An improved deep network for tissue microstructure estimation with uncertainty quantification

An Improved Deep Network for Tissue Microstructure Estimation with Uncertainty Quantification

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An Improved Deep Network for Tissue Microstructure Estimation with Uncertainty Quantification Chuyang Ye, Yuxing Li, Xiangzhu Zeng PII: DOI: Reference:

S1361-8415(20)30017-7 https://doi.org/10.1016/j.media.2020.101650 MEDIMA 101650

To appear in:

Medical Image Analysis

Received date: Revised date: Accepted date:

23 August 2019 26 November 2019 16 January 2020

Please cite this article as: Chuyang Ye, Yuxing Li, Xiangzhu Zeng, An Improved Deep Network for Tissue Microstructure Estimation with Uncertainty Quantification, Medical Image Analysis (2020), doi: https://doi.org/10.1016/j.media.2020.101650

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Highlights • We have proposed an improved deep network for tissue microstructure estimation • We exploit signal sparsity in the spatial-angular domain with a separable dictionary • Quantification of estimation uncertainty is also developed using Lasso bootstrap • The proposed method outperforms competing methods in terms of estimation accuracy • The proposed method also produces reasonable uncertainty quantification results

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Graphical Abstract



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Margin notes indicate the addressed review comments (reviewer# + question#). Revisions are highlighted in red

An Improved Deep Network for Tissue Microstructure Estimation with Uncertainty Quantification Chuyang Yea,∗ , Yuxing Lia , Xiangzhu Zengb a

School of Information and Electronics, Beijing Institute of Technology, Beijing, China b Department of Radiology, Peking University Third Hospital, Beijing, China

Abstract Deep learning based methods have improved the estimation of tissue microstructure from diffusion magnetic resonance imaging (dMRI) scans acquired with a reduced number of diffusion gradients. These methods learn the mapping from diffusion signals in a voxel or patch to tissue microstructure measures. In particular, it is beneficial to exploit the sparsity of diffusion signals jointly in the spatial and angular domains, and the deep network can be designed by unfolding iterative processes that adaptively incorporate historical information for sparse reconstruction. However, the number of network parameters is huge in such a network design, which could increase the difficulty of network training and limit the estimation performance. In addition, existing deep learning based approaches to tissue microstructure estimation do not provide the important information about the uncertainty of estimates. In this work, we continue the exploration of tissue microstructure estimation using a deep network and seek to address these limitations. First, we explore the sparse representation of diffusion signals with a separable spatial-angular dictionary and design an improved deep network for tissue microstructure estimation. The procedure for updating the sparse code associated with the separable dictionary is derived and unfolded to construct the deep network. Second, with the formulation of sparse representation of diffusion signals, we propose to quantify the uncertainty of network outputs with a residual bootstrap strategy. Specifically, because of the sparsity constraint in the signal representation, we perform a Lasso bootstrap strategy for uncertainty quantification. Experiments were performed on brain dMRI scans with a reduced number of diffusion gradients, where the proposed method was applied to two representative biophysical models for describing tissue microstructure and compared with state∗

Address: Room 316, Building 4, 5 Zhongguancun South Street, Beijing 100081, China. Email address: [email protected]

Preprint submitted to Elsevier

January 21, 2020

of-the-art methods of tissue microstructure estimation. The results show that our approach compares favorably with the competing methods in terms of estimation accuracy. In addition, the uncertainty measures provided by our method correlate with estimation errors and produce reasonable confidence intervals; these results suggest potential application of the proposed uncertainty quantification method in brain studies. Keywords: tissue microstructure, deep network, separable dictionary, uncertainty quantification 1. Introduction

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Tissue microstructure probed by diffusion magnetic resonance imaging (dMRI) has been used as biomarkers in various neuroscientific studies, where neurite morphology can be described with tissue microstructure measures inferred from diffusion signals (Alexander et al., 2019). The diffusion tensor model (Basser et al., 1994) is a basic signal model for the quantification of tissue microstructure, where measures such as fractional anisotropy (FA) and mean diffusivity (MD) can indicate the integrity of neuronal tissue. However, these measures obtained from diffusion tensors are known for their lack of specificity, where multiple sources of structural alterations can lead to similar changes in FA and MD (Pasternak et al., 2018). Therefore, more advanced biophysical models, such as the neurite orientation dispersion and density imaging (NODDI) model (Zhang et al., 2012) and the spherical mean technique (SMT) model (Kaden et al., 2016b,a), have been developed. In these models, specific tissue organization is related to diffusion signals, and thus the specificity of the derived tissue microstructure measures is enhanced. In particular, the tissue microstructure measures described by the NODDI model have gained increasing popularity in a variety of brain studies (Batalle et al., 2019; Parker et al., 2018; Gen¸c et al., 2018; Ocklenburg et al., 2018). In the advanced biophysical signal models, the relationship between tissue microstructure and diffusion signals is usually complicated, and the estimation of tissue microstructure involves nonlinear optimization. Reliable estimation of tissue microstructure described by these advanced models could require close to or more than 100 diffusion gradients (Zhang et al., 2012; Kaden et al., 2016b,a), which may be impractical in clinical settings due to the constraint of imaging times. Thus, it is desirable to improve the quality of tissue microstructure estimation when only a reduced number of diffusion gradients can be applied. To improve tissue microstructure estimation for advanced signal models, Nedjati- #2 Q1 Gilani et al. (2014) (and the extended version in Nedjati-Gilani et al. (2017)) first 4

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propose a learning-based estimation approach, where random forest regression is used to predict tissue microstructure for the K¨arger model (K¨arger et al., 1988) using rotationally invariant features computed from diffusion signals, including features computed from diffusion tensors and spherical harmonics. Similarly, in#2 Alexander Q2 et al. (2014) and Alexander et al. (2017), random forests are used to map diffusion tensors to tissue microstructure described by the NODDI model and SMT model. Due to the success of deep learning techniques, Golkov et al. (2016) propose #2 Q1 to estimate tissue microstructure for advanced signal models with a multiple layer perceptron (MLP). Moreover, instead of using features extracted from diffusion signals, the MLP directly maps diffusion signals acquired with a reduced number of diffusion gradients to tissue microstructure measures. Since diffusion signals can be interpreted as measurements in the so-called q-space, this mapping is referred to as q-space deep learning (Golkov et al., 2016). q-Space deep learning is later improved by Ye (2017a) for the NODDI tissue microstructure estimation, where the sparsity of diffusion signals in the q-space is incorporated into the deep network design. Specifically, motivated by AMICO (Daducci et al., 2015), a dictionary-based framework for estimating NODDI tissue microstructure, the method in Ye (2017a) assumes that a sparse representation of diffusion signals exists and tissue microstructure can be computed from the sparse representation. By unfolding an iterative process for solving sparse reconstruction problems, a network structure can be constructed for computing the sparse representation, where the weights are learned instead of precomputed from the dictionary. Such a network is then concatenated to a second stage, which resembles the AMICO procedure and maps the sparse representation to tissue microstructure measures with learned weights. All weights in the whole network in Ye (2017a) are learned jointly, and improved estimation results can be obtained compared with the simple MLP in Golkov et al. (2016). Golkov et al. (2016) and Ye (2017a) use the diffusion signals at each voxel to compute the tissue microstructure measures at that voxel. It is possible to further incorporate the information in the spatial domain for improved tissue microstructure estimation. For example, Ye (2017b) extends the network in Ye (2017a) by taking patches of diffusion signals as input and predicts the tissue microstructure at the center voxel. In Ye (2017b) a smoothing stage is inserted before the network structure in Ye (2017a), where the weights for smoothing are learned together with the other weights, and such incorporation of information in the spatial domain is shown to be beneficial for the quality of tissue microstructure estimation. The use of the information in the spatial domain and q-space—also referred to as the angular domain (Schwab et al., 2016)—is further investigated in Ye et al. (2019a), where the sparsity of diffusion signals jointly in the spatial and angular domains is exploited. 5

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The network in Ye et al. (2019a) is also improved by allowing adaptive incorporation of historical information in the update of sparse codes, and it is achieved with modified long short-term memory (LSTM) units. In addition, unlike Ye (2017a) and Ye (2017b), where the estimation is designed specifically for the NODDI model, the network in Ye et al. (2019a) is not limited to a particular signal model, and this is achieved by using fully connected layers to map the sparse representation to tissue microstructure. Despite the improved network design in Ye et al. (2019a), the spatial-angular sparse representation of the input signals and the adaptive incorporation of historical information together lead to a drastically increased number of network parameters. This could increase the difficulty of network training and limit the estimation performance. Moreover, existing deep learning based approaches that directly map diffusion signals to tissue microstructure only give deterministic prediction. They do not provide the important information about estimation uncertainty, which could inform the treatment or analysis of patients and benefit downstream image processing steps (Tanno et al., 2016, #2 2017, 2019). In this work, we continue Q2,Q5 the exploration of tissue microstructure estimation using a deep network and seek to address the limitations described above. Specifically, based on the spatial-angular sparse representation of diffusion signals with a dictionary that is separable in the spatial domain and angular domain (Schwab et al., 2018a,b), we propose an improved deep network which comprises two functional components for tissue microstructure estimation, and we also develop a strategy for quantifying the uncertainty of the network estimates.1 The separable dictionary allows a more expressive representation of diffusion signals with fewer network weights. We derive the corresponding iterative process for solving the sparse reconstruction problem with the separable dictionary and unfold the process to construct the first network component. This component takes a patch of diffusion signals as input and computes the sparse representation of diffusion signals with learned operations that are separable in the spatial domain and angular domain. The sparse representation is then mapped to tissue microstructure with fully connected layers in the second component, and these fully connected layers are also decomposed into separable learned operations in the spatial domain and angular domain. The network weights in the two components are learned jointly from training tissue microstructure without explicitly specifying the dictionary. In addition, based on the sparse representation formulation, we propose to quantify the uncer1

Demo scripts of the proposed method will be provided at https://github.com/PkuClosed/ after the work is published.

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tainty of network outputs with a Lasso bootstrap strategy (Knight and Fu, 2000), where bootstrap diffusion signals can also be processed by the trained deep network. Since in training the network for tissue microstructure estimation the dictionary is implicitly specified and unknown, we compute the dictionary by minimizing the signal reconstruction loss using the trained estimation network. Then, we can obtain the residuals of sparse representation for test data and apply Lasso bootstrap to quantify the uncertainty of tissue microstructure estimates. Experiments were performed on brain dMRI data, which is associated with a reduced number of diffusion gradients. For demonstration, we applied the proposed method to two representative biophysical signal models, the NODDI model (Zhang et al., 2012) and the SMT model (Kaden et al., 2016a,b), where the performance of tissue microstructure estimation and uncertainty quantification was evaluated. The remaining of the paper is structured as follows. In the next section, existing deep networks for tissue microstructure estimation are introduced, and the proposed method of tissue microstructure estimation and uncertainty quantification is presented. In Section 3, we describe the experimental results on brain dMRI data. In Section 4, the results and future works are discussed. Finally, Section 5 summarizes the paper.

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2. Methods In this section, we first describe the background of existing deep learning based methods of tissue microstructure estimation. The frameworks that exploit sparsity #2 Q3 of diffusion signals are introduced, because they are closely related to the proposed method and their limitations motivate the proposed method. Specifically, the number of network parameters is large for an advanced network structure, which could increase the training difficulty and limit the estimation performance; in addition, these existing methods do not provide the valuable information about estimation uncertainty. Then, we present the proposed approach to tissue microstructure estimation that addresses these limitations. The proposed deep network encodes diffusion signals more efficiently and the number of network parameters can be drastically reduced; moreover, we develop a strategy of uncertainty quantification for the proposed deep network. Finally, the implementation details of the proposed method and the strategies for training and evaluation are given. 2.1. Background: Incorporation of signal sparsity for deep learning based tissue microstructure estimation #2 Q3 7

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q-Space deep learning (Golkov et al., 2016) has been developed to improve the estimation of tissue microstructure from diffusion signals undersampled in the qspace. An MLP is first used to learn the mapping from measurements in the q-space to tissue microstructure in Golkov et al. (2016). It is then improved by Ye (2017a) for the NODDI model, where sparsity of diffusion signals is exploited and a deep network for tissue microstructure estimation is constructed by unfolding an iterative process for solving sparse reconstruction problems. Specifically, suppose the vector of diffusion signals (normalized by the b0 signal acquired without diffusion weighting) at each voxel is y, where y ∈ RK and K is the number of diffusion gradients; it is assumed that the signals in the q-space—i.e., the angular domain (Schwab et al., 2016)—can be sparsely represented with the following dictionary-based framework y = Dx + η.

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(1)

Here, D is a dictionary, x is a sparse vector associated with the dictionary, and η is noise. We have D ∈ RK×N (N is the number of dictionary atoms), x ∈ RN , and η ∈ RK . If the dictionary D is known, then the dictionary coefficients x can be estimated by solving the following `1 -norm regularized least squares problem ˆ = arg min ||Dx − y||22 + β||x||1 , x

(2)

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where β is a weighting constant tuning signal sparsity. The solution to Eq. (2) can be found via an iterative process. For example, if we define F (x) = ||Dx − y||22 and R(x) = ||x||1 , then the general update of x using proximal gradient descent at iteration t is (Combettes and Wajs, 2005) xt = proxτ R (xt−1 − τ ∇x F (xt−1 )).

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(3)

Here, proxτ R (·) represents the proximal operator for R(·) and τ is a positive real number. For the `1 -norm regularization, proxτ R (·) is a soft thresholding operator (Daubechies et al., 2004). This update can also be modified by replacing the soft thresholding operator with a hard thresholding function hλ (·) and setting τ to one half, leading to the iterative hard thresholding (IHT) algorithm (Blumensath and Davies, 2008) 1 xt = hλ (xt−1 − ∇x F (xt−1 )) 2 = hλ (Wy + Sxt−1 ), 8

(4) (5)

where W = DT , S = I − DT D, and

( 0 [hλ (a)]i = ai

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if if

|ai | < λ . |ai | ≥ λ

(6)

Here, λ is a positive threshold related to β and a represents the input vector of hλ (·). With a properly designed dictionary D, the sparse representation x can be used to compute tissue microstructure. For example, in the AMICO algorithm (Daducci et al., 2015), the parameters in the NODDI model are computed from the normalized dictionary coefficients with linear transformation. Motivated by AMICO, Ye (2017a) designs an end-to-end deep network with two functional components specifically for the estimation of NODDI tissue microstructure. The first component computes sparse representation of diffusion signals in the angular domain with learned weights, and it is constructed by unfolding the IHT process. Note that like in AMICO, the sparse dictionary coefficients are assumed to be nonnegative (x ≥ 0) in Ye (2017a). Thus, the hard thresholding operation hλ (·) in Eq. (6) becomes a thresholded rectified linear unit (ReLU) activation function (Konda et al., 2014) h+ λ (·): ( 0 if ai < λ [h+ , (7) λ (a)]i = ai if ai ≥ λ and the update in Eq. (5) becomes t−1 xt = h+ ). λ (Wy + Sx

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By unfolding this iterative process, the t-th iteration becomes the t-th layer in the first component of the network. Note that the weights W and S—shared among layers—are now learned instead of precomputed by the dictionary, and they are no longer related to the dictionary via the relationship specified in IHT (Blumensath and Davies, 2008). In this way, successful sparse reconstruction can be achieved across a wider range of restricted isometry property (RIP) conditions (Xin et al., 2016). The sparse representation is then mapped to NODDI tissue microstructure with the second component that resembles the AMICO computation with learned weights. The weights in the two components are learned jointly from training tissue microstructure without the need of explicit specification of the dictionary, and thus the dictionary is flexible and determined implicitly. The deep network for tissue microstructure estimation can be also improved by 9

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incorporating information in the spatial domain. For example, the deep network in Ye (2017b) takes image patches as input by inserting a learned smoothing component before the network structure in Ye (2017a), and it outperforms the network in Ye (2017a). The incorporation of information in the spatial domain is further extended in Ye et al. (2019a), where, inspired by Schwab et al. (2018b), the network exploits the sparsity of diffusion signals jointly in the spatial and angular domains. Specifically, ˜ Suppose V voxels are the diffusion signals in a patch are concatenated as a vector y. KV included in the patch; then y˜ ∈ R . Similar to Eq. (1), y˜ can be represented by a e and its nonnegative sparse coefficients x ˜ dictionary D ex ˜ + η, ˜ y˜ = D

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where η˜ is a patch of noise. Although it is also possible to unfold the IHT process ˜ Ye et al. (2019a) improve the network to construct a deep network to compute x, by adaptively incorporating historical information—information before the output of the previous iteration or layer—in the iterative process. The adaptive use of historical information allows flexible update of sparse codes in the network, so that better sparse reconstruction can be achieved (Zhou et al., 2018). Specifically, the IHT ˜ and improved by introducing intermediate process in Eq. (8) can be rewritten for x t t terms c and c˜ at each iteration t as follows ˜ t−1 , c˜t = Wy˜ + Sx ct = f t ◦ ct−1 + g t ◦ c˜t , t ˜ t = h+ x λ (c ).

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(9)

(10) (11) (12)

Here, ct−1 encodes the historical information, i.e., the information before the sparse #2 Q4 ˜ t−1 given by the previous iteration t − 1, such as x ˜ t−2 , x ˜ t−3 , and so code estimate x t t on; f and g weight the historical information and the information in the current iteration, respectively; and ◦ represents the Hadamard product. The process in Eqs. (10)–(12) can then be unfolded to construct a deep network for sparse reconstruction, where W and S become weights that are to be learned and shared among layers. To allow adaptive incorporation of historical information in the network, f t and g t are dependent on the input and the sparse code of the previous layer t − 1 ˜ t−1 + Wf y y), ˜ f t = σ(Wf x x t t−1 ˜ + Wgy y). ˜ g = σ(Wgx x

(13) (14)

Here, Wf x and Wf y are matrices to be learned for computing f t , Wgx and Wgy 10

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are matrices to be learned for computing g t , and [σ(a)]i = 1/(1 + exp(−ai )) is a sigmoid function. With this formulation, the update in Eqs. (10)–(12) corresponds to a modified LSTM unit, where no output gate is used. In addition to the improved sparse coding of diffusion signals in the spatialangular domain, the framework in Ye et al. (2019a) is extended for generic tissue microstructure estimation that is not restricted to the NODDI model. This is achieved by using fully connected layers, which are capable of approximating continuous functions on compact subsets of Rn (Hornik, 1991; Sonoda and Murata, 2017), to map the sparse representation to tissue microstructure, instead of using the second component in Ye (2017a). Like in Ye (2017a), all weights in the network in Ye et al. (2019a) are jointly learned from training tissue microstructure without explicit specification of the dictionary.

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2.2. An improved deep network for tissue microstructure estimation with a separable dictionary Because of the sparse representation of the input diffusion signals jointly in the spatial and angular domains and the adaptive incorporation of historical information in the update, the network in Ye et al. (2019a) includes a huge number of parameters, which could increase the difficulty in training and limit the estimation performance. Therefore, motivated by the sparse coding of diffusion signals with a separable dictionary in Schwab et al. (2018a), where the dictionary can be decomposed into two parts that encode the information in the spatial domain and angular domain separately, we reformulate the framework in Eq. (9) and design an improved deep network for tissue microstructure estimation accordingly. This reformulation allows a more expressive spatial-angular sparse representation yet a reduced number of network weights. As in Ye et al. (2019a), the proposed network comprises two functional components and performs end-to-end tissue microstructure estimation. The first component computes the spatial-angular sparse representation of diffusion signals based on a separable dictionary with learned weights, and the second component maps the sparse representation to tissue microstructure with learned operations that are consistent with the use of the separable dictionary. The details of the design are given below.

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2.2.1. Component one: Spatial-angular sparse representation with a separable dictionary We reorganize the diffusion signals in a patch into a matrix Y = (y1 , . . . , yV ) ∈ RK×V , where yv (v ∈ {1, · · · , V }) is the diffusion signal vector for the v-th voxel in the patch. Suppose the diffusion signals in the spatial-angular domain can be

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e sparsely represented by a separable spatial-angular dictionary, i.e, the dictionary D in Eq. (9) is separable in the spatial domain and angular domain; then, according to Schwab et al. (2018b) we can rewrite Eq. (9) as Y = ΓXΨT + H,

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(15)

where Γ and Ψ are decomposed dictionaries that encode the information in the angular domain and spatial domain, respectively, X is a matrix of nonnegative sparse dictionary coefficients, and H represents image noise. The derivation of this relationship can be found in Appendix A. We denote the sizes of Γ and Ψ by NΓ and NΨ , respectively, and have Γ ∈ RK×NΓ and Ψ ∈ RV ×NΨ . Accordingly, we have X ∈ RNΓ ×NΨ . Note that because information#1inQ1 the spatial domain is encoded by the dictionary Ψ, each entry in X does not correspond to a specific spatial location (see Eqs. (A.2) and (A.3)). With the rearrangement in Eq. (15), if Γ and Ψ are known, X can still be estimated via an `1 -norm regularized least squares problem b = arg min ||ΓXΨT − Y||2 + β||X||1 , X F

(16)

X≥0

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where || · ||F is the Frobenius norm and || · ||1 is the element-wise `1 -norm. By analogy to Eqs. (4), (5) and (8), the iterative update for solving Eq. (16) at iteration t is t−1 Xt = h+ − ΓT (ΓXt−1 ΨT − Y)Ψ) λ (X T t−1 = h+ − ΓT ΓXt−1 ΨT Ψ), λ (Γ YΨ + X

where h+ λ (·) is now a thresholded ReLU function for a matrix A ( 0 if Aij < λ [h+ . λ (A)]ij = Aij if Aij ≥ λ

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(17) (18)

(19)

Note that here we have used the fact that the derivative of ||ΓXΨT − Y||2F with respect to X is 2ΓT (ΓXΨT − Y)Ψ (Petersen and Pedersen, 2008). Eq. (18) can be unfolded to construct a deep network for computing the sparse representation, where at layer t we have a s t−1 Xt = h+ − Sa Xt−1 Ss ). λ (W YW + X

(20)

Here, Wa ∈ RNΓ ×K , Ws ∈ RV ×NΨ , Sa ∈ RNΓ ×NΓ , and Ss ∈ RNΨ ×NΨ are all weight 12

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matrices to be learned, and like in Ye et al. (2019a), these weight matrices are shared among layers and not necessarily related to the dictionary as specified in Eq. (18). Wa and Sa can be interpreted as operations for the information in the angular domain for Y and X, respectively; and Ws and Ss can be interpreted as operations for the information in the spatial domain for Y and X, respectively. As shown in Ye et al. (2019a), adaptive incorporation of historical information in the update process can improve the network performance. Thus, we further extend Eq. (20) with adaptive incorporation of historical information. Specifically, by analogy to Eqs. (10)–(12), we have the following update process e t = Wa YWs + Xt−1 − Sa Xt−1 Ss , C e t, Ct = Ft ◦ Ct−1 + Gt ◦ C

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t

=

t h+ λ (C ),

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(22) (23)

e t and Ct are introduced, Ct−1 encodes the historical where intermediate terms C information (the information before Xt−1 in the update process), and Ft and Gt adaptively weight the historical and current information, respectively. As shown in Eq. (20), the operation on the diffusion signals or sparse codes can be decomposed into an angular operation and a spatial operation. Thus, the computations of Ft and Gt are also extended from Eqs. (13) and (14) as a s a s Ft = σ(WFX Xt−1 WFX + WFY YWFY ), t a t−1 s a s G = σ(WGX X WGX + WGY YWGY ),

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(21)

(24) (25)

a s a s a s a s where WFX , WFX , WFY , WFY , WGX , WGX , WGY , and WGY are weight matrices to be learned, and σ(·) is now an element-wise sigmoid function for matrices. Note that given the same input signal patch, it is easy to see that, because of the formulation with a separable dictionary, the number of weights to be learned in Eqs. (21)–(23) can be much smaller than that corresponding to Eqs. (10)–(12), even ˜ (for example, for a if the sparse representation X has many more entries than x typical input patch where (K, V ) = (60, 33 ), consider the case where the numbers of e Γ, and Ψ). dictionary atoms are set to 300 for D, With Eqs. (24) and (25), the steps in Eqs. (21)–(23) form a layer in the first component of the proposed deep network. Like in Ye et al. (2019a), this layer can be interpreted as a variant of the LSTM unit, and we refer to it as LSTM with a separable dictionary (SD-LSTM). SD-LSTM is visualized in Figure 1, where the operations for the information in the angular domain and spatial domain can be represented by left and right matrix multiplication, respectively. Similar to Ye et al.

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Figure 1: The SD-LSTM unit associated with the setting of a separable dictionary. The input signal Y is indicated by the green color.

Figure 2: The proposed deep network structure for tissue microstructure estimation using the SDLSTM unit shown in Figure 1 and modified fully connected layers (H 1 , H 2 , and H 3 ). The network input and output are indicated by the green and orange color, respectively.

(2019a), eight SD-LSTM units are used, as visualized in the first component of the proposed deep network shown in Figure 2.

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2.2.2. Component two: Mapping from sparse representation to tissue microstructure Like in Ye et al. (2019a), the second component in the proposed network maps the sparse representation X to tissue microstructure Z with three fully connected layers. Here, Z is a matrix containing M tissue microstructure measures of interest 0 in a patch with V 0 voxels, and it is organized as Z ∈ RM ×V , where each column is a vector of tissue microstructure measures in a voxel. V 0 is usually smaller than V — the number of input voxels—so that redundant information in the spatial domain can be exploited (Ye, 2017b; Ye et al., 2019a). Because the sparse representation X is a matrix, motivated by the separable operations in the spatial domain and angular domain, we modify each fully connected layer H l (l = 1, 2, 3) in the second component by decomposing it into Hal and Hsl as follows
H l (Ul ) = Hal Hsl (Ul ) , (26) where Ul is the input to H l , and

Hal (Ula ) = σH (Wal Ula + Bla ), Hsl (Uls ) = σH (Uls Wsl + Bls ).

(27) (28)

Here, Ula and Uls represent the inputs to Hal and Hsl , respectively; Wal and Wsl are the corresponding weights to be learned, Bla and Bls are the corresponding bias terms to be learned, and σH (·) is the ReLU activation function (Nair and Hinton, 2010).

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2.2.3. Learning of network weights The modified fully connected layers in the second component are concatenated to the output of the first component, and the complete network is visualized in Figure 2, b where the sparse representation computed by the first component is denoted by X b All weights in the proposed and the estimated tissue microstructure is denoted by Z. network are jointly learned from training tissue microstructure without an explicit design of the dictionary by minimizing the estimation loss Ns 1 X b n ||2 , ||Zn − Z Le = F Ns n=1

(29)

b n are the training and estiwhere Ns is the number of training samples, and Zn and Z mated tissue microstructure for the n-th training sample, respectively. After training, the proposed network can perform end-to-end tissue microstructure estimation. 15

Figure 3: The reconstruction component for dictionary computation.

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2.3. Uncertainty quantification with Lasso bootstrap Information about the uncertainty of the estimates given by deep networks is valuable and informative for many applications (Gal and Ghahramani, 2016; Dalca et al., 2018; Tanno et al., 2017; Zhang et al., 2019), yet uncertainty quantification has not been explored for existing deep learning based methods that map diffusion signals to tissue microstructure. In this work, because Eq. (15) is a rearrangement of the linear formulation in Eq. (9), we propose to perform residual bootstrap to quantify the uncertainty of tissue microstructure estimates. The residual bootstrap strategy also allows the trained deep network to estimate tissue microstructure for bootstrap diffusion signals. However, the proposed network—as well as the networks in previous works (Ye, 2017b; Ye et al., 2019a)—does not require an explicitly designed dictionary, and the dictionary is implicitly learned from training tissue microstructure and unknown. Thus, we first need to compute the dictionary after training the estimation network in Figure 2. This can be achieved by concatenating a reconstruction component to the first component of the trained estimation network. b from the The reconstruction component computes reconstructed diffusion signals Y b estimated sparse codes X according to Eq. (15) with weights that are to be learned. Specifically, the structure of the reconstruction component is shown in Figure 3, where weight matrices Φa ∈ RK×NΓ and Φs ∈ RNΨ ×V are to be learned and provide estimates of Γ and ΨT , respectively. Φa and Φs can be learned from training samples by minimizing the reconstruction loss Ns 1 X b n ||2 , ||Yn − Y Lr = F Ns n=1

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b n represent the input and reconstructed signal patches for the n-th where Yn and Y training sample, respectively. Note that during the minimization of Lr , the learned weights in the first component of the estimation network are fixed. With the computed dictionary, or more directly, using the reconstructed diffusion signals given by the reconstruction component, for each test sample we can perform 16

residual bootstrap to quantify the uncertainty of tissue microstructure estimates. Specifically, the residuals R ∈ RK×V can be computed as

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b R = Y − Y.

(31)

0 ¯ KV , Rkv = Rkv − R

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We assume that the noise associated with each diffusion gradient in each voxel of the input patch is independent and identically distributed and has a Gaussian distribution, which is a reasonable assumption when the signal-to-noise ratio (SNR) is high (Gudbjartsson and Patz, 1995). Consequently, the residuals are assumed to be homoscedastic. Because the dictionary coefficients are obtained by solving an `1 -norm regularized least squares problem in Eq. (16), we perform a Lasso bootstrap strategy (Knight and Fu, 2000) with the residuals. Specifically, the elements in the residuals R are first centered:

0 where Rkv is the centered residual corresponding to the k-th row and v-th column ¯ KV = 1 PV PK Rkv . For convenience, we denote the set of the of R, and R k=1 v=1 KV 0 |k ∈ {1, . . . , K}, v ∈ {1, . . . , V }}. From R, centered residuals by R, where R = {Rkv #2 Q6 b b for each Ykv (the element at the k-th row and v-th column of Y) we can sample with ∗ replacement a centered residual, denoted by Rkv , and compute a bootstrap sample of the diffusion signal ∗ ∗ Ykv = Ybkv + Rkv .

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∗ By repeating the above sampling for each pair of k and v and concatenating Ykv , we ∗ obtain a bootstrap diffusion signal matrix Y . The procedure for generating bootstrap samples Y∗ can be performed multiple times for each test sample, and these bootstrap samples approximate the distribution of input diffusion signal patches. Since the entries in Y∗ correspond to those in Y in terms of diffusion gradients and spatial locations (see Eq. (33)), we can feed each bootstrap sample Y∗ into the estimation network in Figure 2 and compute the corresponding bootstrap estimate of tissue microstructure. Then, the uncertainty of the tissue microstructure estimates given by the estimation network can be quantified with these bootstrap estimates. For demonstration, the uncertainty is measured by the standard deviation of the estimator, which is approximated by the standard deviation of the bootstrap estimates.

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2.4. Implementation details The proposed method is implemented with Keras2 and the Theano3 backend. We set the threshold λ in h+ λ (·) to 0.01 (Ye, 2017b; Ye et al., 2019a). The dictionary sizes of Γ and Ψ are set equal and denoted by N for convenience, i.e., N = NΓ = NΨ . In Ye et al. (2019a) the dictionary size is set to about 300, and here we set N = 300. Note that because of the formulation with a separable dictionary, the dimension of the sparse representation in this work is much greater than that in Ye et al. (2019a) (about 3002 vs 300), yet the total number of network weights is smaller than that in Ye et al. (2019a) (more details will be shown later in Section 3.2.2). For the second component, like in Ye et al. (2019a), the number of hidden units is set to 75 for each decomposed operation (Hal and Hsl ). The sizes of the input and output patches follow the settings in Ye et al. (2019a). Specifically, the size of the input patch is set to 33 , and thus V = 27; we only predict tissue microstructure at the center voxel of the input patch, and thus V 0 = 1. For training the estimation network in Figure 2, the loss function Le is minimized with the Adam algorithm (Kingma and Ba, 2014). The learning rate is 0.0001, the batch size is 128, and 10 epochs are used (Ye et al., 2019a). Also like in Ye et al. (2019a), 10% of the training samples are left out as a validation set to avoid overfitting. The same setting is used for learning the dictionary, where the loss function Lr is minimized. Note that Z contains different tissue microstructure measures of interest, and these quantities can have different magnitude scales. Therefore, like in Ye et al. (2019a), we rescale each training tissue microstructure measure before training the estimation network. Specifically, for a tissue microstructure measure z, we compute its mean value z¯ using all the training samples. Then, the tissue microstructure zn of each training sample n is rescaled as zn → zn /10blog10 z¯c , so that the mean of the rescaled z is between 1 and 10. The z¯ for each tissue microstructure measure is recorded and used to scale back the corresponding network output for test samples. 2.5. Training and evaluation strategies We followed Ye et al. (2019a) to obtain training samples. For each dMRI dataset acquired with a set G of diffusion gradients that undersample the q-space, a set G˜ ˜ comprising a large number of diffusion gradients that densely sample the (G ⊂ G) q-space was applied to the training scans. Training tissue microstructure images were then computed from the diffusion signals associated with G˜ using conventional 2 3

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model-based approaches. The training tissue microstructure measures at each voxel inside the brain were used to train the deep network, where the corresponding patches of diffusion signals associated with G were network inputs. These patches of training diffusion signals were also used for the dictionary computation. For quantitative evaluation, gold standard tissue microstructure was computed for test scans, which is similar to the computation of training tissue microstructure. Specifically, the set G˜ of diffusion gradients was applied to the test scans associated with G, and the gold standard was computed from the diffusion signals corresponding to G˜ using conventional approaches. To evaluate estimation accuracy, the estimation error was computed by measuring the absolute difference between the tissue microstructure estimates and the gold standard. To evaluate the proposed uncertainty quantification method, the relationship between the estimation error and the estimation uncertainty was investigated; in addition, we investigated the relationship between the gold standard and the confidence interval computed from the tissue microstructure estimates and their uncertainty. 3. Results

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In this section, we present the evaluation of the proposed approach. To demonstrate that our method is not limited to a particular signal model, it was applied to two representative biophysical models, the NODDI model (Zhang et al., 2012) and the SMT model (Kaden et al., 2016b,a), and it was evaluated both qualitatively and quantitatively. In particular, we evaluated the accuracy of tissue microstructure estimation and the quality of uncertainty quantification, as described in Section 2.5. We have also investigated the impact of hyperparameters and the number of diffusion gradients on the proposed method. The experiments were performed on a Linux machine with an NVIDIA GeForce GTX 1080Ti GPU, which has about 11 GB memory. 3.1. Data description We selected 25 subjects from the Human Connectome Project (HCP) dataset (Van Essen et al., 2013), where five subjects were selected as training data and the rest 20 subjects were used as test data. The dMRI scans of these subjects were acquired with three b-values, where b = 1000, 2000, and 3000 s/mm2 . Each b-value is associated with 90 gradient directions, and the image resolution is 1.25 mm isotropic. For each dMRI scan, we selected 60 fixed diffusion gradients that represent the diffusion gradient set G undersampling the q-space, and applied the proposed method to the scans associated with these reduced diffusion gradients. These 60 diffusion gradients resemble clinically feasible diffusion gradients and comprise 30 gradient 19

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directions for each of the two shells b = 1000, 2000 s/mm2 . The 30 gradient directions for each b-value were selected using the strategy in Ye et al. (2019a), which gives approximately evenly distributed gradient directions. Specifically, a clinical imaging protocol that has 30 gradient directions was selected, and we used these 30 gradient directions as reference directions. Then, from the HCP data, for each selected shell (b = 1000 or 2000 s/mm2 ) we selected 30 gradient directions that are closest to the reference directions, and obtained the 60 diffusion gradients in G. The full set G˜ of all 270 diffusion gradients was used to compute the training or gold standard tissue microstructure using conventional model-based approaches for the two biophysical models considered in the experiment. 3.2. Application to the NODDI model The proposed method was first applied to the NODDI model, which has become increasingly popular in neuroscientific studies (Pasternak et al., 2018). The three scalar tissue microstructure measures in the NODDI model were considered, which are the intra-cellular volume fraction vic , cerebrospinal fluid (CSF) volume fraction viso , and orientation dispersion (OD). The training and gold standard tissue microstructure images were computed using the AMICO algorithm (Daducci et al., 2015) with the full set of 270 diffusion gradients. Compared with the original NODDI model fitting, AMICO drastically reduces the computation time without degrading the estimation quality (Daducci et al., 2015), and it has been incorporated into the standard processing pipeline for the UK biobank dataset (Miller et al., 2016). 3.2.1. Network training and impact of dictionary sizes and patch sizes We trained the proposed estimation network with the settings specified in Section 2.4, which lead to about 6.90 × 105 weights to be learned. Using the training strategy in Section 2.5, we extracted about 3.4 × 106 training samples and 3.8 × 105 validation samples. The training took about 35 hours and used about 45 GB memory and 5 GB GPU memory. The training and validation losses became stable after 10 epochs (see Figure S1 in the supplementary materials). Then, we investigated the impact of dictionary sizes using the validation loss. We tested different dictionary sizes, where N ∈ {100, 200, 300, 400, 500}. Setting N = 500 led to insufficient GPU memory. The validation losses for the remaining cases are shown in Figure 4. We can see that the validation loss decreases as N increases until N = 400. These results justify our use of N = 300. The impact of input and output patch sizes (V, V 0 ) was also investigated. We considered three different cases, where (V, V 0 ) ∈ {(33 , 13 ), (33 , 33 ), (53 , 33 )}. Note that for the latter two cases, if nonoverlapping tissue microstructure patches were extracted 20

Figure 4: Validation losses after training the proposed estimation network with different dictionary sizes for the NODDI model.

Figure 5: Validation losses per voxel after training the proposed estimation network with different input and output patch sizes for the NODDI model.

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for training, there would be fewer training samples than the case of (V, V 0 ) = (33 , 13 ). Thus, for fair comparison, in the latter two cases we allowed a stride of one in the extraction of training tissue microstructure patches, and randomly selected a subset of the training patches, so that the memory usage in training was approximately the 21

Table 1: The numbers of network weights for the learning-based methods in the estimation of NODDI tissue microstructure.

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same for the three cases. The validation loss per voxel for each of the three cases is shown in Figure 5, where it can be observed that increasing the patch sizes did not lead to improved performance. These results justify our selection of (V, V 0 ) = (33 , 13 ). 3.2.2. Evaluation of estimation quality The estimation network trained with the settings specified in Section 2.4 was then applied to the test scans, and we evaluated the quality of tissue microstructure estimation. For each test scan, the estimation of NODDI tissue microstructure took about 20 minutes. The proposed method was compared with the conventional method AMICO (Daducci et al., 2015) and the learning-based approaches including the MLP in Golkov et al. (2016), MEDN (Ye, 2017a), MEDN+ (Ye, 2017b), and MESC-Net (Ye et al., 2019a). As described in Section 2.1, MLP and MEDN perform tissue microstructure estimation for each voxel using the diffusion signals at that voxel only. The MEDN network exploits the sparse representation of diffusion signals in the angular domain and is constructed by unfolding the IHT process. MEDN+ improves MEDN by allowing patch inputs, where the diffusion signals in the input patch are averaged with learned weights to obtain the input for MEDN. Both MEDN and MEDN+ are applicable only to the NODDI model, because their mapping from the sparse representation to tissue microstructure is designed specifically for NODDI. MESC-Net extends MEDN+ with spatial-angular sparse coding, and the network is improved by incorporating historical information in the update of sparse codes. It also uses fully connected layers to map the sparse representation to tissue microstructure, and thus it is a generic tissue microstructure estimation method and not restricted to the NODDI model. Both MEDN+ and MESC-Net use the same input as the proposed method. A list of the number of network weights is shown in Table 1 for each learningbased method. Note that for each learning-based competing method, the default number of network weights was used, as it has been shown that increasing the number of weights does not necessarily lead to improved performance (Ye, 2017b; Ye et al., 2019a). It can be seen that with the separable dictionary representation, the proposed method uses a smaller number of network weights than MESC-Net, yet allows a much more expressive sparse representation, where the number of entries in the 22

Figure 6: Axial views of the estimated vic maps of a representative test subject. The gold standard is also shown for reference. 525

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sparse representation in the proposed network is about 300 times the number in MESC-Net. We first qualitatively evaluated the results of tissue microstructure estimation. Axial views of the estimated maps of a representative test subject are shown in Figures 6–8, together with the gold standard for reference. The results of learningbased methods appear less noisy than those of the conventional method AMICO. In addition, methods that exploit information in the spatial domain—i.e., MEDN+, MESC-Net, and the proposed method—achieve smoother results than MLP and MEDN, which only process information in a single voxel. Since smoother results do not necessarily indicate better estimation accuracy, we have also compared the error maps between the proposed method and the competing methods in Figure 9, where zoomed views of color-coded errors are shown. We can see that the proposed method has smaller intensities in the error maps, indicating that its results are more similar to the gold standard (note the regions highlighted by circles). To quantitatively compare the methods, we computed the average estimation er23

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Figure 7: Axial views of the estimated viso maps of a representative test subject. The gold standard is also shown for reference. 540

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rors in the brain of each test subject for each method. Note that in the NODDI model, when viso = 1, vic and OD can take arbitrary values. Thus, the CSF is excluded from the quantitative analysis in this work. The means and standard deviations of these average errors4 are shown in the barplots in Figure 10, where the competing methods are compared with the proposed method using paired Student’s t-tests. It can be seen that the performance of the proposed method is highly significantly (p < 0.001) better than those of the competing methods. In addition, we #3 Q1 compared the proposed method with the second-best method MESC-Net by computing the effect size. Specifically, Cohen’s d was computed. The effect size is large (d close to or greater than 0.8) for vic (d = 0.92) and OD (d = 1.38), and it is medium (d close to 0.5) for viso (d = 0.42). 4

The average errors for each test subject (S01–S20) can be found in Figure S2 in the supplemen- #2 Q11 tary materials, where the proposed method consistently achieves smaller estimation errors than the competing methods.

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Figure 8: Axial views of the estimated OD maps of a representative test subject. The gold standard is also shown for reference.

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3.2.3. Evaluation of uncertainty quantification Next, we evaluated the results of uncertainty quantification. The dictionary was computed using the training samples extracted in Section 3.2.1, and the residuals were computed for the test scans. For demonstration, for each test sample ten bootstrap samples were produced, which is a typical number of samples for quantifying the uncertainty of deep network outputs (Gal and Ghahramani, 2016). The uncertainty quantification step took about 3.5 hours for each test subject. We have observed that the distributions of the centered residuals at different #2 Q11 locations of the input patch and associated with different diffusion gradients are approximately Gaussian and have similar variances (see Figure S3 in the supplementary materials for an example). In addition, the covariance between different residuals is much smaller than the variance of each residual (an example of the covariance matrix between the centered residuals at different locations and associated with different diffusion gradients can be found in Figure S4 in the supplementary materials). These observations together justify our assumption of homoscedasticity. 25

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Figure 9: Zoomed maps of estimation errors of a representative test subject. Note the regions highlighted by circles for comparison.

Figure 10: The means and standard deviations of the average estimation errors of the test subjects for the NODDI model. Asterisks (??? ) indicate that the difference between the proposed method and the competing method is highly significant (p < 0.001) using a paired Student’s t-test.

Then, the estimation uncertainty computed with the Lasso bootstrap strategy was evaluated. Sagittal views of the uncertainty maps of a representative test subject are shown in Figure 11. For reference, the gold standard, the tissue microstructure 26

Figure 11: Sagittal views of the estimation error and uncertainty maps of a representative test subject for the NODDI model. The gold standard and the tissue microstructure maps given by the proposed method are also shown for reference. Note the regions highlighted by arrows of the same color where the uncertainty resembles the error.

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maps given by the proposed method, and the corresponding estimation errors are also shown. It can be seen that in many structures, such as the regions highlighted by arrows, the estimation uncertainty resembles the estimation error. We then quantitatively investigated the relationship between the uncertainty and estimation error 27

Figure 12: Boxplots of the correlation coefficients between the estimation uncertainty and estimation error for the test subjects for the NODDI model. The mean values are indicated by the diamonds.

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by computing the correlation coefficient between them. Specifically, the correlation was computed for each test subject and each tissue microstructure measure. The correlation coefficients are shown in the boxplots in Figure 12 for each tissue microstructure measure. In all cases, the correlation is highly significant (p < 0.001). These results indicate that the produced uncertainty maps are sensible and could potentially be developed to suggest estimation errors. In addition, we quantitatively evaluated the proposed uncertainty quantification method by investigating the 95% confidence interval (CI). Specifically, for each voxel, we computed the 95% CI using the tissue microstructure estimate given by the proposed estimation network and the standard deviation of the estimator approximated by the proposed Lasso bootstrap procedure (Fox, 2002). Then, for each test subject we computed the fractions of voxels where the gold standard value lies inside the 95% CI. The fractions are shown for each tissue microstructure measure in the boxplots in Figure 13. The fractions are close to 0.95, which is consistent with the definition of the 95% CI. 3.2.4. Impact of the number of diffusion gradients We also investigated the impact of the number of diffusion gradients on the proposed method. Similarly to the selection of 60 diffusion gradients, we considered two additional numbers of diffusion gradients. Specifically, 36 and 24 diffusion gradients—18 and 12 gradient directions on each of the shells b = 1000, 2000 s/mm2 , 28

Figure 13: Boxplots of the fractions of voxels where the 95% CI contains the gold standard NODDI tissue microstructure value for the test subjects. The mean values are indicated by the diamonds.

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respectively—were selected. Then, the proposed method was applied to the two cases. First, we quantitatively evaluated the estimation accuracy of the proposed approach. The means and standard deviations of the average estimation errors in the brains of test subjects are shown in Figure 14, and the proposed method is compared with the competing methods using paired Student’s t-tests. In all cases, the proposed method achieves higher estimation accuracy than the competing methods, and the difference is highly significant (p < 0.001). We have also compared the proposed method with the second-best method MESC-Net by computing the effect size (Cohen’s d). The result is shown in Table 2. Except for viso when 24 diffusion gradients were used, d is close to or greater than 0.8, indicating a large effect size. For viso when 24 diffusion gradients were used, d is greater than 0.5, indicating a medium effect size. Next, we quantitatively evaluated the results of uncertainty quantification. The correlation between the estimation uncertainty and estimation error was computed for each test subject and is shown in the boxplots in Figure 15. In all cases, the correlation is highly significant (p < 0.001). In addition, by comparing the results in Figures 12 and 15, we can see that the correlation between the estimation uncertainty and estimation error is similar for the different numbers of diffusion gradients. We also computed the fractions of voxels where the gold standard tissue microstructure value is in the 95% CI given by the proposed method for each test subject, and the 29

Figure 14: The means and standard deviations of the average estimation errors of the test subjects for the NODDI model, where different numbers of diffusion gradients were used: (a) 36 diffusion gradients and (b) 24 diffusion gradients. Asterisks (??? ) indicate that the difference between the proposed method and the competing method is highly significant (p < 0.001) using a paired Student’s t-test.

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fractions are shown in the boxplots in Figure 16. Like in Figure 13, the fractions are close to 0.95. In addition, the fractions are similar for the cases of 60, 36, and 24 diffusion gradients. These results indicate the robustness of the proposed uncertainty quantification method to the number of diffusion gradients.

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Table 2: The effect sizes (Cohen’s d) for comparing the proposed method with MESC-Net for the NODDI model when 36 and 24 diffusion gradients were used.

vic viso OD 36 diffusion gradients 1.32 0.74 1.09 24 diffusion gradients 1.30 0.62 0.76

Figure 15: Boxplots of the correlation coefficients between the estimation uncertainty and estimation error for the test subjects for the NODDI model, where different numbers of diffusion gradients were used: (a) 36 diffusion gradients and (b) 24 diffusion gradients. The mean values are indicated by the diamonds.

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Figure 16: Boxplots of the fractions of voxels where the 95% CI contains the gold standard NODDI tissue microstructure value for the test subjects. Here, different numbers of diffusion gradients were used: (a) 36 diffusion gradients and (b) 24 diffusion gradients. The mean values are indicated by the diamonds.

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3.3. Application to the SMT model To demonstrate that the proposed approach is a generic method for advanced biophysical models and not limited to the NODDI model, we performed additional experiments with the 60 diffusion gradients for the SMT model (Kaden et al., 2016a,b). The SMT model includes two tissue microstructure descriptors, which are the intrinsic diffusion coefficient λ and intra-neurite volume fraction vint . The training and gold standard tissue microstructure images were computed using the SMT method 32

Figure 17: Axial views of the estimated λ and vext maps of a representative test subject. The gold standard is also shown for reference. 625

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implemented by the Dmipy software (Fick et al., 2019) with the full set of 270 diffusion gradients. Note that Dmipy gives the extra-neurite volume fraction vext instead of vint , and the estimation of vext is equivalent to the estimation of vint because vext = 1 − vint . Therefore, in this work we follow Dmipy and estimate λ and vext . The computational overhead of the proposed method for the SMT model is similar to that for the NODDI model. We first qualitatively evaluated the results of tissue microstructure estimation. Axial views of the estimated maps of a representative test subject are shown in Figure 17, together with the gold standard and the results of competing methods. Here, the proposed method is compared with the conventional SMT approach implemented in Dmipy using the undersampled diffusion signals—referred to as SMT in method comparison—and the learning-based methods MLP (Golkov et al., 2016) and MESC-Net (Ye et al., 2019a). Note that MEDN (Ye, 2017a) and MEDN+ (Ye, 2017b) are not considered because they are designed specifically for the NODDI model. Compared with the gold standard, SMT produces different contrasts of tissue microstructure maps given the limited number of diffusion gradients. The MLP gives reasonable vext but an all dark λ map with the color mapping in Figure 17. MESC-Net and the proposed estimation network both produce tissue microstructure maps that resemble the gold standard. Then, we quantitatively compared the estimation accuracy between the proposed 33

Figure 18: The means and standard deviations of the average estimation errors of the test subjects for the SMT model. Asterisks (??? ) indicate that the difference between the proposed method and the competing method is highly significant (p < 0.001) using a paired Student’s t-test.

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method and the competing methods. The average estimation error in the brain was computed for each test subject and each tissue microstructure measure. The means and standard deviations of the average estimation errors5 are summarized in the barplots in Figure 18, where the proposed method is also compared with the competing methods using paired Student’s t-tests. The errors of the proposed method are highly significantly (p < 0.001) smaller than those of the competing methods. In addition, we compared the proposed method with the second-best method MESCNet by computing the effect size. Like in Section 3.2, Cohen’s d was computed. The effect size is large for both λ (d = 2.84) and vext (d = 1.76). Next, we evaluated the proposed uncertainty quantification method. Sagittal views of the uncertainty maps of a representative test subject are shown in Figure 19. For reference, the gold standard, the tissue microstructure maps given by the proposed method, and the corresponding estimation errors are also shown. In many structures, such as the regions highlighted by arrows, the uncertainty maps resemble the error maps. We also quantitatively investigated the relationship between the estimation uncertainty and estimation error by computing the correlation coeffi5

The average errors for each test subject (S01–S20) can be found in Figure S5 in the supplementary materials, where the proposed method consistently performs better than the competing methods.

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Figure 19: Sagittal views of the estimation error and uncertainty maps of a representative test subject for the SMT model. The gold standard and the tissue microstructure maps given by the proposed method are also shown for reference. Note the regions highlighted by arrows of the same color where the uncertainty resembles the error.

Figure 20: Boxplots of the correlation coefficients between the estimation uncertainty and estimation error for the test subjects for the SMT model. The mean values are indicated by the diamonds.

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Figure 21: Boxplots of the fractions of voxels where the 95% CI contains the gold standard SMT tissue microstructure value for the test subjects. The mean values are indicated by the diamonds.

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cient between them for each test subject, and the results are shown in the boxplots in Figure 20. Like for the NODDI model, in all cases here the correlation is highly significant (p < 0.001). Lastly, for each test subject we computed the fractions of voxels where the gold standard tissue microstructure value is inside the 95% CI given by the proposed method. The fractions are shown in the boxplots in Figure 21 for each tissue microstructure measure, and they do not deviate far from 0.95. 4. Discussion

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With a dictionary that is separable in the spatial domain and angular domain, we have developed an improved deep network for tissue microstructure estimation. Compared with the previous work in Ye et al. (2019a), the formulation with the separable dictionary allows a more expressive representation of diffusion signals in the spatial-angular domain with a reduced number of network parameters. The experiments with different numbers of diffusion gradients and different biophysical models demonstrate that such a formulation leads to more accurate estimation of tissue microstructure. The dictionary-based signal representation also allows a bootstrap strategy for the quantification of estimation uncertainty. Because of the assumption of sparse dictionary coefficients, a Lasso bootstrap strategy (Knight and Fu, 2000) was adopted to quantify the uncertainty of the output of the proposed estimation network. The 36

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experiments on brain dMRI scans show that these uncertainty measures correlate with estimation errors and produce CIs that are consistent with their definitions. (Also, the uncertainty #2 Q10 expectedly increases when the number of training samples decreases, as shown in Figure S6 in the supplementary materials.) These results suggest potential application of the proposed uncertainty quantification method for brain analysis using tissue microstructure. For example, surrogate errors could be developed from the uncertainty to indicate in what regions tissue microstructure estimates are reliable for downstream analysis; also, the CIs could replace the point estimates in group analysis. The uncertainty information could also be used to improve model training. For example, in classification problems uncertainty can be used to indicate rare classes and improve the generalization ability for imbalanced classes (Khan et al., 2019), and it would be interesting to explore how uncertainty information can be exploited to improve tissue microstructure estimation. Note that for the quantification of estimation uncertainty, because in training the estimation network the dictionary is implicitly determined and unknown, a separate step of dictionary computation is needed. Since the relationship between the dictionary and the weights in the network is no longer as specified in IHT (Blumensath and Davies, 2008) so that successful sparse reconstruction can be achieved across a wider range of RIP conditions (Xin et al., 2016), we choose to compute the dictionary by minimizing the reconstruction loss. In addition, although it may seem also possible to use the estimated tissue microstructure to compute the residuals of the biophysical models for residual bootstrap, the tissue microstructure estimates are not obtained directly from fitting the biophysical models. Thus, although the tissue microstructure estimates can be close to the gold standard, the residuals computed from the biophysical models may not properly represent the noise. In our work, the dictionary computation is performed after the network for tissue microstructure estimation is trained. It is possible to jointly learn the dictionary and the mapping from diffusion signals to tissue microstructure, where the reconstruction loss Lr for dictionary computation could provide additional regularization to improve tissue microstructure estimation. However, we have empirically observed that a direct combination of the estimation loss Le and reconstruction loss Lr could lead to unstable training. Future works could explore more effective training that jointly learns the tissue microstructure estimation and computes the dictionary. The current approach to uncertainty quantification took about 3.5 hours for a test scan on an NVIDIA GeForce GTX 1080Ti GPU. Since the computation of each bootstrap estimate is independent, a straightforward way of acceleration is to compute bootstrap estimates in parallel on different GPUs. Future works could further explore more advanced methods for accelerating the uncertainty quantification. For 37

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example, simplification of the network structure via knowledge distillation (Hinton et al., 2015) could be investigated so that the computation speed can be increased without significant degradation of estimation accuracy; it is also interesting to integrate semi-analytic resampling into the Lasso bootstrap procedure (Obuchi and Kabashima, 2019) so that the computational cost for uncertainty quantification can be reduced. For existing or ongoing datasets, it is possible that training data with diffusion signals densely sampled in the q-space is not available or cannot be acquired. It would be interesting to explore how to perform deep learning based tissue microstructure estimation in such situations. For example, in Qin et al. (2019) we have proposed to interpolate the densely sampled diffusion signals in an existing source dataset in the q-space using the SHORE basis (Merlet and Deriche, 2013), so that undersampled diffusion signals that correspond to the target dataset of interest can be produced. These interpolated undersampled diffusion signals are used to train deep networks together with the high-quality tissue microstructure computed from the densely sampled diffusion signals in the source dataset, and the trained deep networks can be applied to the target dataset. Using such a strategy, methods that are based on deep networks can outperform conventional model-based methods (Qin et al., 2019). Since the performance of deep networks on the target dataset depends not only on the network design, but also on the alignment of the distributions of the training and test data, we plan to further explore domain adaptation techniques (Ganin and Lempitsky, 2015) to deal with the potential distribution difference between the undersampled diffusion signals of the source and target datasets. For example, a signal transformation component can be added to the network to align the signal distribution of the source dataset to that of the target dataset, where loss of distribution difference, such as maximum mean discrepancy (Gretton et al., 2012), can be incorporated into network training. The quality of dMRI scans can be limited not only by undersampling in the #2 q-space, but also by the relatively poor spatial resolution (Alexander et al., 2014, Q2,Q5 2017; Tanno et al., 2016, 2017, 2019; Blumberg et al., 2018). Previous works have designed deep networks to estimate high-resolution tissue microstructure maps from low-resolution diffusion tensor images (Tanno et al., 2017, 2019; Blumberg et al., 2018), and it is interesting to further explore the input of low-resolution diffusion signals directly and exploit the sparsity of diffusion signals in the network design. For example, super-resolved q-space deep learning has been proposed, where deep networks are trained to map low-resolution diffusion signals undersampled in the q-space to high-resolution tissue microstructure and the sparsity of diffusion signals in the angular domain is exploited (Ye et al., 2019b). It is possible to extend the 38

proposed framework to super-resolved q-space deep learning so that signal sparsity in the spatial-angular domain can be incorporated to improve high-resolution tissue microstructure estimation. 5. Summary and conclusion 760

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We have proposed an improved deep network for tissue microstructure estimation, where sparse representation of diffusion signals in the spatial-angular domain is exploited with a dictionary that is separable in the spatial domain and angular domain. In addition, based on the linear sparse representation of diffusion signals, we have developed an approach to quantifying the uncertainty of the estimated tissue microstructure using Lasso bootstrap. The proposed method of tissue microstructure estimation and uncertainty quantification was applied to brain dMRI scans. The results show that the proposed method achieves more accurate tissue microstructure estimation than competing methods and produces reasonable quantification of estimation uncertainty that is potentially useful for brain analysis. Conflicts of interest None Acknowledgment

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This work is supported by the National Natural Science Foundation of China (NSFC 61601461), Beijing Municipal Natural Science Foundation (7192108), and Beijing Institute of Technology Research Fund Program for Young Scholars. Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University. Appendix A. Derivation for the signal representation with a separable spatial-angular dictionary

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In this appendix, we show that Eq. (9) can be rearranged as Eq. (15) when we assume that the patches of diffusion signals can be represented by a dictionary that is separable in the spatial domain and angular domain, as shown in Schwab et al. (2018b). For convenience, we omit the noise term. We denote the element at the 39

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k-th row and v-th column of the signal patch Y ∈ RK×V by Ykv , which represents the diffusion signal associated with the k-th diffusion gradient in the v-th voxel of the patch. Suppose the diffusion signals can be represented by a separable spatialangular dictionary. First, the signal Ykv can be written as a linear combination of Γ NΓ angular basis functions {Γi (·)}N i=1 Ykv =

NΓ X

ai (v)Γi (k),

(A.1)

i=1

where ai (v) is the coefficient of Γi (·) at the v-th voxel. For each i, combining the coefficient ai (v) at each voxel v, we have a 3D patch of coefficients. Then, ai (v) can Ψ be written as a linear combination of NΨ spatial basis functions {Ψj (·)}N j=1 ai (v) =

NΨ X

Xi,j Ψj (v),

(A.2)

j=1

795

where Xi,j is the coefficient of Ψj (·) for the i-th angular basis function Γi (·). Combining Eqs. (A.1) and (A.2), we have Ykv =

NΓ X NΨ X

Xi,j Ψj (v)Γi (k).

(A.3)

i=1 j=1

Since in Eq. (9) y˜ is the concatenated diffusion signals in a patch: y˜ = (Y11 , . . . , YK1 , . . . , Y1V , . . . , YKV )T ,

(A.4)

we have 

  Ψ2 (1)Γ . . . ΨNΨ (1)Γ X1,1   Ψ2 (2)Γ . . . ΨNΨ (2)Γ    X2,1    ..  , .. .. ...  .  . . Ψ1 (V )Γ Ψ2 (V )Γ . . . ΨNΨ (V )Γ XNΓ ,NΨ

Ψ1 (1)Γ  Ψ1 (2)Γ  y˜ =  ..  .

40

(A.5)

where 

Γ1 (1)  Γ1 (2)  Γ =  ..  .

Γ2 (1) Γ2 (2) .. .

... ... .. .

 ΓNΓ (1) ΓNΓ (2)   . ..  .

(A.6)

Γ1 (K) Γ2 (K) . . . ΓNΓ (K)

800

This shows how diffusion signals can be represented with the formulation in Eq. (9) using a separate spatial-angular dictionary. If we further define   Ψ1 (1) Ψ2 (1) . . . ΨNΨ (1)  Ψ1 (2) Ψ2 (2) . . . ΨN (2)  Ψ   Ψ =  .. (A.7)  .. .. . .  .  . . . Ψ1 (V ) Ψ2 (V ) . . . ΨNΨ (V ) and reorganize y˜ as Y, by simple matrix multiplication, Eq. (A.5) becomes Y = ΓXΨT ,

(A.8)

where 

  X= 

805

X1,1 X2,1 .. .

X1,2 X2,2 .. .

... ... .. .

X1,NΨ X2,NΨ .. .

XNΓ ,1 XNΓ ,2 . . . XNΓ ,NΨ



  . 

(A.9)

Thus, Eq. (9) can be rewritten as Eq. (15) if the dictionary is separable in the spatial domain and angular domain. References Alexander, D.C., Dyrby, T.B., Nilsson, M., Zhang, H., 2019. Imaging brain microstructure with diffusion MRI: Practicality and applications. NMR in Biomedicine 32, e3841.

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