Laplacian-Regularized MAP-MRI: Improving Axonal Caliber Estimation

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1 -Regularized MAP-MRI: Improving Aonal Caliber Estimation Rutger Fick, Demian Wassermann, Gonzalo Sanguinetti, Rachid Deriche To cite this version: Rutger Fick, Demian Wassermann, Gonzalo Sanguinetti, Rachid Deriche. -Regularized MAP-MRI: Improving Aonal Caliber Estimation. International Smposium on BIOMEDICAL IMAGING: From Nano to Macro, Apr, Brookln, New York Cit, United States. <hal- > HAL Id: hal- Submitted on 7 Apr HAL is a multi-disciplinar open access archive for the deposit and dissemination of scientific research documents, whether the are published or not. The documents ma come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 -Regularized MAP-MRI: Improving Aonal Caliber Estimation R.H.J. Fick D. Wassermann G. Sanguinetti R. Deriche Athena Project-Team, Inria Sophia Antipolis - Méditerranée, France Centre for Analsis, Scientific computing and Applications, TU/e - Eindhoven, The Netherlands ABSTRACT In diffusion MRI, the accurate description of the entire diffusion signal from sparse measurements is essential to enable the recover of microstructural information of the white matter. The recent Mean Apparent Propagator (MAP)-MRI basis is especiall well suited for this task, but the basis fitting becomes unreliable in the presence of noise. As a solution we propose a fast and robust analtic regularization for MAP-MRI. Using both snthetic diffusion data and human data from the Human Connectome Project we show that () MAP-MRI has more accurate microstructure recover compared to classical techniques, () regularized MAP-MRI has lower signal fitting errors compared to the unregularized approach and a positivit constraint on the EAP and () that our regularization improves aon radius recover on human data. Inde Terms Diffusion MRI, MAP-MRI, Regularization, Corpus Callosum, Aon Radius Recover. INTRODUCTION In diffusion MRI (dmri), the reconstruction of the diffusion signal in q-space allows for the estimation of the water displacement probabilit, known as the Ensemble Average Propagator (EAP) []. This EAP describes the probabilit densit that a particle will move along a certain direction in a given diffusion time. The EAP, or P (r), is related to the diffusion signal b a Fourier transform P (r) = R E(q)e iπq r dq. Here r is a real displacement vector in r-space and E(q) is the normalized diffusion signal where wave vector q = (γδg)/π related to the applied magnetic field gradient magnitude, direction and duration []. The clinicall used b-value is related to q as b = π q ( δ/) mm/s for rectangular pulses. Historicall, the diffusion tensor [] was the first model to describe the EAP b assuming it lies within the famil of Gaussian distributions. However, this assumption limits its abilit to describe comple tissue structures []. We can reconstruct the EAP without prior assumptions b using an inverse Fourier transform on dense acquisition schemes such as Diffusion Spectrum Imaging (DSI) []. However, DSI s dense sampling of q-space limits its clinical applicabilit. As a solution, analtical bases whose functions are eigenvectors of the Fourier transform have been proposed which provide compact representation of the D q-space signal. These bases capture the radial and angular properties of the diffusion signal b fitting a linear combination of orthogonal basis functions as E(q) = N i c i Φ(q). The then describe the EAP as P (r) = c i Ψ(r) where Ψ(r) is the inverse Fourier transform of Φ(q). In particular, the recent Mean Apparent Propagator (MAP)-MRI basis [], which is related to the preceding D Simple Harmonic Oscillator based Reconstruction and Estimation (D-SHORE) basis [], fits the signal etremel well b using anisotropic basis functions. However, without proper regularization the basis fitting becomes highl unreliable in the presence of noise, making accurate EAP estimation impossible. As a solution we propose a fast and robust regularization for the MAP- MRI basis. We derive the regularization functional analticall and from now on refer to our regularized MAP- MRI as LMAP-MRI. We show that () MAP-MRI has more accurate microstructure recover compared to classical techniques, () LMAP-MRI has lower signal fitting errors compared to regular MAP-MRI and a positivit constraint on the EAP [], and () that LMAP-MRI has improved aon radius recover on human data of the MGH Human Connectome Project [].. MATERIALS AND METHODS.. Regularization of MAP-MRI We propose to regularize MAP-MRI s basis fitting b minimizing the norm of its. The MAP-MRI basis functions are given as a product of three D functions Φ Ni (A, q) =φ n(i) (u, q )φ n(i) (u, q )φ nz(i) (u z, q z ) () { φn (u, q) = i n q u with H n n! e π n (πuq) A = Diag(u, u, u z) with basis order N i = (n (i), n (i), n z(i) ). We find the diagonalized scaling factors A = RA R T b fitting a tensor A, where R contains the tensor eigenvectors. We then rotate the data into the frame of reference using R and scale the basis functions using A along each direction. The anisotropic basis functions is what makes MAP-MRI quickl fit anisotropic

3 data found in white matter tissue. MAP-MRI s zeroth order basis function is in fact a DTI tensor [], while setting scaling factors (u, u, u z ) to an isotropic u makes the MAP-MRI basis equivalent to the D-SHORE basis []. We fit the MAP-MRI coefficients c to the data = E(q) using regularized least-squares as c = argmin c Qc + λ R(c) where design matri Q R N data N coef has elements Q ij = Φ Ni (A, q j ), with q j the q-space positions of the data. Parameter λ weights the regularization functional R(c) = E R c (q) dq with E c (q) = i c i Φ Ni (A, q) the of the reconstructed signal. We then rewrite R(c) = c T Rc in quadratic form as R ik = Φ Ni (q, A) Φ Nk (q, A)dq. () R We solve this integral using the orthogonalit of Hermite polnomials with respect to the weight function e. This allows us to describe ever entrn R ij as a function of the anisotropic scaling factors (u, u, u z ) and basis orders ( i,, ) = (n (i), n (i), n z(i) ) as R ik = u S k u u i U k + u u T k z u i T k z + u S k u z u i U k i + u u z T k u i T z k U k i () + u z S z k z u u i U k i U k + u u z T k u i T z k U k with functions S m n, T m n and U m n given as S m n = ( ) n π 7/ (δ m n (n + n + ) + δn+ m ( + n) m!/n! + δn+ m m!/n! + δn m+ ( + m) ) n!/m! + δn m+ n!/m! T m n = ( ) n+ π / (δ m n ( + n) ) + δn m+ n(n ) + δ m n+ m(m ) () () U m n = δ m n ( ) n /(π / ) () with δ m n the Kronecker delta. We finall obtain the MAP-MRI coefficients through c = (Q T Q+λR) Q T. We used generalized cross-validation [7] to find an optimal value of λ =. in nois data and set this value for all our eperiments... Estimation of Aon Radii From the MAP-MRI coefficients c we can describe aon radii distributions using the Return-To-Ais Probabilit (RTAP) []. RTAP is defined directionall, representing the integrated probabilit that a spin diffuses along the ais of the aons, here defined as the main tensor eigenvector R. The RTAP is onl related to the aon radius if () aons are represented as parallel clinders, () onl the intra-aonal signal is considered and () the acquisition has short gradient pulses (δ ) and gradient separation time long enough for restricted diffusion ( δ). If these conditions are met we compute the aon radii as R = /(RTAP π) with RT AP = P (R r)dr (7) Deviation from these criteria will cause misestimation of the aon radius [8], which we quantifn Section... RESULTS We validate our MAP-MRI regularization on snthetic data using Camino [9] and on the MGH Human Connectome project (HCP) data set []... Snthetic Validation and Microstructure Recover We use the Camino diffusion MRI toolkit [9] to simulate the diffusion signal in voels with parallel clindrical aons with radii ranging from.µm to µm. We generate both data respecting all the conditions stated in Section. (δ, δ) (I) and data using HCP settings (δ/ =.9/.8 ms) (II). We simulate according to the HCP parameters at b-values {;,;,;,;,} mm/s with {; ; ; 8; } gradient directions, and sample (I) such that we sample the same q-space positions of (II) bnversel scaling the diffusion gradient strength with δ. We first consider the noiseless case of the optimal data and compute the aon radii with MAP-MRI, SHORE and DTI (Figure a). It is clear that MAP-MRI is able to distinguish smaller aon radii than the other techniques thanks to its anisotropic basis functions. We then take one aon radius of.8µm and appl noise different times at a realistic SNR =. We compute the aon radius with increasing radial order using either the positivit constraint, our or both. We find that both methods come close to the ground truth but our, in blue, has significantl lower standard deviations at higher radial orders (Figures b and c). We then compare the different methods based on the Mean Squared Error (MSE) of the nois signal reconstruction compared to the ground truth. We used radial order and truncated the outer two shells such that bma= mm/s. We show this eperiment in Figures d and e: our regularization, again in blue, has lower MSE values and standard deviations. Finall, we observe the effect of truncating the outer shells on the noiseless HCP simulated data on aon radius estimation (Figure f). In the full data we find that the suboptimal HCP parameters lead to underestimated aon radii until a plateau is reached at around.µm. Truncation of the outer shells leads to higher aon radii estimates. R

4 D-SHORE MAP-MRI + : Estimated Radius HΜmL, None> : 8 (b) MAP-MRI Convergence bma=, Mean Squared Error Standard Deviation HMSEL Least Squares bma =, bma =, SNR (d) MAP-MRI signal fitting bma=, bma =,..... Least Squares. (c) MAP-MRI standard deviation bma=,.. Radial Order... Radial Order (a) Aon Radius Estimation Optimal Settings. Ground Truth =.8Μm Estimated Radius HΜmL Estimated Radius HΜmL DTI Ground Truth Radius HΜmL> Standard Deviation HΜmL SNR (e) MAP-MRI standard deviation bma=, Ground Truth Radius HΜmL (f) MAP-MRI effect of truncation HCP settings Fig. : Results on snthetic Camino eperiments. (a) shows a comparison of aon estimations between MAP-MRI, SHORE and DTI on optimal data. (b) compares the average MAP-MRI aon radius estimate for different radial orders and methods with known ground truth. (c) shows the standard deviation. (d) quantifies the average MAP-MRI signal reconstruction for different SNR levels based on MSE. (e) gives the standard deviations. (f) shows the effect truncating the outer sampling shells on the aon radius estimation in HCP simulated data. shells we find higher aon radius estimates, but both methods remain able to distinguish between aon populations.. DISCUSSION AND CONCLUSION Fig. : Segmentation of the splenium, midbod and genu (red, green and blue) of the corpus callosum... Microstructure on the Human Connectome Project In our second eperiment we estimate aon radii in the corpus callosum data in a subject of the MGH Human Connectome Project data []. In this area the white matter is known to be highl coherent, and that aon radii are larger in the midbod than in the genu and splenium []. We segment these areas (Figure ) and compute the aon radii in each of these areas using our or the positivit constraint (Figure ). For both methods we find larger aon radii in the midbod (green) than in the genu in splenium (red and blue), though we alwas overestimate compared to the real radius of around.µm µm []. When we truncate the outer In this paper we proposed a new regularization method for the recent MAP-MRI basis. We derived the functional analticall, and showed on snthetic Camino data that a -regularized MAP-MRI basis (LMAP-MRI) has more robust and accurate signal reconstruction and aon radius recover than using a positivit constraint on the EAP. We find that this effect is more distinct when dealing with nois, more undersampled acquisition data. Even combining the two methods does not decrease the reconstruction error further. Our regularization method is also fast, having the same speed as ordinar least squares fitting. We also show that MAP-MRI is able to retrieve aon radii on snthetic data until significantl smaller aon radii compared to D-SHORE and DTI. We finall validate our method on human data from the HCP project. We find that LMAP-MRI can actuall distinguish aon populations in the corpus callosum, even in truncated data. However, overall we see aon radii are overestimated. In closing, we find in both our snthetic and human eperiments that MAP-MRI is superior to classical techniques

5 Fig. : Aon radius estimation in the corpus callosum using either regularization or unregularized with a positivit constraint. We correctl find larger aon radii in the midbod (green) than in the genu and splenium (blue and red). The laplacian generall estimates populations with smaller standard deviations and finds slightl smaller radii in the b=, data but larger for b=, data. In terms of computation time our approach is times faster than the positivit contraint. in terms of aon radius recover, and our LMAP-MRI provides both faster and more robust signal fitting and aon radius recover compared to both regular MAP-MRI and using a positivit constraint when facing nois, undersampled data. Acknowledgments Data used in the preparation of this work were obtained from the Human Connectome Project (HCP) database ( The HCP project is supported b the National Institute of Dental and Craniofacial Research (NIDCR), the National Institute of Mental Health (NIMH) and the National Institute of Neurological Disorders and Stroke (NINDS). This work was partl supported b the French ANR MOSIFAH under ANR--MONU-9-.. REFERENCES [] Stejskal. Use of Spin Echoes in a Pulsed Magnetic-Field Gradient Stud Anisotropic Restricted Diffusion Flow. J CHEM PHYS., pp. 97-, 9. [] Basser et al. Estimation of the Effective Self-Diffusion Tensor from the NMR Spin Echo. J MAGN RESON, Series B., pp. 7-, 99. [] Assaf Yaniv and Ofer Pasternak. Diffusion tensor imaging (DTI)-based white matter mapping in brain research: a review. J Mol NEUROSCI., pp. -, 8. [] Wedeen et al. Mapping Comple Tissue Architecture with Diffusion Spectrum Magnetic Resonance Imaging. MAGNET RESON MED., pp. 77-8,. [] Özarslan et al. Mean Apparent Propagator (MAP) MRI: A novel Diffusion Imaging Method for Mapping Tissue Microstructure. NeuroImage 78, pp. -,. [] Setsompop et al. Pushing the limits of in vivo diffusion MRI for the Human Connectome Project. NeuroImage 8, pp. -,. [7] Craven et al. Smoothing Nois Data with Spline Functions. NUMER MATH., pp. 77-, 978. [8] Bar-Shir et al. The effect of the diffusion time and pulse gradient duration ratio on the diffraction pattern and the structural information estimated from q-space diffusion MR: Eperiments and simulations. J MAGN RESON 9., pp. -, 8. [9] Cook et al. Camino: Open-Source Diffusion-MRI Reconstruction and Processing, th Scientific Meeting of the International Societ for Magnetic Resonance in Medicine, vol. 79, Ma. [] Aboitiz et al. Fiber composition of the human corpus callosum. Brain research 98., pp. -, 99.