Relationships between β-amyloid and tau in an elderly population: An accelerated failure time model

Abstract

Using positron emission tomography (PET)-derived amyloid and tau measurements from 1,495 participants, we explore the evolution of these values over time via an accelerated failure time (AFT) model. The AFT model assumes a shared pattern of progression, but one which is shifted earlier or later in time for each individual; an individual’s time shift for amyloid and for tau are assumed to be linked. The resulting pattern for each outcome consists of an earlier indolent phase followed by sharp progression of the accumulation rate. APOE ε4 shifts the amyloid curve leftward (earlier) by 6.1 years, and the tau curve leftward by 2.6 years. Female sex shifts the amyloid curve leftward by 2.4 years and the tau curve leftward by 2.6 years. Per-person shifts (i.e., the individual’s deviation from the population mean) for the onset of amyloid accumulation ranged from 13 years earlier to 13 years later (10th to 90th percentile) than average and 11 years earlier to 14 years later for tau, with an estimated correlation of 0.49. The average delay between amyloid increase and tau increase was 13.3 years.

1. Introduction

The combination of aggregated β-amyloid deposits (in the form of plaques composed of Aβ42 and other peptides, hereafter “amyloid”) and aggregated hyper-phosphorylated tau deposits (in the form of neurofibrillary tangles made up of paired helical filaments containing 3R/4R tau, hereafter “tau”) histopathology has defined Alzheimer’s disease (AD) since the earliest formal diagnostic criteria (McKhann et al., 1984). The two specific proteinopathies, plaques (Glenner and Wong, 1984) and tangles (Brion et al., 1985), were described in the 1980s. The inciting mechanisms for amyloid accumulation remain controversial although age and genetics are established risk factors (Liu et al., 2013). The genetics of autosomal dominant AD pointed to disordered Aβ metabolism as the initiating event in the disease, which led to the amyloid cascade hypothesis (Hardy and Higgins, 1992). The inciting mechanisms for tau accumulation are equally uncertain beyond the presence of elevated amyloid (Ittner and Gotz, 2011). Amyloid and tau positron emission tomography (PET) studies, and inference from neuropathological studies, have confirmed that amyloidosis precedes the development of extensive neocortical tauopathy (Jack et al., 2019). Overt cognitive impairment occurs subsequently (Brier et al., 2016; Johnson et al., 2016).

A descriptive model that ordered the evolution of biomarkers of AD pathophysiology and cognitive impairment as staggered time-dependent functions (Figure 1) (Jack et al., 2013; Jack et al., 2010) evolved from the mechanistic model proposed in the amyloid cascade hypothesis (Glenner and Wong, 1984; Hardy and Selkoe, 2002). By expressing the magnitudes of biomarker and clinical findings relative to time, the descriptive model introduced a quasi-mathematical framework suitable for quantitative in vivo analyses (Jack et al., 2013). Although the mechanistic and descriptive models are not independent of one another, the former makes causal claims while the latter simply describes observable phenomenon. Our emphasis here will be on the temporal aspects of the relationship between amyloid and tau accumulation using PET imaging. The descriptive model assumed that there was a time delay between amyloid accumulation and tau accumulation, but when the model was first formulated (Jack et al., 2010), there were no data on what the time delay was. Are the accumulations of amyloid and tau inevitably linked in time or is their sequential appearance simply an outcome of other forces?

Figure 1.

Hypothetical descriptive biomarker model

The horizontal axis has age as the timescale and the vertical axis represents severity of biomarker and cognitive abnormality from completely normal (Min.) to maximally abnormal (Max.). The grey area denotes the zone in which abnormal pathophysiology lies below the biomarker detection threshold, e.g. due to imaging constraints. Amyloid biomarkers become abnormal first, followed by tau, followed by biomarkers of neurodegentration. Cognitive impairment is the last event in the progression of the disease. All biomarker curves (as well as cognitive impairment) are configured as sigmoids; however, the right side of the sigmoids correspond to a deceleration of abnormality which occurs late in the disease process when many individuals are no longer able to participate in research imaging (and thus are not represented in the data set). For this reason, the right side of the curves are shown as a dashed line. Modified from (Jack et al., 2013).

We addressed these questions using data from participants enrolled in either the Mayo Clinic Study of Aging (MCSA) or the Mayo Clinic Alzheimer’s Disease Research Center (ADRC). All participants were aged 50 or older and had either longitudinal amyloid PET imaging or both amyloid PET and tau PET within a short period, typically the same day. We hypothesized that we could represent the amyloid and tau PET standardized uptake value ratio (SUVR) values in a model by accounting for participant-to-participant variation in age of accumulation of either protein aggregate by a left–right time shift in the accumulation curves. The large collection of amyloid and tau PET scans from participants in the MCSA (1361 participants, age range 50–94 years, 45% women, 89% cognitively unimpaired, 10% with mild cognitive impairment, 1% with dementia) and ADRC (134 participants, age range 51–92 years, 56% women, 100% with dementia) allowed us to empirically fit this model, looking at the two outcomes of amyloid accumulation and tau accumulation.

2. Materials and Methods

2.1. Enrollment and clinical characterization of participants

All individuals in this study were enrolled in one of two studies. The MCSA is a longitudinal population-based study of cognitive aging among a stratified random sample of a geographically defined population (Roberts et al., 2008). Residents of Olmsted County, Minnesota, USA aged 30–89 years old were enumerated using the medical records-linkage system of the Rochester Epidemiology Project (St Sauver et al., 2012b). From this sampling frame, individuals were randomly selected within 10-year age and sex strata such that men and women were equally represented. All individuals without a medical contraindication were invited to participate in imaging studies. The ADRC is a longitudinal research study of individuals recruited from clinical practice.

Evaluations included a medical history review, an interview with the participant and a study partner, a neurologic examination by a physician; and a neuropsychological examination (Roberts et al., 2008). Participants were assigned a diagnosis of cognitively unimpaired (CU, defined as not mild cognitive impairment [MCI] or dementia), MCI (Petersen, 2004), or dementia (American Psychiatric Association, 1994) using established criteria. Among individuals with dementia only those with a clinical diagnosis of Alzheimer’s clinical syndrome (Jack et al., 2018a; McKhann et al., 1984; McKhann et al., 2011) were included in this study. Clinical diagnoses were always made blinded to PET results in the MCSA. In the ADRC, where individuals present with cognitive symptoms, the physician is not blinded to biomarker information.

For inclusion, participants must have been at least age 50 and had a minimum of 2 PET scans, either serial amyloid PET or both an amyloid and tau PET. (Some 96% of all tau PET scans occurred on the same day as an amyloid PET scan.) Many participants had more PET scanning than this minimum inclusion requirement with more than a third having 3 or more longitudinal PET scans.

2.2. Imaging methods

Amyloid PET imaging was performed with Pittsburgh Compound B (Klunk et al., 2004) and tau PET with flortaucipir (Chien et al., 2013) on either GE or Siemens scanners. PET harmonization was performed following the methods developed for ADNI (Joshi et al., 2009). Amyloid and tau PET SUVR values were formed by normalizing composite multi-region target regions of interest (ROIs) to the cerebellar crus gray matter (Jack et al., 2017). The amyloid PET target meta-ROI included the prefrontal, orbitofrontal, parietal, temporal, anterior and posterior cingulate, and the precuneus (Jack et al., 2017). Amyloid PET was also expressed in centiloid units (Klunk et al., 2015). The tau PET target meta-ROI included the amygdala, entorhinal cortex, fusiform, parahippocampal, and inferior temporal and middle temporal gyri (Jack et al., 2017). PET data were not partial volume corrected. MRI was performed at 3T on either GE or Siemens scanners and was used in the PET data processing pipeline, described in previous work (Schwarz et al., 2019). SUVR values ≥1.48 or ≥ 1.29 were considered abnormal for amyloid PET and tau PET, respectively (Lowe et al., 2019a; Lowe et al., 2019b).

2.3. Accelerated failure time model

We used an accelerated failure time (AFT) model (Wei, 1992): y_i(t) = f(t + b_i + X_iβ) + ϵ where t is participant age, y_i(t) is the set of measurements for a participant, f is a smooth increasing function of age, ϵ is a random error, and b_i is the per-participant offset or shift corresponding to whether the participant starts the accumulation process earlier or later than the population average. Per-participant covariates X_i were also added. This AFT model differs from the more common linear mixed effects (LME) model y_i(t) = f(t) + b_i + X_iβ + ϵ in having the per-participant terms inside the parentheses; the effect is to shift a participant’s curve left–right rather than up–down. Our fitted model addresses both amyloid and tau at the same time, so there are two smooth functions f and g, two sets of regression coefficients, and two offsets or shifts per participant: b_i for amyloid and d_i for tau.

The formal model for the amyloid and tau levels is

$$\begin{gathered} \text{log}\left(a_{ij}\left(t_{ij}\right)\right)=f\left(t_{ij}+{\eta }_i\right)+{ϵ}_{ij1} \\ {\eta }_i=b_i+x_{i1}{\beta }_{11}+x_{i2}{\beta }_{12}+x_{i3}{\beta }_{13}+x_{i4}{\beta }_{14} \end{gathered}$$

$$\begin{gathered} \text{log}\left({\tau }_{ij}\left(t_{ij}\right)\right)=g\left(t_{ij}+{\gamma }_i\right)+{ϵ}_{ij2} \\ {\gamma }_i=d_i+x_{i1}{\beta }_{21}+x_{i2}{\beta }_{22}+x_{i3}{\beta }_{23}+x_{i4}{\beta }_{24} \\ {ϵ}_{ij1}~N\left(0,{\sigma }_1^2\right),{ϵ}_{ij2}~N\left(0,{\sigma }_2^2\right) \\ \left(b_i,d_i\right)~t_5\left(m,{\sigma }_3^{2}A\right) \\ A=\left(\begin{array}{ll}1\hfill & \rho \hfill \\ \rho \hfill & 1\hfill \end{array}\right) \end{gathered}$$

where a_ij is the jth amyloid measurement for participant i, measured at age t_ij, τ_ij is the corresponding measure of tau, and ϵ is a Gaussian random error. Onset of the amyloid process for participant i is shifted left–right by the random effect b_i and onset of the tau process is shifted by the random effect d_i. The two random effects have correlation ρ, and were drawn from a t-distribution with 5 degrees of freedom, which allows more flexibility for outliers in the onset time as compared to a Gaussian distribution. The centering values m are the mean of the onset age for amyloid and tau, respectively. The four covariates x for each participant are APOE (ε4− = 0, ε4+ = 1), sex (female = 0, male = 1), years of education, and the cohort (population-based MCSA = 0, referral-based ADRC = 1), with separate regression coefficients for amyloid and tau; the covariates also are assumed to act by inducing a left–right shift in the onset of the processes.

The smooth functions f and g for the amyloid and tau processes, respectively, were chosen as the simplest possible non-linear function: a line with a single change point at zero. The Markov Chain Monte Carlo (MCMC) method used to fit the model requires smooth first derivatives, however, so this was modified to be the hyperbola tangent to the two lines; this smooths out the sharp corner. The equation of the resulting curve is

$$f(x)=\left({\gamma }_1+{\gamma }_{2}x\right)+\frac{{\gamma }_3}{2}\left(x+\sqrt{x^2+k^2}\right)$$

The first two parameters define the intercept and slope of the line segment for x < 0 and γ₃ is the change in slope at 0. (The change point itself is estimated by the parameter m above.) The free parameter k determines how tightly the hyperbola tucks into the corner defined by the two lines; a value of k = 0 gives a hockey stick shape with a sharp corner and larger values result in a smoother transition. A second function g(x) of the same form was used for the tau effect.

2.3.1. Implementation details

The model data has one row per scan, with the participant identifier, scan type (amyloid or tau), age at scan, and the covariates APOE, sex, education, and cohort. Each participant may have a different number of rows. The parameters of the model include an amyloid and tau offset for each participant, along with overall effects (separate for amyloid and tau) for APOE, sex, education, and cohort. If a particular participant has no tau scans, the tau offset for that participant is not connected to any data and the parameter becomes a random draw from the posterior distribution. (This has no detrimental effect on the solution, other than perhaps a slight increase in compute time.) These free parameters, being uninformative, are omitted from the final tabulations and graphs of results.

The model allows for amyloid non-accumulators in a natural way: their posterior maximizes with the change point at an age that is well beyond their last observed amyloid time point. A person who is “due to” begin accumulating amyloid 10+ years from now is indistinguishable from one who will never accumulate amyloid, both in observed data and in the fitted model.

The model was fit using Hamiltonian Markov chain Monte Carlo (MCMC) using the rstan package, R version 3.6.2, using 5000 iterations for burn-in and retaining 5000 iterations in each of 4 chains. We inspected all MCMC traces visually, examined convergence diagnostics, and performed posterior predictive checks to assess model fit and adequacy (Gelman et al., 2013). Code for the fit and further details can be found at https://github.com/therneau/AFTmodel. Tables of model parameters and their interpretation are provided as supplementary material.

2.4. Cohort

We chose to combine MCSA and ADRC participants and treat cohort as a covariate in the model for several reasons. Combining the cohorts allows us to utilize all available PET data and provide more precise estimates. We also consider the cohorts complimentary as inclusion of those with Alzheimer’s clinical syndrome from the ADRC adds to the dynamic range of the amyloid and tau PET values available for analysis. Further, the ADRC participants give information about amyloid and tau dynamics from later in the disease course while the MCSA participants give information about dynamics earlier in the disease course. The use of ADRC participants is particularly valuable when it comes to tau PET, which is closely tied to cognition (Brier et al., 2016; Johnson et al., 2016). By virtue of it being population-based, both cognitive impairment and highly elevated tau are relatively uncommon in the MCSA. Given that MCSA and ADRC participants are systematically different by design, we also fit the AFT model among only MCSA participants.

2.5. Model comparisons

In order to illustrate conceptual differences between the AFT approach and that of linear mixed effects (LME) models we fit an LME model using amyloid PET as the response and using a natural spline to account for non-linear age effects. The two modeling approaches are compared in the discussion. We also discuss other models found in the literature.

3. Results

3.1. Demographics

Table 1 provides demographic summary information for the full PET cohort and the subset with tau PET. Data are shown overall, within the population-based Mayo Clinic Study of Aging (MCSA) cohort, and within the referral-based Alzheimer’s Disease Research Center (ADRC) cohort.

Table 1.

Demographic summaries for the full PET cohort and the subset with tau PET.

Characteristic	PET cohort Combined (n = 1495)	MCSA (n = 1361)	ADRC (n = 134)	Tau PET subset Combined (n = 1141)	MCSA (n = 1039)	ADRC (n = 102)
Age at first amyloid PET, y
Median (IQR)	72 (64, 78)	72 (64, 78)	68 (60, 77)	69 (62, 77)	69 (62, 77)	68 (61, 76)
Range	50 to 94	50 to 94	51 to 92	50 to 93	50 to 93	52 to 89
Age at first tau PET, y*
Median (IQR)	–	–	–	71 (64, 80)	72 (64, 80)	68 (61, 76)
Range	–	–	–	50 to 99	50 to 99	52 to 91
Female sex, n (%)	684 (46%)	609 (45%)	75 (56%)	548 (48%)	490 (47%)	58 (57%)
Education, y†
Median (IQR)	15 (12, 16)	14 (12, 16)	16 (13, 18)	15 (13, 16)	15 (13, 16)	16 (13, 18)
Range	0 to 25	0 to 20	11 to 25	0 to 25	0 to 20	12 to 25
APOE ε4 carrier, n (%)	483 (32%)	393 (29%)	90 (67%)	378 (33%)	308 (30%)	70 (69%)
Diagnosis at first PET, n (%)
CU	1216 (81%)	1216 (89%)	0	947 (83%)	947 (91%)	0
MCI	138 (9%)	138 (10%)	0	87 (8%)	87 (8%)	0
Alzheimer’s clinical syndrome	141 (9%)	7 (1%)	134 (100%)	107 (9%)	5 (0%)	102 (100%)
Duration from first amyloid PET to first tau PET, y*
Median (IQR)	–	–	–	2.4 (0.0, 4.8)	2.4 (0.0, 5.0)	0.0 (0.0, 0.0)
Range‡	–	–	–	−3.1 to 11.4	−0.1 to 11.4	−3.1 to 5.1
Amyloid PET at first amyloid PET
SUVR, median (IQR)	1.41 (1.33, 1.68)	1.39 (1.32, 1.52)	2.46 (2.18, 2.71)	1.40 (1.32, 1.60)	1.39 (1.32, 1.50)	2.46 (2.21, 2.71)
SUVR, range	1.04 to 3.38	1.04 to 3.38	1.44 to 3.18	1.04 to 3.38	1.04 to 3.38	1.44 to 3.17
SUVR ≥ 1.48, n (%)§	540 (36%)	407 (30%)	133 (99%)	380 (33%)	279 (27%)	101 (99%)
Tau PET at first tau PET*
SUVR, median (IQR)	–	–	–	1.20 (1.14, 1.28)	1.20 (1.14, 1.25)	1.96 (1.60, 2.33)
SUVR, range	–	–	–	0.91 to 3.27	0.91 to 2.34	1.10 to 3.27
SUVR ≥ 1.29, n (%)§	–	–	–	257 (23%)	162 (16%)	95 (93%)
Number of amyloid PET scans, (%)
1 scan	357 (24%)	313 (23%)	44 (33%)	357 (31%)	313 (30%)	44 (43%)
2 scans	601 (40%)	548 (40%)	53 (40%)	343 (30%)	311 (30%)	32 (31%)
3 scans	360 (24%)	338 (25%)	22 (16%)	276 (24%)	261 (25%)	15 (15%)
4 + scans	177 (12%)	162 (12%)	15 (11%)	165 (14%)	154 (15%)	11 (11%)
Duration from first to last amyloid PET, y
Median (IQR)	3.9 (2.4, 6.1)	4.1 (2.5, 6.2)	1.4 (1.0, 2.2)	5.1 (2.7, 6.5)	5.2 (3.5, 6.5)	1.8 (1.0, 2.3)
Range	0.8 to 11.7	1.0 to 11.7	0.8 to 6.1	0.9 to 11.7	1.1 to 11.7	0.9 to 4.4
Number of Tau PET scans, (%)*
1 scan, n (%)	–	–	–	722 (63%)	665 (64%)	57 (56%)
2 scans, n (%)	–	–	–	314 (28%)	290 (28%)	24 (24%)
3 + scans, n (%)	–	–	–	105 (9%)	84 (8%)	21 (21%)
Duration from first to last tau PET, y*
Median (IQR)	–	–	–	2.6 (2.0, 3.8)	2.7 (2.3, 3.8)	1.9 (1.1, 2.2)
Range	–	–	–	0.9 to 5.6	1.0 to 5.6	0.9 to 4.4

Results are shown overall, within the population-based Mayo Clinic Study of Aging (MCSA) cohort, and within the referral-based Alzheimer’s Disease Research Center (ADRC) cohort.

^*Summaries for tau related variables are only shown within the tau PET subset.

^†One individual reported no formal education, 32 reported education ranging from 6 to 10 years, and 1 reported 25 years of education. To reduce the potential influence of outliers we analyzed education by setting values less than 11 years to 11 years and the value of 25 to 20 (the next highest observed education) in the accelerated failure time model.

^‡Negative numbers indicate that for some individuals the tau PET scan was performed prior to the amyloid PET scan.

^§Abnormal amyloid PET was defined as SUVR ≥1.48, the cut point beyond which rates of amyloid PET reliably increased. Abnormal tau PET was defined as ≥1.29, the optimal cut point determined using ROC analysis separating individuals with AD spectrum neuropathology diagnoses from non-AD spectrum neuropathology diagnoses.

3.2. Amyloid and tau change points

Figure 2A shows amyloid PET SUVR and centiloid (Klunk et al., 2015) values versus age for all participants (3409 amyloid PET scans from 1495 individuals) with a hypothetical common curve overlaid. Symbols and colors indicate APOE ε4 status. Our hypothesis was that each individual follows such a curve, but with their change point (“amyloid onset”) shifted left–right. Figure 2B shows the results of the fitted model, where the horizontal axis is not age but the model-adjusted amyloid age (t + b_i + X_iβ) for each participant from the fitted model. Panels C and D of Figure 2 show the observed and fitted values based on a model estimated using only MCSA participants (3132 amyloid PET scans from 1361 individuals). While limiting the analysis to MCSA participants results in few individuals with highly elevated amyloid at younger ages, the fitted values are quite similar whether or not ADRC participants are included.

Figure 2.

Relationship between age and amyloid

Panel A shows amyloid PET SUVR versus age for 3409 amyloid PET scans among 1495 individuals in either the MCSA or the ADRC with APOE genotype indicated by color and symbol. The centiloid scale is a standardized scale for reporting amyloid PET (see ref (Klunk et al., 2015) for explanation). The orange curve indicates a prototypical smooth increasing function of age (with implicit censoring of the slowing rate on the right). We assume individuals follow a trajectory that is consistent with this curve after an appropriate left–right shift of their data points. Panel B shows the fitted relationship for amyloid with adjusted amyloid age defined as a participant’s age plus random effect b_i plus covariate effects on the horizontal axis. In this way, each participant’s amyloid data have been shifted left or right based on the model estimates. The population has an estimated amyloid change point at 77.3 years, beyond which the amyloid load increases rapidly. Panel C shows amyloid PET SUVR versus age among the subset of 1361 individuals in the MCSA having a total of 3132 amyloid PET scans. Panel D shows amyloid PET SUVR versus adjusted amyloid age based on a model fit among only MCSA participants. The estimated amyloid change point here is 77.1 years.

Figure 3 shows the observed and fitted tau PET values for the sample as a whole and the subset of MCSA participants. With the relatively recent introduction of tau PET imaging, the tau data are more sparse than the amyloid PET data. Further, there are very few participants with highly elevated tau in the MCSA sample as previously reported (Jack et al., 2019; Lowe et al., 2019a). Still, the general shape of the fitted model is relatively insensitive to whether or not ADRC participants are included in the analysis.

Figure 3.

Relationship between age and tau

Panel A shows tau PET SUVR versus age for 1677 tau PET scans among 1141 individuals in either the MCSA or the ADRC with APOE genotype indicated by color and symbol. The orange curve indicates a prototypical smooth increasing function of age (with implicit censoring on the right). We assume individuals follow a trajectory that is consistent with this curve after an appropriate left–right shift of their data points. Panel B shows the fitted relationship for tau with adjusted tau age defined as a participant’s age plus random effect d_i plus covariate effects on the horizontal axis. In this way, each participant’s tau data have been shifted left or right based on the model estimates. The population has an estimated tau change point at 90.7 years, beyond which the tau load increases rapidly. Panel C shows tau PET SUVR versus age among the subset of 1039 individuals in the MCSA having a total of 1502 scans. Panel D shows tau PET SUVR versus adjusted tau age based on a model fit among only MCSA participants. The estimated tau change point based on the MCSA-only model is 94.9 years.

The parameters of the model fits related to change points or onset times are shown in Table 2. For the model combining MCSA and ADRC participants, the estimated mean age for the amyloid change point in the population is 77.3 years (95% credible interval [CI]: 76.8–77.9). The mean tau change point in the population is 90.7 years (89.2–92.3) and the estimated mean amyloid to tau delay is 13.3 years (11.8–15.0). APOE ε4 carriers in the population have a mean amyloid change which is accelerated by 6.1 years (5.0–7.2), i.e., earlier; APOE ε4 carriage accelerates the age of the tau change point by less: 2.6 years (1.2–4.1). The larger effect of APOE ε4 on amyloid than on tau should not be unexpected: amyloid is a primary driver of tau but not the only driver. This has the surprising consequence, however, that APOE ε4 lengthens the apparent amyloid to tau delay by 3.4 years (2.0–4.9). Female sex lowers the age of the amyloid change point by 2.4 years on average (1.4–3.4) and the tau change point by 2.6 years (1.3–3.9). Education has little effect on either change point. For the amyloid to tau delay, sex and education have no significant effect. The amyloid change point for the participants from the ADRC referral sample is 56.5 years (55.4–57.5) which is shifted left by about 21 years as compared to the population sample; the amount of shift depends on biases introduced by clinical referral processes and will likely differ from study to study and institution to institution. The ADRC sample has an estimated tau change point of 63.6 years (61.7–65.6) and consequently an amyloid to tau delay that is shorter (7.2 years [5.2–9.2] vs 13.3 [11.8–15.0]).

Table 2.

Change point, delay, and covariate effects.

	Amyloid change point	Tau change point	Amyloid to tau delay*
Model using MCSA and ADRC
Population mean	77.3 (76.8, 77.9)	90.7 (89.2, 92.3)	13.3 (11.8, 15.0)
Covariate effect
APOE ε4 +	6.1 (5.0, 7.2)	2.6 (1.2, 4.1)	3.4 (2.0, 4.9)
Female sex	2.4 (1.4, 3.4)	2.6 (1.3, 3.9)	−0.2 (−1.5, 1.2)
Education (1-year)	0.2 (0.0, 0.4)	0.6 (0.3, 0.8)	−0.3 (−0.6, −0.1)
Referral effect	20.9 (19.9, 21.9)	27.0 (25.4, 28.6)	−6.2 (−7.7, −4.6)
Model using MCSA only
Population mean	77.1 (76.4, 77.8)	94.9 (90.0, 101.4)	17.8 (12.9, 24.3)
Covariate effect
APOE ε4 +	7.0 (5.9, 8.2)	1.0 (−0.5, 2.5)	6.0 (4.4, 7.6)
Female sex	2.0 (0.9, 3.0)	1.7 (0.4, 3.0)	0.3 (−1.2, 1.7)
Education (1-year)	0.3 (0.1, 0.6)	0.5 (0.2, 0.8)	−0.2 (−0.5, 0.1)

Values shown are mean (95% credible interval). Estimates are in years. The covariate effects for amyloid and tau report the estimated acceleration. For example, the estimate of 6.1 for APOE ε4+ implies an amyloid accumulation that begins, on average, 6.1 years earlier in APOE ε4 carriers than non-carriers. The rows for population mean correspond to the mean onset for an MCSA participant with mean values for APOE, sex, and education.

^*Defined as the difference between the tau and amyloid change points

We note that the coefficients can be viewed in two equivalent ways: either as accelerating the process (e.g., onset for APOE ε4 carriers begins 6.1 years earlier and therefore the change point is shifted left by 6.1 years) or as accelerating the participant’s age (e.g., APOE ε4 carriers are effectively 6.1 years older). We have largely emphasized the first view as having the more natural clinical interpretation, but the second is equally acceptable.

Estimates from the model including only MCSA participants are quite similar to those from the model including both cohorts. When limiting the data to MCSA participants, the amyloid change point in the population is estimated to be 77.1 years (76.4–77.8). APOE ε4+ accelerates the amyloid change point by 7.0 years (5.9–8.2), and female sex accelerates the amyloid change point by 2.0 years (0.9–3.0). The tau change point is an estimated 94.9 years (90.0–101.4) based on the MCSA-only model and the amyloid to tau delay is 17.8 years (12.9–24.3). The credible intervals for the tau change point and the amyloid to tau delay are much wider when using only the MCSA data but overlap the estimates from the combined group.

The population-based amyloid and tau slope estimates before and after their respective change points are summarized in Table 3. Prior to the change point, amyloid accumulates at a rate of approximately 5% per decade (0.05, 95% CI 0.04–0.05). After the change point, the accumulation is much greater (0.32 per decade, [0.31–0.34]). A parallel pattern is observed for tau with accumulation of again about 5% per decade prior to the change point (0.05, 95% CI 0.04–0.05) followed by a steeper slope (0.46, 95% CI 0.40–0.52). The slope estimates for the MCSA-only model are similar to those from the model based on both MCSA and ADRC participants although there is little precision in the tau slope after the change point (0.56 [0.26–1.04]).

Table 3.

Slope estimates before and after the change point.

	Amyloid	Tau
Model using MCSA and ADRC
Initial slope	0.05 (0.04, 0.05)	0.05 (0.04, 0.05)
Final slope	0.32 (0.31, 0.34)	0.46 (0.40, 0.52)
Model using MCSA only
Initial slope	0.05 (0.04, 0.05)	0.05 (0.04, 0.05)
Final slope	0.35 (0.34, 0.36)	0.56 (0.26, 1.04)

Values shown are mean (95% credible interval). Estimates are shown per decade. Parameters of the overall shape functions f (amyloid) and g (tau) described in section 2.3. Each shows the increase per decade in the initial quiescent period and the increase after the inflection, e.g., 0.05 increase in log(amyloid) levels per decade early, followed by 0.32 per decade after the onset of accumulation. These correspond to exp(0.05) = 5% and exp(0.32) = 38% increase per decade.

3.3. Amyloid and tau at the individual level

Figure 4A shows tau versus amyloid levels for all pairs of scans that occurred at the same visit (1671 pairs on 1139 participants). Figure 4B shows the estimated participant-level amyloid acceleration, b_i, on the x-axis and the estimated participant-level tau acceleration, d_i, on the y-axis from the model; there is a point for each participant with both amyloid and tau imaging. The box plots on the margin show the median, 25th and 75th percentiles. Individuals with positive adjustments (b_i > 0 or d_i > 0) can be thought of as having either an earlier onset/change point or as an accelerated process (they are effectively that much older). Figures 4C and 4D show the comparable information for the model fit using data from MCSA participants alone and are very similar to the panels from the model with both MCSA and ADRC participants, even though the higher end of the tau distribution is far less dense in the MCSA.

Figure 4.

Relationship between amyloid and tau and individual-level adjustments

Panel A shows amyloid PET SUVR on the x-axis and tau PET SUVR on the y-axis for 1671 pairs of scans across 1139 individuals in either the MCSA or the ADRC with APOE genotype indicated by color and symbol. Two individuals whose amyloid and tau PET scans did not occur at the same clinical visit are omitted from the scatter plot. Panel B shows the estimated per-participant amyloid onset adjustment (b_i) on the x-axis and the corresponding estimated per-participant tau onset adjustment (d_i) on the y-axis for individuals in either the MCSA or ADRC who had both amyloid and tau imaging. For panels B and D the axes have been flipped so that earlier age of onset (i.e. more acceleration) is on the left and later age of onset (i.e. less acceleration) is on the right. A positive value indicates an individual with an earlier onset age relative to the population mean (i.e. an older “effective” age); a negative value indicates a later onset age relative to the population mean (i.e. a younger “effective” age). The box plots in the margins show the median, 25^th, and 75^th percentiles. The trend line represents the median tau onset adjustment for a given amyloid onset adjustment. Note that with a slope less than 1, an individual with an earlier-than-average amyloid onset can be expected to have a tau onset that is also earlier than average, but to a lesser degree. This attenuation tends to result in a longer delay between the two events. Panels C and D show the corresponding values based on a model limited to MCSA participants.

For the model using both MCSA and ADRC participants, the range of the adjustments is fairly modest (Figure 4B): 80% of participants have an estimated amyloid onset that is between 13 years earlier and 13 years later than the mean (10^th–90^th percentiles). The 10^th and 90^th percentiles of the estimated tau onset adjustment were 11 years earlier and 14 years later, respectively. Large outlier estimates for the amyloid onset adjustment correspond to unusual individuals, e.g. the data point at −43 on the horizontal axis corresponds to the individual with the lowest amyloid in the sample; the fit assumes that this person will never develop elevated amyloid by assigning an amyloid onset age to be 43 years later than average.

Someone with amyloid onset 10 years earlier than the mean (10 on the x-axis of Figure 4B) will have an expected tau onset adjustment that is approximately 0.13 + 0.66(10) ≈ 6.7 years; the tau onset for this person is expected to be about seven years earlier than the mean. The model-based correlation between an individual’s amyloid and tau adjustments was estimated to be 0.49 (0.43–0.54).

3.4. Goodness of fit

In the model using data from both MCSA and ADRC participants, the final values of the Gelman-Rubin R-hat diagnostics (Gelman and Rubin, 1992) ranged from 1.00 to 1.01 for all parameters. Posterior predictive checks confirmed good model fit in that when randomly generating SUVR values based on the posterior distribution of the model parameters, the generated values were consistent with the observed SUVR values aside from Gaussian error. Similarly, in the model limited to MCSA participants, both MCMC convergence and posterior predictive checks indicated good model fit.

4. Discussion

4.1. Modeling neurodegenerative diseases using the accelerated failure time approach

Using an accelerated failure time approach, we show the long lag between the acceleration of amyloid deposition and tau deposition. The AFT approach is common in the analysis of industrial reliability data and is a mainstay of standard textbooks (Meeker and Escobar, 1998), e.g., a high-temperature environment will make a component age faster. It has been far less prevalent in the biomedical context, though several authors have suggested it as a useful framework for the statistical study of aging (Stroustrup, 2018; Swindell, 2009). In this study, AFT models provide a natural insight into the accumulation process for amyloid and tau, which can be viewed as indicators of failing brain function (Jones et al., 2017; Jones et al., 2016). The underlying model — that each participant will follow the same trajectory but shifted in their change point or onset time — is, like all models, an oversimplification of underlying biology but a useful one. The variation in amyloid onset times between individuals has a standard deviation of 10 years, reflecting the wide variation in the age of onset of amyloid accumulation. The approximately 6 year effect of APOE ε4 carriage on amyloid accumulation (a shift of the change point to earlier age) agrees with prior knowledge of the profound effects of APOE ε4 on amyloid accumulation (Liu et al., 2013).

The sharp bend in the amyloid accumulation curve suggests a loss of homeostatic control of amyloid production and clearance at that level (Jack et al., 2017; Jones et al., 2017). The positive correlation between the individual level amyloid onset adjustment and the tau adjustment suggests that those with an earlier onset of amyloid can expect an earlier tau onset. This finding is consistent with the widely held clinical view that earlier age of onset is commonly associated with a more aggressive clinical disease course (Ho et al., 2002; Koss et al., 1996).

The model also shows why some people with elevated amyloid will not develop elevated tau before they die. For example, the model predicts that an APOE ε4 negative woman will experience an elevation of amyloid (change point) around age 78, but not experience an elevation of tau until around age 90. Because annual all-cause mortality rates increase exponentially after age 30 (St Sauver et al., 2012a), many individuals with elevated amyloid will die from other causes before developing clinically relevant tau burden. The AFT model depicting the hypothetical average trajectory of amyloid and tau PET values with aging is entirely consistent with the more clinically oriented view of rising proportions of individuals with advancing age who exceed a defined cut point for the biomarkers (Jack et al., 2017).

The late onset of tau accumulation in the population predicts that a substantial proportion of individuals with elevated amyloid will die before tau accumulation would have been detectable by PET imaging. The exponential increase in all-cause mortality with age in turn, combined with the clinical consequences of elevated tau, predicts that the age distribution of those with elevated tau will be systematically lower than the projected population average tau change point (age 91). Indeed, the participants who were recruited from the Mayo Clinic behavioral neurology practice had an estimated 21-year leftward shift in amyloid onset relative to the randomly sampled Olmsted County participants, and a 27-year leftward shift for tau. The difference between referral-based participants and those randomly selected from the community highlights how different recruitment strategies necessarily modify the age of onset of amyloidosis and tauopathy but do not fundamentally change the relationship between the two as shown here. Many of the MCSA participants who, by definition, were recruited by random selection from the community have begun to accumulate amyloid but have not yet begun to accumulate tau. In contrast, most of the high tau values in the combined sample were found in the much younger referral based ADRC sample; these individuals developed amyloid many years before the population average amyloid change point of age 77.

Tau accumulation (in the setting of amyloid elevations) in the medial temporal lobe is associated with overt but mild declines in memory function, but it is not until tau accumulations appear in the isocortex that more severe cognitive decline occurs. Although we did not model cognition here, the relationships between tau elevations and cognitive impairment are quite clear in showing that high tau SUVRs in isocortical regions is almost invariably associated with significant cognitive impairment (Hanseeuw et al., 2019; Harrison et al., 2019; Jack et al., 2018b; Lowe et al., 2018). However, even assuming that the lag between tau accumulation and overt cognitive impairment were on average short and varied little across individuals, the long and variable time lag from the point at which amyloid accumulation occurs to the point when tau accumulation occurs means that predicting when cognitive impairment would begin once amyloid accumulation occurs would involve a very large confidence band.

This modeling should be interpreted as showing temporal links but not necessarily mechanistic ones. It is entirely possible that elevations in brain amyloid are a proxy for some other process more closely linked to neuronal dysfunction and the eventual enabling of abnormal tau processing (Jones et al., 2017; Jones et al., 2016). The same is true for tau elevations; they may not necessarily be in the direct mechanistic path of, despite their strong diagnostic linkage to, symptomatic cognitive impairment (Jack et al., 2013).

4.2. Comparison to other work

The conceptual Jack model (Jack et al., 2013; Jack et al., 2010) has inspired many efforts to validate the concept in observed data. One of the principal limitations of studies of slowly developing chronic diseases of aging is shown in Figure 5. Even though many individuals in this cohort have had long follow-up compared to a clinical trial, up to a decade in fact, this is short compared to the timeline for evolution of the pathologic and clinical timeframe of AD. As a consequence, the predicted age curve is in essence spliced together from many small segments. Our model and others discussed below have each needed to make particular assumptions in order to overcome this; successful models need to strike a balance between flexibility and constraints if they are to avoid implausible results.

Figure 5.

Amyloid PET versus age and versus adjusted amyloid age among MCSA participants with extended follow-up.

Data for the 49 MCSA participants who have 5 or more amyloid PET scans. Panel A shows amyloid PET SUVR versus age. Panel B shows amyloid PET SUVR versus adjusted age. Even with extended imaging follow-up, the timescale of biomarker abnormality means an individual spans a small fraction of the entire curve.

Our analysis was based on three key concepts: (1) a linear predictor $l_i=b_i+x_i^{\text{T}}\beta$ for each participant, which combines a random per-participant intercept b_i and fixed covariates x_i such as APOE; (2) a non-linear transformation between l_i and the outcome; and (3) multiple outcomes (amyloid and tau) for a participant linked through correlated random effects.

The simplest analysis is an LME model; it incorporates the first principle, but not 2 or 3. Figure 6 shows the resultant LME fit for three individuals from a model using MCSA participants with age, sex, and education as covariates. Age is modeled as a natural spline. This illustrates one flaw for models with an up–down shift when applied to these data, which is that the model implicitly assumes that the three participants’ curves have had this separation over all ages, something that is biologically implausible. The model fits the observed data, but not a possible mechanism of the data. Li et al. extend this approach and fit a joint LME model to multiple outcomes at once (amyloid PET, CSF tau, FDG PET, etc.) using principles 1 and 3, after pre-transforming each outcome to a Gaussian shape (Li et al., 2019). Covariates and random effects shift the transformed response up–down, and a random per-subject age shift links the outcomes. To the extent that the pre-defined transforms mimic the data-driven shape of Figure 2B, our results will be similar.

Figure 6.

Linear mixed model estimates for three MCSA participants

Mean amyloid PET versus age for APOE ε4 carriers versus APOE ε4 non-carriers based on a linear mixed effects model. The model uses a natural spline to allow for non-linear age effects.

Koscik et al. derive a concept of “chronicity,” which is an individual’s age relative to the age at which their amyloid curve is predicted to cross a defined positivity threshold (Koscik et al., 2020). This is equivalent to age + the horizontal axis of Figure 2B. They estimate amyloid chronicity beginning with a trajectory analysis approach. Limitations of that particular software package are a single outcome, no covariates other than age, and polynomial curves. The authors overcome these issues by further building off of that base; their resulting Figure 2 is remarkably similar to our Figure 2B. An “effective age” or chronicity is defined as the number of years from a fixed progression point, which translates to (age + b_i) − 77 in our notation. The derived chronicity for each participant is then used as a covariate in further models. Kiddle et al. (Kiddle et al., 2017) point out a potential flaw of trajectory approaches when used with short follow-up, which is that the realized curves may be too sensitive to participants’ initial measurement, leading to unrealistic “always has been high” or “always will be low” trajectories; particularly if subjects are first recruited across a wide range of severity. Indeed, using our data an attempt to fit a trajectory model displayed exactly the artifacts they point out. Kiddle et al. then proceed to fit a model based on principles 1 and 2 to longitudinal Mini Mental State Exam data. Their assumed curve includes the assumption, similar to ours, of an initial steady state followed by a progressive loss in function, with a participant-specific change point. Proust-Lima et al. fit a model most congruent to ours in that it incorporates all 3 principles, though in a different mathematical expression, for multiple cognitive measures in a Bordeaux cohort (Proust-Lima et al., 2016). The fitted curves for both Kiddle et al. and Proust-Lima et al. have a much more gradual bend than the amyloid fit of Koscik et al. or our Figure 2B; this may be due to the greater uncertainty in cognitive measurements or that the cognitive scales themselves are a multidimensional construct which is the sum of many processes, thus smoothing out any sharp transition.

4.3. Limitations

In order for the splicing of Figure 5 to be valid, we must assume that a participant enrolled at 85 will have followed, when they were 55, the same trajectory shape as participants currently enrolled at age 55. For this to hold, the participants need to be sampled from a population that has been stable over a long period with respect to factors that would change the trajectory of tau and amyloid accumulation, which is one of the advantages of studies using the population-based MCSA. There is also the assumption that covariate effects will remain the same over calendar time.

The use of both population and referral participants in this study is both a weakness and strength. A primary assumption of this model is that once recruited, all participants will follow the same trajectory independent of their source. The ADRC preferentially recruits those participants with an earlier age of onset (change point) — i.e., individuals who seek medical care because they have become symptomatic — and this was accounted for as a covariate in the model. However, once recruited the model assumes that population and referral participants will follow the same disease course, i.e., amyloid accumulation and tau accumulation patterns, although the respective change points can depend on the cohort. Because of the sparsity of high tau values in the population-based MCSA sample, these earlier-in-life accumulation participants in the ADRC provide most of the information about the amyloid to tau delay time, while the population sample provides much of the information about the timing of amyloid changes relative to age. Note that the credible interval for the tau change point (and as a consequence, that for the amyloid to tau delay) is much narrower for the combined sample than for MCSA alone.

It is difficult to estimate the amyloid to tau delay in participants with later amyloid onset. For practical reasons, individuals with dementia are rarely imaged in a cohort such as the MCSA where participants are randomly drawn from the population rather than from a clinical referral sample. An older individual with high tau accumulation is very likely to fall into this category, and at best would have a very narrow age window for said imaging. This means that scans of older, high tau participants will systematically be underrepresented.

Data Availability and Code Availability

The data set analyzed for the current study is available from the investigators who can be contacted at mcsaadrcdatasharing@mayo.edu. The programs used to perform the analyses for the current study are available at https://github.com/therneau/AFTmodel. The directory also includes a more detailed comparison of this approach to the other methods.