Variational Physiologically Informed Solution to Hemodynamic and Perfusion Response Estimation from ASL fmri Data

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1 Variational Pysioloically Informed Solution to Hemodynamic and Perfusion Response Estimation from ASL fmri Data Aina Frau-Pascual, Florence Forbes, Pilippe Ciuciu To cite tis version: Aina Frau-Pascual, Florence Forbes, Pilippe Ciuciu. Variational Pysioloically Informed Solution to Hemodynamic and Perfusion Response Estimation from ASL fmri Data International Worksop on Pattern Reconition in NeuroImain, Jun 2015, Stanford, CA, United States. IEEE, pp.57-60, 2015, 2015 International Worksop on Pattern Reconition in NeuroImain, Stanford, CA, USA, June 10-12, < /PRNI >. <al > HAL Id: al ttps://al.arcives-ouvertes.fr/al Submitted on 7 Jan 2016 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacin and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseinement et de recerce français ou étraners, des laboratoires publics ou privés.

2 Variational pysioloically informed solution to emodynamic and perfusion response estimation from ASL fmri data Aina Frau-Pascual 1,3, Florence Forbes 1, Pilippe Ciuciu 2,3 1 INRIA, MISTIS, Grenoble University, LJK, Grenoble, France 2 CEA/DSV/IBM NeuroSpin center, Bât. 145, F Gif-sur-Yvette, France 3 INRIA, Parietal, F Orsay, France Abstract Functional Arterial Spin Labelin (fasl) MRI can provide a quantitative measurement of cerebral blood flow. A joint detection-estimation (JDE) framework as been considered to extract task-related perfusion and emodynamic responses not restricted to canonical response function sapes. In tis work, we provide a variational expectation-maximization (VEM) aloritm for emodynamic and perfusion responses estimation. Tis approac provides a lower computational load compared to previous attempts, and facilitates te incorporation of prior knowlede and constraints in te estimation. Validation on simulated and real data sets as been performed. I. INTRODUCTION Functional Arterial Spin Labelin (fasl) is a functional MRI modality tat is able to provide a quantitative measurement of cerebral blood flow (CBF) and its variations elicited by specific tasks. Altou ASL [1] is mainly used as a static probe of cerebral perfusion, it as also been used in functional MRI as an alternative to te standard blood-oxyenlevel-dependent (BOLD) [2] modality, as it can ive more specific information about brain function. FASL data consists of alternatin pairs of control and manetically taed ( ta ) imaes. Local CBF or perfusion canes can be measured by doin te control-ta difference, so as to et rid of te emodynamic component (BOLD effect) of te sinal contained in bot control and ta imaes. Many pairs (ą 50) of control-ta imaes need to be acquired to compensate for te low Sinal-to-Noise Ratio (SNR) of tis difference. Te standard approac for fasl analysis is eneral linear model (GLM) [3], [4] usin te stimulus-induced canonical emodynamic response function (HRF) as reressor for bot components of te sinal: emodynamic and perfusion components. Te latter one encodes control-ta differences. Altou te canonical HRF is enerally used, it as been just calibrated on BOLD experiments and it reflects simultaneous variations of CBF, cerebral blood volume (CBV) and cerebral oxyen consumption (CMRO 2 ). In contrast, te perfusion response function (PRF) only reflects te variation in CBF and tus departs from te canonical sape. Moreover, tere as been evidence tat tis response can vary between reions and across subjects so tat assumin a constant response sape mit be misleadin. Some flexibility in te response sape as already been considered in BOLD GLM analysis by usin time/dispersion derivatives [5] or finite impulse response models. In a joint detection-estimation (JDE) framework bot taskrelated perfusion and emodynamic responses can be estimated as well as perfusion-related and BOLD-related maps of evoked activity. JDE as been oriinally developed for BOLD data analysis [6], [7] and as been extended in [8] for fasl data analysis, considerin Monte Carlo Markov Cains (MCMC) metods. In tis work, followin te spirit of [7], we provide an alternative solution based on te variational expectation-maximization (VEM) aloritm. Tis framework is more convenient to deal wit constraints (e, normalization or positivity) in te M-step. JDE-VEM provides comparable results to JDE-MCMC for a muc lower computational load. One of te novelties is te introduction of prior knowlede as in [9] trou te relationsip between perfusion and emodynamic responses derived from pysioloical models [10], [11]. Tis relationsip allows us to inform te PRF estimation from te HRF one, as te emodynamic component as a ier SNR tan te perfusion one due to te acquisition procedure. Experiments on simulated and real data sow te ood performance of te metod, wit a fast converence of te parameter estimates. II. JOINT DETECTION ESTIMATION MODEL FOR FASL In fasl te manetically taed imae reflects a perfusion effect besides te emodynamic effect, tat can be subtracted by comparison wit te control imae. Te control imae is a noisy BOLD sinal because of te effect of te manetic tain. Te different ASL sinal components can be explained wit te ASL JDE model [8], [12], a reion-based approac tat considers functionally omoeneous reions. In a reion P comprisin J voxels, te enerative model for j P J: Mÿ y j a looomooon m j X m ` looooomooooon c m j W X m ` lomon P l j ` lomon α j w ` lomon b j m1 paq pbq pcq pdq peq (1) in te presence of M experimental conditions. Te data y j P R N can be decomposed into (a) task-related emodynamic

3 and (b) perfusion components; (c) a drift component P l j ; (d) a perfusion baseline term α j w 1 wic completes te modellin of te perfusion component and (e) a noise term, assumed wite Gaussian of variance σj 2. Te control/ta effect is implicit in te ASL JDE model iven te use of W diapwq. Vectors and represent te D-dimensional (D ă N) unknown HRF and PRF sapes, constant witin P. Te manitudes of activation or response levels for emodynamic and perfusion components are a ( ( a m j and c c m j and denoted as BRLs and PRLs ereafter. X P R NˆD is a binary matrix tat encodes te laed onset stimuli. Te response levels are assumed to follow different spatial Gaussian mixture models but overned by M common binary idden Markov random fields q m wit q m tq mj, j P Pu encodin voxels activation states for eac experimental condition m. BRLs and PRLs are assumed independent conditionally to q tq m, m 1 : Mu. For furter detail, please refer to [8]. Te perfusion component in te ASL sinal as a very low SNR owin to its small size captured by te controlta subtraction. To address tis issue, a conditional prior as been used to inform te PRF estimation from te less noisy HRF recovery durin inference. To link tese two responses, a relationsip Ω was derived in [9] from pysioloical models. In our model, we consider tat HRF and PRF sapes follow prior Gaussian distributions N p0, v Σ q and N pω, v Σ q, wit covariance matrices Σ and Σ encodin a constraint on te second order derivative so as to account for temporal smootness. We also consider constraints on te response functions to enforce teir L 2 -norm to be 1, i.e., P S 2 pr D`1 q, S 2 pr D`1 q bein te L 2 unit ball of R D`1. III. VARIATIONAL EM In our fasl model, tere are missin parameters tat need to be estimated: a P A, P H, c P C, P G, q P Q. In tis work, an Expectation-Maximization (EM)framework is proposed to perform te estimation. EM can be viewed [13] as an alternatin maximization procedure of a function F suc tat for any p P D, bein D te set of all probability distributions on A ˆ H ˆ C ˆ G ˆ Q, F p p, θq E p lo ppy, a,, c,, q ; θq ` Ir ps. Here Ir ps E p lo ppa,, c,, qq is te entropy of p, and E p denotes te expectation wit respect to p. Maximizin function F is equivalent to minimizin te Kullback-Leibler diverence KLp p, ppa,, c,, q yqq, tat ives a measure of te distance between two distributions, in tis case te approximation p and te true ppa,, c,, q yq distributions. Tis view of EM as led to a number of variants in wic te E-step is solved over a restricted class of probability distributions, D. Te variational approac corresponds to D cosen as te set of distributions tat factorize as ppa,, c,, qq p a paq p pq p c pcq p pq p q pqq were p a P D A, p P D H, p c P D C, p P D G and p q P D Q, te sets of probability distributions on A, H, C, G, Q respectively. Note 1 Vector w is N-dimensional suc tat w tn 1{2 if t n is even (control) and w tn 1{2 oterwise (taed). tat te dependence between random variables, as in MCMC, is translated into dependence between statistical moments in VEM. Te E-step becomes an approximate E-step tat can be furter decomposed into five staes updatin te different variables in turn. At iteration prq, wit current estimates denoted by p pr 1q a, p pr 1q, p pr 1q c, p pr 1q, p pr 1q q and θ prq, te updatin formulae are of te form: E-H-step: ar max p prq E-G-step: p prq p PD H F p p pr 1q a p p pr 1q c p pr 1q p pr 1q q ; θ prq q (2) ar max F p p a pr 1q p prq p c pr 1q p p pr 1q q ; θ prq q (3) p PD G wit similar expressions for te oter steps obtained by permutin te roles of te variables. Hereafter, for te ease of presentation, te prq and pr 1q superscripts are omitted. We also consider normalization constraints on and so tat our solution as unit L 2 -norm. For tat matter, we modify te seek variational approximation to p p a δ p c δ p q, were te probabilities on and are replaced by Dirac functions. Tis reduces te searc to pointwise estimates and. Te E-H and E-G steps in Eqs. (2)-(3) yield ten maximizations problems wic are easily constrained to account for normalization: E-H: ar max E p a p c p q lo pp y, a, c,, q; θq (4) E-G: ar max E pa p c p q lo pp y, a,, c, q; θq (5) It is straitforward to see tat (4) and (5) amount to minimizin a quadratic function under a quadratic constraint, namely }} 2 1 and }} 2 1 respectively. Te oter E-steps can be derived from standard expressions replacin expectations over and by and, e..: E-Q: p q pqq9 exp E pa p c lo ppq y, a,, c, ; θq, (6) wit similar expressions for te E-A and E-C steps obtained by permutations of te variables. BRLs and PRLs ave Gaussian distributions as te likeliood and te priors are Gaussian too. Reardin te labels, a furter factorization is needed: p qm pq m q ś p qmj pq mj q. Te labels depend on te BRLs jpj and PRLs as well as on te binary activation states q mj. Te correspondin M-step is iven by: M: θ ar max E pa p c lo ppy a,, c, ; α, l, σ 2 q θpθ ` lo pp ; v q ` lo pp ; v q ` E pa p q lo ppa q; µa, σ a q ` E pc p q lo ppc q; µc, σ c q ` E pq lo ppq; βq j were θ α, l, σ 2, µ a,c, σ a,c, v, v, β (. Given te separability of te prior probability density functions, te M-step can be divided into separate M-steps, as in [7]. Note ere

4 tat a mean-field approximation is used to compute te MRF parameter β. For furter details on tese aspects, te reader is invited to refer to [7]. IV. RESULTS Different data sets ave been analysed to test te performance of tis aloritm. First, some artificial data as been enerated wit te ASL JDE enerative model. Ten, real data acquired on different individuals from te AINSI initiative 2 ave been analysed to validate te proposed approac. A. Artificial data N 288 ASL artificial imaes (i.e. 144 control/ta pairs) ave been simulated usin a realistic low SNR accordin to te observation model in Eq. (1). Different levels of SNR ave been used, in order to sow te performance of te metod dependin on te noise level. To emulate te slow samplin rate of ASL imaes, Eq. (1) was syntesized at t 0.5s and ten down-sampled at a certain time of repetition (TR), wic means tat te temporal resolution of rows (TR) and columns ( t) of X m is different. Here, we considered a fast event-related paradim comprisin two conditions (M 2), wit mean ISI 5s. TR 3s. is considered as a realistic ASL experiment, compared to te TR 1s. tat could be used for a realistic experiment wen usin BOLD sinal. In te experiments, and are enerated as depicted in Fi.2(a)-(b) by dased lines. P is a polynomial basis of order O 4. Drift coefficients and noise realizations were drawn accordin to l j N p0, 10.I O q and b j N p0, 2.I N q, respectively. BRLs were sampled from pa m j qm j 1q N p2.2, 0.3q (for activatin voxels) and from pa m j qm j 0q N p0, 0.3q (for inactivatin voxels). PRLs were enerated wit a lower contrast tan BRLs: pc m j qm j 1q N p1.6, 0.3q and pc m j qm j 0q N p0, 0.3q. PRLs and BRLs were cosen so as to make tis syntetic settin realistic: PRLs lower tan BRLs, and activatin/non-activatin voxels distribution means close. Activation states (assinment variables Q) were set by a and-drawn map. perfusion sinal perfusion sinal (a) BOLD sinal BOLD sinal (b) Fi. 2. Artificial data: Ground-trut response curves (black dased lines) and estimated perfusion (a) and emodynamic (b) response functions. Te first and second rows correspond to te response functions for TR 3s and noise variances 1 (i.e. SNR 2.4 db, top) and 6 (SNR 0.5 db, bottom). simulated T R 1 s. T R 3 s. (a) (b) (c) Fi. 3. Results on artificial data for response levels. Top row: roundtrut maps. Center and bottom rows: estimated maps for T R 1 s and T R 3 s. (a) Hemodynamic response levels. (b) Perfusion response levels. (c) Assinment variables (activation states). noise variance Fi. 1. RMSE of te response functions BRF and PRF estimated in experiments enerated wit different SNRs. Here te difference of te errors in te case of TR 1s and TR 3s is sown. 2 ttp://talie.ujf-renoble.fr/ainsi Fi. 1 sows te root mean squared error (RMSE) of te estimated PRFs and HRFs for experiments enerated wit noise variances from 1 to 6, wic corresponds to decreasin te SNR from 2 down to 0.5 db. Curves are depicted for TR 1s and TR 3s, so as to sow te impact of increasin TR on te performance of te metod and tus mimicin part of te increased difficulty wen movin from BOLD to ASL fmri data. For TR 1s, we observed as expected ood HRF and PRF estimates wit an accurate recovery of bot peaks (Fi. 2 top), as well as ood BRLs and PRLs map estimates (Fi. 3 center). In terms of RMSE (Fi. 1), te values are pretty close to eac oter altou slitly larer for te PRF. However, in te case of TR 3s, we recovered a muc better PRF estimate owin to a lare undersoot in te HRF sapes for very low SNR scenarios (Fi. 2 bottom rit). For te worse SNR scenario, we obtained over-smooted sapes, wit peaks displaced compared to te round trut.

5 sinal (a) (c) Fi. 4. Results on real fasl data for a subject of te AINSI database, for te auditory condition (radioloical convention: Left is rit). (a) BRLs, (b) PRLs, (c) response functions (d) reion of interest of te response functions. In (c), te red and blue curves represent te PRF and HRF estimates. As a reference, we depicted te canonical HRF in black dased line. Tis impacts te recovery of te emodynamic and perfusion activation maps too (see Fi. 3 bottom vs center), muc worse in tis case. Fis. 1-2 sow tat despite te dependence on te SNR level, our sape estimates are pretty reliable. B. Real data Real ASL data were recorded durin an experiment desined to map auditory and visual brain functions, wic consisted of N 291 scans lastin T R 3 s, wit T E 18 ms, FoV 192 mm, eac yieldin a 3-D volume composed of 64 ˆ 64 ˆ 22 voxels (resolution of 3 ˆ 3 ˆ 7 mm 3 ). Te tain sceme used was PICORE Q2T, wit T I ms, T I ms. Te paradim was a fast event-related desin (mean ISI 5.1 s) comprisin sixty auditory and visual stimuli. Ward parcellations were computed in auditory and visual cortices for reionally analysin te evoked activity elicited by auditory and visual stimuli, respectively. Te estimation results are depicted in Fi. 4. First, it is wort notin tat as expected we recovered BOLD response levels wit larer manitude compared to tat of te perfusion response levels (Fi. 4(a)-(b)). Interestinly, in response to auditory stimuli, we retrieved bilateral evoked activity in te auditory cortices for te BOLD and perfusion components, altou not exactly in te same parcels. Subsequent tresoldin of individual posterior probability maps sould allow us to derive reliable statistical maps. Besides, we also recovered plausible sapes for BRF and PRF estimates, as illustrated in Fi. 4(c), were we selected a parcel correspondin to i BRL values. As expected, te HRF sape in te primary auditory cortex (left emispere) is well captured by te canonical sape. Te pysioloical prior elps te parameter estimation, in te perfusion component and enforces te temporal precedence of te PRF estimate over te HRF one. (b) (d) V. DISCUSSION We proposed a variational Expectation-Maximization aloritm to address te issue of jointly detectin activity and estimatin emodynamic and perfusion responses from functional ASL data. Te variational approac as te advantae to provide estimations in analytic form for eac variable of interest. It facilitates te inclusion of additional information from pysioloical models and te incorporation of constraints on te responses to favor stability in te estimations. In particular, we considered a pysioloically informed link between normalized emodynamic and perfusion responses so as to compensate te low sinal-to-noise ratio of te perfusion component. Te results obtained on simulated and real data indicate a ood performance of te approac and suest te variational approac as a potential robust, fast and pramatic metod to analyse callenin ASL data. Addressin perfusion quantification is ten te main perspective of tis work as providin a ood estimation of perfusion modulation effects mit translate into more accurate quantification of perfusion. REFERENCES [1] D. Williams, J. Detre, J. Lei, and A. Koretsky, Manetic resonance imain of perfusion usin spin inversion of arterial water, Proceedins of te National Academy of Sciences, vol. 89, no. 1, pp , [2] S. Oawa, D.W. Tank, R. Menon, J.M. Ellermann, S-G. Kim, H. Merkle, and K. Uurbil, Intrinsic sinal canes accompanyin sensory stimulation: functional brain mappin wit manetic resonance imain, Proceedins of te National Academy of Sciences, vol. 89, pp , [3] L. Hernandez-Garcia, H. Jaanian, and D. B. Rowe, Quantitative analysis of arterial spin labelin fmri data usin a eneral linear model, Manetic resonance imain, vol. 28, no. 7, pp , [4] J. A. Mumford, L. Hernandez-Garcia, G. R. Lee, and T. E. Nicols, Estimation efficiency and statistical power in arterial spin labelin fmri, Neuroimae, vol. 33, no. 1, pp , [5] Karl J Friston, P Fletcer, Oliver Joseps, A Holmes, MD Ru, and Robert Turner, Event-related fmri: caracterizin differential responses, Neuroimae, vol. 7, no. 1, pp , [6] T. Vincent, L. Risser, and P. Ciuciu, Spatially adaptive mixture modelin for analysis of witin-subject fmri time series, IEEE Trans. on Medical Imain, vol. 29, no. 4, pp , Apr [7] L. Caari, T. Vincent, F. Forbes, M. Dojat, and P. Ciuciu, Fast joint detection-estimation of evoked brain activity in event-related fmri usin a variational approac, IEEE Trans. on Medical Imain, vol. 32, no. 5, pp , May [8] T. Vincent, J. Warnkin, M. Villien, A. Krainik, P. Ciuciu, and F. Forbes, Bayesian Joint Detection-Estimation of cerebral vasoreactivity from ASL fmri data, in 16t Proc. MICCAI, LNCS Spriner Verla, Naoya, Japan, Sept. 2013, vol. 2, pp [9] A. Frau-Pascual, T. Vincent, J. Sloboda, P. Ciuciu, and F. Forbes, Pysioloically informed Bayesian analysis of ASL fmri data, in Bayesian and rapical Models for Biomedical Imain, pp Spriner, [10] R. B. Buxton, E. C. Won, and Frank. L. R., Dynamics of blood flow and oxyenation canes durin brain activation: te balloon model, Manetic Resonance in Medicine, vol. 39, pp , June [11] K. J. Friston, A. Mecelli, R. Turner, and C. J. Price, Nonlinear responses in fmri: te balloon model, Volterra kernels, and oter emodynamics, Neuroimae, vol. 12, pp , June [12] T. Vincent, F. Forbes, and P. Ciuciu, Bayesian BOLD and perfusion source separation and deconvolution from functional ASL imain, in 38t Proc. IEEE ICASSP, Vancouver, Canada, May 2013, pp [13] R.M. Neal and G. E. Hinton, A view of te EM aloritm tat justifies incremental, sparse, and oter variants, in Learnin in rapical models, pp Spriner, 1998.