Fitting the epidemiology and neuropathology of the early stages of Alzheimer’s disease to prevent dementia

Introduction Recent research on biomarkers has made possible the diagnosis of pre-dementia and even preclinical Alzheimer’s disease (AD), thus providing the ideal context for prevention. The aim of this study was to investigate the epidemiology of the early stages of AD by fitting neuropathologic and epidemiological data to assess the feasibility of prevention programs. Methods The study addressed primarily the construction of a discrete event simulation model of the stages of dementia. Age was included in the mathematical functions to combine the two competitive risks that determine the epidemiology of AD, that is, time to onset of dementia and time until death by other causes. Subsequently, this model was calibrated to reproduce the prevalence of pathological findings associated with AD. The beginning of the preclinical stage was taken to coincide with Thal phase 1 deposition of amyloid-beta. The duration of the prodromal stage, marked by mild cognitive impairment, was based on a 10% annual conversion rate from this level of impairment to dementia. The validation of prevalence figures also permitted estimation of the incidence and duration of preclinical and prodromal stages. Results In Spain, half of the nearly 10 million people aged more than 60 years are in the early stages of AD; 35.9% are in a preclinical stage, and up to 14.2% are in a prodromal stage. However, dementia will develop in only 38% of this population. The weighted mean time to dementia was 22.0 years from the start of Thal phase 1 and 9.0 years from the start of phase 2. Results of simulation models showed a lack of correlation between clinical and pathological classifications. Conclusions These findings raise questions about the feasibility of drug-based prevention strategies. Currently, screening programs with biomarkers in the early stages of AD cannot be applied to the half of the general population older than 60 years. Hence, intensive research is needed regarding risk factors, so that more affordable strategies may be planned. More efficient criteria are also needed to select those subjects with mild cognitive impairment who have an increased probability of positive screening for biomarkers (prodromal stage). Electronic supplementary material The online version of this article (doi:10.1186/s13195-014-0079-9) contains supplementary material, which is available to authorized users.

-At the age of 60 for instance this individual was in the preclinical phase according to epidemiological classification and Thal phase 2 according to pathological classification.
-The individual would die of other causes at 76 years in the preclinical stage and Thal phase 3 before entering the stage of dementia.

Allocation of individuals and incidence and prevalence measurement
The computation of the entities that pass through each of the phases/stages allowed both the annual incidence and prevalence to be ascertained. We used two variables for this purpose.

SYSTEM PARAMETERS
In the following, we provide a brief description of the main system parameters.

Input population
The focus of the study was the population over 40 years of age. As mentioned above, to avoid a long warming-up period all cohorts were introduced at the same time, assigning them an appropriate health status at that point. This was not, however, sufficient, since the incidence was also analyzed. Therefore, the entry of entities to the system was staggered, with the whole population over 40 years of age being Population data were obtained from the Spanish National Institute of Statistics (INE) [4]. An empirical distribution was used to determine the age at entry of each individual.
Time to death due to other causes Individuals with AD are at the same time exposed to other risks such as unrelated diseases, traffic accidents, etc. Therefore, some may die before the disease developed. We assumed that individuals in the preclinical and prodromal stages had general population mortality rates. On the other hand, once that they enter the stage of dementia it was assumed that they could die either due to the disease or other causes.
The DES makes it necessary to consider time explicitly. Time to an event was modeled by using the Gompertz distribution, which as a function of age, can be expressed as follows [5] (1). This equation includes a uniformly distributed random factor between 0 and 1 (u) and two parameters α and β which define the characteristics of the distribution. These parameters, α and β, are estimated from a linear regression of a logarithmic transformation of the rates of occurrence of the event with respect to age, expressed in Equation 2 (2) where (4).

Time to dementia
Time to dementia was the time between the reference age and the loss of capacity to perform instrumental activities of daily living. It was also the time to be taken as reference for other calculations since dementia is the only known stage of AD. For calculating the time until dementia we took a reference age of 50 years. AD is rarely expressed clinically before the age of 60. However, almost 1% of the population between 60 and 65 years suffer it. Therefore, the reference age had to be set somewhat earlier so that there could be incident cases of AD-dementia by that age. In order to build a Gompertz function, AD-dementia incidence rates given by Fratiglioni were used [6] with the same methodology as previously described [5].
Time from the appearance of Aβ deposits to AD-dementia Currently, it is not possible to measure the Aβ level in living individuals, rather it can only be ascertained by post-mortem brain biopsy, so the duration of this stage is unobservable. However, Braak et al described the prevalence of Thal phases in terms of prevalence by age group [1]. This information allowed us to ascertain this time by calibration, prevalence in general terms being the consequence of the interaction of incidence and duration, and calibration involving the fitting of these two in order to obtain the observed prevalence.
The first step was achieved when the model reproduced the incidence and prevalence of the dementia stage. The second step consisted in estimating the duration of the preclinical stage by adjusting the model until the results matched the pattern observed for Thal phase 1 in the Braak et al study [1].As the parameter Thal phase 1 prevalence couldn't be directly observed, calibration was performed using a random search method [7]. Calibration is the process of determining the values of unobservable parameters by constraining model output to replicate observed data [7]. The epidemiological parameters for Thal phases 2 and 3 were also calculated by applying the same method.
This same procedure was used to determine the beginning of each of the Thal phases.

Time from MCI to AD-dementia
Considering that the rate of conversion is constant over time and applying the exponential function (mean time=1/rate), we obtained the duration of prodromal stage.

Survival in the clinical stage
These values were obtained from Dodge's clinical series [8].
In order to obtain stochastic values, an exponential function was built. The mean value was obtained from a polynomial function (Equation 5), the parameters for which are listed in Table 1.

Statistical assessment of validation and calibration
Calibration was performed using a random search method [7]. For each phase, we established the time from the beginning of that phase to dementia. Thal phase's prevalence was obtained in 2010 and grouped by age and sex. The model was assessed using the following goodness-of-fit statistics: the correlation coefficient (R), normalized mean square error (NMSE), fractional bias (FB), fractional variance (FV) and the fraction of predictions within a factor of two (FAC2) [7].The same approach was applied to assess the validation process for dementia incidence and prevalence.

MODEL RESULTS
In the following, we describe the results in detail.

Validation results
On the one hand, the dementia stage is the one for which results have been validated, since it is the only stage for which data are available, specifically the following: 1) the incidence rate, and 2) the prevalence rate. The incidence rates obtained were compared with the results of the meta-analysis of Fratiglioni et al [6] (Figure 3 and Table 2) and the prevalence with the one held values reported by Lobo et al [9] (Figure 4 and Table 3). Note that incidence is not an input but the result of the interaction between the competing risks of having dementia and death due to other causes. Likewise, the prevalence is not an input; rather it is determined by dementia incidence and dementia survival time.
On the other hand, Thal phases have been validated. For this, we compared the prevalence results obtained by Braak et al [1] and those obtained with the model. This demonstrated that the calibration had been performed correctly ( Figure 5 and Table 4).

Goodness-of-fit results
All the statistics used to assess the goodness-of-fit were within the established criteria as shown in Tables 5 and 6. Time to dementia (Fratiglioni) Time from βA deposit to dementia The time that goes from Aβ deposit to dementia is both an input and a research result obtained by calibration. As Thal stages prevalence is known and its also the result of the iteration between incidence and duration, disease development duration has been calibrated to obtain Thal stages duration.