypes-only and stages-only models so that they were as close as possible to the SuStaIn model, but did not model heterogeneity in disease stage or disease subtype, respectively. This allows us to assess the benefit of accounting for this disease stage or subtype heterogeneity in the SuStaIn model. The subtypes-only model consists of a mixture of Gaussians with unknown mean and variance. The subtypes-only model is fitted to symptomatic mutation carriers for GENFI, and AD subjects for ADNI, so that the subtypes correspond to a single diagnostic group. As done for the SuStaIn model, we evaluated the optimal number of clusters (subtypes) using the CVIC44. The stages-only model is a special case of the SuStaIn model outlined in Mathematical Model, where only a single subtype is modelled, i.e. C = 1.

SuStaIn modelling of GENFI data set

Supplementary Tables 46 provide a summary of the settings of the SuStaIn algorithm. We applied SuStaIn modelling to various subgroups of the GENFI data set: all 172 mutation carriers, 76 GRN mutation carriers, 63 C9orf72 mutation carriers, 33 MAPT mutation carriers. For all mutation carriers we fitted SuStaIn models of up to a maximum of 5 subtypes. For the GRN, C9orf72 and MAPT mutation carriers we fitted SuStaIn models of up to a maximum of 3 subtypes. We chose the z-score events for the GENFI data set to include z-scores of 1, 2 and 3 for each volume, but excluded z-score events where fewer than 10 mutation carriers had values that were greater than that z-score. The maximum z-score, which is reached at the final stage of the progression, was set to be 2, 3 or 5 depending on whether the maximum z-score event was 1, 2 or 3, respectively. We maintained the same z-score events across each of the GENFI experiments.

SuStaIn modelling of ADNI data set

We applied SuStaIn modelling to two largely independent (overlap of 59 individuals) subgroups of the ADNI data set: 793 individuals with 3T MRI scans and 576 individuals with 1.5T MRI scans, for which we tested SuStaIn models of up to a maximum of 5 subtypes. As we did for GENFI, we chose the z-score events to include z-scores of 1, 2 or 3 for each volume, but excluded z-score events where fewer than 10 subjects had values that were greater than that z-score. Again the maximum z-score, which is reached at the final stage of the progression, was set to be 2, 3 or 5 depending on whether the maximum z-score event was 1, 2 or 3 respectively. Full details of the settings of the SuStaIn algorithm can be found in Supplementary Tables 46.

Classification of mutation groups using subtypes

We performed two experiments to compare the ability of subtypes obtained from SuStaIn and the subtypes-only model to classify mutation carriers in GENFI into their different mutation groups. In the first experiment (Tables 1 and 2) we optimised the probability required for assignment to each of the subtypes. This accounts for different amounts of heterogeneity within the different subtypes. In the second experiment (Supplementary Tables 1 and 2) we simply assigned individuals to their most probable subtype and compared their assigned subtype with their mutation group. In both experiments the classification results are reported as out-of-sample accuracies obtained through 10-fold cross-validation.


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# Representation Of A Function

A function is a relation between two sets of variables such that one variable depends on another variable. We can represent different types of functions in different ways. Usually, functions are represented using formulas or graphs. There are four ways for the representation of a function as given below:

• Algebraically
• Numerically
• Visually
• Verbally

Each one of them has some advantages and disadvantages. Let us look at them once and try to understand them.

## Representation of a Function – Algebraic

It is one of the usual representations of functions. In this, functions are explicitly represented using formulas. The functions are generally denoted by lower case alphabet letters. For e.g. let us take the cube function.

Figure 1: Block diagram depicting a cube function

The standard letter to represent function is f. However, it can be represented by any variable. To denote the function f algebraically i.e. using the formula, we write:

$f:x$$x^3$

where x is the variable denoting the input. It can be represented by any variable.

$x^3$ is the formula of function

f is the name of the function

Though one of the easy and understandable ways of representing a function, it is not always easy to get the formula of the function. For those cases, we use other methods of representation.

### Representation of a Function- Visual

This is basically the graphical representation of functions. This way of representation is very easy to understand. The input values are marked along the x-axis. For any input value, the corresponding output value is the vertical displacement from the x-axis. For e.g. at x = a, the output is equal to f(a).

Figure 2: Graph of a function

The graph shows the properties of the functions. For e.g. from figure 2, we can directly tell:

• where the graph is increasing or decreasing
• where the rate of change is more and where it is less
• where are the extreme values

Thus, graphs are very beneficial for studying the behaviour of the function. One drawback is it that we can’t always get the exact values of all the outputs from the graph.

### Representation of a Function- Numerical

This is basically the tabular way of representing a function. The table contains two columns; one with the dependent variable and other with the independent variable. To show with an example, let us take a function f and independent variable as x. The table is given as:

Table 1: Table representing a function f(x) = 2x

 x F(x) -1 -2 – 0.5 -1 0 0 0.5 1 1 2 1.5 3

Though we have the exact value of the outputs, we can only have a finite number of such outputs. The analysis of the function and study of its behaviour hence becomes difficult.

### Representation of a Function- Verbal

In this way of representing functions, we use words. For e.g.

• For the input x, the function gives the largest integer smaller than or equal to x i.e. floor function (see fig. 3).
• For the input x, the function gives the value equal to x i.e. identity function (see fig. 4).

Figure 3: Floor function

Figure 4: Identity function

Each of the representations has its pros and cons. According to the information required, appropriate representation should be adopted. To learn more about functions, visit BYJU’S and fall in love with learning!

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# Book: Combinatorics and Graph Theory (Guichard)

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$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Combinatorics is often described briefly as being about counting, and indeed counting is a large part of combinatorics. As the name suggests, however, it is broader than this: it is about combining things. Questions that arise include counting problems: "How many ways can these elements be combined?'' But there are other questions, such as whether a certain combination is possible, or what combination is the "best'' in some sense. We will see all of these, though counting plays a particularly large role. Graph theory is concerned with various types of networks, or really models of networks called graphs. These are not the graphs of analytic geometry, but what are often described as "points connected by lines''.

This textmap is stilll under construction. Please forgive us.

Thumbnail: Rubik's Cube. Image used with permission (CC BY-SA 3.0 Unported; Booyabazooka).


text: Proportional Integral Derivative (PID) controller is one of the most common control loop algorithms used, as more than 95 $\%$ of all the regulatory controller utilize PID feedback system. Based on the heuristic approach feedback controllers allow for the response to be sufficiently close to the desired value and are therefore ideal for most practical control problems.A control feedback system was first established in the industry by James Watt’s “conical pendulum” governor. This control system was operating with only a proportional gain ($K_p$) resulting in wider oscillations of the Process Variable for bigger values of the $K_p$ or higher load. The history of PID controllers in the form that we know them today started on 1868 when James Clerk Maxwell explored the mathematical basis of controlling systems stability in his famous paper “On Governors”.  In the same year, a pendulum-and-hydro-stat control was introduced in the newly invented Whitehead torpedo being one of the first PID control systems to be used in the industry.  The last step of defining the PID controllers came from Nicolas Minorsky, who presented a formal control law using theoretical analysis. Since then PID controllers are spreading in the industry becoming until today the most used control loop in the industry.egin{figure}Hcenteringincludegraphicswidth=linewidth,height=6cm{pip.png} itle{Figure 1: Step function}end{figure} A step function as seen above, can be used to measure control system performance requirements representing the set point. Then by measuring the process variable response  over time and using waveform characteristics (represented in figure 1 above) such as, rise time, overshoot and steady-state error the response is evaluated.Feedback control allows the correction of errors in a system based on the desired and actual output. One of the most widely used control algorithms is the Proportional Integral Derivative controller, thanks to its effective performance in a variety of operating conditions and its simplicity. A PID consists of 3 coefficients the values of which can be varied in order to achieve the optimal response.The PID controller can be represented by the following equation where, u(t) is the control signal and $e( au)$ the error, which is the difference between the desired result and the actual response of the system.egin{equation}u(t)=K_P(t)+K_I int_{t}^{0}e( au)d au+K_Dfrac{de}{dt}end{equation}The control signal is comprised of 3 terms, proportional, integral and derivative, with control parameters $k_p$, $k_i$ and $k_d$ respectively.The above equation can also be expressed in the Laplace domain, where the roles of each term  is better represented:egin{equation}G_R(s)=frac{U(s)}{E(s)}=K_P(1+frac{1}{T_Is}+frac{T_Ds}{1+frac{T_D}{N} imes s})end{equation}In its Laplace form it easier to manipulate and relate to practical results, especially in higher order problems.egin{figure}Hcenteringincludegraphicswidth=linewidth,height=5cm{Block.png} itle{Figure 2: PID controller block diagram}end{figure}For the proportional component, the error magnitude is multiplied by the proportional gain $K_p$  resulting in the proportional response. Therefore the greater the percentage difference of the output signal from the desired output is, the greater the proportional response will be.Integral:egin{equation} K_P imes e(t) end{equation}The integral component sums the error term over time, multiplying it by the integral coefficient $K_i$ and then integrating.Even a minor deviation from the desired value will result in a large integral component over time. The steady-state errors, the final difference between the process variable and set point, will be driven to zero through the integral component since the integral response will rise until the error is zero.Proportional:egin{equation} K_I imes int_{t_D}^{t} e(t) end{equation}The derivative response is proportional to the rate of change of the process variable. Hence the control system is able to minimize the effect of a disturbance and the time it takes to correct one, through the use of a projection of the process variable value. This allows higher Integral and Proportional coefficients to be used without the system becoming unstable.However since any changes in the error term affect it, the component is highly sensitive to noise in the sensor feedback and can easily render the system unstable if it is set too large.Derivative:egin{equation} K_D imes frac{d}{dt} e(t) end{equation}section{Block Diagram} The effects of a corresponding increase at each individual component of the PID are represented below in Table 1. The term rise time refers to the time taken for the helicopter to rise from 10\% to 90\% of its steady value. Overshoot is when the helicopter goes past the required elevation, desired response signal. Settling time is the time period for the helicopter to remain within a certain margin from the required elevation with no further deviations. Steady State Error (SSE) is the difference between the desired result and the actual response of the system, $u(t)-e(t)$. The term stability refers to the oscillatory behavior of the system.egin{table}ht!centeringcaption{Increasing the independent variable values of a PID control}label{tab:table1}egin{tabular}{|c|c|c|c|c|c|}hlineParameter & Rise time & Overshoot & Settling Time & Steady State Error & Stability \hline$K_P$ & Decrease & Increase & Slight Change & Decrease & Degrade \hline$K_I$ & Decrease & Increase & Increase & Gradually Eliminate & Degrade \hline$K_D$ & Slight Change & Decrease & Decrease & No effect in theory & Improve if $K_D$ is small\hlineend{tabular}end{table}section{Qunaser Helicopter PID control practical results}In this part of the laboratory, the values of $K_P$, $K_D$, $K_I$ where tested individually, by adjusting their values and observing the visual result on the signal. The overall results, satisfy our theoretical background.egin{figure}Hcenteringincludegraphicswidth=1linewidth,height=8cm{kP_-0_2__0__+0_2__+0_4_NEW.jpg} itle{Figure 3: Test values for $K_{P}$, a:0.3, b:0.5, c:0.7, d:0.9}end{figure}When increasing the value of $K_{P}$ we can see that the overall Rise time decreases, the Overshoot increases and there is no particular change Settling time. However, having a high $K_{P}$ results to instabilities, as can be seen from Figure 3: d ($K_{P}$=$K_{Pi}+0.4)$. Furthermore, the benefits than can be obtained by rising $K_{P}$ can be achieved by adjusting $K_{I}$ to a higher value too. Therefore, increasing $K_{p}$ does not ensures us with a gradual and smooth result in steady state. As a result, the final value is decreased compared to the initial one, Figure 3: b ($K_{Pi}=0.5$), to $K_{Pf}=0.27$, to avoid unwanted oscillations.egin{figure}Hcenteringincludegraphicswidth=linewidth,height=9cm{___-0_05__0__0_10__0_20_NEW.jpg} itle{Figure 4: Test values for $K_{I}$, I:0.03, II:0.08, III:0.18, IV:0.28}end{figure}Adjusting $K_{I}$ variable to a higher value, as can be seen from the above Figure, the Rise time and the Steady State Error decreases, while the Overshoot and the Settling time increase. Moreover, the change of the value $K_{I}$, as it is demonstrated in Figure 4, doesn’t affect the stability greatly, like $K_{P}$. Consequently, the value of $K_{I}$ has been raised to $K_{I}=0.23$ from an initial value of $K_{Ii}=0.08$egin{figure}Hcenteringincludegraphicswidth=1linewidth,height=9cm{Kd_-0_14__0__+0_06__+0_14_NEW.png} itle{Figure 5: Test values for $K_{D}$, A:0.20, B:0.34, C:0.40, D:0.48}end{figure}Altering the value of $K_{D}$ in this the overshoot and the Settling Time.  Raising $K_{D}$, gives smaller overshoot and shorter Settling Time with no visual effect on Rise Time or Steady State Error. As a result, the final value of $K_{D}$ is 0.40, +0.06 from the initial value.section{Discussion} In order to achieve the desired response some tuning had to be undertaken, adjusting the proportional, differential and integral gains. For that purpose a trial and error method was followed. A $0.2$ , 0.07 and  0.01, step was used for each of $K_P$,$K_D$ and $K_I$ respectively, hence finding the optimum value for each component and balance between them.The progress can be seen in Figures 3 to 6, as each component is adjusted the rise time, overshoot and settling time are stabilized and the signal deviation from the ideal response is minimized. As seen in Figure 6 the final values were $+0.15$ for $K_I$, $-0.23$ for $K_P$ and $+0.06$ for $K_D$.  egin{figure}Hcenteringincludegraphicswidth=0.8linewidth,height=7cm{+0_15_-0_23_+0_06.jpg}centering itle{Figure 6 Final Results : $K_I +0.15, K_P -0.23, K_D +0.06$}end{figure} section{References} egin{thebibliography}{2} ibitem{1} $Desborough Honeywell. “PID Control.” Caltech.edu, 2000, pp. 301–322., \www.cds.caltech.edu/~murray/books/AM08/pdf/am06-pid_16Sep06.pdf.$ibitem{2}$”PID for Dummies.” PID for Dummies – Control Solutions, www.csimn.com/CSI_pages/PIDforDummies.html.$ibitem{3}”PID Theory Explained.” PID Theory Explained – National Instruments, www.ni.com/white-paper/3782/en/.ibitem{4}Smuts, Jacques. “PID Controllers Explained.” Blog.opticontrols, 7 Mar. 2011, blog.opticontrols.com/archives/344.ibitem{5}Smuts, Jacques F. Process Control for Practitioners: How to Tune PID Controllers and Optimize Control Loops. OptiControls, 2011.ibitem{6}$”PID Controller.” Wikipedia, Wikimedia Foundation, 26 Jan. 2018, en.wikipedia.org/wiki/PID_controller.$end{thebibliography}end{document}


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test image inline w text

List of scratch pages

Unit Attributes
Rank Armor Strength Willpower Exertion Armor Break

$\mathrm{ Damage = \color{red}{STR_{att}} - \color{blue}{ARM_{def}} }$

 Rank is a measure of how honed are a unit's special abilities. Each unit has both passive and active abilities. Passive abilities have no cost and are "always on," i.e., the player automatically receives their benefit whenever appropriate. Active abilities become available at Rank 1 (and higher) and require the unit to spend Willpower (WP) in order to use them. Rank {1,2,3} active abilities cost {1,2,3} WP respectively; any ability Rank from 1 up to unit Rank can be used, by selecting the appropriate amount of WP. Armor (ARM) negates incoming Strength damage. When a Strength attack is made, the damage dealt is equal to STR(att)— ARM(def). Armor can reduced when attacked with Armor Break. (See also the battle page) Strength (STR) is a measure of both a units health and attack-damage. A unit's attack is gradually weakened as he loses STR throughout a fight; units die when their STR reaches 0. Willpower (WP) allows you to use your unit's special (active) abilities or exceed your normal potential in standard actions. In particular, you can spend WP to move farther, hit harder or do more armor-break. WP is the only attribute that can be replenished during a battle, by the resting or using the Horn. Exertion (EX) is an indicator of how much WP a unit can use on a given action. Example: A character with 3EX can move 3 extra squares in one turn at the cost of 3WP, then attack for +3STR damage at the cost of 3WP more. Meanwhile, a character with 1EX can only move 1 extra square for 1WP and/or attack for +1STR damage for 1WP. Armor Break (AB) is an indicator of how much armor damage the unit can do on a single attack. AB can be increased with the use of WP up to the amount of the units EX.
page revision: 15, last edited: 13 Apr 2013 19:37


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# Fuel injector surrounded by a coaxial air stream

1. Aug 26, 2004

### Clausius2

Maybe you, chemical people could answer my question.

Imagine a fuel injector surrounded by a coaxial air stream. The fuel and air flow are parallel, coaxial and axisymmetric. Both fluids obey ideal gas equation. Let's consider steady and compressible flow, i.e, variations of $$\rho$$ are important.

At low Mach numbers, the flow velocity is small enough to allow both reactants to be mixed by means of molecular diffusion and convection, resulting in a diffusion flame that can be seen experimentally. I think chemical reaction takes place in the mixing region of both substances. The thickness of this mixing layer I think it will depend on how important is the diffusion mechanism in the transport phenomena.

But my question is about high Mach numbers, in particular M>>1 (supersonic). What would happen?. Will there be enough time for molecular diffusion?. If not, it would be possible to the reaction to take place?.

Moreover, what do you think it will happen if fuel stream is subsonic and air is supersonic?

Any help is welcomed.

2. Sep 11, 2004

### altered-gravity

Hi spanish collegue,

Interesting. I cannot assure nothing about it.

I would say that yes, even at very high flux velocities in a conctrete direction, there is always a numer of molecules traveling in the transversal direction (in fact in all directions). So there would be allways diffusion mechanisms (i think). If combustion iniciates, it wolud create turbulences enhancing the mixturing process don´t you think so?

Molecular Brownian motions are about 1 km/s, are you talking of flux velotities higher than this? Then perhaps combustion wolud appear only after a long distance from the start of the fluxes. During this distance perhaps some molecules would react, but heat transfer in the transversal direction would be avoided by such a high velocity.

Last edited: Sep 11, 2004
3. Sep 12, 2004

### Clausius2

Thanks for your reply. I was searching for an answer before doing any numerical simulation. I was pretending to employ Euler equations (without the diffusion terms) in order to describe the flow. Euler equations assume that diffusion and heat transfer are negligible compared with convection phenomena. This is the same thing that saying mixing layer has a negligible thickness compared with the longitudinal variations at #Mach>>1.

I do not agree with you have just said:

"I would say that yes, even at very high flux velocities in a conctrete direction, there is always a numer of molecules traveling in the transversal direction (in fact in all directions). So there would be allways diffusion mechanisms (i think)"

There will be a convection of both reactants in transverse directions. It is enough for enhancing a mixing process (isn't it?). :surprised . I've just make up my mind... this is an interesting question: chemical reaction is not possible if diffusion does not exist?. I would say not. The reactive energy term in N-S equations is:

$$D_{a}\theta Y_{F}Y_{O2} e^{\frac{T}{1+T/\beta}}$$ (non-dimensional).

So that, the chemical reaction will take place in a space region with both fuel and oxidizer are present simultaneusly. This can be reached only by a convective process (do you agree??, I'm not sure of that).

Anyway, I was trying to program some simulation of this flow in Matlab. This is very interesting in supersonic combustion (have you ever heard about Scramjet engine?). If not, see the attached file below. I've found it in the web. It's a supersonic flame simulated with Fluent.

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4. Sep 13, 2004

### altered-gravity

Ok, I confess that i´ve never read about such rapid flows. If you´re talking about match 3 or even higher velocities, the flow itself is quicker than the brownian motions of the molecules, that wold mean that almost all the molecules are travelling in the direction of the flow right? then there´s not diffusion to the transverse direction, so you would be right.

5. Sep 13, 2004

### altered-gravity

No, how does it work?

6. Sep 14, 2004

### Clausius2

Ok. It doesn't matter. I really didn't hope anyone to read this thread, because it's a bit complex to solve. Anyway, thanks again for trying something.

The Scramjet engine is a supersonic engine. It breaths the air directly of a supersonic atmosphere. Thus, air into combustion chamber runs supersonically, also in the combustion process. But knowing this process is a very hard task due to flame unstabilities. By the way, NASA's sciencist are almost able now to build such engine at large scale. Some people say the next decade they'll get rid of the space shuttle and use this hypersonic spacecraft.


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Expression of Concentration of Solutions

We always discuss a solution being diluted or concentrated; this is a qualitative way of expressing the concentration of the solution. A dilute solution means the quantity of solute is relatively very small, and a concentrated solution implies that the solution has a large amount of solute. But these are relative terms and do not give us the quantitative concentration of the solution.

So, to quantitatively describe the concentrations of various solutions around us, we commonly express levels in the following way:

Concentration

It is the amount of solute present in one liter of solution. It is denoted by C or S.

$C or S = \frac{Weight\ of\ solute\ in\ grams}{Volume\ in\ liters}$

1. Concentration in Parts Per Million (ppm)

The parts of a component per million parts (106) of the solution.

$ppm(A) = \frac{Mass\ of\ A}{Total\ mass\ of\ the\ solution}\times {10}^6$

2. Mass Percentage (w/w):

When the concentration is expressed as the percent of one component in the solution by mass it is called mass percentage (w/w). Suppose we have a solution containing component A as the solute and B as the solvent, then its mass percentage is expressed as:

Mass % of A = $\frac {Mass \space of \space component \space A ~ in ~the ~ solution}{Total ~ mass ~of~ the~ solution } × 100$

3. Volume Percentage (V/V):

Sometimes we express the concentration as a percent of one component in the solution by volume, it is then called as volume percentage and is given as:

volume % of A = $\frac {Volume ~of~ component~ A~ in~ the ~solution}{Total ~ volume ~ of ~ the ~ solution } × 100$

For example, if a solution of NaCl in water is said to be 10 % by volume that means a 100 ml solution will contain 10 ml NaCl.

4. Mass by Volume Percentage (w/V):

This unit is majorly used in the pharmaceutical industry. It is defined as the mass of a solute dissolved per 100mL of the solution.

5. Molarity (M):

One of the most commonly used methods for expressing the concentrations is molarity. It is the number of moles of solute dissolved in one litre of a solution. Suppose a solution of ethanol is marked 0.25 M, this means that in one litre of the given solution 0.25 moles of ethanol is dissolved.

6. Molality (m):

Molality represents the concentration regarding moles of solute and the mass of solvent. It is given by moles of solute dissolved per kg of the solventThe molality formula is as given-

$Molality (m) = \frac {Moles~ of ~solute}{Mass~ of~ solvent~ in~ kg}$

7. Normality

It is the number of gram equivalents of solute present in one liter of the solution and it is denoted by N.

$N = \frac{Weight\ of\ solute\ in\ grams}{Equivalent\ mass \times Volume\ in\ liter}$

The relation between normality and molarity.

• N x Eq.Wt = Molarity x Molar mass
• N = Molarity x Valency
• N = Molarity x Number of H+ or OH– ion.

8. Formality

It is the number of gram formula present in one litre of solution. It is denoted by F.

$F = \frac{Weight\ of\ solute\ in\ gram}{Formula\ wt \times Volume\ in\ liter}$

It is applicable in the case of ionic solids like NaCl.

9. Mole Fraction:

If the solution has a solvent and the solute, a mole fraction gives a concentration as the ratio of moles of one component to the total moles present in the solution. It is denoted by x. Suppose we have a solution containing As a solute and B as the solvent. Let nA and nB be the number of moles of A and B present in the solution respectively. So mole fractions of A and B are given as:

$~~~~~~~~~~~~~~~$$x_A = \frac {n_A}{n_A + n_B}$
$~~~~~~~~~~~~~~~$$x_B = \frac {n_B}{n_A + n_B}$

The above-mentioned methods are commonly used ways of expressing the concentration of solutions. All the methods describe the same thing that is, the concentration of a solution, each of them has its own advantages and disadvantages. Molarity depends on temperature while mole fraction and molality are independent of temperature. All these methods are used on the basis of the requirement of expressing the concentrations.

Solutions of Solids in Liquids

1. A saturated solution is a solution that remains in contact with an excess of solute.
2. The amount of solute dissolved per 100g of solvent in a saturated solution at a specific temperature represents the solubility of the solute.
3. For exothermic substances such as KOH, CaO, Ca(OH)2, M2CO3, M2SO4, etc, solubility is inversely proportional to temperature.
4. For endothermic substances such as NaCl, KNO3, NaNO3, glucose, etc., solubility is directly proportional to temperature.

Solubility of Gases

The solubility of gases is mostly expressed in terms of the absorption coefficient,k that is the volume of the gas dissolved by unit volume of solvent at 1 atm pressure and a specific temperature.

The solubility of a gas in a liquid depends upon

1. Temperature solubility is inversely proportional to temperature as the dissolution of gas is exothermic in most cases.
2. Nature of gas – Gases having a higher value of van der Waals force of attraction that is gases that are more easily liquefied are more soluble. For example, SO2 and CO2 are more soluble in water than O2, N2, and H2.
3. Nature of solvent – Gases which can ionize in aqueous solution are more stable in water as compared to the other solvents.

How do you change the concentration of a solution?

Sometimes, by modifying the quantity of solvent, a worker would need to modify the concentration of a solution. Dilution is the addition of a solvent that reduces the solute concentration of the solution. Concentration is solvent elimination, which increases the solute concentration in the solution.

What is a high concentration?

A concentration of persons means that in one place there are more of them. A high concentration of a material in a solution means there’s a lot of it compared to the volume: because of the high concentration of salt, the Great Salt Lake has very little fish.

Is dilute acid dangerous?

In general, a mild condensed acid is more harmful than a solid diluted acid. Although concentrated acetic acid is much less reactive, you do not want to touch the skin or mucous membranes because it is corrosive.

What is the concentration of a solution?

A solution concentration is a measure of the quantity of solute that has been dissolved in a given quantity of solvent or solution. One that contains a relatively high volume of dissolved solute is a concentrated solution. That that contains a relatively minimal volume of dissolved solute is a dilute solution.

How do you prepare a solution of known concentration?

Solutions of known concentration can be prepared either by dissolving the known mass of the solvent solution and diluting it to the desired final volume or by diluting it to the desired final volume by diluting the acceptable volume of the more concentrated solution (the stock solution).

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