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276 nips-2012-Probabilistic Event Cascades for Alzheimer's disease

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Author: Jonathan Huang, Daniel Alexander

Abstract: Accurate and detailed models of neurodegenerative disease progression are crucially important for reliable early diagnosis and the determination of effective treatments. We introduce the ALPACA (Alzheimer’s disease Probabilistic Cascades) model, a generative model linking latent Alzheimer’s progression dynamics to observable biomarker data. In contrast with previous works which model disease progression as a ﬁxed event ordering, we explicitly model the variability over such orderings among patients which is more realistic, particularly for highly detailed progression models. We describe efﬁcient learning algorithms for ALPACA and discuss promising experimental results on a real cohort of Alzheimer’s patients from the Alzheimer’s Disease Neuroimaging Initiative. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Accurate and detailed models of neurodegenerative disease progression are crucially important for reliable early diagnosis and the determination of effective treatments. [sent-6, score-0.539]

2 We introduce the ALPACA (Alzheimer’s disease Probabilistic Cascades) model, a generative model linking latent Alzheimer’s progression dynamics to observable biomarker data. [sent-7, score-0.446]

3 In contrast with previous works which model disease progression as a ﬁxed event ordering, we explicitly model the variability over such orderings among patients which is more realistic, particularly for highly detailed progression models. [sent-8, score-0.956]

4 1 Introduction Models of disease progression are among the core tools of modern medicine for early disease diagnosis, treatment determination and for explaining symptoms to patients. [sent-10, score-0.698]

5 Despite their utility, traditional models of disease progression [3, 17] have largely been limited to coarse symptomatic staging which divides patients into a small number of groups by thresholding a crude clinical score of how far the disease has progressed. [sent-14, score-1.036]

6 The models are thus only as precise as these crude clinical scores — although providing insight into disease mechanisms, they provide little beneﬁt for early diagnosis or accurate patient staging. [sent-15, score-0.704]

7 The recent availability of cross sectional datasets such as the Alzheimer’s Disease Neuroimaging Initiative data has generated intense speculation in the neurology community about the nature of the cascade of events in AD and the ordering in which biomarkers show abnormality. [sent-17, score-0.57]

8 Beckett [2] was the ﬁrst, nearly two decades ago, to propose a data driven model of disease progression using a distribution over orderings of clinical events. [sent-20, score-0.565]

9 Examples of events in the model of [8] include (but are not limited to) clinical events (such as a transition from Presymptomatic Alzheimer’s to Mild Cognitive Impairment) and the onset of atrophy (reduction of tissue volume). [sent-24, score-0.597]

10 By assuming a single universal ordering of events within the disease progression, the method of [8] is able to scale to much larger collections of events, thus achieving much more detailed characterizations of disease progression compared to that of [2]. [sent-25, score-1.16]

11 1 The assumption made in [8] of a universal ordering common to all patients within a disease cohort, is a major oversimpliﬁcation of reality, however, where the event ordering can vary considerably among patients even if it is consistent enough to distinguish different diseases. [sent-26, score-1.173]

12 In practice, the assumption of a universal ordering within the model means we cannot recover the diversity of orderings over population groups and can make ﬁtting the model to patient data unstable. [sent-27, score-0.631]

13 Additionally, like [8], our method can achieve the scalability that is required to produced ﬁne-grained disease progression models. [sent-31, score-0.424]

14 • We propose the Alzheimer’s disease Probabilistic Cascades (ALPACA) model, a probabilistic model of disease cascades, allowing for patients to have distinct event orderings. [sent-33, score-0.857]

15 • We develop efﬁcient probabilistic inference and learning algorithms for ALPACA, including a novel patient “staging” method, which predicts a patient’s full trajectory through clinical and atrophy events from sparse and noisy measurement data. [sent-34, score-0.8]

16 2 Preliminaries: Snapshots of neurodegenerative disease cascades We model a neurodegenerative disease cascade as an ordering of a discrete set of N events, {e1 , . [sent-36, score-0.991]

17 These events represent changes in patient state, such as a sufﬁciently low score on a memory test for a clinical diagnosis of AD, or the ﬁrst measurement of tissue pathology, such as signiﬁcant atrophy in the hippocampus (memory related brain area). [sent-40, score-0.844]

18 An ordering over events is represented as a permutation σ which corresponds events to the positions within the ordering at which they occur. [sent-41, score-0.882]

19 In practice, the ordering σ for a particular patient can only be observed indirectly via snapshots which probe at a particular point in time whether each event has occurred or not. [sent-46, score-0.953]

20 We denote a snapshot by a vector of N measurements z = (ze1 , . [sent-47, score-0.49]

21 , zen ), where each zei is a real valued measurement reﬂecting a noisy diagnosis as to whether event i of the disease progression has occurred prior to measuring z. [sent-50, score-0.825]

22 1 Were it not for noise within the measurement process, a single snapshot z would partition the event set into two disjoint subsets: events that have occurred already (e. [sent-51, score-1.035]

23 Where prior models [8] considered data in which a patient is only associated with a single snapshot (taken at a single time point), we allow for multiple snapshots of a patient to be taken spaced throughout that patient’s disease cascade. [sent-62, score-1.511]

24 For example, if σ = e3 |e1 |e4 |e5 |e6 |e2 , then k = 2 snapshots might partition the event ordering into sets X1 = {e1 , e3 }, X2 = {e4 , e5 }, X3 = {e2 , e6 }, reﬂecting that events in X1 occur before events in X2 , which occur before events in X3 . [sent-64, score-1.278]

25 Such partitions can also be thought of as partial rankings over the events (and indeed, we will exploit recent methods for learning with partial rankings in our own approach, [11]). [sent-65, score-0.4]

26 To denote partial rankings, we again use vertical bars, separating the events that occur between snapshots. [sent-66, score-0.307]

27 Instead of reasoning with continuous snapshot times, we use the fact that many distinct snapshot times can result in the same partial ranking, to reason instead with discrete snapshot sets. [sent-70, score-1.446]

28 By snapshot set, we refer to the collection of positions in the full event ordering just before each snapshot is taken. [sent-71, score-1.337]

29 In our running example, the snapshot set is τ = {2, 4}. [sent-72, score-0.464]

30 Given a full ordering σ, the partial ranking which arises from snapshot data (assuming no noise) is fully determined by τ . [sent-73, score-0.842]

31 3 ALPACA: the Alzheimer’s disease Probabilistic Cascades model We now present ALPACA, a generative model of noisy snapshots in which the event ordering for each patient is a latent variable. [sent-76, score-1.142]

32 ALPACA makes two main assumptions: (1), that the measured outcomes for each patient are independent of each other and (2), conditioned on the event ordering of each 1 For notational simplicity, we assume that measurements corresponding to each event are scalar valued. [sent-77, score-0.957]

33 2 patient and the time at which a snapshot is taken, the measurements for each event are independent. [sent-79, score-0.99]

34 In contrast with [8], we do not assume that multiple snapshot vectors for the same patient are independent of each other. [sent-80, score-0.778]

35 Draw an ordering of the events σ (j) from a Mallows distribution P (σ; σ0 , λ) over orderings. [sent-86, score-0.441]

36 Draw a snapshot set τ (j) from a uniform distribution P (τ ) over subsets of the event set. [sent-88, score-0.65]

37 For each element of the snapshot set, τi = τ1 , . [sent-90, score-0.464]

38 , if event e has occurred prior to time τi ), draw zi,e ∼ (j) N (µoccurred , coccurred ). [sent-98, score-0.332]

39 e e e e (j) In the above basic model, each entry of a snapshot vector, zi,e , is generated by sampling from a univariate measurement model (assumed in this case to be Gaussian). [sent-100, score-0.574]

40 If event e has already occurred, (j) (j) the observation zi,e is sampled from the distribution N (µoccurred , coccurred ) — otherwise zi,e is e e sampled from a measurement distribution estimated from a control population of healthy individuals, N (µhealthy , chealthy ). [sent-101, score-0.395]

41 For notational simplicity, we denote the collection of snapshots for patient e e (j) (j) j by z·,· = {zi,e }i=1,. [sent-102, score-0.459]

42 The Kendall’s tau distance penalizes the number of inversions, or pairs of events for which σ and σ0 disagree over relative ordering. [sent-112, score-0.306]

43 Our algorithms are thus applicable for more general classes of distributions over orderings as well as snapshot sets. [sent-117, score-0.537]

44 With respect to the event-based characterization of disease progression, a critical problem is that of patient staging, the problem of determining the extent to which a disease has progressed for a particular patient given corresponding measurement data. [sent-119, score-1.258]

45 ALPACA offers a simple and natural formulation of the patient staging problem as a probabilistic inference query. [sent-120, score-0.465]

46 In particular, given the measurements corresponding to a particular patient, we perform patient staging by: (1) computing a posterior distribution over the event ordering σ (j) , then (2) computing a posterior distribution over the most recent element of the snapshot set τ (j) . [sent-121, score-1.423]

47 To visualize the posterior distribution over the event ordering σ (j) , we plot a simple “ﬁrst-order staging diagram”, displaying the probability that event e has occurred (or will occur) in position q according to the posterior. [sent-122, score-0.87]

48 Two major features differentiate ALPACA from traditional patient staging approaches, in which patients are binned into a small number of imprecisely deﬁned stages. [sent-123, score-0.563]

49 In particular, our method is more ﬁne-grained, allowing for a detailed picture of what the patient has undergone as well as a prediction of what is to come next. [sent-124, score-0.314]

50 Given a collection of K (j) snapshots for a patient j, the critical inference problem that we must solve is that of computing a posterior distribution over the latent event order and snapshot set for that patient. [sent-129, score-1.176]

51 One reason is that it is not obvious how to tractably sample the event ordering σ conditioned on its Markov blanket, given that the corresponding likelihood function is not conjugate prior to the Mallows model. [sent-134, score-0.431]

52 This augmented model is equivalent to the original model, but has the advantage that it reduces the sampling step for the event ordering σ to a well understood problem (described below). [sent-138, score-0.437]

53 1) Observe that since the snapshot set τ is fully determined by the partial ranking γ, it is not necessary to condition on τ in Equation 4. [sent-141, score-0.619]

54 1 (right), since γ is fully determined given both the event ordering σ and the snapshot set τ , one can sample τ ﬁrst, and deterministically reconstruct γ. [sent-144, score-0.873]

55 We now turn to the problem of sampling a snapshot set τ (j) of size K (j) from Equation 4. [sent-157, score-0.492]

56 In the following, we present a dynamic programming algorithm for sampling snapshot sets with running time much lower than the exhaustive setting (even for small K (j) ). [sent-161, score-0.492]

57 Our core insight is to exploit conditional independence relations within the posterior distribution over snapshot sets. [sent-162, score-0.506]

58 As we show in the following, however, we can bijectively associate each snapshot set with a trajectory through a certain grid. [sent-164, score-0.464]

59 With respect to this grid-based representation of snapshot sets, we then show that the posterior distribution can be viewed as that of a particular hidden Markov model (HMM). [sent-165, score-0.506]

60 , given a snapshot set τ , there is a unique staircase walk pτ mapping to τ ). [sent-186, score-0.669]

61 (b): Grid structured state space G for sampling snapshot sets with edges labeled with transition probabilities according to Equation 4. [sent-188, score-0.492]

62 The example path (highlighted) is p = ((2, 3), (2, 2), (1, 2), (1, 1), (0, 1), (0, 0)), corresponding to the snapshot set τ = {4, 2}. [sent-191, score-0.464]

63 Conditioned on σ = σ (j) , the posterior probability P (τ = τ (j) | σ = σ (j) , z·,· ) is equal to the posterior probability of the staircase walk pτ (j) under the hidden Markov model deﬁned by Equations 4. [sent-206, score-0.289]

64 To sample a snapshot set from the conditional distribution in Equation 4. [sent-209, score-0.464]

65 2, we therefore sample staircase walks from the above HMM and convert the resulting samples to snapshot sets. [sent-210, score-0.673]

66 We claim that the complexity of sampling a snapshot set is also O(N 2 ). [sent-214, score-0.492]

67 The one exception is when the size of the snapshot set is one less than the number of events (K (j) = N − 1). [sent-225, score-0.682]

68 2 Note that to have so many snapshots for a single patient would be rare indeed. [sent-230, score-0.459]

69 5 Even when K (j) < N − 1, mixing times for the chain can be longer for larger snapshot sets (where K (j) is close to N − 1). [sent-231, score-0.503]

70 For example, when K (j) = N − 2, it is possible to show that the T th ordering in the Gibbs chain can differ from the (T + 1)th ordering by at most an adjacent swap. [sent-232, score-0.446]

71 For each patient in the cohort, use the inference algorithm described in Section 4. [sent-244, score-0.339]

72 Note that the sampled snapshot sets ({τ (j) }) do not play a role in the M-step described here, but can be used to estimate parameters for the more complex snapshot set distributions described in Section 5. [sent-252, score-0.928]

73 In clinical datasets, it is more conceivable that different biomarkers within a disease cascade change over different timescales, thus leading to higher positional variance for certain events and lower positional variance for others. [sent-259, score-0.756]

74 ([11]), in particular, proved that these hierarchical rifﬂe independent models form a natural conjugate prior family for partial ranking likelihood functions and introduced efﬁcient algorithms for conditioning on partial ranking observations. [sent-264, score-0.31]

75 It is ﬁnally interesting to note that it would not be trivial to use traditional Markov chains to capture the dependencies in the event sequence due to the fact that observations come in snapshot form instead of being indexed by time as they would be in an ordinary hidden Markov model. [sent-265, score-0.65]

76 (Left) ﬁrst order staging diagram, the (e, q)th entry is the probability that event e has/will occur in position q. [sent-287, score-0.369]

77 (Right) posterior probability distribution over the position in the event ordering at which the patient snapshot was taken. [sent-288, score-1.251]

78 Setting 0 ≤ α < 1/2, however, reﬂects a prior bias for snapshots to have been taken earlier in the disease cascade, while setting 1/2 < α ≤ 1 reﬂects a prior bias for snapshots to have been taken later in the disease cascade. [sent-291, score-0.838]

79 Since we are interested in the ability of the model to recover the true central ranking, we evaluate based on the Kendall’s tau distance between the ground truth central ranking and the central rankings learned by our algorithms. [sent-294, score-0.331]

80 2(a) illustrates the results on a problem with N = 10 events and 250 patients (with K (j) set to be 1, 2, or 3 randomly for each patient) as cM AX varies between [0. [sent-297, score-0.341]

81 2(b) shows, as expected, that recovery rates for the central ordering improve as the number of patients increases. [sent-307, score-0.381]

82 2(c) shows example Gibbs trace plots with N = 20 events and varying sizes of the snapshot set, K (j) . [sent-312, score-0.682]

83 [8] (which assumes that all patients follow a single ordering σ ∗ ) by searching exhaustively over the collection of all 7! [sent-320, score-0.346]

84 We use a single Gaussian for each of the healthy and occurred measurement distributions (as described in [8]), assuming that all patients in the control group are healthy. [sent-330, score-0.341]

85 3 The results show that by allowing for the event ordering σ to vary across patients, the ALPACA model signiﬁcantly outperforms the single ordering model (shown in the σ ∗ column) in BIC score with respect to all of the tried settings of α. [sent-331, score-0.653]

87 Observe that the two event orderings are largely in agreement with each other with CSF Aβ42 and CSF tau events shifted to being earlier in the event ordering, which is more consistent with current thinking in neurology [12, 5, 1], which places the two CSF events ﬁrst. [sent-335, score-1.004]

88 It is surprising that the hippocampal volume and atrophy events are inferred in both models to occur before the CSF events [13], but we believe that this may be due to the signiﬁcant proportion of misdiagnosed patients in the data. [sent-337, score-0.741]

89 2(e) shows the patient staging result for an example patient from the ADNI data. [sent-341, score-0.754]

90 The left matrix visualizes the probability that each event will occur in each position of the event ordering given snapshot data from this patient, while the right histogram visualizes where in the event ordering the patient was situated when the snapshot was taken. [sent-342, score-2.303]

91 7 Conclusions We have developed the Alzheimer’s disease Probabilistic Cascades model for event ordering within the Alzheimer’s disease cascade. [sent-343, score-0.957]

92 In its most basic form, ALPACA is a simple model with generative semantics, allowing one to learn the central ordering of events that occur within a disease progression as well as to quantify the variance of this ordering across patients. [sent-344, score-1.158]

93 Our preliminary results show that relaxing the notion that a single ordering over events exists for all patients allows ALPACA to achieve a much better ﬁt to snapshot data from a cohort of Alzheimer’s patients. [sent-345, score-1.117]

94 One of our main contributions is to show how the combinatorial structure of event ordering models can be exploited for algorithmic efﬁciency. [sent-346, score-0.409]

95 There may exist biomarkers for Alzheimer’s which are more effective than those considered in our current work for the purposes of patient staging. [sent-348, score-0.381]

96 Identifying such biomarker events remains an open question crucial to the success of data-driven models of disease cascades. [sent-349, score-0.514]

97 We have discussed several such possible extensions, from more general measurement models to more general rifﬂe independent ordering models. [sent-351, score-0.305]

98 The alzheimer’s disease neuroimaging initiative: progress report and future plans. [sent-376, score-0.318]

99 s disease biomarkers in the alzheimer’s disease neuroimaging initiative cohort. [sent-394, score-0.682]

100 An event-based model for disease progression and its application in familial alzheimer’s disease and huntington’s disease. [sent-415, score-0.723]

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