A new formalism for temporal modeling in medical decision-support systemsConstantin F. Aliferis, M.D., M.Sc., and Gregory F. Cooper, M.D., Ph.D.

Section ofMedical Informatics & Intelligent Systems ProgramUniversity ofPittsburgh, Pittsburgh PA

ABSTRACT

We present a new mathematical formalism, which wecall modifiable temporal beliefnetworks (MTBNs) thatextends the concept ofan ordinary beliefnetwork (BN)to incorporate a dynamic causal structure and explicittemporal semantics. An important feature ofMTBNs isthat they allow portions ofthe model to be abstract andportions of it to be temporally explicit. We show howthis property can lead to substantial knowledgeacquisition and computational complexity savings. Inaddition to temporal modeling, the language ofMTBNscan be an important analytical tool, as well as temporallanguagefor causal discovery.

INTRODUCTION

Representing and reasoning about temporal concepts isan essential part of medical problem solving. All fi'vemain medical tasks (prevention, diagnosis, therapeuticplanning/intervention, prognosis, and medicaldiscovery) involve time modeling and inference [1].A great number of medical decision-support systems

(MDSSs) have been developed throughout the years, butfew of them have incorporated explicitly temporalaspects of their respective domains [2,3]. In general,modeling time is considered to be one of the greatestchallenges in developing MDSSs. We believe that amajor reason for this perceived difficulty is the lack ofmodels for temporal reasoning that simultaneously: (a)are of sufficiently general applicability, and (b) canhelp developers deal with pragmatic constraints such asknowledge availability, computational tractability andease of development.

Accepting this premise means that the "timeproblem" in MDSSs must be attacked at the model levelfirst. In other words, we need to develop methods thatsupport the creation of efficient, flexible andexpressive temporal medical reasoning systems. In thispaper we present a formalism (and sunmarze theassociated theory, algorithms and programs) that canserve as a basis for the conceptual modeling andpractical implementation of time in MDSSs. Ourformalism (which we call modifiable temporal beliefnetworks - MTBNs) can also be used as an analytic tooland a machine-learning model language.

1. MEDICAL TIME-MODELING DESIDERATA

We have developed a number of requirements for anidealized temporal representation and reasoning

architecture for MDSSs, based on the body of work intheoretical artificial intelligence dealing with generaltemporal reasoning [4,5], the literature about systems(and respective problem domains) developed over theyears in medical informatics [1,6], and the analysis of alarge-scale MDSS's temporal concepts and the analysisof a number of real-life patients medical records [3]. Wepropose that an idealized temporal representation andreasoning MDSS architecture should provide:L Expressive temporal representation (rich

ontologies, handling of uncertainty, ability to beintegrated with other fonnalisms, modeling ofcausality).IL Effective temporal reasoning (soundness /

completeness and computational tractability).m Flexible and efficient modeling (availability ofknowledge for model specification, extensibility /shareability-reusability, model specification ease).IV. Formal foundation.A much more detailed exposition of the desiderata

can be found in [6]. In the following section weintroduce a formalism designed specifically to meet theabove requirements.

2. MODIFIABLE TEMPORAL BELIEF NETWORKS

2.1. Intuitive description of belief networksWe chose to base our fonnalism on a probability-basedscheme because probabilistic formalisms can handleuncertainty and decision-theoretic reasoning, while theyare not hindered by the undecidability of first-order-logics [6]. The current state-of-the-art formalism forprobabilistic reasoning is the belief-network model (BN)[7]. BNs are mathematical models combining agraphical representation of probabilistic dependenciesand independences among random variables (in theform of a directed acyclic graph - DAG), with a set ofconditional probability distributions (cpd). A BNimplicitly captures a joint probability distribution (ipd)over the variables. BNs have been used to performtemporal representation and reasoning (TRR) tasksbased on the following basic scheme: the stucture of aBN is replicated n times, each corresponding to one ofndiscrete time points (i.e., they have a temporal range ofn time points). Arcs within a time slice (i.e., onestructure copy) are considered to be instantaneous,while arcs between time slices are time-lagged (i.e.,delayed). We will call a BN used in this manner atemporal BN (TBN) [8,9]. Figure lb presents a TBNcorresponding to a three-fold replication of the BN inFigure la. The basic BN structure that gets replicated

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involves variables A, B and the arc between them. ArcAl to BJ in Figure lb is considered instantaneous,while arcAI to B2 is time-lagged.

2.2.1. Problems arising when BNs are used fortime modelingUnfortunately TBNs present the following problems:(a) Cumbersome model specification andpresentation. Hundreds or even thousands of nodes andconditional probability distributions may have to bedefined one by one, even if they are the same or varysystematically.

63 s s s e3L

d

Figure 1. Replicating a BN structure (a) to form aTBN (b), and problems arising with TBNs (c-d).

(b) Predefmed temporal range of interest. Ideally wewould like to vary the temporal range as a modelparameter. TBNs do not allow us to do so.

(c) The processes modeled are either static orevolving over time in a way that is implicit. That is,the causal processes' temporal evolution is "buried" inthe oftentimes subtle differences between slices, or even

worse, in the conditional probability distributionsassociated with the graph's variables. For example infigure lc, it is not at all clear how the causal structureevolves.(d) Non-integrated temporal/causal semantics.Typically (in their general form) TBNs are used withstandard BN inference algorithms [9]. Thus knowledgeabout time can not be exploited easily to preventserious inconsistencles from happening, or to enhancethe reasoning efficiency. This problem is most serious

in cases where: (i) a dynamic causal structure ismodeled and (ii) a model contains both temporallyindexed and abstracted variables (see Sections 2.2.2 and2.5. for a discussion of the interpretation andsignificance of such variables). An example of suchproblems follows: in Figure Id, variables C, D are nottemporally indexed (i.e., they are abstract), whilevariables Al to An are indexed (i.e., time tagged).Although the graph does not violate the requirement fora DAG, it does violate the trivial requirement for causalprecedence of the cause relative to the effect (an obviousinconsistency in this example is that D should come

after C (because it is causally influenced by C), but alsobefore C, since D precedes (Al and thus) A3, which is acause of C.

2.2.2. MTBN solutions to TBN problemsMTBNs address the problems with TBNs as follows:(a) The MTBN models have two forms: a condensed(specification) and a deployed (run-time) form. Thecondensed form allows a concise description of thedomain while the deployed form is the networkproduced after the structure replication process iscarried out.(b) The temporal range can be dynamically modified,as long as the condensed model has been specified.(c) The most radical departure ofMTBNs from TBNs isthe ability to model explicitly how the generating(i.e., causal process) structure evolves with time. Thisis achieved via the introduction of two new types ofvariables: causal mechanism (arc) variables andassociated time-lag variables. Interactions of thesevariables with ordinary domain variables is allowed inorder to describe complex ways in which diseases andother processes evolve over time. Variables can alsohave arcs to themselves to model persistence over time.(d) MTBNs utilize a well-specified model of time, andassociate with it all variables in a clear andunambiguous manner. Temporal and causal semanticsare well-defined and integrated to the formalism. Bothtemporally indexed variables and non-indexedvariables are allowed. Non-indexed variablescorrespond to variables that are abstractions over

indexed variables, variables that we do not know or

wish to model their exact temporal location, or finallyexternal models (utilized as "procedural calls" fromwithin the MTBN model). Model specification andinference takes into account this temporal semantics.The causal semantics described in [10] are also obeyed.In this paper we discuss single-granularity (i.e., smallesttemporal duration) MTBNs only. The formalspecification ofMTBNs can be found in [11].

Figure 2 shows an example of a MTBN involvingvariables A and B. The condensed form is presented onthe left part of Figure 2, while the deployed form is onthe right. The two variables are causing each other (in a

simple feedback loop).- The numbers in the squaresadjacent to arcs denote time lags.

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a b

Figure 2. A MTBN in compact form (left) anddeployed form (right).

Figure 3 shows the MTBN corresponding to Figure lc.Contrary to the TBN of Figure lc, the MTBNrepresentation makes clear thatA causes B and that thiscausal effect is delayed by L time units and that thevalue of time-lag variable L is determined by the valuesof C.

Figure 3. The MTBN corresponding to Figure lc.2.3. Properties of MTBNs

The following main properties hold for MTBNs [11]:Proposition 2.3.1. MTBNs can be factored according tothe conditional probability distributions of eachvariable's Xi values dependent on its active parents'values. Result 2.3.1. makes it possible to retrieve thejpdfrom the cpds.Proposition 2.3.2. An arbitrary MTBN MI can beconverted to a standard BN Bl such that Bl capturesthe same joint probability distribution as Ml. Thus any

MTBN can be implemented as a BN alternatively. Wewill explore the resulting trade-offs in more detail inSections 2.4 and 2.5.Corollary 2.3.3. Valid MEBNs are defined from just a

temporal graph, temporal range and the conditionalprobability specifications (i.e., we do not have toexplicitly define the full jpd).Corollary 2.3.4. For every joint probability distribution,there exists a MITBN that captures it.Coroilary 2.3.5. Inference with MTBNs can be carriedout with any standard BN inference algorithm and isNP-hard. This result allows us to use any standard BNalgorithm to do inference with MTBNs. It also suggeststhat in the worst case the computational timecomplexity ofMTBNs is intractable. In Secton 2.5 we

will show how to use partial abstractions to cope withthis problem.Proposition 2.3.6. MTBNs are more expressive thancausal BNs. This results follows since MTBNs can

express graphically more precise dependences.

Proposition 2.3.7. In some instances, MTBNs can bemore efficient than BNs, by exploiting knowledge aboutthe causal process they model.

2.4. Inference with MTBNsAs mentioned in Section 2.3., we can carry outinference with MTBNs by converting them to a BN andusing any BN inference algorithm. This approach hasadvantages and drawbacks. Some of the existing BNinference algorithms are fast and convenient from thedevelopers' standpoint. They do not however help us

with the problems outlined in Section 2.2.1. For thisreason we have developed a stochastic simulationinference algorithm that is a variation on the logicsampling algorithm (LS) which we call temporal LS(LST). We have implemented LST in a program calledHARMONY. This program supports a number offeatures including temporally explicit and abstractedvariables in the same model (see Sections 2.2.2, and2.5), case simulation from any MTBN model, andcontinuous variables. Arbitrary code can implement thecpds, allowing external models to be integrated with a

MTBN. HARMONY allows us to specify as part of a

query the causal manipulations of variables (if any).Finally, LST and HARMONY perform simulation onlyon a per temporal slice basis. This allows LST toperform inference with very big models using limitedamounts of memory.

2.5.Using partial abstractions to satisfy practicalmodeling constraints

In this section we show how MTBNs can be a

flexible, expressive and efficient formalism for creatingMDSS models with well-defined temporal aspects. Weuse an example from the domain of endocrinology tocreate multiple MBN and TBN models. Weimplemented models both as MBNs and TBNs. Weperformed inference using HARMONY (for MTBNs),and LS, as well as the recursive decomposition exactalgorithm (for the TBNs). We compare the 8 modelsagainst each other along the following dimensions:Computational tractability (measured by the numberof variables in the model), query completeness(measured by the number of joint probabilities of allinvolved variables that can be derived from eachmodel), model expressiveness and temporal semanticclarity (assessed qualitatively as satisfactory or not),availability of knowledge necessary to instantiate themodel (assessed qualitatively), specification ease

(measured by the number of required conditionaldistributions), and model behavior specification ease

(ease of specifying the high-level model behavior fromthe low-level causal mechanisms - assessedqualitatively). For all qualitative assessments we

provide justifications in the descrinptions of the models.Table 1 summarizes these properties for all 8

different models. Our models capture some basicregulatory features of thyroid function. The productionofthe main thyroid hormone, T4, is stimulated by the

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A W1

Table 1. Comparison of time modeling properties of 8 alternative models for the TRH stimulation example. Thefirst three columns summarze quantitatively assessed properties, while the last three columns summarizequalitatively assessed p s (for explanations see text). Calculations omit variables with constant values.

COMPLEXITy QUERY SPECFICAIION MODEL AVALABITY | MODEL

(NUMBER OF COMPLETENESS EASE EXPRESSIVENESS OF REQUIRED BEHAVIOR

VARIABLES) (NUMBER OF JOINT (NUMBER OF & TEMPORAL KNOWLEDGE SPECIFCATION

PROBABILITIES) CONDITIONAL SEMANTIC EASE

PROBABILITY CLARITYDISTRIBUTIONS)

MODEL MTBN TBN MTBN TBN MTBN TBN MTBN TBN MTBN TBN MTBN TBN

FULLY 11*106 i1*i06 _(3e+1084) '-(3e+1O84) 7 77*106 + - -1-1-1 -EXPLICrTr _

FULLY 3 3 20 20 3 3 + + + + + +

HIYBRIID 1 22 22 10*365 10*365 6 22 + - + + + +

HYBRD2 1 0 10 19.4*103 19.4*103 6 10 + - + + + +

pituitary hormone TSH. TSH production is suppressedby high levels of T4, and stimulated by TRH. Afunctional thyroid adenoma (ETA) is a benign tumorthat produces T4 independently of the rest of the thyroidtissue. This in effect disrupts the feedback loop betweenTSH and T4, resulting in high concentrations of T4 andlow TSH. A paicular cause for thyroid adenoma isexposure to X-radiation. The abnormality in the T4regulation can be detected by a TRH stimulation testthat involves measuring TSH levels over a period oftime after administering TRH to the patient. In anormal person the observed pattern is bell-shaped,while in the presence of FTA, it is flat and closer to 0[12]. Ideally we would like to ask questions of the tpe"In the context of a TRH stimulation test, and giventhat the values ofTSH at times I to n are known, whatis the probability ofa ETA in this patient?".

A first approach towards modeling this problem is

Figure 4. Functional thyroid adenoma interfereswith TSH-T4 loop (fully explicit).

Here we model all variables as temporally explicit (i.e.,indexed). The problem is that the temporal granularitiesof interest for different variables are vastly different.For instance the TSH and T4 interaction is on the orderof seconds or minutes, while X radiation (X) and FTA(D) are on the order of months or years. To incorporateX and D into our model we have to convert them to thesmallest granularity of interest (i.e., minutes) whichleads to an computationally unmgeable model forboth the MTBN and TBN models (table 1, row 1). Also

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we note that it is easy to specify such a model in MTBNform but the available knowledge does not support sucha specification. Finally, the number of questions givenobserved evidence in both models is astronomical, butmost ofthem are ofno clinical interest. All calculationsabout this model in table 1 assume a time lag between Xand D that takes values between 1 and 10 years.

Another approach, shown in Figure 5, involvesabstracting X and D, and replacing variables TRH,TSH, and T4 by an abstraction variable TRHSTMtULATION RESPONSE PATTERN (taking asvalues: normal, abnormal).

Figure 5. Functional thyroid adenoma interfereswith TSH-T4 loop (fully implicit).

This is the method used in most current MDSSs(implicit time modeling [31). This approach yieldseasily specified models for both formalisms, and it iscomputationally tractable. Unfortunately two problemsremain: first these models can answer a very restrictednumber of queries of interest, and second the evidenceand questions comprising these queries are temporallyimplicit. Therefore, in a patient case someone mustabstract over the values of TSH and map theabstraction to one of the values of the TRHSTIMULATION RESPONSE variable.

Both of these difficulties are addressed when wemodel this problem using the hybridl model, which isa combination of temporally explicit (indexed) andabstracted variables (Figure 6). In particular, we modelTSH TRH, T4 as explicit variables so that we canobserve their temporal patterns and reason about them,

while we are "collapsing" D to the present time and Xto involve a small number of past periods, eachspanning years.

Figure 6. Functional thyroid adenoma interfereswith TSH-T4 loop (hybrid] model).

The models are tractable and easy to specify. The rangeof TSH, TRH, and T4 is constrained to the duration of aTRH stimulation test (-30 minutes). Unfortunately, formany domains it is questionable whether we could haveaccess to accurate conditional probability distributionsfor such a model. An equally serious problem isspecifying the probabilities so that the high levelbehavior of the model is consistent with knownempirical patterns of TSH given TRH manipulations(because it requires repeated tuning of the model so thatthe correct system behavior emerges from thespecification of low-level local variable interactions).

Assuming that the difficulties associated with thetype of hybrid model described in the previousparagraph can not be overcome, MTBNs give us theflexibility to change our modeling approach and buildthe model of Figure 7 (hybrid2 model).

T~ ~~TH

RESPONSE PTERN

Figure 7. Functional thyroid adenoma interfereswith TSH-T4 loop (hybrid2 model).

This model instead of involving TSH, TRH, and T4,involves only TSH measurements after TRHstimulation. The model consists of two parts: anabstracted reasoning part (on the left) essentiallyidentical to the one of Figure 5, and an explicit one, (onthe right) corresponding to an abstraction function PRover values of TSHi (i.e., TSH at time I). The output ofPR is assigned to (i.e., determines) the value of the"TRH STIMULATION RESPONSE PATTERN"variable. For details on integrating external models toMTBNs see [6].

DISCUSSION

In addition to the modeling services offered by MTBNsthat were presented in sections 2.2. to 2.5, MTBNs also

allow us to reason spatio-temporally, and to buildtemporal entities such as facts /events /intervals andreason with them , or reference them in multiple ways[6]. Unique features of MTBNs that can not be found intemporal variants of BNs such as TBNs, and actionnetworks [6], include lag and arc variables, integrationof explicit /implicit time modeling, and a more conciserepresentation. In [13] we also explore the use ofMTBNs to study temporal abstractions and temporalcausal discovery methods. Our current work is focusedon: (a) extensions to MTBNs to handle simultaneouslymultiple-granularity models, and (b) applying andevaluating the methods presented here.

ACKNOWLEDGEMENTS

We thank Prof. B. Buchanan, Prof. M. Pollack, Dr.R. Ambrosino, Dr. M. Wagner, Dr. Bonnie Joe, and EdLomax, for providing useful feedback on the ideas ofthis study. This work was supported by grant BES-9315428 from the National Science Foundation and bygrant LM05291-02 from the National Library ofMedicine.

References

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