Calibration of Biophysical Models for tau-Protein Spreading in Alzheimer's Disease from PET-MRI
CCalibration of Biophysical Models for tau-ProteinSpreading in Alzheimer’s Disease from PET-MRI
Klaudius Scheufele , Shashank Subramanian , and George Biros Oden Institute, The University of Texas at Austin, Texas, USA * [email protected] ABSTRACT
Aggregates of misfolded tau proteins (or just “tau” for brevity) play a crucial role in the progression of Alzheimer’s disease(AD) as they correlate with cell death and accelerated tissue atrophy. Longitudinal positron emission tomography (PET) isused to measure the extent of tau inclusions; and PET-based image biomarkers are a promising technology for AD diagnosisand prognosis. Here, we propose to combine an organ-scale biophysical mathematical model and longitudinal PET to extractcharacteristic growth patterns and spreading of tau. The biophysical model is a reaction-advection-diffusion partial differentialequation (PDE) with only two unknown parameters representing the spreading (the diffusion part of the PDE) and growth of tau(the reaction part of the PDE). The advection term captures tissue atrophy and is obtained from diffeomorphic registration oflongitudinal magnetic resonance imaging (MRI) scans. We describe the method, present a numerical scheme for the calibrationof the growth and spreading parameters, perform a sensitivity study using synthetic data, and apply it to clinical scans from theADNI dataset. Despite having only two parameters, the model can reconstruct clinical scans quite accurately.
Introduction
Alzheimer’s disease is the sixth highest leading cause of death in the US (alz.org). Its complex evolution isthought to be closely related to the formation and spreading of abnormal proteinaceous assemblies in the nervoussystem. In particular, the misfolding of β -amyloid (A β ) and tau proteins are believed to be key factors drivingthe progression of Alzheimer’s . These corruptive templates incite a chain reaction of protein misfolding,by imposing their anomalous structure on benign molecules. Subsequent growth, fragmentation and furtherspreading of such toxic proteins hampers proper function of the nervous system, leads to accelerated tissueatrophy, necrosis, and ultimately causes death .Tau aggregates are primarily found in the axon bundle and rapidly spread along neuronal pathways to distantlocations, but also invade the extracellular space . Understanding the distinct spatiotemporal growth patternsand original seeding of corrupted protein templates is imperative in developing new treatment protocols and canreveal important complimentary information to aid diagnosis and overall efficacy of therapy. Longitudinal PETscans (using the F-AV-1451 tracer) can image tau spreading and lead to improved diagnosis and prognosis. Manygroups are designing image-analysis algorithms for this purpose . Here we propose a complementary approach:We employ a PDE model of tau propagation and calibrate its parameters using longitudinal PET scans. Our goal isto estimate such rates of amplification of misfolded tau-protein aggregates and the rate of fragmentation and migration thereofto distant parts of the brain . Our hypothesis is that an informative minimal model can produce biomarkers, whichcan augment imaged-based approaches. Once calibrated, the current and future spatiotemporal spreading of taucan be quantified using our model.
Contributions.
The novelty of our work can be summarized as follows: We solve an inverse problem to estimate patient specific, characteristic growth parameters describing thespatiotemporal evolution of misfolded tau-protein throughout the brain based on longitudinal 3D tau-PET imagingof AD subjects. We model tau progression as reaction-diffusion-advection PDE which accounts for both, atrophyand tau propagation; atrophy is modeled using material transport with a velocity obtained from diffeomorphicimage registration, and is one-way coupled to tau progression. We investigate and demonstrate invertibility and accuracy of estimation for such growth parameters. Weperform a sensitivity study with respect to the relative change in tau signal between scans, the effect of tissueatrophy, and the effect of partial observations. Our results indicate good agreement for future prediction of tauuptake compared to the true data. a r X i v : . [ q - b i o . Q M ] J u l ) We test our method on clinical tau-PET scans, and study the sensitivity of our model to different subjects,and algorithmic hyper-parameters. To our knowledge, this is the first study of its kind.
Related work.
We believe organ-scale biophysical models hold an enormous promise in aiding clinical man-agement of AD. Such models of protein misfolding gained momentum with the advent of the prion-paradigm(misfolding chain-reaction) disease model. Models range from molecular level , to graph models , andkinetic equations . Inspired by the successful application of such models in computational oncology , weuse an organ level PDE-model which reflects the most dominant evolution patterns of tau propagation. ForAD, a similar model to the one we propose here has been recently proposed for prion-like diseases but didnot consider the PET-based calibration problem. To the best of our knowledge, our method is the first one topersonalize a biophysical model of tau propagation. Results
We examine the quality of biophysical model personalization, and the accuracy of subsequent prediction oftau spreading. Specifically, we ask the following two questions: ( i ) How does the solution of (3) depend onthe time horizon T and the effects of tissue atrophy (modeled via material transport)? And ( ii ) how does ourmethod perform on clinical tau-PET scans? For the latter, we use clinical PET data from the Alzheimer’s DiseaseNeuroimaging Initiative (ADNI) database. Synthetic Data.
We generate synthetic tau-PET data based on our physics-based model. Fig. 1 gives an illustrationof the data and test case setup. As underlying brain anatomy, we use (segmentations of) longitudinal T1 MRIscans of an AD subject with significant tissue atrophy (cf. panel A in Fig. 1). The advection velocity v used tocouple the effect of tissue atrophy to tau spreading is obtained from diffeomorphic image registration betweensegmentations of the T1 MRI. Synthetic tau data is generated using our reaction-diffusion-advection model for tauevolution based on the tissue segmentation of the first time point, and ground truth parameters ρ (cid:63) = κ = v (cid:63) = v (see panel B in Fig. 1 for an illustration). Performance Metrics.
For our synthetic studies, we report relative errors (cid:101) ι = ι rec / ι (cid:63) , ι ∈ { ρ , κ } in the reconstructionof the characteristic growth parameters. We further report the relative data misfit µ O = (cid:107)O T ( c T − d T ) (cid:107) / (cid:107) d T (cid:107) of thecalibrated model (at observation points defined by O ), as well as the relative forecast errors µ t = (cid:107) c ( t ) − d t (cid:107) / (cid:107) d t (cid:107) forpredicted tau concentration at future time points t ∈ { T (cid:48) , T (cid:48)(cid:48) } . Accuracy of Model Calibration and Tau Forecast
We study the sensitivity of model calibration and prediction to ( i ) varying time frames of image acquisition, and( ii ) the effect of tissue atrophy, one-way coupled via material transport in a synthetic setting, with known groundtruth values. For these synthetic tests, we set the observation threshold τ obs to zero (i.e., observe all data). Resultsare given in Tab. 1. We also study the sensitivity to a nonzero observation threshold in Tab. 2 Sensitivity to Small Relative Change in tau Signal.
Tau misfolding, and, hence AD progression is an (initially)slow process. In clinical practice, PET imaging is acquired in intervals of 8 to 15 months with varying, but typicallysmall relative change of measured tau SUV. Hence, we study the sensitivity of the model calibration to smallerrelative change in tau uptake. We consider different acquisition times T for the first tau snapshot d , such thatthe relative change in tau SUV between the two time points varies from 99% to 3%. Tab. 1 (runs ρ , and up to 20% for κ . Similarly the accuracy of tau forecast deteriorates marginally, but remains lowwith errors of 3-13% for T (cid:48) , and 5-26% for T (cid:48)(cid:48) (see also panel C in Fig. 1). http://adni.loni.usc.edu/ able 1. Model calibration and tau evolution forecasting for synthetic data.
We study the sensitivity with respect to 1)the time horizon T between consecutive scans, and 2) tissue atrophy (one-way coupled to tau spreading via materialtransport). For 1), we consider varying image acquisition time points T for d with a relative change ( ∆ SUV) in tau signal(between scans d and d ) ranging from 99% to 3%. For 2), we consider three different scenarios: I. Disregard tissue atrophyand use structural MRI of T as material properties for tau-propagation; II. Like before but use MRI of T ; III. Account fortissue atrophy via material transport, governed by a velocity field found from image registration of (segmented) T and T MRI. We report relative errors (cid:101) ρ and (cid:101) κ for the inversion parameters (true values are ρ (cid:63) = , κ (cid:63) = ). µ T denotes therelative data misfit in the inversion; µ T (cid:48) and µ T (cid:48)(cid:48) denote forecast errors at future times T (cid:48) = , and T (cid:48)(cid:48) = . ID T T ∆ SUV ρ κ (cid:101) ρ (cid:101) κ µ T µ T (cid:48) µ T (cid:48)(cid:48) I . T M R I − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − II . T M R I − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − III . A dv ., T M R I − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − igure 1. Qualitative results for model calibration and tau evolution forecasting using synthetic data.
Panel A illustrates structural differences induced bytissue atrophy: a)–1 and a)–5 show original T1 MRI scans of an AD subject at time points T = and T = ; a)–2 through a)–4 show material advection of CSF toaccount for tissue atrophy at intermediate time points; a)–5 and 6 show the magnitude and deformation gradient of the velocity, obtained from registration of T1 MRI (avalue of |∇ v | < ndicates contraction, a value above 1 expansion; the registration is performed from the second to the first snapshot). Panel B shows synthetic tau dataat different time points used for calibration and tau distribution at forecast time points T (cid:48) and T (cid:48)(cid:48) (cf. Tab. 1). Panel C shows the forecast distribution c (cid:48) and c (cid:48)(cid:48) of tau attime points T (cid:48) and T (cid:48)(cid:48) based on calibration for the three scenarios I-III from Tab. 1 and two different time horizons, along with the relative mismatch to the groundtruth (dark black indicates high error). / ensitivity to Effects from Tissue Atrophy (Material Transport). Next, we study the sensitivity of the modelpersonalization with respect to the material transport, coupling the effects of tissue atrophy to our tau evolutionmodel. For this first study, we consider three scenarios:I. Disregard tissue atrophy and use structural MRI of time point T as material properties for tau-propagation;II. Like above but use MRI of time point T ;III. Account for tissue atrophy via material transport, governed by a velocity field found from image registrationof (segmented) MRI.From Tab. 1 we observe that the effect of tissue atrophy is not negligible for the calibration of our model:Disregarding the material transport which simulates effects of cortical thinning in our model, results in muchlarger errors for both scenarios I. and II., compared to the full model in scenario III. Most prominently, smallerrelative change of tau SUV between scans results in a rapid and more drastic decline of accuracy for boththe calibrated growth parameters and the forecast of tau concentration, if the advection of material properties,according to the cortical thinning, is disregarded. The differences in forecast of tau distribution at future timepoints can be seen from!Fig. 1 (panel C) for the different scenarios. We conclude that the proposed model issensitive to the transport of material properties, and the coupling of tissue atrophy to tau spreading is notnegligible for model calibration. Sensitivity to Partial Observations.
We also study the sensitivity of the calibration step to partial observations,meaning that the (synthetic) tau data d and d T is observed only up to 10%, 20%, 30% and 40% of its maximumconcentration. The calibration and forecast results are given in Tab. 2. We observe a strong sensitivity of our modelon the threshold of the observation operator, paired with large errors for the parameter reconstruction and tauforecast for higher thresholds. Furthermore, the error is pronounced if the relative change of tau SUV betweenscans becomes small.From this first exploratory study using synthetic data, we find that small relative changes of tau SUV betweenscans as well as partial observations severely complicate the calibration problem. The solution of the latter isfurther hampered by the noise typically present in actual clinical tau-PET imaging. The presence of noise calls forpartial observations to focus attention to informative regions with change of tau SUV, and, consequently requirecareful calibration of the observation threshold parameter. Application to Clinical Data
We report reconstruction results in Tab. 3 and Figs. 2 and 2. PET scans are acquired 1-2 years apart; the number ofdays between image acquisition is given by T . Not all cases have T1 MRI for the second time point; we do notaccount for atrophy changes for these subjects. Discussion
Perhaps the most interesting result in our study is that the estimated growth parameters show significant variabilityacross the different subjects. For example, we obtain values for ρ between 2 and 7, and κ ranges between 1.70E − −
4. This demonstrates that our model is sensitive to patient-specific information and provides an indicationthat the methodology could be clinically useful for differential diagnosis. Our model is minimal. Despite havingonly two parameters, the model can reconstruct clinical scans quite accurately. The reconstructed parameters κ and ρ could be used as biomarkers. Finally, the model can be run forward in time to predict tau propagation andestimate AD progression, which can serve as an overall validation. Indeed, having more than two scans wouldallow us to validate our model by using the first two scans to calibrate and the third scan to test the prediction.This task is our first priority.Second, the choice of the observation threshold τ obs is critical, and strongly affects the calibration and predictionresult. We study the calibration result for three different choices of τ obs . Figs. 2 and ?? show normalized PET scansfor two subjects with indicated observation contours in black for various values of τ obs (panels a ) – b ) ). For lowervalues of τ obs , larger regions of the tau scans are observed. This limits the flexibility of the model to account forlocalized regional SUV variations (cf. localized intensity changes in tau-PET, panels a ) – b ) in Figs. 2 and ?? ). Asa result, the estimated proliferative parameter ρ decreases, and the diffusive parameter κ increases, for smallervalues of τ obs . Finding an optimal choice for the hyper parameter τ obs requires calibration to a large patient cohort.For this first exploratory study, we opted for a simple model that captures the dynamics of tau propagationand allows us to establish a calibration baseline upon which more complex models can be evaluated able 2. Sensitivity to Observation Threshold.
We study the sensitivity with respect to the observation threshold τ obs .Synthetic data d and d T is observed only up to , , and of the maximum tau concentration, whencalibrating the model (compare rows 1–3 in figure; the red contour line indicates the observation threshold). We report relativeerrors (cid:101) ρ and (cid:101) κ for the inversion parameters (true values are ρ (cid:63) = , κ (cid:63) = ), the calibration error µ O at observationpoints, and forecast errors µ T (cid:48) and µ T (cid:48)(cid:48) . We consider two different time horizons with and relative change in tauSUV, respectively. The last two rows of the figure show c (cid:48) tau concentrations at forecast time point T (cid:48) , and correspondingresidual. ID ∆ SUV τ obs ρ κ (cid:101) ρ (cid:101) κ µ O µ T (cid:48) µ T (cid:48)(cid:48) − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − τ obs = O d d w it h ∆ S UV = % τ obs = O d τ obs = O d τ obs = O d τ obs = O d
01. –0.0 – τ obs = O d d w it h ∆ S UV = % τ obs = O d τ obs = O d τ obs = O d τ obs = O d
01. –0.0 – τ obs = O d d T τ obs = O d C a li b r a t i o n τ obs = O d C a li b r a t i o n τ obs = O d C a li b r a t i o n τ obs = O d C a li b r a t i o n
1. –0.0 – axial slice 99 residual c c f o r ∆ S UV = % axial slice 99 residual P re d i c t i o n c axial slice 99 residual P re d i c t i o n c axial slice 99 residual P re d i c t i o n c axial slice 99 residual P re d i c t i o n c axial slice 99 residual c c f o r ∆ S UV = % axial slice 99 residual c axial slice 99 residual c axial slice 99 residual c axial slice 99 residual c able 3. Model Personalization for Clinical Data.
We report the calibrated growth parameters ρ and κ for seven ADpatients, taken from the ADNI database. We investigate sensitivity to the threshold τ obs in the observation operator andconsider three different values τ obs ∈ { } . The time T between scans is given in days. If available, we register T1MRIs of longitudinal snapshot, and account for atrophy (via material advection) in the personalization step (indicated by“Adv”). For each applied threshold value, we report the relative mismatch µ O at observation points, as well as the relativeerror µ in the entire brain. Observation Threshold τ obs = Observation Threshold τ obs = Observation Threshold τ obs = Subject ID T [ d ] Adv ρ κ µ µ O ρ κ µ µ O ρ κ µ µ O yes − − − − − − − − − yes − − − − − − − − − no − − − − − − − − − no − − − − − − − − − yes − − − − − − − − − no − − − − − − − − − yes − − − − − − − − − Subject 035_S_4114 sagittal slice 158 d PET a) τ o b s = . sagittal slice 158 d T PET b) c = O d SIM c) c T SIM d) axial slice 132 d PET a) axial slice 132 d T PET b) c = O d SIM c) c T SIM d) sagittal slice 158 d PET a) τ o b s = . sagittal slice 158 d T PET b) c = O d SIM c) c T SIM d) axial slice 132 d PET a) axial slice 132 d T PET b) c = O d SIM c) c T SIM d) sagittal slice 158 d PET a) τ o b s = . sagittal slice 158 d T PET b) c = O d SIM c) c T SIM d) axial slice 132 d PET a) axial slice 132 d T PET b) c = O d SIM c) c T SIM d) Subject 033_S_4179 sagittal slice 156 d PET a) τ o b s = . sagittal slice 156 d T PET b) c = O d SIM c) c T SIM d) axial slice 141 d PET a) axial slice 141 d T PET b) c = O d SIM c) c T SIM d) sagittal slice 156 d PET a) τ o b s = . sagittal slice 156 d T PET b) c = O d SIM c) c T SIM d) axial slice 141 d PET a) axial slice 141 d T PET b) c = O d SIM c) c T SIM d) sagittal slice 156 d PET a) τ o b s = . sagittal slice 156 d T PET b) c = O d SIM c) c T SIM d) axial slice 141 d PET a) axial slice 141 d T PET b) c = O d SIM c) c T SIM d) Figure 2.
Model Personalization for Clinical Data, ADNI subjects 035_S_4114 and 033_S_4179.
The three rowscorrespond to different observation thresholds τ obs = (first row), τ obs = (second row), and τ obs = (third row).Shown are saggital and axial cuts of PET data and reconstruction. Panels a ) and b ) show the normalized PET for bothsnapshots; the observation threshold is indicated by black contour lines. Panels c ) and d ) show the initial seeding andpredicted tau concentration, overlaid with the T1 MRI. imitations and Future Directions . More complex models could be used for tau propagation. An important (andrelatively easy to implement) model improvement is to include anisotropic diffusion, especially for data withlonger time horizon T . Also in our model, the atrophy is computed from image registration and it is only one-waycoupled to tau. In our future work, we will evaluate such models . We remark that more complex models alsorequire more informative data; either more time snapshots of tau or larger time intervals between scans.Another challenge is the normalization of longitudinal tau PET, especially for longitudinal studies whererelative change is small. The choice for a good reference region is not trivial, depends on the employed PET tracer,and is sensitive to, e.g., registration errors . PET normalization is ongoing research . Other challenges arelinked to lack of standardization in PET image acquisition protocols and subsequent normalization. Inconsistentintensity changes, can be caused by several bio-physical factors such as age, weight change, and blood glucose levelof the patient; or by other imaging factors such as varying scanner models or image reconstruction algorithms .Dealing with such challenges would require cross-validation of our method with several subjects in order to selectthe hyper-parameters in our scheme. Methods
Models and Materials
Forward Model.
To model the spatiotemporal evolution of misfolded tau-protein, we adopt the Fisher-Kolmogorovequation coupled with an advection equation for material transport: ∂ t c + ∇ · ( c v ) − κ D c − ρ R c =
0, in Ω B × ( T ] , (1a) c ( ) = c in Ω B , (1b) ∂ t m + ∇ mv = , in Ω × ( T ] , (1c) m ( ) = m in Ω . (1d)This simple model captures the basic dynamics of the problem: ( i ) The growth and formation of tau aggregates,and ( ii ) their subsequent fragmentation and spatial propagation. It further represents AD specific characteristicssuch as slow early stage progression with a rapid acceleration after symptom onset, and the inevitable progressionof the disease: even a single corruptive protein will spread, and ultimately cause disease .Our model follows , but differs in a new image-driven transport term, that captures atrophy without amechanical model. In our case c = c ( x , t ) ∈ [
0, 1 ] is the tau concentration with initial seeding c . Its evolutionover time t ∈ [ T ] in the three-dimensional brain domain Ω B ⊂ Ω = [
0, 2 π ] is governed by the coefficients of thereaction and diffusion terms in (1a): The nonlinearity R c = ρ m c ( − c ) with growth rate ρ provides a saturationterm expressing the maximal concentration of toxic proteins. ρ m is a spatially variable coefficient, dependet on theunderlying material properties m = ( m i ( x , t )) i = W , G , F , a vector of voxelwise probabilities for white matter ( W ) , graymatter ( G ) , and cerebrospinal fluid with ventricles ( F ) . Spatial spreading of tau is driven by extracellular diffusion and axonal transport . This is modeled by the diffusion operator D c = ∇ · κ m ∇ c , where κ m = κ I + ( κ i / κ − ) T definesthe inhomogeneous (anisotropic) diffusion tensor. κ captures the inhomogeneous diffusion based on the materialproperties m , while T expresses preferential direction of diffusion along the axon bundle weighted by κ i , andenables anisotropy. Tau does not invade the cerebrospinal fluid, and spreading occurs primarily in white matter .We approximate no-flux boundary conditions ∂ Ω B via a penalty approach , and use periodic boundaryconditions on ∂ Ω . Atrophy: Advection of Material Properties.
Over time, tau aggregates disrupt cell function and ultimately causecell death and tissue atrophy (thinning of white and gray matter) . We model tissue atrophy as image-driventransport equation (1c)–(1d) of the spatiotemporal material properties m . As a result, tissue atrophy is one-waycoupled to tau spreading via the definition of the spatially variable diffusion and reaction coefficients.The velocity field v is found via large-deformation diffeomorphic image registration of longitudinal MRI data,and can be computed solving the inverse problemmin (cid:126) v (cid:107) (cid:126) m ( ) − (cid:126) m T (cid:107) L ( Ω ) + β S ( v ) s.t. (1c)–(1d). (2)Here, (cid:126) m and (cid:126) m T denote segmentations of T1 MRI, corresponding to the acquisition times t = t = T ofthe tau-PET time-series. The regularization term S ( v ) is given as an H -seminorm (cid:82) Ω ∑ i − |∇ v i ( x ) | d x . For thesolution of (2) we use the registration software CLAIRE . The proposed atrophy model is distinct from other pproaches in the literature. In the authors couple a reaction-diffusion equation to a mechanical deformationmodel for atrophy, based on nonlinear elasticity.Solving (1) defines the forward problem F ( c , c T , (cid:126) v , ρ , κ ) =
0. Next, we discuss the image-driven calibration ofthis model.
Model Calibration and Extraction of Tau-Spreading Characteristics
The model is personalized based on patient specific, tau-PET imaging data. That is, we estimate biophysicalgrowth parameters g = ( ρ , κ ) in (1) based on measurements of tau uptake value ratios (SUVR) from longitudinalPET by solving a parameter estimation problem min g =( ρ , κ ) (cid:107)O T ( c T − d T ) (cid:107) L ( Ω ) s.t. F ( c , c T , (cid:126) v , ρ , κ ) =
0, from (1), (3)minimizing the discrepancy between predicted tau concentration c T = c ( T ) (based on the seeding c ( ) ) and thetarget data d T . The observation operator O i is defined as O i c : = [ d i ( x ) > ε ] c with threshold ε = τ obs · max ( d i ) for i = T . In other words, points with uptake values below τ obs percent of the maximum concentration of d and d T ,respectively, are not observed. This is critical due to the low signal-to-noise ratio in PET imaging. Observation inthe entire brain would disregard localized relative changes of SUVR, and drive the growth parameters close tozero. The seeding concentration c ( ) is given by the observation points of the data d from the first time point O d = : c . As a proof of concept, we only consider isotropic diffusion for the model calibration. The extension toanisotropy is straightforward and will be realized in future work. Next, we discuss the numerical scheme for thesolution of (3). Numerical Scheme
Numerical Solution of the Forward Problem.
We employ a pseudo-spectral Fourier approach on a regular grid forspatial discretization. That means all spatial differential operators are computed via a 3D fast Fourier Transform .For numerical solution of the forward problem, we employ a first-order operator-splitting method to split the tauprogression equation (1a) into a reaction, diffusion, and advection part. For the diffusion split, we use an implicitCrank-Nicholson method, and solve the linear system with a preconditioned matrix-free CG method. The reactionsub-steps are solved analytically. The hyperbolic transport equations are solved using an unconditionally stablesemi-Lagrangian time-stepping scheme to avoid stability issues and small, CFL restricted time-steps ? ,36,37 . Thismethod requires evaluations of the space-time fields at off-grid locations, defined by the characteristic associatedwith v . We compute off-grid evaluations using a cubic Lagrange polynomial interpolation model. We use n t = ∆ t = Numerical Optimization.
The optimization problem (3) is solved using a bound constrained, limited memoryBFGS quasi-Newton solver globalized with Armijo line-search. The gradient is computed using a first-orderaccurate finite difference scheme with (cid:126) h = √ (cid:101) mach (cid:126) g , for (cid:126) g = ( ρ , κ ) T . We terminate the optimization after a gradientreduction of 4 orders of magnitude (relative to the initial guess). To keep the optimizer within feasible bounds,and prevent bad local minima, we define bound constraints κ min = −
4, and κ max = ρ min = ρ max = Workflow Summary
For the evaluation on clinical data, we use seven AD subjects from the ADNI database as outlined in Tab. 3.Imaging data is processed using FSL by applying the following steps: ( i ) longitudinal T1 MRI are rigidlyregistered to the first time point using FLIRT ; ( ii ) longitudinal tau-PET images are rigidly registered to thefirst time point T1 MRI; ( iii ) aligned T1 images are skull-stripped using BET ; ( iv ) registered T1 brain masks areapplied in PET space to obtain the skull-stripped PET image; ( v ) skull-stripped T1 images are segmented usingFAST ; ( vi ) PET images are individually normalized with average tau uptake value in the cerebellum. For the laststep, the cerebellum is extracted from a registered labeled standard template. References Jucker, M. & Walker, L. C. Self-propagation of pathogenic protein aggregates in neurodegenerative diseases.
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