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A Fuzzy Approach to Temporal Model-Based Diagnosis

for Intensive Care Units

Jos´ e Palma1and Jos´ e M. Ju´ arez1and Manuel Campos1and Roque Mar´ ın1

Abstract.

In the Intensive Care Unit (ICU) domain, temporal evolution of

diseases and patients’ contextual information are critical pieces of

knowledge that must be considered in the design of a diagnosis task.

The uncertainty inherent in the description of temporal information

associated to diseases requires a temporal representation and reason-

ing framework. This temporal framework has to be flexible enough

to facilitate its integration in a behavioral model. This paper pro-

poses a Temporal Behavioral Model (TBM) that makes this integ-

ration possible and permits the specification of contextual inform-

ation (that may modify the TBM). A diagnosis process is also pro-

posed.ThisprocessusestemporalmodelbasedtechniquesandFuzzy

Temporal Constraints Networks (FTCN) as the underlying temporal

framework. Some heuristics, which affect not only the temporal reas-

oning dimension but also the causal, have been designed in order to

compute solutions efficiently.

1INTRODUCTION

Physicians at intensive care units (ICUs) have to deal with an over-

whelming amount of data provided not only by on-line monitoring

but also collected from patients’ records (e.g., laboratory results),

which are, in most cases, collected manually at different time in-

stants. In order to provide efficient decision support systems and

medical research tools in the ICU domain, it is necessary to integrate

and analyse the information provided from these different sources.

These tools are focused on the analysis of patient’s evolution over

time. This kind of analysis may provide valuable information for

making decisions about patient treatments and for improving clin-

ical guidelines.

A good analysis of patient evolutions lies in an efficient dia-

gnosis process. The use of deep causal models together with model-

based diagnosis techniques has proved its efficiency in the develop-

ment of intelligent diagnosis systems [17]. Moreover, the ICUs do-

main reveals the importance of the temporal component modelling

in capturing the temporal information associated to patient evolution

[13]. However, the inclusion of temporal representation techniques

in MBD has increased the complexity of the diagnosis process. Dif-

ferent formalisms have been proposed to represent time in MBD,

ranging from totally qualitative approaches [11], based on Allen’s

interval logic [1], to totally quantitative approaches [9, 15, 16]. A

serious attempt to provide a general framework for temporal MDB

can be found in [3, 8], which presents a general characterization of

temporal MDB at knowledge level.

Our goal, therefore, is to present a general framework for Tem-

poral MDB, along the lines of [3] but using an algebraic approach

1University of Murcia. Murcia, Spain. Contacting author jpalma@dif.um.es

based on Fuzzy Temporal Constraints Network framework (FCTN)

for temporal dimension representation.

The structure of the paper is as follows: the underlying temporal

framework is described in a concise manner in section 2. Section 3

presents the temporal behavioral model. The elements that constitute

the inputs and outputs of the algorithm are introduced in section 4.

The diagnosis process is analysed in section 5. Section 6 shows some

experimental results provided by a performance analysis. Finally, we

provide conclusions and future works.

2 TEMPORAL FRAMEWORK

In some proposals for Temporal MBD, the temporal dimension is

modelled by means of the so-called Fuzzy Temporal Constraint Net-

work (FTCN) formalism [14]. A FTCN is a pair N =< T ,L >

consisting of a finite set of temporal variables, T = {T0,T1,...,Tn},

and a finite set of binary temporal constraints, L = {Lij, 0 ≤ i,j ≤

n} defined on the variables of T . A FTCN can be represented by

means of a directed constraint graph, where nodes represent temporal

variables and arcs represent binary temporal constraints.

Each binary constraint Lijon two temporal variables Tiand Tjis

defined by means of a convex possibility distribution πLij(π(v?) ≥

min{π(v),π(v?)};v ≤ v?≤ v??), whose discourse universe is Z,

and which restricts the possible values of the time elapsed between

both temporal variables. In the absence of other constraints, the as-

signments Ti = tiand Tj = tj are possible if πLij(tj − ti) > 0 is

satisfied.

An n-tuple S = (t1,...,tn) ∈ τnis a σ-possible solution of a

FTCN network N if πSN= σ, where πSN= min{πLij(tj −

ti), 0 ≤ i,j ≤ n}. The possibility distribution πSNdefines the

fuzzy set SN of the σ-possible solutions of the network, with σ ≥ 0.

A FTCN network N is consistent if and only if SN is greater than

a previously established threshold α, where α ∈ [0,1], with α = 1

being equivalent to the crisp case. The value of α is conditioned by

the context and is set up arbitrarily by the user.

This model has been implemented and extended in FuzzyTIME

[4], a general purpose temporal reasoner that provides high level

language and reasonings capabilities on fuzzy temporal constraints

between temporal variables which can represent intervals or time in-

stants.

3TEMPORAL BEHAVIORAL MODEL

In this proposal, we opt for a Temporal Behavioral Model, TBM,

an abnormal behavioral model in which only the causal and tem-

poral relations between hypotheses (diseases) and abnormal obser-

vations caused by them are represented. These relations are defined

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by Diagnostic Fuzzy Temporal Patterns (DFTPs). Apart from the

abnormal behavioral model, a DFTP includes knowledge about

how the context affects the temporal behavioral model, referred to as

Contextual Meta-knowledge (CTX). Hence, TBM = {DFTPk}.

Each DFTP can be formally defined by the tuple DFTP =

?H,IM,IH,Rdftp,CTX? where:

• H is the diagnostic hypothesis described by DFTP.

• IM = {imk|k = 1,...,nim}, is the set of abnormal manifesta-

tions implied by the hypothesis H.

• IH = {ihk|k = 1,...,nih} is the set of hypotheses implied by

H (in medical domains, ihkis a disease caused by H).

• Rdftp= ?Tdftp,Ldftp? is a consistent FTCN, where tem-

poral variables in Tdftpare associated to H, IM and IH,

Tdftp

poral constraints between them are defined in Ldftp, where

Ldftp= C(tH,tim

those constraints defined by the expert are instantiated, and a

subsequent process computes the minimal network of constraints

between all temporal variables.

• CTX = {CTXi} is the set of temporal contexts. A context

describes how the DFTP definition is modified when a con-

text factor occurs (temporal or atemporal concepts). Formally

CTXi = ?ACi,TCi,Rct

– ACi is the set of possible atemporal concepts described in the

context (e.g. patient age, smoker).

=

{tH,tim

1 ,···,tim

nim,tih

1,···,tih

nih} and the tem-

1 ,···,tim

nim,tih

1,···,tih

nih). Furthermore, only

i,MFi? where:

– TCi is the set of possible temporal concepts described in the

context (e.g. a drug was given at a certain time).

– Rct

with the temporal concepts TCi

– MFi = {mf1,...,mfm} is the set of modification functions

(mfi) that describes the DFTP modifications. These functions

create, delete and modify elements of the IM, IH sets, and the

Rdftpnetwork.

i is a FTCN that includes constraints in the hypothesis H

Theoretical descriptions of diseases are clearly shown in medical

manuals, however those descriptions are deeply conditioned by the

present situation of each particular patient. Temporal contexts are

therefore important aspects of diagnosis. The presence or absence

of manifestations can be explained by a given disease, but this could

change depending on patient contextual conditions. These conditions

affect existing manifestations, but they also could justify new symp-

toms not gathered in the original DFTP. Other possible represent-

ations of TBM are possible. However, the representation of context

knowledge and the behavior in medical environments is easily rep-

resented by the model previously proposed in this work.

As an example, we present a (simplified) description of the acute

myocardial infarction (AIM) according to the TBM presented: The

AIM (Root Hypothesis: (AIM, t1)) is manifested by a precordial

pain, and moderate values of the ST levels (implied manifestations:

(pain, location,precordial,t2), (ST-level, intensity, moderate,t3)), the

second one more or less two minutes after the infarction (tem-

poral constraint: t3 APPROX 2 MINS AFTER t1). The AIM could

also produce a mixed shock syndrome (implied hypothesis: (Mixed-

Shock-Syndrome, t4)).

4DIAGNOSIS ALGORITHM INPUTS AND

OUTPUTS

In order to provide a solution, the diagnosis process requires as in-

puts (apart from the TBM) the patient’s observations (EV T−=

{evti|i = 1,...,nobs}), the contextual observables (CTXobs =

{ctxi|i = 1,..,nctx}), and a consistent temporal network (Rinput),

whose temporal variables are associated to elements in EV T−and

CTX. In most cases, these temporal variables are specified as abso-

lute time instants, which makes the reasoning process more efficient.

In our proposal, the diagnostic process output (i. e., the explan-

ation provided) is composed not only of a set of abducibles, such

as in [3, 11], but by all the elements that conform the final instanti-

ated causal network (physiopathological and ethiological diagnosis,

in medical domains). This kind of diagnosis explanation is neces-

sary from the point of view of decision support system development.

Therefore, the diagnosis algorithm output can be formally defined

as the tuple EXP = ?CNexp,Rexp,DFTPexp,BLexp,ABexp?

where:

• CNexprepresents a directed graph describing the final causal net-

work, where nodes represent observables and hypotheses in the

final explanation.

• Rexpis a FTCN where the temporal variables are associated to

the CNexpnodes.

• DFTPexpis the set of contextualized DFTP selected for explan-

ation.

• BLexprepresents the binding list. It is a set of links between the

hypothesis nodes of the causal network and their corresponding

temporal patterns definitions.

• AB ⊂ DFTPexp is the set of abducibles generated by the dia-

gnosis process.

In MBD, different interpretations of temporal diagnosis ex-

planation have been proposed. On the one hand, there is totally

consistency-based diagnosis [10, 12], in which the explanation

provided should be consistent with all observations. On the other

hand, there is totally abduction-based diagnosis [7, 13], in which the

explanation should logically entail all the observations.

The same considerations can be made for temporal dimension. In

[3], a generic knowledge level model for temporal MDB is proposed

in which the definition of explanation has been parameterized. This

parameterization allows the definition of explanation to be moved

on the continuous line defined between totally consistent diagnosis

and totally abductive diagnosis. In our proposal we opt for an inter-

mediate model in which an abductive component is applied to ab-

normal events EV T−, and components consistency is applied for

thetemporaldimension.Thisintermediateinterpretationofdiagnosis

explanation can be formally stated as follows:

Definition 1 (Temporal Diagnosis). Given a Temporal Diagnostic

Problem TDP =< TBM,EV T,CTXevt,Rinput>, EXP =<

CNexp,Rexp,DFTPexp,BL,AB > is a possible explanation for

TDP iff:

1. DFTPexpSCTXevtSCTX |= EV T−,

2. RinputSRexpis consistent.

5THE DIAGNOSIS PROCESS

Thediagnosisprocessinthisworkisdescribedbyanalgorithmbased

on the TBM described in Section 3. The following assumptions

have been made:

Multiple cardinality solution. Several hypotheses may be found

in a solution, which represent alternative or complementary solu-

tions. Furthermore, different instances of the same hypothesis (the

same hypothesis located at different time instants) are possible in a

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solution. However, all hypotheses should be consistent with the con-

text information.

Parsimonious covering based diagnosis. The proposed al-

gorithm explains the abnormal event set EV T−

nious covering. New hypotheses are included in the final explana-

tion if and only if events cannot be explained by the hypotheses

already instantiated. Of course, the solutions provided do not con-

tradict either temporal or atemporal contextual concepts.

Acceptable efficiency of the process. Despite the fact that the

algorithm presents an exponential time execution, the algorithm in-

cludes some heuristics (subsumption and temporal shifting) to im-

prove efficiency. Experimental results,as we will see in Section 6,

point to an acceptable time response.

newthrough parsimo-

5.1Subsumption

The aim of the subsumption process is to avoid an excessive prolifer-

ation of temporally nearby hypotheses. Thus, before creating a new

instantiated pattern to explain a given event evti, the subsumption

process tries to include it in one of the already instantiated patterns,

particularly those patterns in DFTPexp which match with the pat-

terns in TBM and which explain evti.

In order to subsume a given evti, with dftpk = evoke(evti), in

DFTP ∈ DFTPexp, the subsumption process checks if the tem-

poral constraints defined in dftpk in which evti takes part are con-

sistent with the temporal constraints of Rexp(the temporal constraint

network of the solution). In other words, the event evtiis included in

the solution and all temporal constraints related to this event in dftpk

are added to Rexp. After that, if Rexpis temporally consistent (that

is, if the consistency degree is greater than the previous established

threshold), the event evtiis subsumed. Furthermore, this event is ex-

plained by the hypothesis of dftpkthat already explains other events

of the solution. This process is carried out by a temporal query to the

temporal reasoner using local propagation of the fuzzy constraints,

similar to the technique defined in [2].

Figure 1 shows how evt3 is subsumed in DFTP. Thus, on the

right-hand side, the causal network of the pattern is represented,

where evt3is a cause of H1. On the left-hand side a part of the tem-

poral constraint network of the solution is represented. Hence, the

new temporal constraints on the network can be observed, due to the

temporal variable associated to evt3.

The diagnosis algorithm tries to subsume an event in a pattern

when contextualization is not possible. We consider that contextu-

alization is a process of characterizing a pattern for a given event in

a given context. Therefore, there is no sense in associating this event

within an existing pattern of the solution, because is is always as-

sumed that any solution that can be framed in a context is better than

any other that can not.

However, subsumption usefulness refers to the time execution

factor. The subsumption process slows down the growth of instan-

tiated hypotheses, which is exponential. Subsumption allows events

to be explained by instantiated hypotheses of the solution, avoiding

temporal nearby instances of hypotheses.

5.2Temporal Shifting

When subsumption is not possible, is a new pattern instantiation

enough?. The answer is no. When a given event cannot be sub-

sumed, it is due to temporal inconsistencies in the instantiated pat-

tern, DFTP. However, DFTP could possibly explain the new

event if temporal conditions were different.

Thus, we therefore propose including a new instance of the same

pattern and associating the event to it. If we reconsider the failed

subsumption, we will notice that only a few of the associated events

subsumed into it do not allow the new subsumption. According to

this, some of these events (already subsumed) can be subsumed by

the new instance of the pattern. In conclusion, a temporal shifting

process will produce two instances of the same pattern (at different

time instants), whose hypotheses explain at least one different event

and, perhaps, some common events.

evt1

evt1evt2evt3

H1

H1

evt3

evt4

evt4

evt2

H2

H2

R CN

exp

exp

new causal relation

old temporal constraints

new temporal constraints

Figure 1.

Subsunction.

In Figure 2, the temporal shifting of H1(H1) is represented when

the evt3 is explained. On the right-hand side, the causal network is

represented. There, H1hypothesis can explain evt3when the hypo-

thesis is shifted. On the left-hand side the temporal constraint net-

work (Rexp) is represented. Note that a temporal constraint inhibits

the subsumption of evt3 in H1. Due to this, H1 is temporally shif-

ted (H1). This new instance of the hypothesis does explain evt3.

Moreover, evt2could also be subsumed in H1.

consistent relation before temp.shift.

inconsistent relation

consistent relation after temp.shift.

evt1

H1

evt2evt3

H1

H1

H1

evt1evt2evt3

exp

exp

R CN

Figure 2.

Temporal Shifting.

The temporal shifting process is used when subsumption is not

possible.However,thisprocessreducesthealgorithm’sefficiencybe-

cause of the large amount of calculi for temporal consistence check-

ing, in spite of the local propagation process. Furthermore, this pro-

cess could imply new subsumptions. In our opinion, this problem

could be partially reduced using some heuristics, which determine

whether the hypothesis must be shifted or not. In this work, we sug-

gest the application of this shifting technique only with the latest

temporal instance of the pattern (last(D), line 10 of Algorithm 1).

This heuristic increases the probability of finding at least one hy-

pothesis that can explain the event. Moreover, it avoids the combin-

atorial explosion of explaining the hypothesis, because a single tem-

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poral shifted hypothesis per event is guaranteed. We are currently

considering working with temporal intervals which provide a higher

level of abstraction. The use of intervals shows us how to associate a

concrete persistence to a DFTP. The definition of hypotheses per-

sistence will allow the aggregation of nearby temporal hypotheses,

e.g. describing temporal influence interval on pattern instances. In

this case, it could be possible to aggregate those temporally shifted

hypotheses allocated in the same influence interval, so that a single

hypothesis substitutes all of them.

5.3The Diagnosis Algorithm

Once the selected event is explained and removed from EV Tnew

(initially EV Tnew = EV T−), its explaining hypothesis (hypo-

theses) will be a new event to be explained, and therefore will be

included in EV Tnew. The algorithm finishes when it is not possible

to find a higher level hypothesis, abducibles of the solution, that can

explain any of the EV Tnew events. The diagnosis process can be

described, as follows:

1. An event e is selected from EV Tnew. The event e is possibly

associated to an evidence or a hypothesis.

2. The algorithm searches (evoke(), line 5) all possible patterns

D from TDM that can explain event e.

3.Finally, the algorithm tries to include each pattern of D found

in the solution as follows: 1)The algorithm considers temporal and

atemporal concepts from the context information input (CTXobs),

then the algorithm tries to contextualize (contextualized(), line 7)

the pattern. 2)If contextualization is not possible, the diagnosis pro-

cess tries to subsume (subsume(), line 9) the event in any of the

already instantiated patterns that exist in the partial solution (see Sec-

tion 5.1). 3)When subsumption is not possible either, the temporal

shifting process (see Section 5.2) is applied (temporal shifting(),

line 12). Due to the computational cost of this procedure, this process

mustfulfilsomeheuristicconditionslikethatproposedinsection5.2.

4)If non previousactions are possible, the diagnosis process will gen-

erate an instance of the new pattern in the solution (generate new(),

line 17).

Function COVER (TBM, EV T−, CTXobs, Rinput) return EXP

1 : subsumed = FALSE

2 : EV Tnew= EV T−

3 :

while EV Tnew ?= ∅ do

4 :

for each evti ∈ EV Tnewdo

5 :D = evoke(evti)

6 :

for each dftpi ∈ D do

7 :

if not contextualized(dftpi,evti,EXP) then

8 :

if dftpi ∈ DFTPexpthen

9 :

if( not subsume(evti,dftpi,EXP) and

10:

not subsumed and last(D) = dftpi)then

11: subsumed = TRUE

12:

dftpnew= temporal shifting(evti,dftpi,EXP)

13:

evth= associateevent(dftpnew)

14:

EV Tnew= EV Tnew∪{evth}

15:

endif

16:

else

17:

dftpnew= generate new(evti,dftpi,EXP)

18:

evth= associate event (dftpnew)

19:

EV Tnew = EV Tnew∪ {evth}

20:

endif

21:

endif

22:

23:

24:

25:

endFunction

endfor

EV Tnew = EV Tnew?{evti}

endfor

endwhile

Algorithm 1: Parsimonious covering algorithm.

6 PERFORMANCE ANALYSIS.

In this work, we have focused our analysis on time execution factors,

trying to find the most relevant variables that influence the perform-

ance of the diagnosis process. The objective of this analysis is to

ascertain the influence of different parameters of the input space on

the overall performance. The following factors are considered:

c)

d)

b)

a)

5000

10000

15000

20000

25000

30000

35000

40000

1.5 2 2.5 3

time (ms)

mean explain level

mean connectivity vs. time

"explainlevelmean.dat"

0

mean explain level vs. time

1

18000

20000

22000

24000

26000

28000

30000

32000

34000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

time (ms)

"connectivitymean.dat"

mean connectivity

0.8

0

5000

10000

15000

20000

25000

30000

35000

0 1 2 3 4 5 6 7 8

"dftpnumber.dat"

time (ms)

dftp number vs. time

dftp number

5000

10000

15000

20000

25000

30000

35000

0 4 6 8 12 14

evidence number

evidence number vs. time

"evidencenumber.dat"

0

10 2

time (ms)

Figure 3.

Experimental results of the algorithm. Prototype: Java.

Proc:AMDAthlonXP Freq:1.53GHz RAM:256MB

• The Number of Input Events. Far from reducing the number

of conjectures, the increase of evidences, in medical domains

increases their possible explanations. This is mainly due to the

nature of clinical hypothesis, where the same evidence could be

explained by several hypotheses. The increase of evidences re-

quires more time to complete the whole explanation process and,

therefore, the study considers only the time used to explain the

first hypothesis of each evidence. As is shown in Figure 6.a, the

executiontime presents an exponential behaviorof time, but this is

expected if we consider that the time spent for checking temporal

consistence (in temporal shifting and subsunctions) increases be-

cause of the growth of temporal variables and events at each iter-

ation.

• The Number of Patterns required to find a solution (Figure 6.b).

This parameter shows the influence of the number of DFTPs

considered in finding the solution, that is, how the depth of the

causal network affects the process performance.

• Mean Pattern Connectivity Degree (Km). Let us define con-

nectivity (K) of a pattern (DFTP) in the TBM as follows:

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KDFTP = |IH|, that is, the number of implied hypotheses of

patterns. Then Kmcan be defined as:

P

• Mean Pattern Explanation Degree (Em): Explanation degree

(E) of a pattern (DFTP) in the TBM is defined as EDFTP =

|IM| + |IH| with IM ∧ IH ∈ DFTP, that is, the number of

causal links. The Emis defined as:

P

Km =

DFTP∈TBMKDFTP

| TBM |

(1)

Em =

∀DFTP∈TBMEDFTP

| TBM |

(2)

These last two factors, Kmand Em, can be considered as a meas-

ure of the TBM complexity. Emcan be considered as an indication

of the complexity in covering the initial observations (Figure 6.c)

whereas Kmcan be associated to the complexity in building CNexp

upwards from the first hypotheses to abducibles (Figure 6.d).

This work is focuses on the capacity of the presented model to

represent causal and temporal knowledge, and the study of the per-

formance analysis. Today, we are at the knowledge acquisition step,

so a causal and temporal knowledge acquisition tool (CATEKAT)

has been implemented for elaborating a complete TBM [5]. How-

ever, this step is not finished yet. Thus, the input data set used in

this work has not been validated by the expert. In any case, we have

taken into account that the complexity of causal and constraints net-

work in the testing bench is similar to a small set of already validated

patterns.

7 CONCLUSIONS AND FUTURE WORKS

This paper describes a general framework for temporal MDB which

tackles the problems of modelling complex interaction between deep

causal models and context knowledge and structure of explanations

(solutions) provided. The proposed framework demonstrates the suit-

ability of FTCNs for time management. Following the general

framework proposed in [3], our proposal can be characterized in the

following terms: (a) the temporal phenomenon described in this pa-

per can be considered a temporal behavior one in which the con-

sequences of the fact that the system is in a given state (normal or

faulty) are observed after some time; (b) time is modelled by means

of a metric time-ontology in which temporal information is repres-

ented by FTCNs [4, 14]; and (c) with regard to the definition of the

explanation chosen, we demand that the explanation provided logic-

ally entails all abnormal observations, and that its temporal inform-

ation is consistent with the one observed. Therefore, we propose an

abductive approach for observations and a totally consistent-based

approach for temporal dimension.

The use of diagnostic temporal patterns proposed in this paper is

similar to that defined in [9], but our proposal makes it possible to

model causal relations between diagnostic patterns. Causal relations

between diagnostic patters allow us to define a causal network of dia-

gnostic patterns. Another difference lies in the temporal representa-

tion framework, since we use the Fuzzy Temporal Constraints Net-

work formalism, while the diagnostic patterns defined in [9] make

use of a quantitative interval based approach.

One of the main differences between our approach and Brusoni et

al. [3] is related to the way that contextual knowledge is integrated in

the model. In Brusioni’s approach, contextual knowledge is defined

as a set of maximal episodes that can be used in the antecedent of

the logical formulae which conform the temporal behavioral model.

In our model, contextual knowledge is defined as a set of logical for-

mulae which includes knowledge about temporal relations between

antecedents components, thus conforming a meta-knowledge base

which defines how the context knowledge affects disease evolution

definition. In our model, therefore, contextual information is ortho-

gonal to temporal behavior.

Future works will focus on the integration of this model with a

possibility theory based evaluation of hypothesis consistency (in or-

der to provide a consistent explanation), and on the logical formula-

tion in terms of temporal logic, like the one defined in [6].

8 Acknowledgement

This work is supported by the Spanish MCYT under the MEDICI

project (project number TIC2003-09400-C04) and by the Spanish

MECD, under the FPU national plan (grant ref. AP2003-4476)

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