{"title": "DPSCREEN: Dynamic Personalized Screening", "book": "Advances in Neural Information Processing Systems", "page_first": 1321, "page_last": 1332, "abstract": "Screening is important for the diagnosis and treatment of a wide variety of diseases. A good screening policy should be personalized to the disease, to the features of the patient and to the dynamic history of the patient (including the history of screening). The growth of electronic health records data has led to the development of many models to predict the onset and progression of different diseases. However, there has been limited work to address the personalized screening for these different diseases. In this work, we develop the first framework to construct screening policies for a large class of disease models. The disease is modeled as a finite state stochastic process with an absorbing disease state. The patient observes an external information process (for instance, self-examinations, discovering comorbidities, etc.) which can trigger the patient to arrive at the clinician earlier than scheduled screenings. The clinician carries out the tests; based on the test results and the external information it schedules the next arrival. Computing the exactly optimal screening policy that balances the delay in the detection against the frequency of screenings is computationally intractable; this paper provides a computationally tractable construction of an approximately optimal policy. As an illustration, we make use of a large breast cancer data set. The constructed policy screens patients more or less often according to their initial risk -- it is personalized to the features of the patient -- and according to the results of previous screens \u2013 it is personalized to the history of the patient. In comparison with existing clinical policies, the constructed policy leads to large reductions (28-68 %) in the number of screens performed while achieving the same expected delays in disease detection.", "full_text": "DPSCREEN: Dynamic Personalized Screening\n\nKartik Ahuja\n\nElectrical and Computer Engineering Department\n\nUniversity of California, Los Angeles\n\nahujak@ucla.edu\n\nWilliam R. Zame\n\nEconomics Department\n\nUniversity of California, Los Angeles\n\nzame@econ.ucla.edu\n\nMihaela van der Schaar\n\nElectrical and Computer Engineering Department, University of California, Los Angeles\n\nEngineering Science Department, University of Oxford\n\nmihaela.vanderschaar@oxford-man.ox.ac.uk\n\nAbstract\n\nScreening is important for the diagnosis and treatment of a wide variety of diseases.\nA good screening policy should be personalized to the features of the patient\nand to the dynamic history of the patient (including the history of screening).\nThe growth of electronic health records data has led to the development of many\nmodels to predict the onset and progression of different diseases. However, there\nhas been limited work to address the personalized screening for these different\ndiseases. In this work, we develop the \ufb01rst framework to construct screening\npolicies for a large class of disease models. The disease is modeled as a \ufb01nite state\nstochastic process with an absorbing disease state. The patient observes an external\ninformation process (for instance, self-examinations, discovering comorbidities,\netc.) which can trigger the patient to arrive at the clinician earlier than scheduled\nscreenings. The clinician carries out the tests; based on the test results and the\nexternal information it schedules the next arrival. Computing the exactly optimal\nscreening policy that balances the delay in the detection against the frequency of\nscreenings is computationally intractable; this paper provides a computationally\ntractable construction of an approximately optimal policy. As an illustration, we\nmake use of a large breast cancer data set. The constructed policy screens patients\nmore or less often according to their initial risk \u2013 it is personalized to the features\nof the patient \u2013 and according to the results of previous screens \u2013 it is personalized\nto the history of the patient. In comparison with existing clinical policies, the\nconstructed policy leads to large reductions (28-68%) in the number of screens\nperformed while achieving the same expected delays in disease detection.\n\nIntroduction\n\n1\nScreening plays an important role in the diagnosis and treatment of a wide variety of diseases,\nincluding cancer, cardiovascular disease, HIV, diabetes and many others by leading to early detection\nof disease [1]-[3]. For some diseases (e.g., breast cancer, pancreatic cancer), the bene\ufb01t of early\ndetection is enormous [4] [5]. Because screening \u2013 especially screening that requires invasive\nprocedures such as mammograms, CT scans, biopsies, angiograms, etc. \u2013 imposes \ufb01nancial and\nhealth costs on the patient and resource costs on society, good screening policies should trade off\nbene\ufb01t and cost [6]. The best screening policies should take into account that the trade-off between\nbene\ufb01t and cost should be different for different diseases \u2013 but also for different patients \u2013 patients\nwhose features suggest that they are at high risk should be screened more often; patients whose\nfeatures suggest that they are at low risk should be screened less often \u2013 and even different for the\nsame individual at different points in time, as the perceived risk for that patient changes. Thus the\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fbest screening policies should account for the disease type and be personalized to the features of the\npatient and to the history of the patient (including the history of screening) [32]. This paper develops\nthe \ufb01rst such personalized screening policies in a very general setting.\nA screening policy prescribes what tests should/should not be done and when. Developing person-\nalized screening policies that optimally balance the frequency of testing against the delay in the\ndetection of the disease is extremely dif\ufb01cult for a number of reasons. (1) The onset and progression\nof different diseases varies signi\ufb01cantly across the diseases. For instance, in [7] the development of\nbreast cancer is modeled as a stationary Markov process, in [36] the development of HIV is modeled\nusing a non-stationary survival process and, in [46] the development of colon cancer is modeled as a\nSemi-Markov process. The test outcomes observed over time may follow a non-stationary stochastic\nprocess that depends on the disease process upto that time and the features of the patient [35][36].\nExisting works on screening [7] [9] are restricted to Markov disease processes and stationary Markov\ntest outcome models, while this is not the case for many diseases and their test outcomes [10][35]-[37].\n(2) The cost of not screening is the delay in detection of disease, which is not known. Hence the\ndecision maker must act on the basis of beliefs about future disease states in addition to beliefs about\nthe current disease state. (3) Patients can arrive at the scheduled time but may also arrive earlier on\nthe basis of external information so the decision maker\u2019s beliefs must take this external information\ninto account. For instance, external information can be the development of lumps on breasts [25][26],\nor the development of a comorbidity [33][41]. (4) Given models of the progression of the disease and\nof the external information, solving for that policy is computationally intractable in general.\nThis paper addresses all of these problems. We provide a computationally effective procedure that\nsolves for an approximately optimal policy and we provide bounds for the approximation error (loss\nin performance) that arises from using the approximately optimal policy rather than the exactly\noptimal policy. Our procedure is applicable to many disease models such as dynamic survival models\n[11]-[13][36]-[37], \ufb01rst hitting time models [7][9][14]-[17].\nEvaluating a proposed personalized screening policy using observational data is challenging. Obser-\nvational data does not contain the counterfactuals: we cannot know what would have happened if a\npatient had been screened more often or an additional test had been performed. Instead, we follow an\nalternative route that has become standard in the literature [7]-[10]: we learn the disease progression\nmodel from the observational data and then evaluate the screening policy on the basis of the learned\nmodel. We also account for the fact that the disease model may be incorrectly estimated. We show\nthat if the estimation error and the approximation error are small, then the policy we construct is very\nclose to the policy for the correctly estimated model.\nIn this work, use of a large breast cancer data set to illustrate the proposed personalized screening\npolicy. We show that high risk patients are screened more often than low risk patients (personalization\nto the features of the patient) and that patients with bad test results are screened more often than\npatients with good test results (personalization to the dynamic history of the patient). The effect of\nthese personalizations is that, in comparison with existing clinical policies, the policy we construct\nleads to large reductions (28-68%) in screening while achieving the same expected delays in disease\ndetection. To illustrate the impact of the disease on the policy, we carry out a synthetic exercise across\ndiseases, one for which the delay cost is linear and one for which the delay cost is quadratic. We\nshow that the regime of operation (frequency of tests vs expected delay in detection) for the policies\nfor the two costs are signi\ufb01cantly different, thus highlighting the importance of choice of costs.\n\n2 Model and Problem Formulation\nTime Time is discrete and the time horizon is \ufb01nite; we write T = {1, ..., T} for the set of time slots.\nPatient Features Patients are distinguished by a (\ufb01xed) feature x. We assume that the features of a\npatient (age, sex, family history, etc.) are observable and that the set X of all patient features is \ufb01nite.\nDisease Model We model the disease in terms of the (true physiological) state, where the state space\nis S. The disease follows a \ufb01nite state stochastic process; S T is the space of state trajectories. The\nprobability distribution over trajectories depends on the patient\u2019s features; for (cid:126)s \u2208 S T , x \u2208 X we\nwrite P r((cid:126)s|x) for the probability that the state trajectory is (cid:126)s given that the patient\u2019s features are\nx. We distinguish one state D \u2208 S as the disease state; the disease state D is absorbing.1 Hence\n\n1The restriction to a single absorbing disease state is only for expositional convenience.\n\n2\n\n\fP r(s(t) = D, s(t(cid:48)) (cid:54)= D) = 0 for every time t and every time t(cid:48) > t. The true state is hidden/not\nobserved.2\nOur stochastic process model of disease encompasses many of the disease models in the literature,\nincluding discrete time survival models. The (discrete time) Cox Proportional Odds model [11], for\ninstance, is the particular case of our model in which there are two states (Healthy H and Disease\nD) and the probability distribution over state trajectories is determined from the hazard rates. To\nbe precise: if (cid:126)s is the state trajectory for which the disease state \ufb01rst occurs at time t0, so that\ns(t) = H for t < t0 and s(t) = D for t \u2265 t0, \u03bb(t|x) is the hazard at time t conditional on x, then\nP r((cid:126)s|x) = [1\u2212 \u03bb(1|x)]\u00b7\u00b7\u00b7 [1\u2212 \u03bb(t0 \u2212 1|x)][\u03bb(t0|x)] and P r((cid:126)s|x) = 0 for all trajectories not having\nthis form. Similar constructions show that other dynamic survival models [14]-[17] [10][37] also \ufb01t\nin the rubric of the general model presented here.3\nExternal Information The clinician performs tests that are informative about the patient\u2019s true\nstate; in addition, external information may also arrive (for instance, patient self-examines breasts for\nlumps, patient discovers comorbidities, etc.). The patient observes an external information process\nmodeled by a \ufb01nite state stochastic process with state space Y; the information at time t is Y (t) \u2208 Y\n(for instance, Y = {Lump, No Lump}). If the patient visits clinician at time t, then this external\ninformation Y (t) arrives to the clinician. Y (t) may be correlated with the patient\u2019s state trajectory\nthrough time t and the patient\u2019s features; we write P r(Y (t) = y|(cid:126)s(t), x) for the probability that\nthe external information at time t is y \u2208 Y, conditional on the state trajectory through time t and\nfeatures x. We assume that at each time t the external information Y (t) is independent of the past\nobservations conditional on the state trajectory through time t, (cid:126)s(t), and features x.\nArrival The patient visits the clinician at time t if either (a) the information process Y (t) exceeds\nsome threshold \u02dcy or (b) t is the time for the next recommended screening (determined in the Screening\nPolicies described below). The \ufb01rst visit of the patient to the hospital depends on the screening\npolicy and the patient\u2019s features (See the description below). If the patient visits the clinician at time\nt, the clinician performs a sequence of tests and observes the results. For simplicity of exposition,\nwe assume that the clinician performs only a single test, with a \ufb01nite set Z of outcomes. We write\nP r(Z(t) = z|(cid:126)s(t), x) as the probability that test performed at time t yields the result z, conditional\non the (unobserved) state trajectory and the patient\u2019s features. We assume that the current test result\nis independent of past test results, conditional on the state trajectory and patient features. We also\nassume that current test result is independent of the external information conditional on the state\ntrajectory through time t and the patient features. These assumptions are standard [7] [36]. We\nadopt the convention that z(t) = \u2205 if the patient does not visit the clinician at time t so that no test\nis performed. If the test outcome z \u2208 Z + \u2282 Z, then the patient is diagnosed to have the disease.\nWe assume that there are no false positives. If a patient is diagnosed to be in the disease state, then\nscreening ends and treatment begins.\nScreening Policies The history of a patient through time t consists of the trajectories of external\ninformation, test results and screening recommendations through time t. Write H(t) for the set of\nt=0 H(t) for the set of all histories. By convention H(0) consists\nonly of the empty history. A screening policy is a map \u03c0 : X \u00d7H \u2192 {1, . . . , T}\u222a{D} that speci\ufb01es,\nfor each feature x and history h either the next screening time t+ or the decision that the patient is in\nthe disease state D and so treatment should begin. A screening policy \u03c0 begins at time 0, when the\nhistory is empty, so \u03c0(x,\u2205) speci\ufb01es the \ufb01rst screening time for a patient with features x. (For riskier\npatients, screening should begin earlier.) Write \u03a0 for the space of all screening policies.\nScreening Cost We normalize so that the cost of each screening is 1. (We can easily generalize to\nthe more general setting in which the clinician decides from multiple tests [50], and different tests\nhave different costs.) The cost of screening is a proxy for some combination of the monetary cost, the\nresource cost and the health cost to the patient. We discount screening costs over time so if Ts is the\n\u03b4t, where \u03b4 \u2208 (0, 1).\n\nset of times at which the patient is screened then the screening cost is(cid:80)\n\nhistories through time t and H =(cid:83)T\n\nt\u2208Ts\n\n2For many diseases, it seems natural to identify states intermediate between Healthy and Disease. For\ninstance, because breast lumps [26] or colon polyps [9] that are found to be benign may become malignant, it\nseems natural to distinguish at least one Risky state, intermediate between the Healthy and Disease states.\n\n3We can encompass the possibility of competing risks (e.g., different kinds of heart failure) [13] simply by\n\nallowing for multiple absorbing states.\n\n3\n\n\fDelay Cost\nIf disease \ufb01rst occurs at time tD (the incidence time) but is detected only at time\ntd > tD (the detection time) then the patient incurs a delay cost C(td \u2212 tD; tD). If the disease\nis never detected the delay cost is C(T \u2212 tD; tD). We assume that the delay cost function C :\n{1, . . . , T} \u00d7 {1, . . . , T \u2212 1} \u2192 (0,\u221e) is increasing in the \ufb01rst argument (the lag in detection)\nand decreasing in the second argument (the incidence time). The cost of delay is 0 if disease never\noccurs or occurs only at time t = T . Note that as soon as the disease is detected screening ends\nand treatment begins; in particular, there is a single unique time of incidence and a single unique\ntime of detection. We allow for general delay costs because the impact of early/late detection on the\nprobability of survival/successful treatment is different for different diseases.\nExpected Costs If the patient features are x \u2208 X then every screening policy \u03c0 \u2208 \u03a0 induces a\nprobability distribution P r(\u00b7|x, \u03c0) on the space H(T ) of all histories through time T and in particular\ninduces probability distributions \u03c3 = P r(\u00b7|x, \u03c0) on the families Ts \u2282 2{1,...,T\u22121} of screening times\nand \u03b2 = P r((\u00b7,\u00b7)|x, \u03c0) on the pairs (tD, td) of incidence time and detection time. The expected\nscreening cost is E\u03c3\ngraphical model for the entire setup in the Appendix B of the Supplementary Materials.\nOptimal Screening Policy The objective of the screening policy is to minimize a weighted sum of\nthe screening cost and the delay cost; i.e. the optimal screening policy is de\ufb01ned by\n\n\u03b4t(cid:3) and the expected delay cost is E\u03b2\n(cid:110)\n\n(cid:2)C(td \u2212 tD, tD)(cid:3). We provide a\n\n(cid:2)(cid:80)\n\n(cid:111)\n\nt\u2208Ts\n\n(1)\n\narg min\n\u03c0\u2208\u03a0\n\n(1 \u2212 w) E\u03c3\n\n+ wE\u03b2[C(td \u2212 tD, tD)]\n\n\u03b4t(cid:105)\n\n(cid:104)(cid:88)\n\nt\u2208Ts\n\nBy de\ufb01nition, a belief is a function b : S T \u00d7 {0, 1} \u2192 [0, 1] such that(cid:80)\n\nThe weight w re\ufb02ects social/medical policy; for instance, w might be chosen to minimize cost subject\nto some accepted tolerance in delay (Further discussion on this is in Section 4).\nComment The standard decision theory methods [18]-[21] used in screening [7][9] cannot be used\nto solve the above problem. In standard POMDPs, the interval between two decision epochs (in this\ncase, screening times) is \ufb01xed exogenously; in standard POSMDPs, the time between two decision\nepochs is the sojourn time for the underlying core-state process. In our setting, the time between two\ndecision epochs depends on the action (follow-up date), the external information process, and the\nstate trajectory. In standard POMDPs (POSMDPs) the cost incurred in a decision epoch depend on\nthe current state, while in the above problem the delay cost depends on the state trajectory. Moreover,\nin our setting the disease state trajectory is not restricted to a Markovian or Semi-Markovian process.\n3 Proposed Approach\nBeliefs By a belief b we mean a probability distribution over the pairs consisting of state trajectories\nand a label l for the diagnosis: l = 1 if the patient has been diagnosed with the disease, l = 0 otherwise.\n(cid:126)s,l b((cid:126)s, l) = 1 but it is often\nconvenient to view a belief as a vector. Beliefs are updated using Bayesian updating every time there\nis a new observation (test outcomes, patient arrival, external information). Knowledge of beliefs\nwill be suf\ufb01cient to solve the optimization problem (1); see the Appendix C in the Supplementary\nMaterials. We write B for the space of all beliefs.\nBellman Equations To solve (1) we will formulate and solve the Bellman equations. To this end,\nwe begin by de\ufb01ning the various components of the Bellman equations. Fix a time t. The cost \u02dcC\nincurred at time t depends on what happens at that time: i) if the patient (with diagnosis status l = 0\nbefore the test) is tested and found to have acquired the disease, the cost is the sum of the cost of\ntesting and the cost of delay, ii) if the patient has the disease and is not detected, then the cost of delay\nis incurred in the time slot T , and iii) if the patient does not have the disease, then the cost incurred in\ntime slot t depends on whether a test was done in time slot t or not. We write these cases below.\n\n\uf8f1\uf8f2\uf8f3wC(t \u2212 tD; tD) + (1 \u2212 w)\u03b4tI(z (cid:54)= \u2205)\n\nwC(T \u2212 tD; tD)\n(1 \u2212 w)\u03b4tI(z (cid:54)= \u2205)\n\n\u02dcC((cid:126)s, t, z, l) =\n\nt \u2264 T, l = 0, z \u2208 Z +\nt = T, l = 0\notherwise\n\n(2)\n\nA recommendation plan \u03c4 : Z \u2192 T maps the observation z at the end of time slot t to the next\nscheduled follow-up time. Note that the recommendation plan is de\ufb01ned for a time t and is different\nthan the policy. Denote the probability distribution over the observations (test outcome z, duration to\nthe next arrival \u02dc\u03c4, and the external information at the next arrival time y) conditional on the current\n\nbelief b and the current recommendation plan \u03c4 by P r(z, y, \u02dc\u03c4(cid:12)(cid:12)b, \u03c4 , x). The belief b is updated to\n\n4\n\n\f\u02c6b in the next arrival time \u02dc\u03c4 based on the observations, current recommended plan and the current\nbeliefs using Bayesian updating as \u02c6b((cid:126)s, l) = P r((cid:126)s, l|b, \u03c4 , y, z, \u02dc\u03c4 , x).\nThe optimal values for the objective in (2) starting from different initial beliefs can be expressed in\nterms of a value function V : B \u00d7 {1, ..., T + 1} \u2192 R. The value function at time t when the patient\nis screened solves the Bellman equation:\n\n(cid:104)(cid:88)\n\n\u2212b((cid:126)s, l)P r(z|(cid:126)s, x)(cid:2) \u02dcC((cid:126)s, t, z, l)(cid:3) +\n\n(cid:88)\n\nP r(z, y, \u02dc\u03c4(cid:12)(cid:12)b, \u03c4 , x)V(cid:0)\u02c6b, t + \u02dc\u03c4(cid:1)(cid:105)\n\n(3)\n\nV (b, t) = max\n\n\u03c4\n\n(cid:126)s,l,z\n\nz,\u02dc\u03c4 ,y\n\nWe de\ufb01ne V (b, T + 1) = 0 for all beliefs. Note that the computation of the \ufb01rst term in the RHS\nof (3) has a worst case computation time of |S|T . Therefore, solving for exact V (b, T ) that satis\ufb01es\n(3) is computationally intractable when T is large. Next, we derive a useful property of the value\nfunction. (The proof of this and all other results are in the Appendix D-F of the Supplementary\nMaterial.).\nLemma 1 For every t, the value function V (b, t) is the maximum of a \ufb01nite family of functions that\nare linear in the beliefs b. In particular, the value function is convex and piecewise linear.\nThe above property was shown for POMDPs in [39], we use the same ideas to extend it to our setup.\n\n3.1 Constructing the Exactly Optimal Policy\nEvery linear function of beliefs is of the form \u03b1\u2217b for some vector \u03b1. (We view \u03b1, b as column\nvectors and write \u03b1\u2217 for the transpose.) Hence Lemma 1 tells us that there is a \ufb01nite set of vectors\n\u0393(t) such that V (b, t) = max\u03b1\u2208\u0393(t) \u03b1\u2217b. We refer to \u0393(t) as the set of alpha vectors. In view of\nLemma 1, to determine the value functions we need only determine the sets of alpha vectors. If we\nsubstitute the expression V (b, t) = max\u03b1\u2208\u0393(t) \u03b1\u2217b into (3), then we obtain a recursive expression\nfor \u0393(t) in terms of \u0393(t + 1). By de\ufb01nition, the value function at time T + 1 is identically 0 so\n\u0393(T + 1) = {0}, where 0 is the |S T \u00d7{0, 1}| dimensional zero vector, so we have an explicit starting\npoint for this recursive procedure. There is an optimal action associated with each alpha vector.\nThe action corresponding to the optimal alpha vector at a certain belief is the output of the optimal\naction given that belief, and so constructing the sets of alpha vectors yields the optimal policy; the\ndetails of the algorithm are in the Algorithm 3 in the Appendix A of the Supplementary Materials.\nUnfortunately, the algorithm to compute the sets of alpha vectors is computationally intractable (as\nexpected). We therefore propose an algorithm that is tractable to compute an approximately optimal\npolicy.\n\n3.2 Constructing the Approximately Optimal Policy\n\nPoint-Based Value Iteration (PBVI) approximation algorithms are known to work well for standard\nPOMDPs [18]. These algorithms rely on choosing a \ufb01nite set of belief vectors and constructing alpha\nvectors for these belief vectors and their success depends very much on the ef\ufb01cient construction of\nthe set of belief vectors. The standard approaches [18] for belief construction are not designed to\ncope with settings like ours when beliefs lie in a very high dimensional space; in our setup belief has\n|S T \u00d7 {0, 1}| dimensions. In Algorithm 1 (pseudo-code in the Appendix A of the Supplementary\nMaterials), we \ufb01rst construct a lower dimensional belief space by sampling trajectories that are more\nlikely to occur for the disease and then sampling the set of beliefs in the lower dimensional space that\nare likely to occur over the course of various screening policies. The key steps for Algorithm 1 are\n1. Sample typical physiological state trajectories Sample a set \u02dcS \u2282 S T of K physiological\ntrajectories from the distribution P r((cid:126)s|x).\n2. Construct the set of reachable belief vectors Say that a belief vector b2 is reachable from\nthe belief vector b1 if it can be derived by Bayesian updating on the basis of some underlying\nscreening policy. We construct the sets of belief vectors that can be reached under different screening\npolicies. For the \ufb01rst time slot, we start with a belief vector that lies in the space \u02dcS \u00d7 {0, 1} given as\nP r((cid:126)s|x)/P r( \u02dcS|x), \u2200(cid:126)s \u2208 \u02dcS, l = 0. For subsequent times, we select the beliefs that are encountered\nunder random exploration of the actions (recommendation of future test dates). In addition to using\nrandom exploration, we can choose actions determined from a set of policies such as the clinical\npolicies used in practice [27] [28] [47] to construct the set of reachable belief vectors.\n\n5\n\n\fDenote the set of belief vectors constructed at time t by \u00afB[t] and the set of all such beliefs as\n\u00afB = { \u00afB[t],\u2200t}. We carry out point-based value backups on these beliefs \u00afB (see Algorithm\n2 in the Appendix A of the Supplementary Materials), to construct the alpha vectors and thus\nthe approximately optimal policy. Henceforth, we refer to our approach (Algorithm 1 and 2) as\nDPSCREEN.\nComputational Complexity\n\nO(cid:0)T (B)2T 2K|Y||Z|(cid:1) steps, where B = maxt | \u00afB[t]| is the maximum over the number of\n\nthe policy requires\n\ncomputation of\n\nThe worst\n\ncase\n\npoints sampled by the Algorithm 1 for any time slot t. The complexity can be reduced by restricting\nthe space of actions; e.g. by bounding the amount of time allowed between successive screenings.\nMoreover, the proposed algorithms can be easily parallelized (many operations carried inside the\niterations in Algorithm 2 can be done parallel), thus signi\ufb01cantly reducing computation time.\nApproximation Error Because we only sample a \ufb01nite number of trajectories, the policy we\nconstruct is not optimal but we can bound the loss of performance in comparison to the exactly optimal\npolicy and hence justify the term \u201capproximately optimal policy.\u201d De\ufb01ne the approximation error to\nbe the difference between the value achieved by the exact optimal policy (solution to (1)) and the value\nachieved by the approximately optimal policy (output from Algorithm 2). As a measure of the density\nof sampling of the belief simplex we set \u2126( \u00afB) = \u03b6 maxt\u2208T maxB minb\u2208 \u00afB[t] ||b \u2212 b\n(cid:48)||1, where \u03b6 is\na constant that measures the maximum expected loss that can occur in one time slot. We make a few\nassumptions for the proposition to follow. The cost for delay is C(td \u2212 tD; tD) = c(td \u2212 tD)\u03b4tD,\nwhere c(d) is a convex function of d. The test outcome is accurate, i.e. no false positives and no\nfalse negatives. The maximum screening interval is bounded by W < T . The time horizon T is\nsuf\ufb01ciently large. We show that the loss of performance is bounded by the sampling density.\nProposition 1 The approximation error is bounded above by \u2126( \u00afB).\n3.3 Robustness\nEstimation Error To this point, it has been assumed that the model parameters are known. In prac-\ntice, the model parameters need to be estimated using the observational data. In the next section, we\nwill give a concrete example of how we estimate these parameters using observational data for breast\ncancer. Here we discuss the effect of error in estimation. Suppose that the model being estimated\n(cid:48) \u2208 M, where M is the space of all the possible models (model parametrizations)\n(true model) is m\nunder consideration. (We assume that the probability distribution of the physiological state transition,\nthe patient\u2019s self-observation outcomes, and the clinician\u2019s observation outcomes are continuous on\nM.) Write L = M \u00d7 B for the joint space of models and beliefs. Let the estimate of the model be\n\u02c6m. Let us assume that for every model in M the solution to (1) is unique. Therefore, we can de\ufb01ne\na mapping \u03c4 \u2217 : L \u00d7 Z \u00d7 T \u2192 T |Z|, where \u03c4 \u2217(l, z, t) is the optimal recommended screening time\nat l, at time t following z. For a \ufb01xed model m, \u03c4 \u2217((m, b), z, t) is the maximizer in (3).\nTheorem 1. There is a closed lower dimensional set E \u2282 L such that the function \u03c4 \u2217 is locally\nconstant on the complement of E.\nTheorem 1 implies that, with probability 1, if the model estimate \u02c6m and the true model m(cid:48) are\nsuf\ufb01ciently close, then the actions recommended by the exactly optimal policies for both models are\nidentical. Therefore, the impact of estimation error on the exactly optimal policy is minimal. However,\nwe construct approximately optimal policies. We can combine these conditions with Proposition 1 to\nsay that if the approximation error \u2126( \u00afB) goes to zero, then the approximately optimal policy (for \u02c6m)\n(cid:48)\nwill also converge to the exactly optimal policy for true model m\nPersonalization: Figure 1 provides a graphical representation of the way in which DPSCREEN\nis personalized to the patients. We consider three Patients. The disease model for each patient\nis given by the ex ante survival curve (the probability of not becoming diseased by a given time).\nAs shown in the graphs, the survival curves for Patients 1, 2 are the same; the survival curve for\nPatient 3 begins below the survival curve for Patients 1, 2 but is \ufb02atter and so eventually crosses\nthe survival curve for Patients 1, 2. All three patients are screened at date 1; for all three the test\noutcome is z = Low. Hence the belief (risk assessment) for all three patients decreases. As a result,\nPatients 1, 2 are scheduled for next screening at date 4 but Patient 3, who has a lower ex ante survival\nprobability, is scheduled for next screening at date 3. Thus, the policy is personalized to the ex ante\nrisk. However, at date 2, all three patients experience an external information shock which causes\nthem to be screened early. The test outcome for Patient 1 is z = Medium so Patient 1 is assessed\nto be at higher risk and is scheduled for next screening at date 3; the test outcome for Patient 2 is\n\n.\n\n6\n\n\fFigure 1: Illustration of dynamic personalization\n\nz = Low so Patient 2 is assessed to be at lower risk and is scheduled for next screening at date 5.\nThus the policy is personalized to the dynamic history. The test outcome for Patient 3 is z = Low\nand Patient 3\u2019s ex ante survival probability is higher so Patient 3\u2019s risk is assessed to be very low, and\nPatient 3 is scheduled for next screening at date 6. Thus the policy adjusts to time-varying model\nparameters.\n\n4\n\nIllustrative Experiments\n\nHere we demonstrate the effectiveness of our policy in a real setting: screening for breast cancer.\nDescription of the dataset: We use a de-identi\ufb01ed dataset (from Athena Health Network [22])\nof 45, 000 patients aged 60-65 who underwent screening for breast cancer. For most individuals\nwe have the following associated features: age, the number of family members with breast cancer,\nweight, etc. Each patient had at least one mammogram; some had several. (In total, there are 84,000\nmammograms in the dataset.) If the patients had a positive mammogram, a biopsy is carried out.\nFurther description of mammogram output is in the Appendix G of the Supplementary Materials.\nModel description We model the disease progression using a two-state Markov model: S = {H, D}\n(H = Healthy, D = Disease/Cancer). Given patient features x, the initial probability of cancer is\npin(x) and the probability of transition from the H to D is ptr(x). The external information Y\nis the size (perhaps 0) of a breast lump, based on the patient\u2019s own self-examination. In view\nof the universal growth law for tumor described in [23], we model Y (t) = g(t) + \u0001(t), where\ng(t) = (1 \u2212 e\u2212\u03b9(t\u2212ts))I(t > ts) is the size of the tumor and ts is the time at which patient actually\ndevelops cancer (the lump exists), \u0001(t) is a zero mean white noise process with variance \u03c32 and I() is\nthe indicator function. If the lump size Y exceeds the threshold \u02dcy, then the patient visits the clinician,\nwhere tests are carried out. The set of test outcomes is Z = {\u2205, 1, 2, 3}, where z = \u2205 when no test is\ndone, z = 1 when the mammogram is negative and no biopsy is done, z = 2 when the mammogram\nis positive and the biopsy is negative, z = 3 when both mammogram and biopsy is positive.\nModel Estimation We use the speci\ufb01city and sensitivity for the mammogram from [7]. Each patient\nhas a different (initial) risk for developing cancer; we compute the risk scores using the Gail model\n[24], which we use as the feature x. We assumed pin(x) and ptran(x) are logistic functions of x. We\nuse standard Markov Chain Monte Carlo methods to estimate these functions pin(x) and ptran(x)\n(further details in the Appendix G of the Supplementary Materials). We assume that each woman has\none self-examination per month [25] [26]. We use the value \u03b9 = 0.9 as stated in [23]. We estimate\nthe parameters for the self-examinations \u03c3 = 0.43 and \u02dcy = 1 on the basis of the values of sensitivity\nand speci\ufb01city for the self-examination from the literature [43]. In the comparisons to follow, we\n\n7\n\n\ud835\udc67=\ud835\udc3f\ud835\udc5c\ud835\udc64TimeBelief\tDiseasePatient\t112345\ud835\udc67=\ud835\udc3f\ud835\udc5c\ud835\udc64TimeBelief\tDisease\ud835\udc67=\ud835\udc3f\ud835\udc5c\ud835\udc64\ud835\udc61\u2019=512345\ud835\udc67=\ud835\udc3f\ud835\udc5c\ud835\udc64Time\tBelief\tDisease\ud835\udc67=\ud835\udc3f\ud835\udc5c\ud835\udc6412345\ud835\udc61\u2019:prescribed\tnext\tarrival\ttime666\ud835\udc61\u2019\t=4\t\ud835\udc61\u2019\t=4\t\ud835\udc61\u2019\t=3\t\ud835\udc61\u2019\t=3\t\ud835\udc61\u2019\t=6\tPatient\t1\tand\t2:\tPersonalization\tthrough\thistoriesSame\tfeatures,\tdifferent\thistories\t\u2192\tdifferent\tscreeningPatient\t2\tand\t3:\tPersonalization\tthrough\tfeaturesSame\thistory,\tdifferent\thazard\trates\t\u2192\tdifferent\tscreening\ud835\udc3f\ud835\udc40\ud835\udc3b\ud835\udc3f\ud835\udc40\ud835\udc3b\ud835\udc3f\ud835\udc40\ud835\udc3b\ud835\udc67:test\toutcomes\ud835\udc67=\ud835\udc40\ud835\udc52\ud835\udc51\ud835\udc56\ud835\udc62\ud835\udc5aSurvival\tprobabilitySurvival\tprobabilitySurvival\tprobabilityPatient\t2Patient\t3\fwill also analyze the setting when there are no self-examinations. We divide the population into two\nrisk groups; the Low risk group consists of patients whose prior estimated risk of developing cancer\nwithin \ufb01ve years is less than 5%; the High risk group consists of patients whose prior estimated risk\nexceeds 5%.\nPerformance Metrics, Objective and Benchmarks: Our objective is to minimize the number\nof screenings subject to a constraint on expected delay cost. We assume the delay cost is linear:\nC(td \u2212 tD, tD) = td \u2212 tD. To derive the solution to this constrained problem from construction,\nwhich minimizes the weighted sum of screening cost and delay cost, we solve the weighted problem\nfor some weight w, and then tune w to select the policy that minimizes the number of screenings\nsubject to a constraint on expected delay cost. For comparison purposes, we take the constraint\non expected delay cost to be the expected delay that arises from current clinical practice (annual\nscreening in the US [27][28], biennial screening in some other countries [29]). (Because our objective\nis to minimize the number of screenings, we take the cost of each screening to be 1, whether or not a\nbiopsy is performed.)\nComment At this point, we remind that existing frameworks [7][9][10] cannot be used to solve for\nthe optimal screening policy in the above setup because: i) the costs incurred (delay) depends on\nthe state trajectory and not just the current state, and ii) the lump growth model and the patient\u2019s\nself-examination of the lump is not easy to incorporate in these works.\nComparisons with clinical screening policies: We compare our constructed policies (for the two\ngroups), with and without self-examination, in terms of three metrics: i) E[N|R]: the expected\nnumber of tests per year, conditional on the risk group; ii) E[\u2206|R]: the expected delay, conditional\non the risk group; iii) E[\u2206|R, D]: the expected delay, conditional on the risk group and the patient\nactually developing cancer. Because E[\u2206|R] is the expected unconditional delay, it accounts for\npatients who do not develop cancer as well as for patients who do have cancer; because most patients\ndo not develop cancer, E[\u2206|R] is small. We show the comparisons with the annual policies in Table 1;\nwe show the comparisons with biennial screening in the Appendix G of the Supplementary Materials.\nIn Table 1 we compare the performance of DPSCREEN (with and without self-examination) for Low\nand High risk groups against the current clinical policy of annual screening. For both risk groups,\nthe proposed policy achieves approximately the same expected delay as the benchmark policy while\ndoing many fewer tests (in expectation). With self-examinations, the expected reduction in number of\nscreens is 57-68% (depending on risk group); even without self-examinations, the expected reduction\nin number of screens is 28-45% percent (depending on risk group).\nIn Table 2 we contrast the difference in DPSCREEN across the two risk groups. To keep the\ncomparison fair, we \ufb01x the tolerance in the delay to a \ufb01xed value. The proposed policy is personalized\nas it recommends signi\ufb01cantly fewer tests to the low risk patients in contrast to the high risk patients.\nImpact of the type of disease: We have so\nfar considered breast cancer as an example\nand assumed linear delay costs. For some dis-\neases (such as Pancreatic cancer [30][5]) the\nsurvival probability decreases very quickly with\nthe delay in detection and therefore it might\nbe reasonable to assume a cost of delay that\nis strictly convex (such as quadratic costs) in\ndelay time for some disease. In Figure 2, we\nshow that for a \ufb01xed risk group and for the same\nweights the policy constructed using quadratic\ncosts is much more aggressive in testing. More-\nover, the regime of operation of the policy (the\npoints achieved by the policy in the 2-D plane\nE[N|R, Cost] vs E[\u2206|R, Cost]) can vary a lot depending on the choice of cost function even though\nthe same weights are used. Therefore, the cost should be chosen based on the disease.\n\nFigure 2: Impact of the type of disease\n\n5 Related Works\nIn Section 2, following the equation (1), we compared our methods with frameworks to some general\nframeworks in decision theory [18]-[21]. Next, we compare with other relevant works.\n\n8\n\nE[\"jR;Cost]months00.20.40.60.81E[NjR;Cost]00.511.522.53Highrisk,LinearcostHighrisk,QuadraticcostLowrisk,LinearcostLowrisk,Quadraticcostw=0.5w=0.3w=0.3w=0.9w=0.5w=0.9w=0.3w=0.5w=0.9w=0.3w=0.9w=0.5\fTable 1: Comparison of the proposed policy with annual screening for both high and low risk group.\n\nRisk\nGroup\nLow\nHigh\n\nMetrics\nE[N|R], E[\u2206|R], E[\u2206|R, D]\nE[N|R], E[\u2206|R], E[\u2206|R, D]\n\nDPSCREEN with\nself-examination\n0.32, 0.23, 9.2\n0.43, 0.50, 6.7\n\nDPSCREEN w/o\nself-examination\n0.55, 0.23, 9.2\n0.72, 0.52,7.07\n\nAnnual\n\n1, 0.24, 9.6\n1, 0.52, 7.07\n\nTable 2: Comparison of the proposed policies across different risk groups\n\nRisk\nGroup\n\nLow\nHigh\n\nDPSCREEN with self-examination DPSCREEN w/o self-examination\nE[N|R], E[\u2206|R], E[\u2206|R, D]\n0.12, 0.33, 13.7\n0.80, 0.35, 4.73\n\nE[N|R], E[\u2206|R], E[\u2206|R, D]\n0.32, 0.33, 13.7\n1.09, 0.35, 4.73\n\nScreening frameworks for different diseases in operations research: Many works have focused\non optimizing population-based screening schedules, which are not personalized (See [42] and\nreferences therein). In [7] [9] the authors develop personalized POMDP based screening models.\nThe underlying disease evolution (breast and colon cancer) is assumed to follow a Markov process.\nExternal information process such as self-exams and the test outcomes over time are assumed to\nfollow a stationary i.i.d process given the disease process. In [10] authors develop personalized\nscreening models based on principles of Bayesian design for maximizing information gain (based\non [40]). The underlying disease model (cardiac disease) is a dynamic (two-state) survival model\nand the cost of misdetection is a constant and does not depend on the delay. The test outcomes are\nmodeled using generalized linear mixed effects models, and there is no external information process.\nTo summarize, all of the above methods rely on very speci\ufb01c models for their disease, test outcomes,\nand external information, while our method imposes much less restrictions on the same.\nScreening frameworks for different diseases in medical literature: The Medical research litera-\nture on screening (e.g., Cancer Intervention and Surveillance Modelling Network, US preventive\nservices task force, etc.) relies on stochastic simulation based methods: \ufb01x a disease model and a\nset of screening policies to be compared; for each policy in the set, simulate outcome paths from\nthe model; compare across the set of policies [44]-[48]. The clinical guidelines for screening issued\nby the US preventive services task force [47][49] for colon cancer cancers are created based on the\nMISCAN-COLON [46] model for colon cancer. Simulations were carried out to compare different\nscreening policies suggested by experts for that speci\ufb01c disease model- MISCAN-COLON. This\napproach allows more realistically complex models but it only compares a \ufb01xed set of policies, all of\nwhich may be far from optimal.\nControlled Sensing: In controlled sensing [21][34][38] the problem of sensor scheduling requires\ndeciding which sensor to use and when; this problem is similar the personalized screening problem\nstudied here. In these works [21][34][38], the main focus is to exploit (or derive) structural properties\nof the process being sensed and the cost functions such that the exactly optimal sensing schedule is\neasy to characterize and compute. The structural assumptions such as the process that is sampled is\nstationary and Markov make these works less suited for personalized screening.\n\n6 Conclusion\n\nIn this work, we develop a novel methodology for constructing personalized screening policies that\nbalance the cost of screening against the cost of delay in detection of disease. The disease is modeled\nas an arbitrary \ufb01nite state stochastic process with an absorbing disease state. Our method incorporates\nthe possibility of external information, such as self-examination or discovery of co-morbidities, that\nmay trigger arrival of the patient to the clinician in advance of a scheduled screening appointment.\nWe use breast cancer data to develop the disease model. In comparison with current clinical policies,\nour personalized screening policies reduce the number of screenings performed while maintaining\nthe same delay in detection of disease.\n\n9\n\n\f7 Acknowledgements\n\nThis work was supported by the Of\ufb01ce of Naval Research (ONR) and the National Science Foundation\n(NSF) (Grant number: 1533983 and Grant number: 1407712).\n\nReferences\n\n[1] Siu, A. L. (2016). Screening for breast cancer: US Preventive Services Task Force recommenda-\ntion statement. Annals of internal medicine, 164(4), pp.279-296.\n[2] Canto, M. et.al. (2013). International Cancer of the Pancreas Screening (CAPS) Consortium\nsummit on the management of patients with increased risk for familial pancreatic cancer, Gut, 62(3),\npp.339-347.\n[3] Wilson, J. et.al. (1968). Principles and practice of screening for disease.\n[4] Jemal, A. et.al. (2010). Cancer statistics, CA: a cancer journal for clinicians, 60(5), pp.277-300.\n[5] Rulyak, S. J. et.al. (2003). 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Annals of Internal\nMedicine, 149(9), pp.638-658.\n[50] Alaa, A.M. et.al. (2016). Con\ufb01dentCare: A Clinical Decision Support System for Personalized\nBreast Cancer Screening, accepted and to appear in IEEE Transactions on Multimedia-Special Issue\non Multimedia-based Healthcare, 18(10), pp.1942-1955.\n\n12\n\n\f", "award": [], "sourceid": 870, "authors": [{"given_name": "Kartik", "family_name": "Ahuja", "institution": "University of California, Los Angeles"}, {"given_name": "William", "family_name": "Zame", "institution": "UCLA"}, {"given_name": "Mihaela", "family_name": "van der Schaar", "institution": "UCLA and Oxford University"}]}