Health Care Management Science4: 13–19, 2001. 2001Kluwer Academic Publishers. Printed in the Netherlands.

13

Data requirements in a model of the natural historyof Alzheimer’s disease

Thierry J. Chaussalet∗ and Wayne A. Thompson

Department of Mathematics, Cavendish School of Computer Science, University of Westminster,115 New Cavendish Street, London W1M 8JS, UK

E-mail: chausst@wmin.ac.uk

The “natural history” of Alzheimer’s disease (AD) is discussed in terms of the data required in a general, discrete-time, non-homogeneous Markov model. The proposed model differs from similar models reported in the literature because mortality is disaggregatedinto AD-specific mortality and competing mortality due to other causes. Data are reviewed from the literature for AD incidence, and rates ofdisease progression and mortality. We conduct a preliminary sensitivity analysis using the reviewed data as base-case. The model shows thatsurvival is sensitive to the modelling assumptions concerning mortality. This observation could have important consequences for studiesthat assess the cost of care following therapeutic interventions.

Keywords: Alzheimer’s disease, natural history, incidence, mortality, Markov model

1. Introduction

Providing supportive care for older people with Alzhei-mer’s disease (AD), and respite for caregivers, places con-siderable financial commitments on the health and socialservices budget [1–5]. This commitment will become in-creasingly burdensome as the effects are felt of changingdemographic trends in the number of older people, the, so-called, “greying of the population”. Since the identifica-tion of Apolipoprotein E (ApoE) as a genetic risk factorfor late-onset AD, several authors have advocated screen-ing for AD [6]. Besides improving quality of life for bothpatient and caregiver, it is anticipated that such a screeningprogramme will confer other benefits. Early detection of-fers the possibility that early therapeutic intervention willdelay neurodegeneration, and, thus, avoid the high cost ofinstitutionalised care that is often incurred as the diseaseprogresses [1]. A screening programme comprising a se-quence of genetic and psychometric tests increases the dis-criminating power and accuracy of diagnosis [7]. Thus, sucha screening programme might be an effective means of caseascertainment and mitigates the financial cost of misdiagno-sis by not treating those who are cognitively intact. In ad-dition, patient treatment and care is better co-ordinated andcan be managed more cost-effectively [8,9].

The management of AD is problematic because of thelack of effective therapies. Recently, however, severalplacebo-controlled trials [10–13] have demonstrated thesafety and efficacy of 5 mg and 10 mg daily doses ofdonepezil, a cholinesterase inhibitor used for the treatmentof mild and moderate AD. In particular, donepezil has a sig-nificant impact on the decline in cognitive function as mea-sured by ADAS-cog (Alzheimer’s Disease Assessment Scale– cognitive sub-scale score), MMSE (Mini-Mental State Ex-

∗ Corresponding author.

amination), CDR-SB (Clinical Dementia Rating – Sum ofthe Boxes), and CIBIC (Clinician’s Interview-Based Impres-sion of Change). Studies in the UK [14], Canada [15],Sweden [16], and the US [17] have used Markov modelsto assess the cost–effectiveness of donepezil for the treat-ment of patients with mild or moderate AD during a 5-yeartime horizon. These studies showed that donepezil is cost-effective, although the model assumptions and sensitivityanalyses revealed some potential limitations of the modelsthat may restrict their applicability. A sensitivity analysisin the UK study showed that the cost–effectiveness dependson the mortality rate [14]. In [15], the model assumes thatmortality risk is independent of severity. In [16,17], themodels assume that changes in transition probabilities arenot correlated with age. Clearly, a cost–effectiveness modelshould distinguish between AD-specific mortality (hereafterreferred to as “the excess mortality”) and competing mortal-ity due to other causes (“the natural mortality”). Each sourceof mortality will depend on age and possibly other attributes.Moreover, the transition probabilities between states mayalso depend on age and other attributes.

In this paper, we propose a general, discrete-time, non-homogeneous Markov model of the natural history of AD,which disaggregates mortality into excess and natural mor-tality. Here, “natural history” refers to disease progressionin terms of cognitive and functional decline, and is describedby epidemiological measures including rates of disease on-set, progression, and mortality. We review data for use inthe natural history model, which, subsequently, provides abase-case from which to measure the effect of therapeuticor behavioural interventions. The natural history model is anecessary precursor to defining goals, planning, and settingpolicy for a proposed screening programme.

We use the natural history model to conduct a sensitivityanalysis using the reviewed data as the base-case. Such an

14 T.J. CHAUSSALET, W.A. THOMPSON

Figure 1. The natural history model. Absorbing Markov-states for deaths due to natural and excess mortality are indicated by a double outline. TransientMarkov-states are labelled by an indexi, denoting the level of cognitive impairment. Arcs, representing allowable transitions during a Markov cycle, are

labelled with the annual rate of transition. See text for details.

analysis is useful in quantifying the potential benefits of im-plementing a screening programme for AD. Development ofthe natural history model to incorporate efficacy and cost–effectiveness of a given therapeutic intervention will be re-ported elsewhere.

2. The natural history model and its data requirements

The state-transition diagram for the natural history modelof AD is shown in figure 1. Data are required for the follow-ing parameters: age-severity-specific rates of disease onset(hereafter referred to as “incidence”, and denoted byν0i (a),i = 1,2,3); severity-specific rates of disease progression(denoted byνij , 0 6 i < j 6 3); age-specific, natural-mortality rate (denoted byµ(a)); and age-specific, excess-mortality rates (denoted byδi(a), i = 1,2,3).

Data for these parameters are reviewed below.

2.1. Incidence of AD

Most studies of the incidence of AD do not present age-specific data. Table 1 shows the age-specific incidenceof AD estimated from two population-based studies and ameta-analysis.

The Framingham Study is population-based, and de-scribes the age-gender-specific incidence of AD in a co-hort of 2611 cognitively intact subjects, where AD was as-sessed using a biennial screening MMSE test [18]. The PiteåDementia Project describes age-specific incidence basedon hospital referrals in a defined population during 1990–95 [19]. Gao et al. [20] report findings from a meta-analysisbased on age-specific incidence in eight population-basedstudies with sample sizes ranging between 347 and 2792. Inall studies in table 1, the incidence of AD increases markedlywith age. In those aged 75 or above in the FraminghamStudy, the incidence was higher in women than in men. Theodds ratio for developing AD was higher in women than menin the Piteå Study (OR= 1.30, 95% CI: 0.99–1.69), and inthe meta-analysis (OR= 1.56, 95% CI: 1.16–2.10).

In all age groups in table 1, the incidence of AD washighest in the meta-analysis although there was some het-erogeneity between the eight studies, particularly in the olderage groups where there were few data. Several factors couldexplain the variation in incidence between the studies in ta-ble 1, and these factors might define possible subgroups inthe Markov model. First, the incidence depends on the casedefinition of AD (that is, the cognitive and functional testsand the diagnostic criteria used), and the method and accu-

DATA REQUIREMENTS IN A MODEL OF OF ALZHEIMER’S DISEASE 15

Table 1Annual, age-specific incidence per 100 population at risk of AD in selected studies.

Study characteristics Annual, age-specific incidence per 100Source Place and time Type and size 60–64 65–69 70–74 75–79 80–84 85–89 90–94

Seshadriet al. [18]a

Framingham, USA.1965–95

Population-based cohort(1061 men aged>65)

– 0.060 0.120 0.340 0.460 0.560 0.800

Population-based cohort(1550 women aged>65)

– 0.040 0.080 0.420 0.780 1.260 1.040

Andreasenet al. [19]

Piteå, Sweden.1990–95

Hospital referrals inpopulation-based cohort(619 men and womenaged>40)

0.013b 0.102 0.326 0.481 0.521 0.575 0.363c

Gao et al. [20] Sweden (3 studies),USA (2 studies),UK, France, Japan

Meta-analysis of 8 studies(347–2792 men andwomen)

0.058 0.186 0.506 1.174 2.310 3.585 5.488

aAnnual incidence estimated from five year risk.b Age 40–64.c Age>90.

racy of case ascertainment (for example, population screen-ing or hospital referrals). Second, the incidence of ADcould be overestimated if there is high mortality due to othercauses [18]. In the model proposed herein, we distinguishbetween excess mortality due to AD and natural mortality.Third, the incidence may vary due to other attributes, for ex-ample, history of head trauma or thyroid disease, family his-tory of dementia [21], and other extrinsic factors such as ge-ographical location, nutrition, and ingestion of aluminium.

2.2. Rates of AD progression

The annual, AD-progression rate is measured by themean annual change in score according to a particular psy-chometric or functional test. Quantifying the diagnostic ac-curacy of these criteria is the subject of much current re-search, for example, [22] gives estimates of test sensitivityand specificity, and possible confounding variables for theMMSE score. Table 2 shows estimates of the rate of ADprogression, as measured by several tests, drawn from threesources. In the studies shown in table 2, the range of themean annual rate of decline is 1.8–4.5 for MMSE, 2.6–5.3for IMCT, 1.5–4.2 for BDS, 2.6–4.5 for BIMC, and 12.3–13for CAMCOG. Some of this inter-study variation might bedue to small sample sizes and short follow-up periods. Inaddition, some measures of progression have poor repeata-bility (within-individual variation) which may give rise tovariation between studies [23,24]. It is possible that somevariation in AD progression is due to attributes such as age,gender, and ApoE genotype. However, in their review ar-ticles, Bracco et al. [23] and Agüero-Torres et al. [24] re-port that many studies of predictors of decline give incon-sistent results. One study found no significant predictors ofthe mean annual change in ESD score in 66 AD subjects,where the examined attributes included age, age at onset,gender, severity of AD, education, and family history [25].In contrast, Piccini et al. [26] and Goldblum et al. [27] showdifferent rates of AD progression in slow and fast decliners(IMCT: slow, 6.8; fast, 2.9, and MMSE: slow, 1; fast, 2.9),

and Frisoni et al. [28] show variation in rates of progressionby ApoE genotype. Variation in progression according tosuch attributes could be taken into account in a natural his-tory model.

Table 2 shows annual rates of AD progression that arenot correlated with age. Table 3 shows estimates of severity-dependent transition probabilities for AD progression usingdata drawn from three sources. The definition of diseaseseverity (minimal, mild, moderate, and severe) depends onthe psychometric or functional test used. Source I, citedin [14,15], refers to data on changes in MMSE scores over aperiod of 24 weeks in the placebo arm of a US randomisedcontrolled trial of donepezil [10]. The placebo group con-sisted of 318 AD subjects aged>75 years. Source II, citedin [16], uses data on changes in MMSE scores in the Kung-sholmen Project, a Swedish population-based cohort studyof 206 AD subjects aged>75 years who were followed upfor a mean of 3.32 years. Source III, cited in [17], usesdata on annual changes in CDR (Clinical Dementia Ratingscale) in the CERAD (Consortium to Establish a Registryfor Alzheimer’s Disease) study, a US cohort of 1145 ADsubjects. The transition probabilities presented in table 3 areconditioned on being alive at the end of the Markov cycle.

2.3. Natural- and excess-mortality rates

Table 4 shows age-gender-specific mortality rates for allcauses of death in England and Wales in 1984 and 1994 [29].Age-specific, all-cause mortality was higher in men thanwomen, and has decreased over time. Although some con-tribution to mortality is attributable to AD, it is assumed thatthis contribution is small, and, hence, the all-cause-mortalityrate is a good approximation to the natural-mortality rate.The study by Robinson [30] reports annual survival by agesince referral when AD is a contributing cause of death. Ta-ble 4 shows the one-year, age-severity-specific, case-fatalityrate following referral. The excess mortality rate is thedifference between the case-fatality rate and the natural-mortality rate. These rates are at least 10% for the data in

16 T.J. CHAUSSALET, W.A. THOMPSON

Table 2Studies providing estimates of rates of AD progression.

Source Type and date of Place Sample size and follow-up Psychometric test Annual rate ofstudy AD

progression(points/year)

Bracco et al. Systematic review Europe 2 studies MMSE 3.5–4.32[23] of longitudinal (n = 90–110, follow-up= 1 yr)

studies in clinical 3 studies IMCT 2.6–4.4case series, (n = 31–110, follow-up= 0.25–1 yr)1992–Apr. 1996 1 study BDS 3.5

(n = 60, follow-up= 0.25 yr)USA 8 studies MMSE 1.8–4.5

(n = 30–430, follow-up= 1–4 yr)7 studies IMCT 3.2–5.3

(n = 40–430, follow-up= 1–8 yr)2 studies BDS 1.5–2.1

(n = 53–430, follow-up= 4 yr)

Agüero-Torres Review of studies 9 studies MMSE 2.7–4.5et al. [24] including the (n = 21–373)

Kungsholmen 7 studies BIMC 2.6–4.5project, 1987 (n = 40–190)

3 studies BDS 1.8–4.2(n = 56–186)

2 studies CAMCOG 12.3–13(n = 53–85)

Bowler et al. Prospective, Canada 1 study ESD 30.8[25] longitudinal study (n = 66)

MMSE, Mini-Mental State Examination; IMCT, Information Memory Concentration Test; BDS, Blessed Dementia Scale; BIMC,Blessed Information Memory Concentration; CAMCOG, Cambridge cognitive examination (part of the Cambridge examination formental disorders of the elderly, CAMDEX); ESD, Extended Scale for Dementia.

Table 3Studies providing estimates of transition probabilities for AD progression by severity status.

Transition from severity status Source of data Transition probabilities per Markov cycle to severity status(see text) Minimal Mild Moderate Severe

Minimal I 0.778 0.222 0 0II 0.182 0.273 0.364 0.182III – – – –

Mild I 0.096 0.578 0.307 0.019II 0 0.500 0.250 0.250III – 0.628 0.329 0.043

Moderate I 0 0.166 0.300 0.533II 0 0 0.364 0.636III – 0.045 0.597 0.358

Severe I 0 0 0 1II 0 0 0 1III – 0 0 1

Sources I and II diagnose severity according to the Mini-Mental State Examination, Source III employed the ClinicalDementia Rating scale and recorded no individuals who had a Minimal score. In each source, the Markov cycle was6 months, 3.32 years, and 1 year, respectively.

most age-severity groups in table 4. However, some esti-mates of the case-fatality rates are based on small numbersof referrals, and, therefore, in subsequent calculations, weassume that the excess mortality rate is approximately 10%.

The studies in [16,17] provide data on excess mortality,but these data are not age-specific. In [16], the probability ofdying during a Markov cycle of 3.32 years is 0.463, 0.363,0.738, and 0.886, respectively, for increasing AD severity

(minimal to severe). In [17], the annual probability of dyingis 0.021, 0.053, and 0.153, respectively, for increasing ADseverity (mild to severe).

3. Description of the natural history model

Markov models have been used in many applicationsof medical decision making and economic evaluation, see,

DATA REQUIREMENTS IN A MODEL OF OF ALZHEIMER’S DISEASE 17

Table 4Age–gender-specific, all-cause-mortality rate per 100 population at risk of AD in England and Wales in 1984 and 1994, and one-year,

age-severity-specific, case-fatality rate.

Study characteristics Annual, age-specific mortality rate per 100Source Type and Year of Cause of Stratified 65–69 70–74 75–79 80–84>85

place of study mortality bystudy

Office for England and 1984 All Male 3.49 5.44 8.47 12.97 21.36National Wales Female 1.84 2.91 4.81 8.14 16.73Statistics [29] national vital 1994 All Male 2.77 4.54 7.31 11.30 18.85

statistics Female 1.60 2.63 4.33 7.25 14.66Robinson Cohort of 51 1973–80 ADa Mildb 19.1 29.3 41.0(personal men and 167 (n = 18) (n = 58) (n = 39)communication, women Moderate 0.0 25.0 46.7see also [30]) referrals in (n = 22) (n = 48) (n = 15)

Oxfordshire, Severe 40.0 25.0 –UK (n = 5) (n = 8)

aOne-year mortality rate following referral (number referred).b AD severity according to Mini-Mental State Examination score: Mild, 22–30; Moderate, 10–21; Severe, 0–9.

for example [31]. The proposed non-homogeneous Markovmodel provides a framework that facilitates discussion of theprocesses describing the natural history of AD. In the first in-stance, our goal is to provide a framework for the natural his-tory model, and use this to gain understanding of the naturalhistory of AD in the absence of specific interventions. Theflow model in figure 1 shows the annual transition rates be-tween the Markov states:No AD (i = 0), Mild AD (i = 1),Moderate AD(i = 2), Severe AD(i = 3), natural mor-tality, and AD-specific mortality for each state of cognitiveimpairment. The Markov states taken in this order comprisethe state vector, which depends on the age of the cohort ofindividuals considered in the model. Leta0 be the age (inyears) of the cohort initially. As time proceeds, the age ofthe cohort at timet (in years) isa(t) = a0 + t , t > 0. Thetransition matrix for the Markov model at time t is written inthe following conventional block form [32]

[Q(a(t)) M(a(t))

0 I

],

in which 0 andI are the null and identity matrices of order4, respectively, and matricesQ(a) andM(a) are given by

Q(a) =

S0(a)

(1−∑3

i=1F(ν0i ))

S0(a)F (ν01)

0 S1(a)(1− F(ν12)− F(ν13))

0 0

0 0

S0(a)F (ν02) S0(a)F (ν03)

S1(a)F (ν12) S1(a)F (ν13)

S2(a)(1− F(ν23)) S2(a)F (ν23)

0 S3(a)

,

M(a) =

F0(a) 0 0 0

F0(a) F1(a) 0 0

F0(a) 0 F2(a) 0

F0(a) 0 0 F3(a)

.

The following notation is used to define the elements of theabove matrices:

F(νij ) = 1− e−νij 1t, 06 i < j 6 3,

F0(a) = 1− S0(a) = 1− exp

[−∫ a+1t

a

µ(α) dα

],

a > a0,

Fi(a) = 1− exp

[−∫ a+1t

a

δi(α) dα

],

i = 1,2,3; a > a0,

whereF andS denote the distribution and survival function,respectively. The Markov cycle is of duration1t (in years).Annual rates of transition are as defined previously, and areshown in figure 1. In the model, it is assumed that the transi-tion probabilities are independent, conditioned on survivingduring a Markov cycle. The probability of surviving duringa Markov cyle when cognitively impaired is

Si(a) = 1− F0(a)− Fi(a), i = 1,2,3; a > a0.

We are guided in formulating the model by the availabilityof data, and, thus, the AD-progression rates are assumed tobe independent of age. In addition, we have not modelledtransition from the stateModerate ADto Mild AD. Althoughthis transition is evidenced by data in table 3, these data areat variance with the degenerative nature of AD when thereis no intervention, and such data might represent variation incase definition. The natural history model has been imple-mented using widely available spreadsheet software.

18 T.J. CHAUSSALET, W.A. THOMPSON

Figure 2. Yearly survival and probability of developing severe AD from time of mild-AD onset at age 65. See text for details.

4. Results and discussion

Figure 2 shows the survival (solid lines) and probabilityof developing severe AD (dotted lines) under four assump-tions (survival curves labelled a–d) on the excess mortality-rate for an individual who is correctly diagnosed with mildAD at age 65. The difference between a survival curve andthe corresponding curve showing the probability of severeAD is the probability of having either mild AD or moder-ate AD. Natural mortality data are for a male in 1994 (ta-ble 4), and rates of progression areν12 = 0.7, ν13 = 0.04,ν23 = 1.4, which are consistent with the transition prob-abilities from source I, shown in table 3. Calculations arebased on a Markov cycle of 6 months. When there is noexcess mortality, so that the only source of mortality is nat-ural mortality, the median survival is 14–14.5 years (curvea). If the annual, excess mortality-rate is 0.1, and is inde-pendent of age and AD-severity status, then the median sur-vival is 5–5.5 years (curve b). A more realistic assumptionconcerning excess mortality might be that only severe ADis associated with an increased burden of mortality, that is,δ1(a) = δ2(a) = 0. Curves c and d are obtained respec-tively whenδ3(a) = 0.1 in all age groups (median survival7–7.5 years), and whenδ3(a) is 0.05, 0.1, and 0.15 in the agegroups 65–74, 75–84, and>85, respectively (median sur-vival 9–9.5 years). Figure 2 shows that the median survivalis appreciably higher when excess mortality is not modelled(curve a), and this might have an impact on the costing ofpost-intervention care. When severe AD only is associatedwith an additional burden of mortality, the median survival issensitive to the form of the excess mortality (curves c and d).

The present model has focused attention on the data re-quired in order to gain an understanding of the natural his-tory of AD before consideration is given to modelling thecost–effectiveness of specific therapeutic interventions. Wefind that the model is sensitive to the form of excess mor-tality. More extensive data on age-severity-specific survivaland disease progression would enable us to establish a ro-bust base-case from which to assess the cost–effectiveness ofsuch interventions. In future work, we will modify the nat-ural history model to include a post-intervention treatmentarm following a screening programme, with the aim of re-ducing the age-specific incidence between screening events.The treatment arm will model the loss of follow-up (for ex-ample, due to non-compliance arising from psychologicaland social factors, and drug-tolerability), which, by analogywith the excess mortality, might affect the cost–effectivenessof the screening programme.

Acknowledgements

The authors thank Professor Peter Millard for commentson the manuscript, and Dr John Robinson for allowing themto use previously unpublished data.

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