Bingnan Gao, Zhiwen Liu

Abstract: Diffusion magnetic resonance imaging (dMRI) is a noninvasive method to capture the anisotropic pattern of water displacement in the neuronal tissue.The soma and neurite density imaging (SANDI) model introduced soma size and density to biophysical model for the first time.In addition to neurite density, it can achieve their joint estimation non-invasively using dMRI.In the traditional method, parameters of the SANDI are estimated in a maximum likelihood framework, where the nonlinear model fitting is computationally intensive.Also, the present methods require a large number of diffusion gradients.Efficient and accurate algorithms for tissue microstructure estimation of SANDI is still a challenge currently.Consequently, we introduce deep learning method for tissue microstructure estimation of the SANDI model.The model comprises two functional components.The first component produces the sparse representation of diffusion signals of input patches.The second component computes tissue microstructure from the sparse representation given by the first component.The deep network can produce not only tissue microstructure estimates but also the uncertainty of the estimates with a reduced number of diffusion gradients.Then, multiple deep networks are trained and their results are fused for the final prediction of tissue microstructure and uncertainty quantification.The deep network was evaluated on the MGH Connectome Diffusion Microstructure Dataset.Results indicate that our approach outperforms the traditional methods in terms of estimation accuracy.

Keywords: diffusion magnetic resonance imaging (dMRI); tissue microstructure; soma and neurite density imaging (SANDI); deep learning

1 Introduction

Diffusion magnetic resonance imaging (dMRI)offers a unique and valuable tool for mapping brain tissue microstructure and structural connectivity noninvasively in the living human brain[1].It is sensitive to the displacement of water molecules affected by the microstructure of the tissue.At present, microstructure imaging technology is developing from technical research to clinical application research, facing challenges such as long fitting time, reduced quality of dispersion gradient estimation, and difficulty in adjustment due to multiple parameters.

Diffusion tensor imaging (DTI) is a basic dMRI technology [2].It uses the Gaussian hypothesis to simulate the stochastic process of water molecule displacement in biological tissues,where fractional anisotropy (FA) and mean diffusivity (MD) can be calculated to describe the tissue microstructure.However, these measurements provided by DTI lack specificity as different biological changes may lead to similar changes in the two parameters.Therefore, more advanced biophysical models such as AxCaliber[3], ActiveAx [4] and neurite directional dispersion and density imaging (NODDI) have been proposed [5], which provide more specific tissue microstructure measurements.

The soma and neurite density imaging(SANDI) model [6] assessed the effect of the size density of the soma cell on direction-averaged dMRI signals with high b-values.It challenges the existing standard models, regards the water dispersion in gray matter and white matter as the restricted dispersion in neurite, and embeds the “stick” modeling into the blocked extra-cellular water.Inspired by recent research indicating the failure of this “stick” hypothesis in gray matter, it can be assumed that a reasonable explanation for this failure is that the soma in gray matter is more abundant than those in white matter.

SANDI is the first imaging model considering the soma cells.This model proposes a tissue microstructure model based on three nonexchange compartments, which includes the contribution of the soma signals as a diffusion pool in a restricted geometries of non-zero size, i.e.not a dot-compartment, but rather a restricted water pool, whose MR signal has a strong dependence on b values and diffusion time (td), and numerical simulations can identify that the time regime of validity for such a simple compartment model to be at relatively short diffusion times.

Tissue microstructure estimation of biological models usually requires a substantial number of diffusion gradients which need a long imaging time.This may be difficult to implement in clinical situations.It is still an open issue to allow tissue microstructure estimation from a reduced number of diffusion gradients.In recent years,deep learning methods have been successfully applied and shown unique advantages in the field of computer vision research [7].They also have rapidly developed in the fields of medical image processing and analysis [8].

At present, research has applied machine learning and deep learning algorithms to brain tissue microstructure imaging and achieved good results [9].In [10], a method based on machine learning is proposed, which uses random forest regression to estimate brain tissue microstructure from the diffusion tensor, and can reliably calculate the diffusion tensor from the reduced diffusion gradients.Then, a multi-layer perceptron (MLP) method is used to estimate the parameters of the NODDI model.MLP has achieved improved results in estimating the microstructure of tissues with reduced diffusion gradients [11].

A multidimensional deep learning method is then developed that simultaneously generates NODDI model and generalized anisotropic fractional images from highly subsampled q-space data [12].In [13], a framework is proposed for evaluating dMRI biomarkers using deep learning and mapping.This framework relies on nonlinear diffusion tensor registration to calculate the biomarker map and estimate the map reliability.The study also uses nonlinear tensor registration to align the atlas with the object, and estimates the error of this alignment.[14] proposes a spherical convolutional neural networks (CNN) model with full spectral domain convolution and nonlinear layers.The model provides rotation invariance, and performs well in the estimation of NODDI parameters.

Strategies based on optimization learning have demonstrated advantages in solving inverse problems [15,16].For example, when solving the problem of regularized least squares, a network can be constructed according to the conventional iterative optimization process of expansion and truncation.It is unnecessary to preset weights,and model learns the weights through the network, which has a fast operation speed and high accuracy [17,18].

In the field of brain tissue microstructure reconstruction, researchers have applied optimization learning algorithms.[19,20] propose a deep network-based method for end-to-end estimation of NODDI microstructure.Subsequently,[21] introduces an improved long short term memory unit to network design, adaptively incorporating historical information in the iterative optimization process to reduce the error of sparse reconstruction.[22] focuses on the value of CNN in uncertainty modeling for tissue microstructure estimates.

Although the existing deep learning methods have worked effectively, it is unclear if deep learning is applicable to the more complicated SANDI model.Therefore, in this work, we investigated how the deep learning method estimates the SANDI tissue microstructure.We select the method proposed in [23] for demonstration.The network contains two functional components.The first component computes the sparse representation of diffusion signals for each voxel in the input patch with learned weights.The second component computes tissue microstructure from the sparse representation using sequential convolutional layers.The weights in the two components are learned jointly by minimizing the mean squared error.Finally, an ensemble of deep networks is trained and their results are fused for the final prediction of tissue microstructure and uncertainty quantification.

2 Method

2.1 SANDI for Tissue Microstructure Estimation

The SANDI model can distinguish water diffusion under the following microstructure environment: intra-cellular compartment and extra-cellular compartment.In details, the intra-cellular compartment is composed of intra-neurite compartment and intra-soma compartment [6].The intra-neurite compartment and intra-soma compartment can be considered as two environments that do not exchange with each other.The direction-averaged signalA, also know{n as the powder averaged signal, can be defined as

Here,fecandficare the relative signal fraction of the extra-cellular and intra-cellular compartments respectively.finandfisare the relative signal fraction of the intra-neurite and intrasoma compartments, respectively.Ain,AisandAecrepresents the normalized signal of the intra-neurite, intra-soma and extra-cellular compartments.b, is the diffusion weighting factor.

The standard model assumes that the extracellular compartment is fully-connected.Then the diffusion of water molecules of this compartment,Aec, is modelled as the isotropic Gaussian diffusion as

Decis the effective diffusion constant of the extra-cellular compartment which is a scalar.

The diffusion of water molecules associated with the intra-cellular compartment is modelled as

Dinis the diffusion constant of intra-cellular compartment.erf(·) computes the Gaussian error of the square root of the product ofbDin.

Assuming that the signal from the soma is limited to a diffusion pool of water molecules in a sphere with a radius ofRs.Under the assumption, it can be considered that the exchange between diffusing water molecules confined within the space of intra-cellular compartment and diffusing molecules in the soma is negligible.The soma is modeled as a closed impermeable sphere, and the normalized signal in soma can be calculated through GPD approximation [24, 25]

In Eq.(4),Disis the bulk diffusivity of water in soma compartment.rsrepresents a single radius for all the soma in a given MRI voxel.? andδare the width and separation of the gradient pulse of diffusion signals, respectively, andgis the magnitude.αmis them-th root of

Here,Jn(x) is the Bessel function of the first kind.

SANDI model has six parameters to describe tissue microstructure, that is,rs,fis,fin,fec,DinandDec.

2.2 Deep Network

Like in [15], we used a patch-based strategy endto-end deep network to estimate the SANDI model.The network comprises two functional components.The first component computes the sparse representation of the input patches and provide the results to the second component.The second component takes the output of the first component and computes the estimation results.The structure of the network is shown in Fig.1.

Fig.1 Structure of the deep network

Fig.2 Structure of the first component

Mathematically, we denote the input patch of diffusion signals acquired withκdiffusion gradients byY, where its spatial size isM3withκchannels; thus,Y ∈RM×M×M×κ.We denote the output patch of tissue microstructure estimates byC.The spatial size of output patch isN3.Its number of channels as well as the number of tissue microstructure estimates of interest isc; thusC ∈RN×N×N×c.

The first component is constructed by unfolding an iterative optimization process and computes the sparse representation of diffusion signals for each voxel in the input patch using learned weights.The unfolded iterative process takes low-quality patches as inputs and computes the sparse representation of diffusion signals.The structure of the first component is shown in Fig.2.We assume that the diffusion signals at each voxel can be represented by a dictionary and its nonnegative sparse coefficients[26].Then, the diffusion signal can be denoted byy, andy=?x, wherexis the vector of the coefficients for the dictionary?.If?is known,xcan be estimated by solving anl1-norm regularized least squares problem.From [13], the estimates can be updated iteratively by

Here,tis the iteration index,WandSare matrices determined by the dictionary?, andh?(·)is a thresholding function parameterized by?

whereIdenotes the indicator function andαis an arbitrary vector input.

The second component of the network uses convolutional layers to compute tissue microstructure from the sparse representation given by the first component.The network has two parallel structures to compute both the predicted mean and variance, which can represent the tissue microstructure estimates and estimation uncertainty, respectively.Each structure is constructed by three convolutional layers and the corresponding kernel sizes are 33, 13, and 33.There is no padding.The first two layers use ReLU activation and the numbers of channel of the first two layers are set to be 200 and 400,respectively.The last layer hasr3Cchannels to compute the tissue microstructure, whereris the subsampling rate andCis the number of the parameters of the SANDI model.The structure of the second component is shown in Fig.3.

Fig.3 Structure of the second component

The deep network uses a scoring rule that assigns higher numerical scores to better calibrated predictions and select the likelihood function to maximize.This leads to the following negative log-likelihood criterion to minimize

Here,θrepresents the parameters of the deep network.We denote the input patch of diffusion signal byYand a tissue microstructure measure at a certain voxel in the output patch byci, wherei ∈{1,...,6}.ui(Y;θ) andσi(Y;θ)are considered functions that takeYas input with parameterθand represent the tissue microstructure estimate and the corresponding estimation uncertainty, respectively.

In the deep ensemble strategy, a number of deep networks are trained, so that their results can be fused to improve the final prediction of the mean and variance [27].Therefore, we train ten deep networks and combine their outputs in the test phase.

2.3 Implementation Details

We use Adam algorithm [28] for optimization in the training process.The learning rate was set to 0.000 1 and the batch size was 128.In the second component, dropout with a probability of 0.1 was applied to the first two convolutional layers.Ten percent of the training data was split as validation set.Ten models were ensembled in the probabilistic deep learning prediction.The training process had twenty epochs for each model to make sure that the training is stable.The experiment is based on Keras with a TensorFlow backend.

3 Experiments and Analysis

3.1 Data Description

Our study used the MGH Connectome Diffusion Microstructure Dataset (CDMD).The dataset presents a comprehensive diffusion MRI dataset of 26 healthy participants acquired on the MGHUSC 3 T Connectome scanner equipped with 300 mT/m maximum gradient strength and a custom-built 64-channel head coil.For each subject, the one-hour long acquisition systematically sampled the accessible diffusion measurement space, including two diffusion times, that is, 19 and 49 ms, eight gradient strengths linearly spaced between 30 mT/m and 290 mT/m for each diffusion time, and 32 or 64 uniformly distributed directions [29].The imaging parameters are shown in Tab.1.

Tab.1 Imaging parameters of the MGH CDMD dataset

There are sixteen different b-values, (i.e., 50,350, 800, 1 500, 2 400, 3 450, 4 750, and 6 000 s/mm2for ? = 19 ms ; 200, 950, 2 300, 4 250, 6 750,9 850, 13,500, 17,800 s/mm2for ? = 49 ms).Gold standard tissue microstructure are computed using conventional methods from these high-quality dMRI scans.We selected four b-values, that is, 800 and 1 500 for ? = 19 ms and 950 and 2 300 for ? = 49 ms for gradient-subsampling and to generate low-quality dMRI scans with a limited number of diffusion gradients.These low-quality dMRI scans are used as input signal to train the deep network.

In our experiment, five subjects were used to train the models and the rest twenty-one subjects were used for test.

3.2 Results and Analysis

We compared the results of the deep network with the results of the traditional model (referred to as AMICO) and a multilayer perceptron based deep learning method.For convenience, we refer to multilayer perceptron based deep learning network and our deep network as MLP and qs-DL,respectively.The six parameters that describe tissue microstructure,rs,fis,fin,fec,DinandDec,were estimated in our work.

We computed the average estimation errors in the brain of each test subject for each method.The corresponding average estimation errors of the six parameters of the SANDI model are shown in the following Tab.2.The qs-DL achieves much more accurate results than AMICO and MLP.The error ofrs,fis,fin,fec,DinandDecis reduced by 1.669 μm, 0.020 78, 0.084 81, 0.056 1,0.339 0 μm2/ms, 0.089 8 μm2/ms, respectively,comparing with AMICO.And comparing with MLP, the error ofrs,fis,fin,fec,DinandDecis reduced by 0.372 0 μm, 0.005 17, 0.008 33, 0.008 78,0.039 3 μm2/ms, 0.015 0 μm2/ms, respectively.

Tab.2 Estimation error of qs-DL and competing methods

We have selected a representative test subject to show the axial views of the tissue microstructure estimation to compare the competing methods and qs-DL.The axial views of results are shown in Figs.4–6.Our qs-DL better estimates the tissue microstructure than the competing methods, where its results are much more similar to the gold standard and the results of the competing methods are very noisy.

For a more comprehensive evaluation of the estimation quality, we computed PSNR in the brain of each test subject for each method.The means of these metrics are shown in Tab.3.Qs-DL achieves high PSNR than the competing methods, which indicates that qs-DL has better performance.

Fig.4 The axial views of rs and fis of the selected subject

Fig.5 The axial views of fin and fec of the selected subject

Tab.3 PSNR of qs-DL and competing methods

Also, the processing time of the qs-DL is shorter.AMICO took 40 min on average for each subject while qs-DL needed only 20 min.

4 Conclusions

In this work, we used a deep learning model to estimate tissue microstructure of the SANDI model.The deep network can compute accurate estimation of the SANDI model from low-quality dMRI scans acquired with a reduced number of diffusion gradients.In the deep network, the sparse presentation of diffusion signals is computed and integrated with convolutional operations for improved estimation accuracy.We have shown that deep learning method can provide tissue microstructure estimation with considerably improved accuracy than the traditional method.We also show that the processing time is much shorter than the traditional method.