New Method of Predicting

A new method for predicting functional recoveryof stroke patients with hemiplegia: logarithmicmodellingTetsuo Koyama, Kenji Matsumoto, Taiji Okuno Department of Rehabilitation Medicine, Nishinomiya KyoritsuRehabilitation Hospital and Department of Physical and Rehabilitation Medicine, Hyogo College of Medicine andKazuhisa Domen Department of Physical and Rehabilitation Medicine, Hyogo College of Medicine, Nishinomiya, Hyogo,Japan

Received 25th August 2004; returned for revisions 14th December 2004; revised manuscript accepted 3rd January 2005.

Objective: To examine the validity and applicability of logarithmic modelling forpredicting functional recovery of stroke patients with hemiplegia.

Design: Longitudinal postal survey.Subjects: Stroke patients with hemiplegia staying in a long-term rehabilitation facility,who had been referred from acute medical service 30/60 days after onset.Methods: Functional Independence Measure (FIM) scores were periodicallyassessed during hospitalization. For each individual, a logarithmic formula that was

scaled by an interval increase in FIM scores during the initial 2/6 weeks was used forpredicting functional recovery.

Results: For the study, we recruited 18 patients who showed a wide variety ofdisability levels on admission (FIM scores 25/107). For each patient, the predictedFIM scores derived from the logarithmic formula matched the actual change in FIM

scores. The changes predicted the recovery of motor rather than cognitive functions.

Regression analysis showed a close fit between logarithmic modelling and actual

FIM scores (across-subject R2/0.945).Conclusions: Provided with two initial time-point samplings, logarithmic modellingallows accurate prediction of functional recovery for individuals. Because the

modelling is mathematically simple, it can be widely applied in daily clinical practice.

Introduction

In the rehabilitative treatment of stroke patientswith hemiplegia, prediction of functional recoveryis crucial. Accurate prediction facilitates properdefinition of goals of intervention for individualpatients, thus improving the quality and efficiencyof rehabilitation service.1 For providers of services

and for those paying for it, accurate predictionenables effective use of resources by allowing betterestimation of such factors as length of hospitaliza-tion.2 Thus, for both individual patients and healthcare administrators, accurate prediction of func-tional recovery would provide crucially importantinformation.

For predicting functional recovery, variousmathematical modelling and other methods havebeen employed.312 Multivariable linear regressionmodelling has proved the most popular.12,13

This type of linear modelling has been usefulfor predicting outcome at a specific time-point

Address for correspondence: Tetsuo Koyama, Department ofRehabilitation Medicine, Nishinomiya Kyoritsu RehabilitationHospital, Jurinji-Minamimachi 2-13, Nishinomiya, Hyogo,Japan 662-0002. e-mail: [email protected]

Clinical Rehabilitation 2005; 19: 779/789

# 2005 Edward Arnold (Publishers) Ltd 10.1191/0269215505cr876oa

(e.g., six months after stroke). Stroke patientstypically, however, show nonlinear recovery pat-terns.14,15 In most stroke cases, patients show rapidrecovery during the initial few months, after whichthe pace of recovery to six months from onsetslows towards the final outcome.16 Consequently,linear modelling is not up to the task of accuratelypredicting the prospective outcome. To simulatethe nonlinear aspects of functional recovery, neu-ral network modelling,17 logistic modelling,18,19

and other types of nonlinear modelling have beenproposed. Although more successfully predictive,these modelling methods are not widely appliedbecause of their mathematical complexity. Thus,for general clinical applicability, there has been aneed for a simpler means of accurately predictingthe progress of recovery.

To explore more simple modelling methods, weinvestigated mathematical powers, logarithms,double-logarithms, and other simple mathematicalfunctions. Of those, we focused on natural loga-rithmic functions (ln) because they displayed threeadvantages for modelling functional recovery.First, the progress curves (Figure 1) resembledactual recovery patterns: if the recovery target isset at 180 days from onset and assigned a value of100%, approximately 70% of recovery is registeredat 90 days and subsequent progress occurs at areduced rate.1,16,20 Similarly, a logarithmic func-tion fitted the recovery patterns of upper limbfunction of stroke patients.21 Second, based onscores sampled on two days separated by aninterval, using simple mathematical procedures(Figure 1), the modelling formula can easily be

Figure 1 Model formula and predictive curve. (A) shows a generic structure; (B) shows mathematical procedures to tailor thegeneric structure to fit individual degree of recovery. For this, actual FIM scores recorded at two time points (Day A and Day B)are required. DFIM indicates change in FIM scores between Day A and Day B. Constant in (A) is countervailed in this procedure.(C) shows the final form of the model formula. Predicted value for Day X can be calculated with this form. FIM, FunctionalIndependence Measure; ln, natural logarithm.

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scaled to fit each individuals magnitude of recov-ery. Third, owing to mathematical specificity oflogarithms, the model formula can easily becalculated (e.g., ln(90)/ln(30)/ln(90/30)/ln(3),see Figure 1B). To evaluate the practical usefulnessof logarithmic modelling we carried out a long-itudinal study.

Methods

PatientsStroke patients with hemiplegia who were ad-

mitted to our long-term rehabilitation hospitalduring August 2003 to April 2004 were recruitedinto the study. Criteria for inclusion were:no past history of hemiplegia; capable of indepen-dent ADL (activities of daily life) before stroke;wheelchair required for locomotion at admission.As a result of Japanese health insurance proce-dures, patients were referred from local communityacute medical services, typically 30/60 days afterthe stroke occurred, and received inpatient care inour long-term rehabilitation hospital for 30/180days. During the prior period of acute medicalhospitalization they received physical therapy.During long-term rehabilitation hospitalizationthey received physical therapy, occupational ther-apy and speech therapy for a joint total of 120 minevery day. To minimize the influence of variabilityof therapeutic regimen, we also limited recruitmentto patients who received treatment from the samerehabilitation team directed by a single physiatrist(first author of this article). The protocol wasreviewed and approved by our hospitals ethicalcommittee and informed consent was obtainedfrom all patients.

Assessment of functional recoveryTo assess functional recovery, we employed the

Functional Independence Measure (FIM), whichhas been widely used in rehabilitation medicine.22

The FIM is derived from scoring 18 items accord-ing to a seven-point scale (1/totally dependent,7/completely independent) to assess functionalindependence in ADL. These 18 items arecategorized as self-care (6 items), sphinctercontrol (2 items), transfers (3 items), locomotion(2 items), communication (2 items), and socialcognition (3 items). The first four categories

involve motor functions (FIM-motor) and othertwo concern cognitive functions (FIM-cognition).The total scores score for all 18 items (FIM-total) is commonly used to assess functionalindependence in rehabilitation medicine (totallydependent in ADL/18, completely independentin ADL/126).

Using FIM scores, nursing staff assessed thefunctional recovery of patients in terms of ADL.Evaluations were typically recorded a few daysafter admission, again at two to six weeks afteradmission, and then once a month during hospi-talization. In our study, to assure reliability of theevaluations, FIM scores were reviewed at weeklyconferences.

Modelling and evaluationA generic structure of modelling was given in a

simple natural logarithmic formula (independentvariable/days from onset) (Figure 1A). To tailorthe generic structure to fit each individuals degreeof functional recovery, we performed calculationson the total FIM scores at the first two time-pointsafter admission. For each patient, the increase intotal FIM scores between these two time-points(DFIM) was used as the basis for scaling a co-efficient (b) in the generic structure (Figure 1B).The introduction of this countervailed the constantin the generic structure. Thus, using the scores atthe initial two sampling points, a generic struc-ture could be tailored to forecast each patientsfunctional recovery (model formula shown inFigure 1C).

To assess the fit of the time-course of themodel, FIM scores (FIM-total, FIM-motor andFIM-cognition) were, on an individual basis,longitudinally plotted with predicted values foreach patient derived from the model formula. Toassess the general applicability of logarithmicmodelling, using data from all patients, a conven-tional linear regression analysis was performed tocompare the total FIM scores that were actuallyobtained and the predicted values that werederived from the model formula. For this analysis,we excluded the scores obtained at the firsttwo sampling points (indicated by arrowheadsin Figures 2 and 3) to determine the particular bcoefficients.

Logarithmic modelling in hemiplegic stroke 781

Figure 2 Time-course of actually obtained and predicted FIM scores for patients with left hemisphere lesions (cases 1/10).Closed circles show actually obtained FIM-total scores, closed triangles show actually obtained FIM-motor scores, closedsquares show actually obtained FIM-cognition scores and open circles show predictive values derived from the model formula(Figure 1). Arrowheads indicate initial two sampling time-points for data to tailor the model formula for each individual.FIM, Functional Independence Measure.

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Results

PatientsWe collected and manipulated data for 18

patients (12 male, 6 female; 10 left, 8 right,

hemisphere lesion; age 33/78 (median 67.5) yearsold). For both motor and cognitive functions,these patients showed widely varying levelsof disability on admission (Table 1, Figures 2and 3). Total FIM scores ranged from 25 to 107

Figure 3 Time-course of actually obtained and predicted FIM scores for patients with right hemisphere lesions (cases 11/18).Closed circles show actually obtained FIM-total scores, closed triangles show actually obtained FIM-motor scores, closedsquares show actually obtained FIM-cognition scores and open circles show predictive values derived from the model formula(Figure 1). Arrowheads indicate initial two sampling time-points for data to tailor the model formula for each individual.FIM, Functional Independence Measure.

Logarithmic modelling in hemiplegic stroke 783

(median, 63.5), motor FIM scores ranged from 14to 74 (median, 36), and cognition FIM scoresranged from 6 to 35 (median, 25). Initial FIMscores were sampled at from 32 to 77 days (median50) after occurrence of stroke and the second set

of FIM scores were sampled at from 46 to 104 days(median 72) after occurrence (indicated by arrow-heads in Figures 2 and 3). The interval betweenthese two time-points ranged from 13 to 44 days(median 32).

Table 1 Patients profiles

Case Age Gender Hemisphere Lesion Cause of stroke Ope. Intervention Comorbidity

No. 1 82 M Left Corona radiata Infarct (/) OT, PT CADNo. 2 63 F Left Putamen Hemorrhage (/) OT, PT, ST HTNo. 3 53 M Left Corona radiata Infarct (/) OT, PT, ST HTNo. 4 33 F Left Putamen Hemorrhage (/) OT, PT, ST HT, HLNo. 5 62 M Left Putamen Hemorrhage (/) OT, PT, ST (/)No. 6 73 F Left Putamen Infarct (/) OT, PT, ST DM, HTNo. 7 78 F Left Corona radiata Infarct (/) OT, PT (/)No. 8 74 F Left MCA Infarct (/) OT, PT, ST Af, CAD, HTNo. 9 74 M Left Putamen Hemorrhage (/) OT, PT, ST HCCNo. 10 65 M Left Prefrontal cortex Hemorrhage (/) OT, PT (/)No. 11 74 M Right Prefrontal cortex Hemorrhage (/) OT, PT, ST (/)No. 12 50 M Right Corona radiata Infarct (/) OT, PT DM, HTNo. 13 73 M Right Corona radiata Infarct (/) OT, PT DM, HTNo. 14 70 F Right MCA Infarct (/) OT, PT (/)No. 15 70 M Right MCA Infarct (/) OT, PT CAD, DMNo. 16 54 M Right Thalamus Hemorrhage (/) OT, PT, ST HTNo. 17 65 M Right MCA Infarct (/) OT, PT HTNo. 18 54 M Right Putamen Hemorrhage (/) OT, PT HT

CAD, coronary artery disease; DM, diabetes mellitus; HCC, hepatic cell carcinoma (post operation); HL, hyperlipidaemia; HT,hypertension; MCA, middle cerebral artery; Ope., operation (open-skull) during acute medical hospitalization; OT, occupationaltherapy; PT, physical therapy; ST, speech therapy.

Figure 4 Scatterplots showing the relationships between actually obtained FIM-total scores and predicted values derivedfrom the model formula (see Figures 2 and 3). Data from the two initial sampling time-points for each patient (indicated byarrowheads in Figures 2 and 3) were excluded from the scatterplots. FIM, Functional Independence Measure.

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Assessment of model fitFor each individual, the pattern of increase in

the predicted values that were derived from themodel formula was very similar to the total FIMscores that were actually obtained: so close in fact,that the correspondence in some cases (3, 6, 8, 10,14, 15 and 16) was almost identical (Figures 3 and4). Actual total FIM scores comprised two maincomponents: measures of motor and cognitiveability. Close observation of the time-courses ofthese subcomponents showed that the main con-tribution to the growth patterns of total FIMscores was mainly from the motor subcomponents.In sharp contrast, changes in cognitive subcompo-nents were, in most cases, minimal (Figures 3 and4). This finding indicates that the model formulasimulates the recovery pattern of motor ratherthan cognitive components.

For cases 1, 4 and 12, the predicted valuesexceeded the actually obtained total FIM scores.Dissociation between actual and predicted valuestended to be greater towards the high end (/120)of the total FIM range. The predicted values forcase 11 also exceeded the actual FIM scores. Thiscase was exceptional, being the only patient whoscored large changes in FIM-cognition soon afteradmission. Nonetheless, even in this case, thegrowth pattern of FIM-motor scores was compar-able to other cases and similar to logarithmiccurve.

Model fit was then assessed using group data.Regression analysis comparing actual data andpredicted values revealed that the model formulaaccurately predicted actually obtained FIM-totalscores (Figure 4; R2/0.945).

Discussion

A logarithmic function was applied to simulate thetime-course of functional recovery of stroke pa-tients with hemiplegia. Based on this, we developeda new model formula that, using FIM resultssampled from two points in time during recovery,could be applied accurately to predict the patternof functional recovery in individual recovery.Among patients with a wide variety of motor andcognitive disability, the model formula accuratelypredicted actual functional recovery during hospi-

talization. Thus, the new model formula based onlogarithmic function could be a powerful tool forpredicting functional recovery of stroke patientswith hemiplegia.

Modelling using raw FIM-total scoresFIM assessment was originally based on an

ordinal rather than an interval scale. Subsequently,Rasch analysis has provided a model for convert-ing the ordinal scale of raw FIM-total scores intoan interval scale.23 Although after Rasch conver-sion the data showed shows a logistic curve, rawFIM-total scores tend to show an almost linearrelationship with converted values within therange from 25 to 120. Within this range, rawFIM-total scores have been widely employed asinterval values in many previous studies.24,25

Accordingly, to keep our model simple, we emp-loyed raw FIM scores as the basis for mathema-tical modelling.

Validity of logarithmic modelling using twotime-point samplings

During actual treatment, naturally each patientand those giving them care are intensely interestedin the particular prospects of functional recovery.Few studies, however, have focused on individualtime-course and degree of functional recovery.1 Inthis study, for each patient, we longitudinallysampled FIM scores at 4/7 time-points duringhospitalization. Observation of this data indicatedthat recovery patterns assessed by FIM scorescould be modeled as logarithmic function. More-over, using data for individual patients that weresampled at two time-points during recovery, the

Clinical messages

/ Logarithmic modelling accurately predictsfunctional recovery of stroke patients withhemiplegia.

/ Provided with two initial time-point sam-plings, logarithmic modelling can be tai-lored to forecast each patients functionalrecovery.

/ The modelling is mathematically simpleenough to be adopted in daily clinicalpractice.

Logarithmic modelling in hemiplegic stroke 785

model accurately predicted the actual results laterobtained for the individuals. We know of no otherprediction modelling studies that provide useful,simple, individual-based mathematical modelling.

When forecasting the functional recovery of anindividual stroke patient, a single physiatrist oftentakes many clinical parameters into consideration.These include: initial motor and cognitive impair-ment levels,26,27 initial day of rehabilitation,28

recovery rate,29 site and size of lesion,30,31 age,32

psychological status,33 unilateral spatial neglect,34

co-morbidities35 and other factors.36 Most of theseprevious prediction studies have attempted tointegrate multiple factors into the model. Ourstudy, however, uses only FIM scores sampled ondifferent days with an interval of 2/6 weeksbetween them. The results show that, processedthrough our logarithmic equation, these dataenable powerful and accurate forecasting of func-tional recovery. Since initial patterns of recoverycould be affected by any of the multiple factorsmentioned above, the FIM scores of individualpatients are likely to be influenced by some or allof these factors.

Simplicity of logarithmic modellingWith the goal of developing a new forecasting

technique to predict functional recovery, we testedseveral mathematical functions in our preliminaryanalyses. Taking a lead from a previous study thatemployed logarithmic transformations of FIM-total scores to model functional outcome,12 wetested, among other manipulations, various dou-ble-logarithmic functions. In fact, in some cases,preliminary models using double-logarithmic func-tions did fit actual data slightly better than themodel formula that we are presenting here. Evenso, we preferred not to employ double-logarithmicmodelling because of its complexity.

Focusing on logarithmic modelling, we at-tempted to adjust the clause for days from onsetto improve the model fit. We attempted adjustmentbased on the clinical observation that the start offunctional recovery varies from case to casedepending on site, size, and age of lesion. Ourpreliminary analyses, however, revealed that thecontribution of such adjustments was minimal.Thus, to keep things simple, we applied logarithmicmodelling without any adjustments (Figure 1).

The model formula is simple and structurallyflexible (Figure 1C). For consistency, in this study,we used data from the first and second FIMsamplings after admission. Any pair of periodicsamplings, however, are suitable for defining thecoefficient (b) of the model formula. The flexibilityof the model formula enables easy re-estimation ifpredictive and actual values deviate. This simpli-city and flexibility means that the model formula issuitable for wide clinical application.

Possible limitations of logarithmic modellingIn this study, we customized the individuals

model formula by using scores from two FIMsamples: based on results of assessment done withan intervening period of from 13 to 44 days, datafrom this sampling pair were collected at between32 and 104 days after the occurrence of stroke. Themodel was effective within these sampling para-meters. Further studies are needed to find out thelimits of applicability to FIM data collected atearlier or later phases of affliction.37 It is promisingthat case 15 (Figure 3), using data collectedrelatively soon (33 days) after stroke occurrenceand with a short sampling interval (13 days),provided accurate prediction. This modellingmight be useful even at earlier stages of illnessand during shorter periods of hospitalization.

Close observation of the time-course dataplotted for each individual revealed that factorsfor change in the predictive model were the motorcomponents rather than cognitive components.Thus the model may not be applicable for patientswhose clinical manifestations are mainly cognitiverather than motor (e.g., patients with subara-chnoid haemorrhage).38,39 Time-course plottingalso revealed a tendency for predicted values toexceed the actual data towards the high end of theFIM-total range. In view of the linearity of rawFIM-scores (as discussed above) this might implythat the model formula is best utilized whenpredictive values range from 25 to 120.

Applicability of logarithmic modellingOur study samples yielded data on patients who

varied widely in age, lesion characteristics, andlevels of motor and cognitive disabilities. Theresults that we obtained indicate that the logarith-mic model formula (Figure 1C) could be effectivelyapplied for various types of hemiplegic stroke

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patients. Our new model is valuable for itssimplicity and applicability on an individual basis.Using FIM scores that were sampled at twodifferent time-points, using a regular pocket calcu-lator (without a log function) and a logarithmlook-up table (see Appendix), within minutes it ispossible to come up with a prediction for eachindividuals functional status for a particular day.Thus, the model formula, based on simple loga-rithmic function, could be adopted in everydayclinical practice for predicting the functionalrecovery of stroke patients with hemiplegia.

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Appendix / A quick reference for logarithmic function

x Ln (x )

1.0 0.000

1.1 0.095

1.2 0.182

1.3 0.262

1.4 0.336

1.5 0.4051.6 0.470

1.7 0.531

1.8 0.588

1.9 0.642

2.0 0.693

2.2 0.788

2.4 0.875

2.6 0.9562.8 1.030

3.0 1.099

4.0 1.386

5.0 1.609

6.0 1.792

7.0 1.946

8.0 2.079

9.0 2.19710.0 2.303

ln, natural logarithm.

Logarithmic modelling in hemiplegic stroke 789

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