José E Romero

Pierrick Coupe

Enrique Lanuza

Gwenaelle Catheline

José V Manjón

Towards a unified analysis of cerebellum maturation and aging across the entire lifespan: A MRI analysis

Keywords: aging, maturation, lifespan, MRI segmentation, patch-based processing, cerebellum trajectory

Previous literature about the structural characterization of the human cerebellum is related to the context of a specific pathology or focused in a restricted age range. In fact, studies about the cerebellum maturation across the lifespan are scarce and most of them considered the cerebellum as a whole without investigating each lobule. This lack of study can be explained by the lack of both accurate segmentation methods and data availability. Fortunately, during the last years, several cerebellum segmentation methods have been developed and many databases comprising subjects of different ages have been made publically available. This fact opens an opportunity window to obtain a more extensive analysis of the cerebellum maturation and aging. In this study, we have used a recent state of the art cerebellum segmentation method called CERES and a large dataset (N=2831 images) from healthy controls covering the entire lifespan to provide a model for 12 cerebellum structures (i.e., lobules I-II, III, IV, VI, Crus I, Crus II, VIIB, VIIIA, VIIIB, IX and X). We found that lobules have generally an evolution that follows a trajectory composed by a fast growth and a slow degeneration having sometimes a plateau for absolute volumes, and a decreasing tendency (faster in early ages) for normalized volumes. Special consideration is dedicated to Crus II, where slow degeneration appears to stabilize in elder ages for absolute volumes, and to lobule X, which does not present any fast growth during childhood in absolute volumes and shows a slow growth for normalized volumes.

Introduction

The human cerebellum, located below the cerebrum, is an important bilateral neuroanatomical structure connected to the cerebrum and the brain stem by means of two white matter fiber tracts called the cerebellar peduncles. At the opposite end of the peduncles, a series of grey matter folds called foliations can be grouped in left, right and vermal (a central portion of the cerebellum) lobes. These lobes can also be classified into lobules. In this study we used a cerebellum lobule classification consisting in 12 bilateral structures: lobules I-II, III, IV, VI, Crus I, Crus II, VIIB, VIIIA, VIIIB, IX and X [START_REF] Park | Derivation of highresolution MRI atlases of the human cerebellum at 3T and segmentation using multiple automatically generated templates[END_REF]. Figure 1 shows an example of this segmentation protocol. Lobules are located surrounding the white matter starting by lobules I-II and ending with lobule X following the order mentioned beforehand. Note that lobules I-II are barely seen in Figure 1 since they represent a small portion of the cerebellar volume containing just a few voxels while the rest of the lobules have much larger volumes.

Figure 1: Cerebellum segmentation protocol

The cerebellum has been related to motor tasks for a long time [START_REF] Kase | Cerebellar infarction. Clinical and anatomic observations in 66 cases[END_REF][START_REF] Compston | Multiple sclerosis[END_REF][START_REF] Davie | Persistent functional deficit in multiple sclerosis and autosomal dominant cerebellar ataxia is associated with axon loss[END_REF][START_REF] Klockgether | The clinical diagnosis of autosomal dominant[END_REF]. However, more recent studies showed that the cerebellum is also involved in many cognitive functions [START_REF] Middleton | Cerebellar output channels[END_REF]) such as language, learning and memory [START_REF] Stoodley | Functional topography in the human cerebellum: A meta-analysis of neuroimaging studies[END_REF][START_REF] Desmond | Dissociation of Frontal and Cerebellar Activity in a Cognitive Task: Evidence for a Distinction between Selection and Search[END_REF][START_REF] Riva | The cerebellum contributes to higher functions during development: evidence from a series of children surgically treated for posterior fossa tumours[END_REF].) and in emotional process [START_REF] Schmahmann | The cerebellar cognitive affective syndrome[END_REF]. In fact, patients with cerebellar lesions limited to lobule VI experienced minimal motor impairment [START_REF] Stoodley | Evidence for topographic organization in the cerebellum of motor control versus cognitive and affective processing[END_REF]. Specific cerebellum dysfunction have also been associated to cognitive impairment in various pathological conditions such as schizophrenia [START_REF] Okugawa | Selective reduction of the posterior superior vermis in men with chronic schizophrenia[END_REF][START_REF] James | Cerebellar, prefrontal cortex, and thalamic volumes over two time points in adolescent-onset schizophrenia[END_REF], Nenadic eta., 2010[START_REF] Laidi | Cerebellar parcellation in schizophrenia and bipolar disorder[END_REF]), Alzheimer's disease [START_REF] Thomann | The cerebellum in mild cognitive impairment and Alzheimer's disease -a structural MRI study[END_REF], Multiple Sclerosis [START_REF] Moroso | Posterior lobules of the cerebellum and information processing speed at various stages of multiple sclerosis[END_REF] and neurodevelopmental disorders Attention Deficit/Hyperactivity Disorder and Autism [START_REF] Courchesne | Abnormality of cerebellar vermian lobules VI and VII in patients with infantile autism: identification of hypoplastic and hyperplastic subgroups with MR imaging[END_REF][START_REF] Bishop | Cerebellar abnormalities in developmental dyslexia: cause, correlate or consequence?[END_REF][START_REF] Seidman | Structural brain imaging of attentiondeficit/hyperactivity disorder[END_REF][START_REF] Ramnani | The primate cortico-cerebellar system: anatomy and function[END_REF].

In neuroimaging studies, different cerebellum lobules have been demonstrated to be involved in particular sensorimotor tasks and/or higher functions [START_REF] Diamond | Close interrelation of motor development and cognitive development and of the cerebellum and prefrontal cortex[END_REF][START_REF] Habas | Distinct cerebellar contributions to intrinsic connectivity networks[END_REF][START_REF] Krienen | Segregated fronto-cerebellar circuits revealed by intrinsic functional connectivity[END_REF][START_REF] O'reilly | Distinct and overlapping functional zones in the cerebellum defined by resting state functional connectivity[END_REF]. Finger tapping activates lobules IV, V, VIIIA and VIIIB.

Observing rotated letters involves activity in lobules VI, VIIB and Crus II. Viewing emotional images produces activations in lobule VI. For memory tasks, activation peaks appeared in lobules V, VI and Crus I. (Stoodley el al., 2012). Far beyond its initial role in motor skills, a growing body of literature supports the role of the cerebellum in a large panel of complex behaviors. Functional classification of cerebellum substructures based on either subjects presenting lesions (Stoodley and Schmamhann, 2010) or on functional connectivity recorded at rest (Guel et al., 2018) revealed functional subdivisions of the cerebellum networks: double motor representation of the sensorimotors (lobules I-VI and VIII) and the triple non-motor representations (lobules VI-Crus I, Crus II-VIIB, IX-X). But its mode of operation is still far from being elucidated. Its role in learning processes and adaptive plasticity suggests that it would play a key role during the development stages. In accordance, clinical studies indicate that cerebellum injuries occurring in childhood have much more dramatic functional consequences than when they occur in adulthood [START_REF] Stoodley | The Cerebellum and Neurodevelopmental Disorders[END_REF][START_REF] Badura | Normal cognitive and social development require posterior cerebellar activity[END_REF].

To gain a better understanding of how neurological structures are related to normal and pathological functions, the study of the evolution of their volume represents a valuable tool. For this reason, several previous works analyzed the evolution of different cerebrum and cerebellum structures at different age ranges. [START_REF] Luft | Patterns of age-related shrinkage in cerebellum and brainstem observed in vivo using threedimensional MRI volumetry[END_REF] studied age-related shrinkage patterns in subjects from 19 to 73 years old using 11 cerebellar structures. [START_REF] Sullivan | Cerebellar Volume Decline in Normal Aging, Alcoholism, and Korsakoff's Syndrome: Relation to Ataxia[END_REF] studied the effect different degrees of alcoholism in subjects between 23 and 72 years old using cerebellar hemispheres and 4 vermis regions. Jerningan et al. ( 2001) conducted a study with patients from 30 to 99 years old using cerebellar grey matter, white matter and cerebrospinal fluid. [START_REF] Raz | Regional brain changes in aging healthy adults: General trends, individual differences, and modifiers[END_REF] studied the evolution of the whole cerebellum volume on healthy adults over 49 years old. [START_REF] Fan | Sexual dimorphism and asymmetry in human cerebellum: an MRI-based morphometric study[END_REF] also found evidences of sexual dimorphism in a study with patients between 18 and 33 years old using 10 lobules and [START_REF] Tiemeier | Cerebellum development during childhood and adolescence: a longitudinal morphometric MRI study[END_REF] showed that cerebellum maturation process extends to late adolescence presenting an inverted U-shaped trajectory using images from patients with ages comprised between 5 and 24 years old and dividing the cerebellum into 3 main lobes and corpus medullare. [START_REF] Wierenga | Typical development of basal ganglia, hippocampus, amygdala and cerebellum from age 7 to 24[END_REF] focused in cerebellar cortex and an age range from 7 to 24 years old. Most recently, [START_REF] Sullivan | Disturbed cerebellar growth trajectories in adolescents who initiate alcohol drinking[END_REF] studied the effect of alcoholism in adolescence in healthy adolescents from 12 to 21 years old using 10 cerebellar structures. All these previous studies considered narrow age range in moderate samples size and they are based on various level of segmentation (global, lobes, lobules, tissue type). All these differences make it difficult to clearly describe a lifelong evolution of the cerebellum. Recently, [START_REF] Han | Longitudinal analysis of regional cerebellum volumes during normal aging[END_REF]) conducted a study using 2,023 images from 822 participants comprising ages from 55 to 90 years old. They found that cerebellum volume trajectories vary spatially over time. Also found significant differences, both longitudinal and cross-sectional, and five of the 28 studies cerebellar regions presented significant sex effects at baseline.

In this paper, we present a series of models of the trajectory of the cerebellum lobules based on a large set of 2,515 images. This set is covering a wide age range (from 1 to 94 years old) to produce robust and valuable normal growth models across the entire lifespan, as previously done by [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: A MRI analysis[END_REF] Obtaining unbiased or unrestricted conclusions applicable to the entire lifespan is challenging. We have addressed this in two ways. First, we took advantage of the new Big Data sharing paradigm in neuroimaging [START_REF] Poldrack | Making big data open: Data sharing in neuroimaging[END_REF] which gave us access to a large number of subjects (N = 2,831 before our quality control) comprising ages from 9 months to 94 years. And second, the images (gathered from different freely available datasets) have been processed using the same cerebellum segmentation pipeline [START_REF] Romero | CERES: A new cerebellum lobule segmentation method[END_REF]. These images and their segmentation results were examined through a demanding multi-stage quality control to avoid poor quality acquisitions and segmentation errors that could affect the trajectories. To analyze the results and produce new knowledge, we modeled every structure growth with the best model between several options. We compared six models from simplest (linear) to most complex (hybrid) model as done in (Coupé at al., 2017) that cope with fast growth during development and slow atrophy during aging.

Material and methods

Image data

To perform this study, we used 2,515 from the total set of 2,832 after quality control. The dataset consists of 3D MRI T1-weighted images from healthy controls obtained from 9 freely available datasets. The subjects have an age range from 1 to 94 years. Figure 2 shows the age distribution.

Image processing

All the images used in this study were processed using CERES pipeline [START_REF] Romero | CERES: A new cerebellum lobule segmentation method[END_REF] (part of the volBrain platform: http://volbrain.upv.es). volBrain is an open access online platform that provides Brain MRI processing including preprocessing and segmentation in a fully automated manner for academic and research purposes [START_REF] Manjón | volBrain: an online MRI brain volumetry system[END_REF]. Since it was launched in 2015, it has deployed 5 different segmentation pipelines (freely accessible to the scientific community) and currently is being used by more than 4500 users (from more than 1700 academic and research centers around the world) and more than 245,000 cases have been already processed online.

The CERES preprocessing pipeline consists of a series of steps aimed to both enhance the image quality and transform the image to a common coordinate and intensity space. This preprocessing also aims to harmonize image features across individuals, sites and manufacturers. The steps are the following: 1) Denoising using the spatially adaptive non-local means filter [START_REF] Manjón | Improved estimates of partial volume coefficients from noisy brain mri using spatial context[END_REF] to reduce noise. 2) Inhomogeneity correction in native space using N4 bias field correction [START_REF] Tustison | N4ITK: improved N3 bias correction[END_REF] to improve the registration. 3) Affine registration to the MNI (Montreal Neurological Institute) space using ANTs [START_REF] Avants | Advanced normalization tools[END_REF] and the MNI152 template. 4) Region of interest cropping around cerebellum area to reduce the computational burden. 5) Low dimensional nonlinear registration estimation using ANTs [START_REF] Avants | Advanced normalization tools[END_REF]. CERES uses a registration strategy to move the whole template library to the case to be segmented space estimatig only one transformation. The details of this step are described in the original paper [START_REF] Romero | CERES: A new cerebellum lobule segmentation method[END_REF]. 6) Intensity normalization based on ROIs as done in [START_REF] Asman | Muti-Atlas Segmentation using Non-Local STAPLE[END_REF] to achieve that all the cerebellum tissues have the same intensity across the dataset.

The segmentation step consists of a patch-based algorithm called OPAL [START_REF] Ta | Optimized patchMatch for near real time and accurate label fusion[END_REF], Giraud et al., 2015) which uses an optimized version of the non-local label fusion [START_REF] Coupé | Patch-based Segmentation using Expert Priors: application to Hippocampus and Ventricle Segmentation[END_REF] using Patch Match [START_REF] Barnes | PatchMatch: a randomized correspondence algorithm for structural image editing[END_REF]. A dataset of 10 (5 left-right flipped) T1-weighted high-resolution (0.5 x 0.5 x 0.5 mm after preprocessing) images manually segmented by experts into 26 structures (right and left lobules I-II, III, IV, VI, Crus I, Crus II, VIIB, VIIIA, VIIIB, IX, X and white matter) was used to construct the template library. The HR images were obtained from the CoBrALab site (http://cobralab.ca/atlases/Cerebellum.html). For more details about the image data see [START_REF] Park | Derivation of highresolution MRI atlases of the human cerebellum at 3T and segmentation using multiple automatically generated templates[END_REF]. Also, to obtain the total intracranial volume (TIV) we used the segmentation method NICE (Manjón et al., 2014).

CERES segments the cerebellum in 26 structures, 13 on the left 13 on the right. One of the 13 structures is the white matter. Since we did not consider lateralization here, right and left volumes have been combined and finally 12 grey matter regions have been studied. For white matter segmentation, results have been presented with the whole cerebellum analysis.

Statistical analysis

It is common to find studies that use low order polynomial function to model structure volume trajectories as done in [START_REF] Walhovd | Consistent neuroanatomical agerelated volume differences across multiple samples[END_REF] where linear and quadratic models are used to model basal ganglia and brainstem aging over adulthood. In (Temeier et al., 2010) the author used cubic function to model cerebellum lobes trajectory across adolescence. Some studies have used Poisson curve [START_REF] Lebel | Diffusion tensor imaging of white matter tract evolution over the lifespan[END_REF] or Gompertz-like function [START_REF] Makropoulos | Regional growth and atlasing of the developing human brain[END_REF] to model fast brain development in childhood. Combining fast development occurring during childhood and slow aging occurring over adulthood requires more advanced models. [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: A MRI analysis[END_REF] proposed to model fast growth in childhood and slow atrophy using hybrid models. We have tested several functions to model the trajectories for each structure under study and selected the best one. We compared them to decide which fits better the evolution of the cerebellar volumes across the lifespan as done in [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: A MRI analysis[END_REF] To select the best general model (no gender separation) for each structure, the six models are evaluated following three criteria. First, they are compared to the constant model using the Fstatistic (ANOVA) to determine if there exist significant differences (P < 0.05). Then, we require that all its coefficients are significant using a t-statistic (p < 0.05). Finally, the Bayesian information criterion (BIC) is used to select the best model from the remaining candidates. BIC allow us to select the simplest model that represent most of the information.

Once a general model is selected, gender interactions in the trajectories are tested, 𝛽 𝑖 𝑆𝑒𝑥 + 𝛽 𝑗 𝑆𝑒𝑥 • 𝐴𝑔𝑒 in terms of volume and shape. All the statistic tests (t-statistic, F-statistic, BIC and R2) performed in this study were calculated using Matlab©.

Quality control

It has been demonstrated that quality control (QC) has a significant impact on trajectory results [START_REF] Ducharme | Trajectories of cortical thickness maturation in normal brain development-The importance of quality control procedures[END_REF]. In this study we used a three stage QC to select adequate subjects consisting in the following steps. First, we removed 219 images (7.7 % from the total) by visual inspection based on its image quality (this selection is shared with (Coupé at al., 2017) as we used the same dataset). The second step was a visual inspection of the processing results of every subject done using the CERES PDF reports generated by volBrain platform. At this stage we validated preprocessing (denoising, intensity normalization, inhomogeneity correction, etc.), MNI registration, intracranial cavity extraction, cerebellum tissue classification and lobule segmentation. In this step 492 images (15.1 % from the total) were removed. Finally, we defined as possible outliers every case with a volume higher or lower than 2 standard deviations of the estimated model for the total cerebellum volume model. After carefully checking this potential outlier cases by displaying its segmentation map using ITK Snap software [START_REF] Yushkevich | Userguided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability[END_REF] we removed 70 more subjects (2.5 % from the total). After the full QC process 2515 remained from the 2831 initial subjects (88.8 % from the total).

Results

A detailed explanation of the trajectories obtained for every structure and the justification of the selected model is listed below. We show both absolute volume models (cm 3 ) and normalized volume models (absolute volume/Total Intracranial Volume in percent). As handedness is not available for all the cases in this study we decided to not consider lateralization.

Cerebellum

Cerebellum development and maturation, in absolute values, presents a rapid growth from 0 to 17 years for the general model (18 years for female and 21 years for male) followed by a slow decrease during the rest of the lifespan (See Figure 3). This is consistent with the model selection which resulted in a hybrid model combining an exponential cumulative distribution to represent the fast growth with a linear model to explain the slow decrease (p < 0.0001). In terms of normalized volumes, we found that the trajectory is divided in 3 stages. First, a slow decrease (faster for male) until 25 years, followed by a plateau until 60-65 years and a slow decrease from this point (See Figure 3). This is consistent with Cubic model selected (p < 0.0001).

Cerebellum grey matter and white matter trajectories

In absolute values, we observe that WM presents a fast growth stage during youth that reaches its maximum around 10 years (See Figure 4). After this, WM shows an inverted U-shape as it keeps growing slowly until 30-40 years. This trajectory is confirmed by the hybrid model selection (p < 0.0001) combining exponential cumulative distribution for growth and quadratic model for aging (See table 2). For GM we have a similar fast growth during approximately the first 10 years followed by a constant slow decrease and a selected hybrid model combining an exponential cumulative distribution with a linear model (p < 0.0001). Focusing on the normalized volumes as percentage of the TIV, we see that GM shows a decreasing volume represented by a three-stage trajectory, having a central plateau with slightly faster decreases in early years (until 25) and after the 60s which is modeled by a Cubic function (p < 0.0001) (see table 3). For WM, despite the normalization, we still observe an inverted U-shape with its maximum around 50 and a slightly faster degeneration than growth. The selected model for this was a cubic function (p < 0.0001).

Lobules I and II

Lobules I and II, which are treated as one structure in our segmentation protocol, showed an almost constant volume during the entire lifespan in both absolute and normalized volumes (see Figures 5 and6) and this was supported by a selected linear model (p < 0.0001). It is important to note that, in our experience, the number of (visible) lobules can vary between subjects at 1 mm resolution. Indeed, the very difficult delineation task at 1 mm resolution of the smallest lobules may produce volumes near to zero. Therefore, segmentation errors can have huge effect when measuring small structures since the volume of a single voxel (1x1x1mm 3 ) start to be significant as it represents a greater percentage of the total volume compared to bigger structures. This may affect the trajectories of small structures as we are summarizing a whole structure in a few voxels. Lobules I and II are the smallest considered structures in this study, consequently these are the most impacted structures by using 1x1x1 mm 3 resolution image.

Lobules III, IV, V, VI, Crus I, Crus II, VIIB, VIIIA, VIIIB and IX

Analyzing the absolute values, lobules III, IV, V, VI, Crus I, VIIB, VIIIA, VIIIB and IX presented a fast growth in early ages and a slow decrease during the rest of the lifespan (see Figure 5). Lobule IX showed the fastest growth reaching the maximum at around 5 years. The rest of the lobules (except for lobule VIIB) reached their maximums between 12 and 16 years. This was assessed by a hybrid linear model combing an exponential cumulative distribution during the fast growth and a linear function during aging (p < 0.0001). Lobule VIIIB showed a different trajectory for males presenting an inverted U-shape which had its maximum around 30 years, being slightly asymmetrical with a faster aging than growth (see Figure 5) (p < 0.0001). Moreover, for the normalized values, lobules III, Crus I and VIIA showed basically the same decrease ratio during the whole lifespan (see figure 6) being represented by a selected linear model (p < 0.0001) while lobules IV, VI and IX presented a more complex three-stage trajectory composed by a slow decrease until around 20 years, followed by a plateau until 60-65 years, ending with a slow decrease. These trajectories were represented by a selected cubic model (p < 0.0001). Lobules V and VIIIB presented an inverted U-shape trajectory adjusting to the pattern mentioned before (slightly faster growth) finding its maximum volume around 40 years (see Figure 5). These trajectories were modeled using a quadratic function (p < 0.0001). Finally, lobule VIIB presented a more complex trajectory composed by 3 stages starting by a fast decrease from 0 to 25 years followed by a plateau until 65-70 years to finish with another fast decrease (see Figure 5) which resulted in a selected cubic model (p < 0.0001). Crus II absolute values showed a rapid growth from 0 to around 7 years and then a slow decrease of the volume which is not linear neither an inverted U-shape. It described a curve which reaches a plateau around 80 years and keep its volume constant from that point (see Figure 5). This is consistent with the model selection which results in a hybrid model (see Table 2) combining an exponential cumulative distribution with a quadratic function (p < 0.0001). For the normalized volume, Crus II presented a 3-stage trajectory starting with a fast decrease until 25 years following with a plateau and a second slightly slower decrease, starting at 60-66 years (see Figure 6), which corresponded to a cubic model selection (p < 0.0001).

Lobule X

At the global scale (absolute volumes), lobule X presented an inverted U-shape trajectory that has a slow growth from 0 to around 30 years and a slow decrease from this point which tends to a plateau at 90 years (see Figure 5). A cubic model was selected (p < 0.0001) explaining this trajectory.

For the normalized values the trajectory changed very little showing again an inverted U-shape, this time with a slow growth until 40 years and a slow decrease (slightly faster than the growth) from this point (see Figure 6) resulting in a selected quadratic model (p < 0.0001).

Sexual dimorphism

Analyzing the absolute values, trajectories showed that males tend to have higher volumes. The whole cerebellum, the grey matter and the lobules IV and VIIIA showed significant sex differences (sex effect with p < 0.0001). Lobules V, VI, Crus I, VIIB, VIIIB and IX also showed significant sex effect with p-values under 0.006 (see Table 2). For the remaining structures sex effect did not reach statistical significance. As expected, relative volumes for male and female were not statistically different. Visual observation of Crus II and VIIB models suggests possible sex differences, one at the early ages and other for the elder cases.

Discussion

In our study, we developed cerebellum maturation models covering the entire lifespan on a large sample size that lead to a better understanding of the cerebellum development and aging processes.

According to our results, the cerebellum exhibits lifespan evolution principles common to cortex: an exponential growth starting for both compartments when raw volumes were considered and decline of grey matter and U-shaped time course for white matter when normalized [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: A MRI analysis[END_REF]. Moreover, we did not observe different lifespan pattern for the motor and the non-motor counterparts described in functional classification of cerebellum substructures based on either subjects presenting lesions (Stoodley and Schmamhann, 2010) or on functional connectivity recorded at rest (Guel et al., 2018). Total cerebellum models for absolute values in the literature showed a descending linear trajectory from 19 to 73 years old [START_REF] Luft | Patterns of age-related shrinkage in cerebellum and brainstem observed in vivo using threedimensional MRI volumetry[END_REF]. For adolescence ranges, trajectories followed an inverted U-shape having its peak at 17 for males and 12 form females (Temeier et al., 2010). The same inverted U-shape can be seen for males in [START_REF] Wierenga | Typical development of basal ganglia, hippocampus, amygdala and cerebellum from age 7 to 24[END_REF] but not for females which showed a descending linear trajectory. These results (excepted for female models in [START_REF] Wierenga | Typical development of basal ganglia, hippocampus, amygdala and cerebellum from age 7 to 24[END_REF] are in line with our models which described a fast growth from 0 to 17 followed by a slow decrease for absolute values and a fairly constant decrease which accelerates at the end of the age interval. Regarding to normalized values, [START_REF] Raz | Regional brain changes in aging healthy adults: General trends, individual differences, and modifiers[END_REF] reported decreasing volumes in a baseline age representation for healthy participants over 49 years old for normalized volumes which is consistent with our findings. The same trajectories described for absolute values in (Temeier et al., 2010) are reported for normalized values. This is not in agreement with our models as we observed a slight decrease in normalized volumes during the childhood followed by a plateau during the adolescence.

Previous works reported descending trajectories for cerebellar grey matter. Sullivan et al. showed a linear descending trajectory from 19 to 73 years old [START_REF] Sullivan | Cerebellar Volume Decline in Normal Aging, Alcoholism, and Korsakoff's Syndrome: Relation to Ataxia[END_REF] and obtained a quadratic descending curve from 12 to 21 years old for absolute values [START_REF] Sullivan | Disturbed cerebellar growth trajectories in adolescents who initiate alcohol drinking[END_REF]. These models are in line with our curves which showed a fast growth until 10 years old and a slow decrease for absolute values.

We showed a fast growth until 10 years old and a slow decrease for absolute values. This is in line with Sullivan et al. showing a linear descending trajectory from 19 to 73 years old [START_REF] Sullivan | Cerebellar Volume Decline in Normal Aging, Alcoholism, and Korsakoff's Syndrome: Relation to Ataxia[END_REF] and obtained a quadratic descending curve from 12 to 21 years old for absolute values [START_REF] Sullivan | Disturbed cerebellar growth trajectories in adolescents who initiate alcohol drinking[END_REF].

Regarding to cerebellar white matter trajectories, Sullivan et al. obtained a slowly ascending linear trajectory from 19 to 73 years old for absolute values [START_REF] Sullivan | Cerebellar Volume Decline in Normal Aging, Alcoholism, and Korsakoff's Syndrome: Relation to Ataxia[END_REF] and reported a slow growth quadratic curve for the range of 12 to 21 years old [START_REF] Sullivan | Disturbed cerebellar growth trajectories in adolescents who initiate alcohol drinking[END_REF]. Our absolute volume models showed a fast growth of white matter until 10 years old followed by a flat U-shape and a slow decrease from 40 years old for absolute values. This trajectory does not match perfectly with previous works trajectories but we think it is in good agreement if we take into account our larger age range which have an effect on the trajectory estimation.

( Temeire et al., 2010) obtained an inverted U-shape trajectory for theach inferior posterior lobe, superior posterior lobe and anterior lobe for absolute values. The inferior posterior lobe corresponds with our Crus II, VIIB, VIIIA, VIIIB and IX. Comparing this trajectory with our results for the same age range we found that our models for lobules VIIB, VIIIB and VIIIB are consistent with Temeier et al. (2010). However, our models for Crus II and lobule IX are slightly different as we obtained a linear decrease for IX and a slightly faster decrease for Crus II. We think that the fact that in (Temeier et al., 2010) consider the entire inferior posterior lobe could be hiding the differences between lobules. The superior posterior lobe corresponds to our VI, Crus I. The trajectory shown in (Temeier et al., 2010) is in line with our models for lobule VI and Crus I at the corresponding age range. Our models showed and inverted U-shape following the fast growth. Finally, the anterior lobe corresponds to our lobules III, IV and V. Temeier et al. ( 2010) results for anterior lobe matches with our models for lobules III, IV and V that showed the same slightly curved inverted U-shape.

Also, by using a segmentation protocol that divides the cerebellum in 12 structures, we provide specific results about sexual dimorphism (in absolute values) distinguishing lobules IV, V, VI, Crus I, VIIB, VIIIA and VIIIB that significantly presented gender effect while the rest of lobules were not affected (Lobule IX showed marginal differences with a p = 0.03). Also we obtained that the whole cerebellum volume shows evidences of sexual dimorphism which is consistent with (Temeier et al., 2010).

We observed that the whole cerebellum volume shows evidences of sexual dimorphism which is consistent with (Temeier et al., 2010). Also, by using a segmentation protocol that divides the cerebellum in 12 structures, we provide specific results about sexual dimorphism (in absolute values) distinguishing lobules IV, V, VI, Crus I, VIIB, VIIIA and VIIIB that significantly presented gender effect while the rest of lobules were not affected (Lobule IX showed marginal differences with a p = 0.03). Finally, none of the structures showed significant interaction between sex and age, either in absolute or normalized volumes

Our results are also in partial agreement with [START_REF] Fan | Sexual dimorphism and asymmetry in human cerebellum: an MRI-based morphometric study[END_REF] in which only absolute values presented significant differences across gender. Grey matter volume showed significant differences in lobules V, right Crus I, VII, VIIB and left IX, in agreement with our findings. However, Fan's study did not show any significant differences for lobules IV, VI and VIIA while reported significant differences for Crus II in contrast with our results. These inconsistencies could be explained by the use of a different segmentation protocol, the age range limited to 18 and 33 years old and the size of the dataset of 112 images.

The trajectories we generated showed different evolution patterns. Lobules III, IV, V, VI, Crus I, VIIB, VIIIA, VIIIB and IX showed a fast growth in the early ages represented by an exponential cumulative distribution followed by a slow decrease that adjusted to a linear function or slightly curved second order model. Crus II described a curve instead of an inverted U-shape that fitted a quadratic model after the fast growth. Finally, lobule X followed an inverted U-shape represented by a cubic function without the presence of a fast growth stage. We observed a late peak maturation at 30 years old for VIIIB and lobule X, a slow volume increase until 40-50 years old as for hippocampus and amygdala (Coupe et al., 2017). Since the cerebellum is involved in learning and experience based-plasticity, the delayed peak maturation of these structures may support the implication of these two subparts in lifespan adaptation abilities.

From the functional point of view, the anterior lobe (lobules I to V) is mainly involved in the wellrecognized sensorimotor function of the cerebellum [START_REF] Stoodley | Functional topography in the human cerebellum: A meta-analysis of neuroimaging studies[END_REF], and the developmental peak showed at about 12 and 16 years (in absolute values) is probably related with the maturation of the control of movement in kids. However, lobule V is somewhat distinct, having an ample peak during late adolescence, which in the relative trajectory translates into a plateau extending along the second and third lifespan decades. This suggests that lobule V may be the principal locus of the plasticity related to motor learning in early and middle adulthood. Several studies have shown that lobule V is associated with hand movements [START_REF] Bushara | Multiple tactile maps in the human cerebellum[END_REF][START_REF] Grodd | Sensorimotor mapping of the human cerebellum: fMRI evidence of somatotopic organization[END_REF], including finger tapping [START_REF] Stoodley | Functional topography of the cerebellum for motor and cognitive tasks: an fMRI study[END_REF], which is consistent with a role of this lobule in learning skilled hand movements.

The superior posterior lobe (our lobules VI and Crus I) is mainly related to cognitive functions, such as working memory and executive functions (motor planning and tool use) [START_REF] Stoodley | Chapter 4 -Functional topography of the human cerebellum[END_REF], and these functionality is likely related with the connectivity of this lobe (mainly Crus I) with prefrontal cortical areas (Balters et al., 2013). Our results show that these lobules VI and Crus I have maturation peaks (in absolute values), slightly delayed with respect to that of the anterior lobe. This suggests that the maturation peak could be influenced by the later development of the prefrontal cortex (Tefer and Semendeferi, 2012).

With regard to the inferior posterior lobe (lobules Crus II, VIIB, VIIIA, VIIIB and IX), it has been associated with a variety of cognitive performance [START_REF] Stoodley | Chapter 4 -Functional topography of the human cerebellum[END_REF]. However, there is heterogeneity in the roles in which the different lobules are involved. Lobule VIII is a secondary sensorimotor structure [START_REF] Bushara | Multiple tactile maps in the human cerebellum[END_REF][START_REF] Debas | Brain plasticity related to the consolidation of motor sequence learning and motor adaptation[END_REF], and in fact the trajectory we found for VIIIB is similar to that of lobule V, which is more involved in motor function. In contrast, Crus II, VIIB and IX are the lobules with more strong interconnections with association cortices such as the prefrontal cortex [START_REF] Kelly | Cerebellar Loops with Motor Cortex and Prefrontal Cortex of a Nonhuman Primate[END_REF][START_REF] O'reilly | Distinct and overlapping functional zones in the cerebellum defined by resting state functional connectivity[END_REF], and in fact they show a trajectory similar (in absolute values) to that of lobules VI and Crus I (the superior posterior lobe).

The lobule X (or flocculonodular lobe) corresponds to the vestibulocerebellum, and is mainly involved in maintaining balance, posture and eye movements [START_REF] Schniepp | Gait ataxia in humans: vestibular and cerebellar control of dynamic stability[END_REF]. The lifespan trajectory we found is somewhat surprising, since for such basic functions it would be expected an early maturation in infancy and a long plateau along most of the lifespan. In contrast, we found a late maturation (around the third decade of life, in absolute values). This suggests maintained plasticity levels along the first four decades of life. Consistent with this hypothesis, experimental studies in animals have revealed the existence of long-term potentiation and long-term depression phenomena underlying motor learning in the vestibulo ocular reflex [START_REF] Broussard | Motor learning in the VOR: the cerebellar component[END_REF]. Thus, it is possible that the functions of lobule X require a high level of plasticity along most of the lifespan.

Trajectories presented in [START_REF] Han | Longitudinal analysis of regional cerebellum volumes during normal aging[END_REF] are in agreement with our results. The anterior lobe (corresponding to lobules I, II, III, IV and V) shows a slightly curved descending trajectory that perfectly accommodates with our lobules I, II, III, VI and V trajectories between 55 and 90 years old for normalized volumes. The posterior lobe (corresponding to lobules VI, VIIA, VIIIA, VIIIB, Crus I, Crus II and IX) presented a constant descending straight line. This is in contrast with our finding where lobules VI, VIIA, VIIIB, Crus II and XI showed a descending curve begin lobules VIIIA and Crus I the only ones fitting a descending straight line. This differences could be explained by the averaging effect in in the posterior lobe where Lobule VIIIA and Crus I have a significant higher volume than the other lobules. Finally, lobule X presents as well a slightly curved descending trajectory similar to our lobule X trajectory which follow a slightly steeper descent. This slope difference can be explained as our models can benefit from a wider age range. Sexual dimorphism findings in [START_REF] Han | Longitudinal analysis of regional cerebellum volumes during normal aging[END_REF] are no comparable to our results as we are considering the whole lifespan while [START_REF] Han | Longitudinal analysis of regional cerebellum volumes during normal aging[END_REF] is only considering participants over 55 years old.

This study has some limitations. First, image resolution of 1 mm 3 does not allow us to extract reliable conclusions about lobules I-II (due to their small size) as it presents a linear trajectory close to constant. Image inhomogeneities affect cerebellum, especially the inferior lobules that can be mislabeled as CSF due to the signal loss. Fortunately, the N4 bias corrector can cope with most of the inhomogeneities and the rest of cases were removed from the study during the quality control. Also, gathering several datasets can be a limitation due to the introduction of artificial differences. However, we used at least two overlapping datasets for each 5 periods and CERES preprocessing pipeline is designed to mitigate the negative impact of the use of different acquisition protocols.

Finally, as occurred in [START_REF] Coupé | Towards a unified analysis of brain maturation and aging across the entire lifespan: A MRI analysis[END_REF], after the quality control we were left with no subjects younger than 9 months and the results over this period might be less accurate. As Coupé et al. explained in their work, very few studies are performed focusing on newborn since the acquisition is difficult and the low contrast before 6 months and the fast myelination progress during the first 2 years makes the analysis challenging.

Conclusion

In this work we have performed a cerebellum volumetric analysis at structure and substructure scales covering the entire lifespan, using a very large number of subjects. We have addressed several limitations of previous studies and provided new knowledge about lobules cerebellum development and maturation. We used absolute and relative cerebellum volumes to better understand the cerebellum evolution in different stages for both males and females.

As a result, we provided robust models that represent the evolution of cerebellum lobules along the whole lifespan of the human cerebellum. These estimated models are very useful to clinical studies by providing normal boundaries of volumetric subparts of the cerebellum. In the near future these models will be available through our online platform volBrain.

The data that support the findings of this study are available in corresponding websites previously listed in the materials and methods section.

Figure 2 :

 2 Figure 2: Age distribution of cases used after quality control. Left chart shows age distribution for all the considered subjects. Right chart shows age distribution for child under 10 years old. Legend indicates the number of images for each dataset after the quality control.

Figure 3 :

 3 Figure 3: Cerebellum trajectories for absolute values (left) and normalized values (right). General model is shown in black, female model in magenta and male model in blue. Dots color correspond to the different datasets used. Male cases are represented by "+"s and female cases are represented by "o"s.

Figure 4 :

 4 Figure 4: First row: Cerebellum grey matter and white matter trajectories for absolute values. Second row: Cerebellum grey matter and white matter trajectories for normalized values. General model is shown in black, female model in magenta and male model in blue. Dots color correspond to the different datasets used. Male cases are represented by "+"s and female cases are represented by "o"s.

Figure 5 :

 5 Figure 5: Volume trajectories for absolute volumes in cm 3 (y-axis) for cerebellum lobules over age in years (x-axis). General model is shown in black, female model in magenta and male model in blue. Dots color correspond to the different datasets used. Vertical axis shows volume in cm 3 . Horizontal axis shows age in years. Male cases are represented by "+"s and female cases are represented by "o"s.

Figure 6 :

 6 Figure 6: Volume trajectories for normalized volumes (y-axis) for cerebellum lobules over age in years (x-axis).. General model is shown in black, female model in magenta and male model in blue. Dots color correspond to the different datasets used. Vertical axis shows volume in percentage to the TIV. Horizontal axis shows age in years. Male cases are represented by "+"s and female cases are represented by "o"s.

Table 1 :

 1 C-MIND (N = 266, after QC N = 182): The images from the C-MIND dataset (https://research.cchmc.org/c-mind/) used in this study consist of 266 control subjects. All the images were acquired at the same site on a 3T scanner. The MRI are 3D T1-weighted MPRAGE highresolution anatomical scan of the entire brain with spatial resolution of 1 mm 3 acquired using a channel SENSE head-coil. //adni.loni.ucla.edu/research/protocols/mri-protocols) and a custom MPRAGE sequence was used on the 1.5T scanner. Dataset summary

	NDAR (N = 147, after QC N = 135): The Database for Autism Research (NDAR) is a national database
	funded by NIH (https://ndar.nih.gov). This database included 13 different cohorts acquired on 1.5T
	MRI and 3T scanners. In our study, we used 147 images of control subjects from the Lab Study 19 of
	National Database for Autism Research. For the NIHPD, T1-weighted images were acquired at six
	different sites with 1.5 Tesla systems by General Electric (GE) and Siemens Medical Systems. The
	MRI are 3D T1-weighted spoiled gradient recalled (SPGR) echo sequence with following parameters:
	TR = 22-25 ms, TE = 10-11 ms, flip angle = 30°, FoV = 256 mm IS × 256 mm AP, matrix size = 256 ×
	256: 1 × 1 × 1 mm 3 voxels, 160-180 slices of sagittal orientation. The participants chosen from the
	Lab Study 19 of National Database for Autism Research (NDAR) were scanned using a 3T Siemens
	Tim Trio scanner at each site. The MRI are 3D MPRAGE sequence (voxel dimensions: 1.0 × 1.0 × 1.0
	mm 3 ; image dimensions: 160 × 224 × 256, TE = 3.16 ms, TR = 2,400 ms).
	ABIDE (N = 528, after QC N = 450): The images from the Autism Brain Imaging Data Exchange (ABIDE)
	dataset (http://fcon_1000.projects.nitrc.org/indi/abide/) used in this study consist of 528 control
	subjects acquired at 20 different sites on 3T scanner. The MRI are T1-weight MPRAGE image and
	the details of acquisition, informed consent, and site-specific protocols are available on the website.
	ICBM (N = 308, after QC N = 289): The images from the International Consortium for Brain Mapping
	(ICBM) dataset (http://www.loni.usc.edu/ICBM/) used in this study consist of 308 normal subjects
	obtained through the LONI website. The MRI are T1-weighted MPRAGE (fast field echo, TR = 17 ms,
	TE = 10 ms, flip angle = 30°, 256 × 256 matrix, 1 mm 2 in plane resolution, 1 mm thick slices) acquired
	on a 1.5T Philips GyroScan imaging system (Philips Medical Systems, Best, The Netherlands).
	OASIS1 (N = 315, after QC N = 290): The images from the Open Access Series of Imaging Studies
	(OASIS1) database (http://www.oasis-brains.org) used in this study consist of 315 control subjects.
	The MRI are T1-weighted MPRAGE image (TR = 9.7 ms, TE = 4 ms, TI = 20 ms, flip angle = 10 degrees,
	slice thickness = 1.25 mm, matrix size = 256 × 256, voxel dimensions = 1 × 1 × 1.25 mm 3 resliced to
	mm 3 , averages = 1) acquired on a 1.5-T Vision scanner (Siemens, Erlangen, Germany).
	IXI (N = 588, after QC N = 519): The images from the Information eXtraction from Images (IXI)
	database (http://brain-development.org/ixi-dataset/) used in this study consist of 588 normal
	subjects. The MRI are T1weighted images collected at 3 sites with 1.5 and 3T scanners (FoV = 256
	mm × 256 mm, matrix size = 0.9375 × 0.9375 × 1.2 mm 3 ).
	ADNI1 (N = 228, after QC N = 218): The images from the Alzheimer's Disease Neuroimaging Initiative
	(ADNI) database (http://adni.loni.usc.edu) used in this study consist of 228 control subjects from
	the 1.5T baseline collection. These images were acquired on 1.5T MR scanners at 60 different sites
	across the United States and Canada. A standardized MRI protocol to ensure cross-site

  𝑉𝑜𝑙 = 𝛽 4 (1 -𝑒 -𝐴𝑔𝑒/𝛽 5 ) + 𝛽 0 + 𝛽 1 𝐴𝑔𝑒 + 𝛽 2 𝐴𝑔𝑒 2 + 𝛽 3 𝐴𝑔𝑒 3 + 𝜀

	which are listed below:
	1. Linear model
	𝑉𝑜𝑙 = 𝛽 0 + 𝛽 1 𝐴𝑔𝑒 + 𝜀
	2. Quadratic model
	𝑉𝑜𝑙 = 𝛽 0 + 𝛽 1 𝐴𝑔𝑒 + 𝛽 2 𝐴𝑔𝑒 2 + 𝜀
	3. Cubic model
	𝑉𝑜𝑙 = 𝛽 0 + 𝛽 1 𝐴𝑔𝑒 + 𝛽 2 𝐴𝑔𝑒 2 + 𝛽 3 𝐴𝑔𝑒 3 + 𝜀
	4. Linear hybrid model: exponential cumulative distribution for growth with linear model aging
	𝑉𝑜𝑙 = 𝛽 4 (1 -𝑒 -𝐴𝑔𝑒/𝛽 5 ) + 𝛽 0 + 𝛽 1 𝐴𝑔𝑒 + 𝜀
	5. Quadratic hybrid model: exponential cumulative distribution for growth with quadratic model
	aging
	𝑉𝑜𝑙 = 𝛽

4

(1 -𝑒 -𝐴𝑔𝑒/𝛽 5 ) + 𝛽 0 + 𝛽 1 𝐴𝑔𝑒 + 𝛽 2 𝐴𝑔𝑒 2 + 𝜀 6. Cubic hybrid model: exponential cumulative distribution for growth with cubic model aging

Table 2 :

 2 Model selection and statistics of model analysis for absolute volumes

	Absolute	Selected	F-Statistic	R 2	Model	vs.	Gender	Age × gender
	volume	Model			constant		interaction	interaction
					model
					p-value of the	p-value of the	p-value of the
					F-statistic		t-statistic on	t-statistic on
					based	on	the coefficient	the
					ANOVA			coefficient
	Cerebellum
	Global	Hybrid	479	0.27 p < 0.0001		p < 0.0001	P > 0.05
	Male	linear	223	0.25 P < 0.0001
	Female		193	0.25 P < 0.0001
	Lobule I-II
	Global	Linear	44.5	0.02 P = 0.001		P > 0.05	P > 0.05
	Male		26.7	0.02 P < 0.0001
	Female		5.55	0.004 P = 0.019
	Lobule III
	Global	Hybrid	128	0.09 P < 0.0001		P > 0.05	P > 0.05
	Male	linear	66.7	0.09 P < 0.0001
	Female		36.8	0.06 P < 0.0001
	Lobule IV
	Global	Hybrid	303	0.19 P < 0.0001		P = 0.0002	P > 0.05
	Male	linear	163	0.19 P < 0.0001
	Female		79.1	0.12 P < 0.0001
	Lobule V
	Global	Hybrid	101	0.07 P < 0.0001		P = 0.006	P > 0.05
	Male	linear	46.3	0.06 P < 0.0001
	Female		29	0.05 P < 0.0001
	Lobule VI
	Global	Hybrid	313	0.2	P < 0.0001		P = 0.019	P > 0.05
	Male	linear	141	0.17 P < 0.0001
	Female		118	0.17 P < 0.0001
	Crus I
	Global	Hybrid	313	0.2	P < 0.0001		P = 0.002	P > 0.05
	Male	linear	138	0.17 P < 0.0001
	Female		111	0.16 P < 0.0001
	Crus II
	Global	Hybrid	192	0.19 P < 0.0001		P = 0.007	P > 0.05
	Male	Second	87.5	0.16 P < 0.0001
	Female	order	63.7	0.14 P < 0.0001
	Lobule VIIB
	Global	Hybrid	192	0.13 P < 0.0001		P = 0.006	P > 0.05
	Male	linear	72.5	0.1	P < 0.0001
	Female		70.4	0.11 P < 0.0001
	Lobule VIIIA
	Global	Hybrid	199	0.14 P < 0.0001		P = 0.0003	P > 0.05

Table 3 :

 3 Model selection and statistics of model analysis for normalized volumes

	Normalized	Selected	F-	R 2	Model	vs.
	volume	Model	Statistic		constant
					model
					p-

value of the F-statistic based on ANOVA Gender interaction p-value of the t- statistic on the coefficient Age × gender interaction p-value of the t- statistic on the coefficient

	Global	Third	143	0.15 P < 0.0001	P > 0.05	P > 0.05
	Male	order	120	0.21 P < 0.0001
	Female		40.8	0.09 P < 0.0001
	Lobule V
	Global	Second	31.1	0.02 P < 0.0001	P > 0.05	P > 0.05
	Male	order	29	0.04 P < 0.0001
	Female		12.1	0.02 P < 0.0001
	Lobule VI
	Global	Third	135	0.14 P < 0.0001	P > 0.05	P > 0.05
	Male	order	94.5	0.17 P < 0.0001
	Female		61.9	0.14 P < 0.0001
	Crus I
	Global	Linear	397	0.14 P < 0.0001	P > 0.05	P > 0.05
	Male		283	0.17 P < 0.0001
	Female		143	0.11 P < 0.0001
	Crus II
	Global	Third	142	0.14 P < 0.0001	P > 0.05	P > 0.05
	Male	order	112	0.2	P < 0.0001
	Female		48.5	0.11 P < 0.0001
	Lobule VIIB
	Global	Third	81.3	0.09 P < 0.0001	P > 0.05	P > 0.05
	Male	order	58.6	0.11 P < 0.0001
	Female		31.8	0.07 P < 0.0001
	Lobule VIIIA
	Global	Linear	234	0.09 P < 0.0001	P > 0.05	P > 0.05
	Male		109	0.07 P < 0.0001
	Female		103	0.08 P < 0.0001
	Lobule VIIIB
	Global	Second	100	0.07 P < 0.0001	P > 0.05	P > 0.05
	Male	order	41.6	0.06 P < 0.0001
	Female		51.7	0.08 P < 0.0001
	Lobule IX
	Global	Third	79.2	0.09 P < 0.0001	P > 0.05	P > 0.05
	Male Cerebellum Global Female Male Lobule X Female Global Male Lobule I-II Global Female Male White matter Female Global Male Lobule III Global Female Male Grey matter Female Global Male Lobule IV Female	order Third order Second order Linear Second order Linear Third order	53.3 237 32.2 169 96.9 92.7 50.6 24.2 56.3 31 3.9 157 58.2 142 125 119 48.5 303 209 108	0.1 0.22 P < 0.0001 P < 0.0001 0.08 P < 0.0001 0.27 P < 0.0001 0.2 0.07 P < 0.0001 P < 0.0001 0.07 P < 0.0001 0.01 P < 0.0001 0.09 P < 0.0001 0.02 P < 0.0001 0.11 P < 0.0001 0.003 P < 0.0001 0.08 P < 0.0001 0.05 P < 0.0001 0.18 P < 0.0001 0.08 P < 0.0001 0.27 P < 0.0001 0.04 P < 0.0001 0.31 P < 0.0001 0.21 P < 0.0001	P > 0.05 P > 0.05 P > 0.05 P = 0.05 P > 0.05 P > 0.05	P > 0.05 P > 0.05 P > 0.05 P > 0.05 P > 0.05 P > 0.05

Acknowledgements

This research was supported by the Spanish DPI2017-87743-R grant from the Ministerio de Economia, Industria y Competitividad of Spain. This work benefited from the support of the project

DeepvolBrain of the French National Research Agency (ANR-18-CE45-0013). This study was achieved within the context of the Laboratory of Excellence TRAIL ANR-10-LABX-57 for the BigDataBrain project. Moreover, we thank the Investments for the future Program IdEx Bordeaux (ANR-10-IDEX-03-02, HL-MRI Project), Cluster of excellence CPU and the CNRS/INSERM for the DeepMultiBrain project.

Also, we want to thank the investigators that collected and made available all the data in the datasets we used: The C-MIND Data Repository created by the C-MIND study of Normal Brain Development. Conducted by Cincinnati Children's Hospital Medical Center and UCLA. https://research.cchmc.org/c-mind. The NDAR data were obtained from the NIH-supported National Database for Autism Research (NDAR). NDAR is a collaborative informatics system created by the National Institutes of Health to provide a national resource to support and accelerate research in autism. Conducted by the Brain Development Cooperative Group. http://pediatricmri.nih.gov/nihpd/info/participating_centers.html. The ADNI data used were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI). The ADNI is funded by the National Institute on Aging and the National Institute of Biomedical Imaging and Bioengineering and through generous contributions from the following: Abbott, AstraZeneca AB, Bayer Schering Pharma AG, Bristol-Myers Squibb, Eisai Global Clinical Development, Elan Corporation, Genentech, GE Healthcare, GlaxoSmithKline, Innogenetics NV, Johnson & Johnson, Eli Lilly and Co., Medpace, Inc., Merck and Co., Inc., Novartis AG, Pfizer Inc., F. Hoffmann-La Roche, Schering-Plough, Synarc Inc., as well as nonprofit partners, the Alzheimer's Association and Alzheimer's Drug Discovery Foundation, with participation from the U.S. Food and Drug Administration. Private sector contributions to the ADNI are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study was coordinated by the Alzheimer's Disease Cooperative Study at the University of California, San Diego. ADNI data are disseminated by the Laboratory for NeuroImaging at the University of California, Los Angeles. The OASIS1 data used were obtained from the OASIS1 project (http://www.oasis-brains.org/). The AIBL data used were obtained from the AIBL study of ageing (www.aibl.csiro.au). The ICBM data used and the IXI data used were supported by http://www.brain-development.org/. The ABIDE data used were supported by ABIDE funding resources listed at http://fcon_1000.projects.nitrc.org/indi/abide/. ABIDE primary support for the work by Adriana Di Martino. Primary support for the work by Michael P. Milham and the INDI team was provided by gifts from Joseph P. Healy and the Stavros Niarchos Foundation to the Child Mind Institute. http://fcon_1000.projects.nitrc.org/indi/abide/. This manuscript reflects the views of the authors and may not reflect the opinions or views of the database providers.


Craig Stuart Carlson

David Milton Rubin

Vilma Heikkilä

Michiel Postema

Extracting transmission and recovery parameters for an adaptive global system

Keywords: system dynamics, coronavirus model, SIR model, parameter extraction

I. INTRODUCTION

The coronavirus disease (COVID-19) pandemic threw the world into a state of turmoil from late 2019, with 160 074 267 confirmed cases reported on 13 May 2021 [START_REF]Coronavirus disease (COVID-19) outbreak[END_REF]. Due to the highly contagious nature and rapid spread of COVID-19 [START_REF] Sanche | High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus 2[END_REF], a need for tools to forecast the spread of the virus has emerged.

Being able to evaluate the number of infections could aid governments to determine necessary precautions to slow down the spread of the disease and assess the required capacity of healthcare facilities to treat the disease. Epidemic characteristics have been described by logistic growth models [START_REF] Zou | Outbreak analysis with a logistic growth model shows COVID-19 suppression dynamics in China[END_REF], [4], natural growth models [START_REF] Huang | A data driven time-dependent transmission rate for tracking an epidemic: a case study of 2019-nCoV[END_REF], auto regressive integrated moving average models [START_REF] Abuhasel | Analyzing and forecasting COVID-19 pandemic in the Kingdom of Saudi Arabia using ARIMA and SIR models[END_REF], Susceptible-Infected-Recovered (SIR) compartment models [4], [START_REF] Abuhasel | Analyzing and forecasting COVID-19 pandemic in the Kingdom of Saudi Arabia using ARIMA and SIR models[END_REF], [START_REF] Nguemdjo | Simulating the progression of the COVID-19 disease in Cameroon using SIR models[END_REF], and derivations thereof [START_REF] Alsayed | Prediction of epidemic peak and infected cases for COVID-19 disease in Malaysia[END_REF]- [START_REF] Zhao | Modeling the epidemic dynamics and control of COVID-19 outbreak in China[END_REF]. The simplest modelling method is the SIR model as it described by two parameters. Such a simplistic model is not to be used to give explanations about the virus itself. Key modelling characteristics include the rate of transmission, the rate of recovery and the basic reproduction number. For COVID-19, the basic reproduction number lies on the range 1.00-3.15 [START_REF] Al-Raeei | The basic reproduction number of the new coronavirus pandemic with mortality for India, the Syrian Arab Republic, the United States, Yemen, China, France, Nigeria and Russia with different rate of cases[END_REF], [START_REF] He | Estimation of the basic reproduction number, average incubation time, asymptomatic infection rate, and case fatality rate for COVID19: meta-analysis and sensitivity analysis[END_REF].

Whilst any number of parameters can be added, the challenge in designing a mathematical model is estimating the actual parameter values to achieve accurate results. This requires the fitting of the model to actual data to extract parameter values. As model parameters are fitted for a geographic region of interest and compared to a geographic data subset there is great variation of parameters in a model. These region-specific parameters may not describe the spread of a disease on a global scale.

In this study an SIR model was implemented to represent the spread of COVID-19 on a global scale, with a simple set of parameters considered. The set of transmission and recovery parameters was reverse-engineered from aggregated global data. Special attention was paid to the deduction of the initial incidence. This is highly relevant, as undertesting in epidemics leads to false parameter estimations and consequently to false predictions. Any effects of mutations and countermeasures were not included in the model.

II. THEORY

The SIR epidemic model is based on earlier epidemics research that propose the division of the population into compartments [START_REF] Ross | An application of the theory of probabilities to the study of a priori pathometry. -Part I[END_REF]- [START_REF] Bertozzi | The challenges of modeling and forecasting the spread of COVID-19[END_REF]. This model simplifies the dynamics of the disease to three compartments -those at risk for infection; those infectious and infected with the disease; and those no longer infected due to recovery. These compartments are named Susceptible; Infected; and Recovered, respectively.

Each compartment, called a stock, represents an accumulation of people at time t. There are two flow rates associated with the stocks in the basic SIR model, namely the infection rate and the recovery rate. The infection rate is the rate of susceptible people becoming infected, and the recovery rate is the rate of infected people recovering from the disease.

The model is governed by a set of three ordinary differential equations which describe how the three stocks change over time [START_REF] Ma | Dynamical Modeling And Analysis Of Epidemics[END_REF]. The rate of change of the susceptible stock is

dS dt = -β I N S, (1)
where I is the infected population size, N is the total population size, S is the susceptible population size and β is the transmission parameter of disease.

Moreover, the rate of change of the infected stock is

dI dt = β I N S -γI, (2)
where γ is the recovery parameter.

The rate of change of the recovered stock is where R is the recovered population.

dR dt = γI, (3)
Figure 1 shows the graphical representation of the basic SIR model described by equations ( 1)-( 3). The graph shows the number of people in each compartment on a given day. This basic model can be extended by including other compartments such as death, quarantine, and hospitalisation. Adding parameters to a model inherently increases the complexity of the solution. The influence of lockdowns, closing of borders, and rolling out a vaccination programme on the susceptibility might be included in a model [START_REF] Seyedalinaghi | Reinfection risk of novel coronavirus (COVID-19): a systematic review of current evidence[END_REF], whilst a mutation of the virus might change the transmission parameter [START_REF] Plante | The variant gambit: COVID-19's next move[END_REF].

For convenience, the potential severity of an infectious disease is expressed by the basic reproduction number, R 0 [START_REF] Breban | Theory versus data: how to calculate R 0 ?[END_REF]. If the susceptible population is equal to the total population, the basic reproduction number is approximated by

R 0 ≈ β γ . (4)
If R 0 > 1, the epidemic is sustained.

III. METHODS

The basic SIR model used in this study is shown in Figure 1A. The model was implemented in the Ventana ® simulation environment Vensim ® (Ventana Systems, Inc., Harvard, MA, USA). Solutions were computed for a 65-week duration. The starting conditions of the simulations were those of the well-documented situation on 1 March 2020 [START_REF] Worldometer | COVID-19 coronavirus pandemic[END_REF].

The transmission and recovery parameters were reverseengineered from aggregated global data of infection and recovery numbers [START_REF] Worldometer | COVID-19 coronavirus pandemic[END_REF], assuming a constant world population size N = 7.8 × 10 9 people, and taking S = N -I -R. These were substituted into equations ( 1)-(3). The input set of SIR parameters, normalised by N , is shown in Figure 2.

The reverse-engineered vectors β(t) and γ(t) were subjected to five-degree polynomial fitting using the fit.m routine of MATLAB ® (The MathWorks, Inc., Natick, MA, USA), yielding β(t) = 5 i=0 β i t i and γ(t) = 5 i=0 γ i t i . Both polynomial representations were subsequently used in an adaptive global system dynamics model for long-term prediction of the COVID-19 pandemic spread. For comparison, parameters were extracted from literature. To find relevant articles for the first dataset, we used the PubMed ® database (National Library of Medicine, Bethesda, MD, USA) using the query (COVID-19) ∩ (SIR model). Only articles published between 1 January 2020 and 28 February 2021 were included. Of the initial selection of 60 SIR-like models, only 21 stated values for the transmission and recovery parameters. A full analysis of the query outcome has been presented by Heikkilä [START_REF] Heikkilä | A quantitative analysis of susceptible-infected-removed models of the coronavirus[END_REF]. The values for the transmission parameter in these 21 articles were averaged, and so were the values for the recovery parameter.

IV. RESULTS AND DISCUSSION

Figure 3 demonstrates graphical representations of the transmission parameter β(t), the recovery parameter γ(t), and the basic reproduction number R 0 (t). While their average values are β = 0.33 week -1 and γ = 0.23 week -1 , it is evident that there is a high degree of variability over the course of the epidemic. This suggests the value of incorporating timevarying transmission and recovery parameters into the SIR model.

The values extracted from articles containing regional data extend over the ranges 0.13 ≤ β ≤ 0.70 week -1 and 0.04 ≤ γ ≤ 0.29 week -1 . Thus, the average global values lie within these ranges.

The polynomial coefficients of the transmission and recovery parameters are shown in Table I. Clearly, the coefficients become negligible beyond third degree. The forward modelling using the transmission and recovery coefficients from Table I is shown in Figure 4, for an interval of 65 weeks, and an adjusted initial infection incidence I(0), which was presumed three times the reported initial infection incidence. The simulated data visually aligns with the real incidence data. We noted that the simulated output became computationally unstable beyond 15 weeks after the reported data. Although a lower initial incidence yields a graph of similar shape (not shown), the peak value is only a third. We attribute this discrepancy with the reported value to lack of testing facilities on the starting date of 1 March 2020. Our parameter extraction suggests a novel methodology to quantify undertesting retrospectively in epidemics.

V. CONCLUSION

The average transmission and recovery parameters were computed to be 0.33 week -1 and 0.23 week -1 , respectively. These values lie well within the range of reported values on COVID-19 determined from geographically different regions.

Fig. 1 .

 1 Fig. 1. A graphical representation of the basic SIR model (A) and associated graphs of the solutions of the ordinary differential equations (B), demonstrating the behaviour of a viral epidemic with starting conditions S(0) = 999 people, I(0) = 1 person, R(0) = 0 people, and model parameters β = 0.5 week -1 , γ = 0.3 week -1 .

Fig. 2 .

 2 Fig. 2. Input SIR values used to compute β and γ.

Fig. 3 .

 3 Fig. 3. Transmission parameter β (-), recovery parameter γ (--), and basic reproduction number R 0 (-), all as a function of time.

TABLE I POLYNOMIAL

 I COEFFICIENTS FOR β AND γ CURVE FITTING.

	Polynomial
	degree i	β i	γ i
	0	0.7993	0.2507
	1	-0.0899	-0.03592
	2	0.005827	0.00471
	3	-0.0001697	-0.0002116
	4	2.306 × 10 -6	3.983 × 10 -6
	5	-1.246 × 10 -8	-2.705 × 10 -8

This work has been based on research supported in part by the National Research Foundation of South Africa, Grant Number 127102, and by the Academy of Finland, Grant Number 340026.* Correspondence to: Craig.Carlson@wits.ac.za

We demonstrate that the extraction of transmission and recovery parameters from real SIR data is feasible for an adaptive global system dynamics model of the COVID-19 pandemic. This study suggests a novel methodology for forward prediction and for the quantification of undertesting early in the epidemic. This model presented in this paper is the first to retrospectively deduce the initial incidence.