3.1 Overall Study Design and Concept
Most longitudinal MRI datasets only cover a few years of an individual’s lifespan. For such a short period, when using years of follow-up as the time variable, a linear trend may be the best fit to the data, even though true brain atrophy over lifespan is non-linear. However, when using the actual age in years as the time variable, the model will look very different. For an entire sample, age has a wide coverage for the lifespan, but for each individual, age only covers a small fragment of the lifespan. This data structure can be conceptualized as a “fish bone” (Fig. 1), where the constructed spline curve can be considered the “back bone” and the straight lines (representing observed longitudinal data) can be considered “rib bones”. By using cross-sectional data or the intercept from a longitudinal model with age as the time variable, we should be able to construct the “back bone" of the spline. Adding the “rib bones” from large number of individuals in different age categories can enhance the shape of the spline.
Generally speaking, to obtain the lifespan trajectory, we should observe the “rib bones” in our typical longitudinal datasets and then attempt to construct the “back bone.” In this work, we attempt the converse; using the “back bone”, we attempt to grow the “rib bones”. In other words, our approach is to first fit an accurate spline model from cross-sectional data to model the non-linear trajectory across the lifespan. We then use the age slope to augment the longitudinal data for each normal aging subject, given that a linear model can suffice to model brain atrophy over a short (5-year) period of time.
The study design includes the following steps: 1) identify a well-fitted spline model using cross-sectional data from normal aging populations; 2) augment longitudinal data from this well-fitted spline model using a linear slope at a given age point; 3) compare 12 different mixed spline models through simulated data and identify the mixed spline model that fits the “fish bone” data structure; 4) combine augmented longitudinal normal aging data with longitudinal MS data to fit mixed spline model and compare across 12 mixed spline models; 5) use a manual forward then backward model building strategy to select the covariates from 52 covariate structures; 6) identify the individual age of onset of progressive brain tissue loss with associated 95% confidence interval using a 10-fold cross validation procedure through 1000 bootstrapping samples.
3.2 Study Sample
Our dataset was assembled from the following three sources (Table 1): 1) The Human Connectome Project (HCP: http://www.humanconnectome.org), 2) Alzheimer’s Disease Neuroimaging Initiative (ADNI: http://www.adni-info.org) and 3) a single-center, prospective case-control cohort MS study conducted from January 2005 through December 201028. Normal aging samples were from HCP, ADNI and 89 healthy control cases from the single-center study. Age at scan date and sex were extracted from each of the data sources. Healthy control subjects (N = 2053) had an overall mean age of 44 ± 21 years (Q1: 27, Q3: 62) with 56% female, while MS subjects (N = 520) had a mean age of 43 ± 10 years (Q1: 36, Q3: 50) with 70% female and an average of 4 ± 1.5 annual scans per subject. Most of the normal aging sample only had one MRI scan, but 228 of them had repeated measures (2.9 ± 1 scans in 2.5 ± 1.4 years). Subjects with age > 90 or < 16 (to avoid brain growth confounding) were excluded.
3T/3D T1-weighted volumetric gradient echo images were processed with FreeSurfer v6.0 to extract thalamic and intracranial volumes. Thalamic volumes were normalized by total intracranial volume and multiplied by 1000. Healthy control subjects had normalized thalamic volumes of 9.7 ± 1 (Q1: 9.1, Q3: 10.4) at study entry, while MS subjects had normalized thalamic volumes at study entry of 9.3 ± 1 (Q1: 8.7, Q3:9.9).
Table 1
Demographic Distribution of Health Control Cohorts
Dataset | # of Subjects | # of MRI Time Points | Age (Mean ± SD) | Age Range | % Female |
HCP-D | 178 | 178 | 19 ± 2 | 16–22 | 55.1 |
HCP | 865 | 865 | 29 ± 4 | 22–37 | 56.3 |
HCP-A | 676 | 676 | 58 ± 14 | 36–90 | 56.8 |
Single Center | 87 | 152 | 40 ± 11 | 22–65 | 67.8 |
ADNI | 247 | 614 | 75 ± 7 | 56–89 | 52.2 |
Total | 2053 | 2485 | | | |
3.3 Longitudinal Normal Aging Data Augmentation from Large Cross-Sectional Data
Multivariate Adaptive Regression Splines (MARS) was used to fit a cross-sectional spline so that we could augment the longitudinal data. MARS was chosen because of its robustness to outliers and its ability to auto-search non-linear associations with high dimensional interactions40. For this demonstration study, only age at scan, intracranial volume (ICV), and sex were used, with three-way interactions among them, as predictors of thalamic volume (percent of total brain volume). ICV and sex were treated as constant for each individual subject when augmenting the longitudinal data. Longitudinal thalamic volumes were augmented at ± 2 years from age at scan. We reserved 433 repeated measurements from 229 individuals as independent testing. ICC two-way mixed with absolute agreement and repeated measure correlation were used to assess the agreement/correlation between MARS model-augmented longitudinal data vs. observed testing longitudinal data. SAS9.4 ADAPTIVEREG was used to fit the MARS model.
3.4. Mixed Spline Model of Thalamic Atrophy Trajectory
After data augmentation, we fitted the mixed spline model. Let n be the number of subjects. For the ith participant, denote ti as the age, denote \({Y_{ij}}(t)\) as the thalamus volume at the jth measurement for subject i, and denote Xij as other predictors such as sex. To model the age effects accurately and efficiently, we use a semiparametric model of the form given below:
$${Y}_{ij}\left(t\right)={\mu }_{ij}\left(t\right)+{X}_{ij}\beta +{\upsilon }_{i}\left(t\right)+{ϵ}_{ij}\left(t\right), i=1, \dots ,n, j=1, \dots , k$$
where\({\mu }_{ij}\left(t\right)\) is the unspecified aging trajectory for subject i at the jth time evaluated at age t, and β are the regression coefficients of the other predictors at the jth time.\({\upsilon }_{i}\) is the random effect of each subject. The measurement errors ϵij are assumed to follow a normal distribution N(0,R), where R is the covariance matrix. This semiparametric regression model is a parsimonious way to both capture the potential nonlinear age trajectory and investigate the effects of other predictors. The simplest special case of this model is the linear mixed model where \({{\mu }_{i}}_{j}\left(t\right)\) =\({\beta _{0i}}+{\beta _{1i}}{t_{ij}}\). Regression splines are a broader class of models and could be fitted under this framework, which can be based on truncated power function (TPF) basis, B-spline basis or natural spline basis. These models vary by the choice of the spline basis and tuning parameters (the number of knots and the knot positions) that have an impact on the estimated shape of a spline function. Parameter-function estimation contains two major steps: (i) approximation using basis functions (e.g., TPF, B-Spline) which allows to fit lower-order polynomials within very small interval partitions (based on knots) and (ii) smoothing the approximation via penalty (e.g., random SPLINE coefficients, TOEPLIZ G-side matrix, RSMOOTH G-side matrix). The smoothing could be done via generalized cross-validation (GCV)41 or mixed effects approaches42,43, which are known to facilitate the choice of the knot positions in spline modelling44. They also allow a penalty to be applied directly to the model coefficients (P-spline penalty penalizes the squared differences between adjacent model coefficients, which in turn penalizes wiggles).
We then compared penalized splines (P-spline) with B-spline basis and truncated power function (TPF) basis with different random effect structures such as P-SPLINE and RSMOOTH (radial smoothing). For the P-spline, the unspecified function \({\mu _{ij}}(t)\)is approximated with a cubic B-Spline or TPF basis. Following Ruppert, Wand and Carroll (2003)45, the cubic spline can be represented as:
$${\mu }_{i}={\beta }_{0}+{\beta }_{1}{x}_{i}+{\beta }_{2}{{x}_{i}}^{2}+{\beta }_{3}{{x}_{i}}^{3}+{\sum }_{j=1}^{K}{\beta }_{3+j}({x}_{i}-{t}_{j}{)}^{3}$$
\(\) \((x-t)=\left\{\begin{array}{c}x-t\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }x>t\\ 0\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{0}\end{array}\right.\)
Estimation of parameters is made by minimizing the penalized log-likelihood function using proc GLIMMIX in SAS 9.4 with smoothing implemented using P-SPLINE smoothing (Random x/type = pspline) or radial smoothing (Random x/type = rsmooth). This mixed model formulation of spline smoothing has the advantage that the smoothing parameter is selected automatically45 and is shown to be more robust with misspecification of error dependence structure, compared to GCV-based approach46.
For the 12 spline structures described above, model comparison was made using four criteria: i) Akaike information criterion (AIC) and Bayesian information criterion (BIC) criterion, with lower values indicating better fit; ii) repeated measure correlation coefficient47,48 and intraclass correlation two-way mixed for longitudinal data between model-predicted vs. observed data from reserved 10% testing dataset; iii) visual inspection of the expected shape for projected lifespan spline (the normal curve must inherit the shape of the spline based on cross-sectional normal aging, and the MS curve must followed the shape of observed spaghetti plots as in Fig. 3A & Supplemental Fig. 10); and iv) both MS and normal aging trajectory curves must have narrow predicted interval along age points.
3.4. Simulation Study Design
The purpose of the simulation study is to compare spline models to choose the most appropriate spline model for the “fish bone” data structure. The simulated data mimic the fish bone data structure by combining 10 sets of data from 10 different age blocks (k = 1 to 10) with age range from 30 to 80 by 5-year intervals (e.g., 30–34, 35–49). Each simulated data set was based on the covariance parameters estimated from a linear mixed model (with random intercept and slope). Block-specific weights (Wk, Vk) were added to the fixed effects of intercept and slope respectively for block k. Wk and Vk were altered to mimic a spline shape (“back bone” as shown in Fig. 1). The final mixed effects model was as follows:
\({Y}_{kij}={W}_{k}{\beta }_{k00}+{{V}_{k}\beta }_{k10}{\left(Year\right)}_{kij}+{\beta }_{k01}MS+{\beta }_{k11}\left(MS\right)*{\left(Year\right)}_{kij}+{b}_{k0j}+{b}_{k1j}{\left(Year\right)}_{kij}+{ϵ}_{kij}\) , \({ϵ}_{kij} \tilde iid N\left(0, {\sigma }^{2}\right)\)
The training sample was a combination of 10 datasets with 50 MS subjects each (age span 30 to 80 years). Each MS subject had 5 longitudinal MRI data points within each block, simulated using the linear mixed model above. Therefore, we simulated 500 subjects total in the training data. Wk and Vk started with small values in younger age, e.g. 1% decrease from the previous age block, but larger in middle age, e.g. 5% decrease, then became smaller again in older age, e.g. 1% decrease. The testing data followed the same simulation procedure, but we used the same subject ID across the 10 blocks; thus, the testing data contained 50 MS subjects, and each subject had 50 simulated age points. Because our ultimate goal is to predict the thalamic volume at an age that is younger than the observed age, the testing data included 4 more younger age points: 26, 27, 28, and 29, in addition to the 50 age points.
We considered twelve different models with three G-side covariance types (TOEPLIZ, P-SPLINE and radial smoothing) and four basis functions (Cubic-B-Spline, Cubic -TPF, Natural-TPF, Natural-B-Spline). To estimate the prediction accuracy from the spline model, we made comparisons using AIC/BIC with 500 iterations. The testing data were scored through each of the 12 spline models. We then obtained the estimated thalamic volume with associated 95% confidence interval at each age point. We took the average of the 500 replicates for the model-predicted thalamic volume and associated 95% CIs, then graphed the spline plots to visually inspect the overlap between true spline curve and the model-estimated spline curves, as well as the width of 95% confidence band.
3.5 Real-Life Data Application
We applied 12 different scenarios of spline models (listed in Table 2) to a real-life dataset with 520 MS subjects and 2053 normal aging subjects. For the normal subjects, we used augmented longitudinal data with 5 follow-up years (actual scan year at the middle). Among the 520 MS subjects, we randomly selected and reserved 52 MS subjects as the independent testing data. We repeated this iteration 10 times with 10 mutually exclusive independent testing data. For the first iteration, we selected the optimal spline structure and covariates using the criteria defined in 3.3. For covariate selection, we used a forward then backward strategy. Age spline, MS status, and age spline × MS status interaction were mandatory for each model. Other covariates included sex, baseline thalamic volume (Thalamus0), baseline ICV (ICV0), age of clinical onset (set as 0 for normal control), and cumulative years of exposure to MS disease modifying therapies at the first scan (DMT0; set as 0 for normal control). Each covariate entered the model first as the main effect, then as interaction terms with MS status or/and age spline. We categorized the covariate structure as the following: a) only the main effect from each covariate; b) two-way interactions with MS status, each covariate interaction term one by one, then multiple interaction terms together; c) three-way interactions for each covariate one by one with MS status and age spline except DMT0 and age of clinical onset; d) select any two covariates with the three-way interactions; e) select any three covariates with the three-way interactions; f) select any four covariates with the three-way interactions; g) backwards selections with the terms showing model improvement from a-f. The same criteria defined in 3.3 were used for model selection. Once the final model was selected, we used the same model structure in the other 9 training datasets to obtain estimated coefficients, then applied them to independent testing data at each fold of iteration.
For each independent testing MS patient, we constructed the hypothetical individualized normal aging trajectory curve (Health Digital Twin) and MS lifespan trajectory curve using the patient-specific covariates, which included sex, Thalamus0, ICV0, age of clinical onset (set as 0 for normal control), DMT0 (set as 0 for normal control) and sequential age points from 15 to 75 with an interval of 1. The age at which the MS trajectory curve began to depart from the HDT trajectory curve (or when both curves crossed in young age) was defined as the age of onset of progressive brain tissue loss. A bootstrapping procedure with 1000 iterations was used to determine the 95% confidence interval of this brain atrophy-defined age of onset. The bootstrapping procedure was conducted at the patient level, i.e., once a patient ID had been selected, all longitudinal scans associated with this patient were selected as a completed block. We repeated this procedure 10 times with 10 mutually exclusive independent validation data (10-fold cross validation procedure with 10% testing data each fold). Thus, each patient had an age of onset of progressive brain tissue loss (PBTL) with 95% CI estimated from their exclusive training dataset. In the end, we identified two groups of MS patients: earlier onset (i.e., the upper 95% CI limit of PBTL onset is younger than clinical onset age; in other words, the age of onset of PBTL was statistically significantly earlier than the age of clinical onset); and simultaneous onset (clinical onset did not differ statistically from the age onset of PBTL). We examined the different patterns of the onset age gap (age of clinical onset minus the age of onset of PBTL) between the earlier onset and simultaneous onset groups used Bland-Altman plots.