# Diffusion-Weighted MRI of the Liver in Patients With Chronic Liver Disease: A Comparative Study Between Different Fitting Approaches and Diffusion Models

Jiqing Huang, Benjamin Leporq, Valérie Hervieu, Jérôme Dumortier, Olivier

Beuf, Hélène Ratiney

► To cite this version:

Jiqing Huang, Benjamin Leporq, Valérie Hervieu, Jérôme Dumortier, Olivier Beuf, et al. Diffusion-Weighted MRI of the Liver in Patients With Chronic Liver Disease: A Comparative Study Between Different Fitting Approaches and Diffusion Models. Journal of Magnetic Resonance Imaging, 2023, 10.1002/jmri.28826. hal-04211925

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# Diffusion-Weighted MRI of the Liver in Patients With Chronic Liver Disease: A Comparative Study Between Different Fitting Approaches and Diffusion Models

Jiqing Huang, MS,¹ ID Benjamin Leporq, PhD,¹ ID Valérie Hervieu, MD,²

Jérôme Dumortier, MD, PhD,³ Olivier Beuf, PhD,¹* ID and Hélène Ratiney, PhD¹ ID

**Background:** Diffusion-weighted imaging (DWI) has been considered for chronic liver disease (CLD) characterization. Grading of liver fibrosis is important for disease management.

**Purpose:** To investigate the relationship between DWI's parameters and CLD-related features (particularly regarding fibrosis assessment).

**Study Type:** Retrospective.

**Subjects:** Eighty-five patients with CLD (age: 47.9 ± 15.5, 42.4% females).

**Field Strength/Sequence:** 3-T, spin echo-echo planar imaging (SE-EPI) with 12 b-values (0–800 s/mm²).

**Assessment:** Several models statistical models, stretched exponential model, and intravoxel incoherent motion were simulated. The corresponding parameters ($D_s$, $\sigma$, DDC, $\alpha$, f, D, $D^*$) were estimated on simulation and in vivo data using the nonlinear least squares (NLS), segmented NLS, and Bayesian methods. The fitting accuracy was analyzed on simulated Rician noised DWI. In vivo, the parameters were averaged from five central slices entire liver to compare correlations with histological features (inflammation, fibrosis, and steatosis). Then, the differences between mild (F0–F2) or severe (F3–F6) groups were compared respecting to statistics and classification. A total of 75.3% of patients used to build various classifiers (stratified split strategy and 10-folders cross-validation) and the remaining for testing.

**Statistical Tests:** Mean squared error, mean average percentage error, spearman correlation, Mann–Whitney U-test, receiver operating characteristic (ROC) curve, area under ROC curve (AUC), sensitivity, specificity, accuracy, precision. A P-value <0.05 was considered statistically significant.

**Results:** In simulation, the Bayesian method provided the most accurate parameters. In vivo, the highest negative significant correlation ($D_s$, steatosis: $r = -0.46$, $D^*$, fibrosis: $r = -0.24$) and significant differences ($D_s$, $\sigma$, $D^*$, f) were observed for Bayesian fitted parameters. Fibrosis classification was performed with an AUC of 0.92 (0.91 sensitivity and 0.70 specificity) with the aforementioned diffusion parameters based on the decision tree method.

**Data Conclusion:** These results indicate that Bayesian fitted parameters may provide a noninvasive evaluation of fibrosis with decision tree.

**Evidence Level:** 1

**Technical Efficacy:** Stage 1

J. MAGN. RESON. IMAGING 2023.

Chronic liver disease (CLD) represents a broad spectrum of diseases involving different etiologies (eg, nonalcoholic steatosis, viral hepatitis, and hemochromatosis). These diseases

are characterized by histological features such as inflammation, fibrosis, steatosis, ballooning, or iron overload. Among them, liver fibrosis is a key feature since it affects the prognosis and
guides the treatment strategy.¹ Overall, estimated 1.5 billion people worldwide have CLD, and 2 million people die from cirrhosis and liver cancer per year.²,³ Although liver biopsy carries an approximate 2% mortality with a high risk of major bleeding, followed histological analysis is still the reference method for the diagnosis of CLD, but its invasiveness limits its clinical use.⁴ Therefore, an alternative noninvasive, sensitive and specific method for population screening and monitoring remains an unmet medical need.

Diffusion-weighted imaging (DWI) derived from MRI has been increasingly used in liver examinations because of the excellent insight offered by the quantitative measurement of water molecule movement in different microenvironments.⁵ The respective water diffusion is intrinsically linked to the degree of liver fibrosis.⁵ Such methods have been investigated extensively in the literature.⁵⁻¹² Specifically, CLD patients have lower apparent diffusion coefficient (ADC) values than healthy individuals since the abnormal liver parenchyma increase the liquid impedance.¹³ However, the diagnostic value of ADC in the evaluation of fibrosis remains relatively controversial.¹⁴,¹⁵ Annet et al demonstrated that the lower ADC values in fibrotic livers were caused mostly by perfusion alterations rather than decreased extra-vascular diffusion.¹⁴ Meanwhile, Yoshimaru et al showed ADC-values were undervalued in cirrhotic tissue and tumors due to the restriction of molecular movement in a high cellularity structure.¹⁵ In consequence, the simple ADC model shows difficulty in describing such multifactorial synergy mechanisms. Hence, in recent diffusion MRI studies, several advanced diffusion models were proposed to solve this problem, such as intravoxel incoherent motion (IVIM),¹⁶ stretched exponential model (SEM),¹⁷ statistical diffusion model (Stat_D).¹⁸

In 1988, Le Bihan et al proposed the IVIM model, which quantified diffusivity separated into diffusion (D), pseudo-perfusion (D*) effects, and perfusion fraction (f).¹⁶ Leporq et al demonstrated that D and D* significantly decreased between early and advanced fibrosis groups.⁵ Lefebvre et al also reported that f was strongly correlated with inflammation and fibrosis after accounting for fibrosis and steatosis.⁶ However, an incomplete understanding of the IVIM mechanism in the tissue structure led to difficulties in separating the pseudo-perfusion and diffusion effects.⁷

In 2003, Bennett et al introduced SEM in the brain that expanded the bi-exponential compartments to multi-compartments with fewer parameters than IVIM.¹⁷ The SEM model evaluated the average diffusion rate through distributed diffusion coefficient (DDC) and diffusion heterogeneity (α).¹⁷ Park et al reported that DDC in SEM presented a higher diagnostic function for hepatic fat staging compared with parameters in IVIM and ADC.⁸ Moreover, SEM may be regarded a promising method for fibrosis staging requiring fewer b-values than IVIM, but the physical relevance of α remains unclear.⁹

In 2003, Yablonskiy et al proposed a phenomenological Stat_D that describes the ADC (D), and its distribution (σ).¹⁷ Zhou et al explored the application of non-Gaussian models for the prognosis of colorectal cancer.¹² Significantly lower values of σ were observed in the response group before treatment.¹² Although this model has not yet been applied to study the liver, the model's effectiveness is still in doubt because one parameter is hard to estimate while capturing both perfusion and the non-Gaussian phenomenon.

In this context, the aim of the present study was to investigate the effects of different fitting methods on the measurement of diffusion parameters from IVIM, SEM, and Stat_D models¹⁶⁻¹⁸ and to study their relationship with diffuse parenchyma liver disease-related features, particularly for fibrosis assessment.

# Materials and Methods

## Numerical Simulation

To evaluate numerically the accuracy of fitted parameters with different noise level conditions according to the diffusion model function and the different fitting methods, we simulated diffusion MRI signals with multiple b-values for all the aforementioned diffusion models. All simulations were implemented in MATLAB R2019b (The MathWorks Inc., Natick, MA, USA). For each voxel, the diffusion parameters can be represented as Sᵢ = S₀ × L(c), where Sᵢ represents the signal intensity at a given b-value, S₀ represents the signal intensity without a diffusion gradient, and c and L(c) are the model parameters and model function, respectively. First, the Gaussian distribution randomly generated the coefficient with a certain mean and standard deviation, as shown in Table 1. Then, the advanced diffusion model signals were calculated with their function, as described in Table 2.

The noise levels were impacted differently according to the different b-values. We thus considered adding various inherent noises that we characterized through the Rician-STD method¹⁹ from 85 patients DWI detailed in MRI acquisition.²⁰ The simulated diffusion signals were then corrupted by Rician noise, which was closer to the actual noise power than the Gaussian distribution since magnitude MRI data were used. The probability distribution for the noisy signal can be calculated as follows:

$$NS_i = \sqrt{S_i + \sigma R_i \times \text{randn}(n)^2 + \sigma I_i \times \text{randn}(n)^2}, i = 1, \dots, 12, \quad (1)$$

where NS and S represent the noisy signal and pure, noiseless signal, respectively, i is the index of b-values, and σRᵢ and σIᵢ are averaged standard deviation at specific b-values for the real and imaginary parts, respectively. In practice, the real and imaginary parts shared the same values simplifying the noise model. After generating the ground truth and noisy signal, the parameters were estimated using NLS, segmented NLS, and Bayesian fitting methods. The performance of the three estimation methods was validated by mean squared error on fitted parameters and mean absolute percentage error on signal.
<table><caption>TABLE 1. Major parameter settings in the simulation</caption><thead><tr><th></th><th>Model</th><th>Parameters</th><th>Values</th></tr></thead><tbody><tr><td rowspan="10">b-values set (s/mm<sup>2</sup>)</td><td>-</td><td>-</td><td>0, 10, 20, 40, 60, 80, 100, 200, 300, 400, 600, 800</td></tr><tr><td rowspan="2">STD</td><td>D<sub>s</sub>(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>1.34 ± 0.81</td></tr><tr><td>σ(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>0.37 ± 0.18</td></tr><tr><td rowspan="2">SEM</td><td>DDC(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>1.353 ± 0.268</td></tr><tr><td>α</td><td>0.569 ± 0.073</td></tr><tr><td rowspan="3">IVIM</td><td>f(%)</td><td>24.6 ± 7.28</td></tr><tr><td>D(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>0.91 ± 0.19</td></tr><tr><td>D*(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>76.2 ± 7.98</td></tr></tbody></table>

Stat_D = statistical diffusion model; SEM = stretched exponential model; IVIM = intravoxel incoherent motion. In the simulation, the same b-values were used as in the in vivo study. The parameters were assumed to follow Gaussian distribution. The data are represented as median ± standard deviation.

<table><caption>TABLE 2. Diffusion-based MRI model parameters and their structural proxy</caption><thead><tr><th>Diffusion MRI Model</th><th>Parameters</th><th>Definition (Structural Proxy)</th><th>Function</th></tr></thead><tbody><tr><td rowspan="2">Stat_D</td><td>D<sub>s</sub> (mm<sup>2</sup>/s)</td><td>Position of distribution maxima ADC (limited isotropic diffusion)</td><td>S<sub>l</sub>/S<sub>0</sub> = exp[-(b × D<sub>s</sub> + 1/2 × b<sup>2</sup>σ<sup>2</sup>)]</td></tr><tr><td>σ (mm<sup>2</sup>/s)</td><td>Distribution width of ADC (non-Gaussian diffusion)</td><td></td></tr><tr><td rowspan="2">SEM</td><td>DDC (mm<sup>2</sup>/s)</td><td>Distributed diffusion coefficient (limited isotropic diffusion)</td><td>S<sub>l</sub>/S<sub>0</sub> = exp[-(b × DDC)<sup>a</sup>]</td></tr><tr><td>α</td><td>Diffusion (heterogeneity perfusion)</td><td></td></tr><tr><td rowspan="3">IVIM</td><td>f</td><td>Vascular volume fraction (isotropic diffusion fraction)</td><td>S<sub>l</sub>/S<sub>0</sub> = (1 - f) exp[-(b × D) + f × exp[-(b × D*)]</td></tr><tr><td>D (mm<sup>2</sup>/s)</td><td>Pure diffusion (limited isotropic diffusion)</td><td></td></tr><tr><td>D* (mm<sup>2</sup>/s)</td><td>Perfusion-related diffusion (perfusion)</td><td></td></tr></tbody></table>

Stat_D = statistical diffusion model; SEM = stretched exponential model; IVIM = intravoxel incoherent motion.

## Patient Population

In this study, the patients were retrospectively enrolled from a database of 85 patients with CLD,²⁰ including alcoholic hepatitis, NASH, and viral hepatitis. This database included conventional diagnostic MRI acquisitions and multiple b-values diffusion acquisition, hepatic blood biomarkers, fibroscan results, and histology. This database was built in our institution.²⁰ Our institutional review board approved this retrospective study and the requirement to obtain informed consent was waived. Information on the patient population is summarized in Table 3. Inclusion criteria were age

superior to 18 years; CLD based on clinical history or prior examination; need for liver biopsy and liver MRI imaging. Exclusion criteria included liver transplantation within the last 6 months, acute liver disease, pregnancy and patient with claustrophobia, metallic implant, pacemaker or coagulation disorders.

## Diffusion-Weighted MRI Acquisition

All patients underwent an MRI examination with a 3-Tesla system (GE MR750, GE Healthcare, Chicago, IL, USA). The DWI sequence was a spin echo-echo planar imaging (SE-EPI) enhanced
<table><caption>TABLE 3. Patient characteristics</caption><thead><tr><th></th><th></th><th>Values</th></tr></thead><tbody><tr><td rowspan="2">General information</td><td>Gender</td><td>Male: N = 49<br/>Female: N = 36</td></tr><tr><td>Age (years)</td><td>47.9 ± 15.5<br/>(20–80)</td></tr><tr><td rowspan="3">Laboratory parameters</td><td>Height (m)</td><td>1.70 ± 0.11</td></tr><tr><td>Weight (kg)</td><td>76.8 ± 17.99</td></tr><tr><td>ASAT(U/L)</td><td>51.25 ± 39.28</td></tr><tr><td rowspan="5">Clinical etiology</td><td>ALAT(U/L)</td><td>83.80 ± 89.48</td></tr><tr><td>ISHAK score</td><td>F0: 9, F1: 20, F2: 15<br/>F3: 15, F4: 16, F5: 8, F6: 2</td></tr><tr><td>Inflammation</td><td>A0: 28<br/>A1: 47, A2: 10</td></tr><tr><td>Alcohol</td><td>23</td></tr><tr><td>NASH</td><td>23</td></tr><tr><td rowspan="3">Viral hepatitis</td><td>B: 24</td></tr><tr><td>C: 21</td></tr><tr><td>Others: 4</td></tr><tr><td>Liver transplant</td><td>7</td></tr></tbody></table>

ASAT = aspartate transaminase; ALAT = alanine transaminase; NASH = nonalcoholic fatty liver disease. Data are represented as median ± standard deviation, with the unit in parentheses and range in brackets.

DWI (eDWI) acquired in the transverse plane using a tetrahedral gradient encoding (i.e., four combinations of x, y, and z gradients within 50 mT/m⁻¹).²¹ The acquisition parameters of the diffusion MRI protocols are listed in Table 4. The MRI signal was acquired with 128 × 96 matrix, a field of view (FOV) of 400 × 300 mm², and joined slices had a slice thickness of 8 mm and were reconstructed in 256 × 256 by zero filling to a FOV of 400 × 400 mm². No motion correction or fat saturation pulses were applied.

## Diffusion Parameter Fitting Methods

Under the additional magnetic field gradients, diffusion signal attenuation is caused by the phase dispersion of transverse magnetization.⁵ This diffusion decay can be decomposed into three compartments: perfusion (eg D*, α, σ), limited isotropic diffusion (eg D, Ds, DDC), and non-Gaussian diffusion (eg σ), depending on the water proton surroundings (microcirculation, tissue fluid, and intra- or extracellular membranes) (Supplementary Material). The contribution of the parameters of each model in the triangular behaviors is listed in Table 2. The MRI diffusion-weighted signals were acquired for the pre-defined b-values indexed with i = 1, 2, 3,

<table><caption>TABLE 4. MRI acquisition parameters</caption><thead><tr><th>Parameters</th><th>Sequence</th></tr></thead><tbody><tr><td>SE-EPI-eDWI</td><td></td></tr><tr><td>b-values<br/>(s/mm²)</td><td>0, 10, 20, 40, 60, 80, 100, 200,<br/>300, 400, 600, 800</td></tr><tr><td>TE (msec)</td><td>54.3</td></tr><tr><td>TR (msec)</td><td>2050</td></tr><tr><td>Acquisition<br/>matrix size</td><td>128 × 96</td></tr><tr><td>Matrix size</td><td>256 × 256</td></tr><tr><td>FOV (mm²)</td><td>400 × 300</td></tr><tr><td>Signal average</td><td>2, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9</td></tr><tr><td>Slice thickness<br/>(mm)</td><td>8</td></tr><tr><td>Acquisition<br/>time (s)</td><td>312</td></tr></tbody></table>

SE-EPI-eDWI = spin echo-echo planar imaging-enhanced diffusion-weighted imaging; TE = echo time; TR = repetition time; FOV = field of view. Adjacent slices were joined with no gap between the slices.

..., 12. The parameters (Table 2) were fitted using a nonlinear least squares (NLS) method, a combination of an NLS fit with a segmented approach as a first step to initiate NLS guess values (segmented NLS), and Bayesian methods.²²,²³ The following equation describes the process for the IVIM model (Table 2). In this case, the model function is given by equation:

$$ \frac{S_i}{S_0} = (1-f) \times \exp(-b \times D) + f \times \exp(-b \times D^*), \quad (2) $$

where $S_i$ represents the signal intensity at a given b-value and $S_0$ represents the signal intensity when no pulsed diffusion gradient is applied.

**NLS METHOD.** The NLS method iteratively finds the best fit for the given nonlinear data by minimizing an objective function, such as in Equation (3) constituted by the residual sum of squares, i.e., the sum of the squares of residuals (deviation of estimated model from actual acquired data):

$$ \sum_{i \in b \text{ values set}} (S_i - \hat{S}_i)^2, \quad (3) $$

where $\hat{S}_i$ is the reconstructed signal with the estimated parameters in each iteration. The parameters are updated by the gradient descent until convergence or until reaching the maximum number of allowed iterations.
**SEGMENTED NLS METHOD.** For the NLS methods to work properly, the parameters need to be weakly correlated.²² However, in the diffusion models studied, most of the model parameters are highly correlated, which leads to an often ill-posed problem and results in a high noise level dependence.²³ To help distinguish between the water motion in diffusion from perfusion behavior, the segmented NLS method adopts a piecewise fitting strategy. The *D*\* is a hundred times higher than *D*.²³ Thus, the *D*\* can be negligible when the *b*-value is high enough (*b* ≥ *b*<sub>th</sub>). In this case, the signals are considered as nearly pure diffusion. In the first step, the IVIM model is simplified to a mono-exponential model:

$$ 
\begin{aligned}
S_i &= S_{b_{\text{th}}} \exp [(-b + b_{\text{th}}) \times \hat{D}] = S_0 (1 - f) \exp (-b \times \hat{D}) \\
    &= \hat{A} \exp (-b \times \hat{D}) (b \ge b_{\text{th}}).
\end{aligned} 
\quad (4) $$

The perfusion fraction is calculated as $\hat{f} = 1 - \hat{A}/S_0$. The parameters are underestimated ($\hat{D} < D, \hat{f} < f$). Therefore, the parameter values fitted in this piece can be used as lower bounds for the next steps.

Then, *f* and *D*\* were estimated in *b* ≤ *b*<sub>th</sub> with fixed *D* as follows:

$$ S_i = S_0 \{ \exp[(1-f)(-b \times D)] + \exp[f(-b \times D^*)] \}, \quad (5) \\ (f > \hat{f}, b \le b_{\text{th}}). $$

**BAYESIAN METHOD.** The NLS-based methods find the optimized parameters with respect to the residuals. By contrast, the Bayesian approach maximizes the posterior probability with the observed signal, an infinite parametric distribution, and noise distribution.²⁴ The objective function is defined as:

$$ \operatorname{argmax} p(P, \beta, \Sigma_{\beta} | S) \propto p(S | P, b, \beta, \Sigma_{\beta}) p(P | \beta, \Sigma_{\beta}), \quad (6) $$

where *P* is a vector of the IVIM's parameters, including, *f*, *D*, *D*\*, *S*<sub>0</sub>, and *β*, *Σ*<sub>*β*</sub>, which are the mean values of *P* and their covariances. In practice, to estimate the full parameters with the Bayesian method in a finite time, a Monte Carlo Markov chain (MCMC) is built, and Metropolis-Hastings sampling with a shrinkage strategy is used to accelerate the MCMC convergence.²⁵ In Bayesian estimation, the parameters are initialized by the NLS results, and their priors are assumed to be of Gaussian form, except for *f*, which is supposed to have a uniform distribution.²³

## In Vivo Data Analysis

Based on the MRI acquisition, the different models were used to compute diffusion parameters with the three fitting methods (NLS, segmented NLS and Bayesian) for each voxel within Regions of interest (ROIs). The ROIs were placed in the entire right hepatic lobe to minimize the heartbeat effect in MATLAB R2019b (The MathWorks Inc., Natick, MA, USA). For each patient, five slices were located in the middle of the liver by J.H. with the guidance of B.L. (12 years' experiences) ensuring the largest view of the liver

### a Diffusion parameters accuracy

<figure><img src="image_3.png" /></figure>

### b Signal accuracy

<figure><img src="image_4.png" /><figcaption>FIGURE 1: Estimation of accuracy in parameters and signal. In the simulation, all parameters fitted by NLS, segmented NLS, and Bayesian methods were compared with the ground truth. (a) Comparison of three estimators for different parameters between ground truth and fitted-parameters based on mean squared error. (b) Comparison of three estimators for signal between ground truth and reconstructed signal based on mean average percentage error. NLS = nonlinear least squares method.</figcaption></figure>

lobe. For post-processing, MR signals were arranged in ascending order into multiple vectors (1 × number of *b*-values) at each pixel. The mean values in ROI for all parameters were calculated for each patient.

## Histology

All patients underwent liver biopsy under ultrasound guidance. A 15G needle was inserted slightly below the bottom right rib cage to obtain the tissue samples. The mean biopsy length was 13.08 ± 2.35 mm (range: 7–18 mm) and the cross-sectional area was 9.14 ± 3.15 mm² (range 2.08–18.72 mm²). Tissue samples were embedded in paraffin after being fixed in buffered formalin. All histological slides were stained with hematoxylin-eosin-saffron, Sirius red, and Perls stain. Histopathological assessment was achieved blinded to any clinical information or MRI result by two
a Mild group

<figure><img src="image_5.png" /></figure>

b Severe group

<figure><img src="image_6.png" /><figcaption>FIGURE 2: Representative diffusion parameters (mm²/s) fitted by three methods for two fibrosis groups. NLS = nonlinear least squares method.</figcaption></figure>

pathologists (pathologist 1: 20 years of experience and H.V. [co-author]: 10 years of experience) according to Ishak, non-alcoholic fatty liver disease activity score (NAS), and meta-analysis of histological data in viral hepatitis (METAVIR) for fibrosis, steatosis, and inflammation, respectively.²⁶,²⁷ The histopathologic analysis was reviewed by H.V. and the discrepancies were co-validated. The area

of steatosis and fibrosis (in %) were evaluated by H.V. The percentage of steatosis was morphological and semi quantitative (NAS) and the percentage of fibrosis was morphological and quantitative. Image analysis was performed by using a light microscope (Eclipse E400, Nikon France, Champigny, France) equipped with a tri-CDD video camera (Sony, Tokyo, Japan) was determined by morphometric
a Steatosis

<figure><img src="image_7.png" /></figure>

b Fibrosis area

<figure><img src="image_8.png" /><figcaption>FIGURE 3: Correlation of diffusion parameters with chronic liver disease histology biomarkers for (a) steatosis and (b) fibrosis (the color of the circle reflects the relevance, and the area of the circle reflects the significance; \*P < 0.05; \*\*P < 0.01), when no r-value is given, there is no significant correlation. A P-value <0.05 was considered statistically significant. NLS = nonlinear least squares method.</figcaption></figure>

analysis (Hisolab, Microvision Instruments, Evry, France). Inflam-
mation was measured by combining the degree of portal necrosis
and lobular necrosis based on METAVIR.

**Statistical Analysis**

Statistical analyses were performed in GraphPad Prism (version 5.0 for Windows, GraphPad Software, San Diego, CA, USA) and MATLAB R2019b (The MathWorks Inc., Natick, MA, USA). To evaluate diffusion parameters in assessing liver fibrosis, we analyzed the results of the three fitting approaches by computing the Spearman correlation coefficients between the diffusion parameter mean values and the fibrosis area. We also evaluated associations between liver steatosis and the diffusion parameters by computing Spearman correlation coefficients between the diffusion parameters

and the steatosis area. Based on the Ishak score, the patients were
divided into two groups according to fibrosis severity (mild fibrosis
[F0–F2] vs. severe fibrosis [F3–F6]) and according to the activity
score (no inflammation [A0] vs. hepatitis [A1–A2]). Mann–
Whitney *U*-test was used to compare averaged diffusion parameter
values for the different groups. A statistical significance level of
0.05 was used, such that any *P*-value below this threshold was
deemed significant.

The parameters showing significant differences with *P* < 0.01
between the groups for a given fitting method were combined to build
classifiers based on several conventional machine learning methods
(Supplementary Material), including linear regression, decision tree
(DT), quadratic discriminant (QD), logistic regression, naïve Bayesian
(NB), support vector machines (SVMs), and k-nearest neighbor
<table><caption>TABLE 5. Diagnostic values of diffusion parameters in diagnosing mild and severe fibrosis</caption><thead><tr><th></th><th></th><th>NLS</th><th>Segmented</th><th>Bayesian</th></tr></thead><tbody><tr><td rowspan="3">D<sub>s</sub>(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>Mild</td><td>3.11 ± 0.59</td><td>3.37 ± 0.58</td><td><strong>2.63 ± 0.60</strong></td></tr><tr><td>Severe</td><td>2.78 ± 0.76</td><td>3.03 ± 0.77</td><td><strong>2.18 ± 0.58</strong></td></tr><tr><td>P</td><td>**</td><td>**</td><td>***</td></tr><tr><td rowspan="3">σ(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>Mild</td><td>1.06 ± 0.11</td><td>1.08 ± 0.10</td><td><strong>0.85 ± 0.20</strong></td></tr><tr><td>Severe</td><td>1.00 ± 0.12</td><td>1.01 ± 0.12</td><td><strong>0.71 ± 0.16</strong></td></tr><tr><td>P</td><td>**</td><td>**</td><td>***</td></tr><tr><td rowspan="3">DDC(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>Mild</td><td>2.30 ± 0.57</td><td>1.37 ± 0.23</td><td>2.28 ± 0.56</td></tr><tr><td>Severe</td><td>2.04 ± 0.69</td><td>1.30 ± 0.25</td><td>2.03 ± 0.65</td></tr><tr><td>P</td><td>*</td><td>0.13</td><td>*</td></tr><tr><td rowspan="3">α</td><td>Mild</td><td>0.58 ± 0.07</td><td>0.49 ± 0.08</td><td>0.58 ± 0.07</td></tr><tr><td>Severe</td><td>0.62 ± 0.10</td><td>0.54 ± 0.10</td><td>0.62 ± 0.09</td></tr><tr><td>P</td><td>*</td><td>*</td><td>0.06</td></tr><tr><td rowspan="3">f</td><td>Mild</td><td>0.22 ± 0.05</td><td>0.23 ± 0.05</td><td>0.22 ± 0.04</td></tr><tr><td>Severe</td><td>0.20 ± 0.06</td><td>0.20 ± 0.07</td><td>0.20 ± 0.06</td></tr><tr><td>P</td><td>*</td><td>*</td><td>*</td></tr><tr><td rowspan="3">D(×10<sup>-3</sup> mm<sup>2</sup>/s)</td><td>Mild</td><td>1.34 ± 0.24</td><td>1.30 ± 0.22</td><td>1.36 ± 0.23</td></tr><tr><td>Severe</td><td>1.24 ± 0.25</td><td>1.21 ± 0.23</td><td>1.27 ± 0.24</td></tr><tr><td>P</td><td>*</td><td>0.07</td><td>*</td></tr><tr><td rowspan="3">D*(×10<sup>-1</sup> mm<sup>2</sup>/s)</td><td>Mild</td><td>1.18 ± 0.37</td><td>0.98 ± 0.33</td><td><strong>1.03 ± 0.34</strong></td></tr><tr><td>Severe</td><td>1.05 ± 0.37</td><td>0.89 ± 0.33</td><td><strong>0.84 ± 0.26</strong></td></tr><tr><td>P</td><td>0.11</td><td>0.18</td><td>**</td></tr></tbody></table>

NLS = nonlinear least squares method. The values are mean values and standard deviation for the average parameters in two groups. A *P*-value <0.05 was considered statistically significant.

(KNN) methods (MATLAB R2019b Classification Leaner; The MathWorks Inc., Natick, MA, USA). The classification task consisted in determining if a patient belongs to a severe or mild fibrosis group using diffusion parameters. The dataset was divided into training (75%) and test subsets (25%) by stratified sampling strategies to guarantee completeness (involving all possibilities) and identities (same proportions as the population). In training, 10-folders cross-validation was used for generalization to avoid the high model bias or overfitting problem. In order to tune the hyperparameters of the aforementioned classification methods, the hyperparameter optimization option was activated in the Hyperparameter Optimization Options toolbox of MATLAB. After training the classifiers, receiver operating characteristic (ROC) analysis with the area under the curve (AUC) was performed on the test subset to evaluate the diagnostic performance in classifying mild or severe fibrosis.

## Results

### Simulation

The diffusion signals were calculated as the ground truth according to the fixed parameters described in Tables 1 and 2. The average standard deviations of noise given through the Rician-STD method were found to be 6.1, 4.2, 4.2, 4.1, 3.7, 3.6, 3.4, 2.9, 2.6, 2.4, 2.3, 2.5 for *b*-values ranging from 0 to 800 s/mm².

The relative errors between the reconstructed signal or fitted parameters and their corresponding ground truth are displayed in Fig. 1 with their mean ± standard deviation plots in order to assess the accuracy performance. The Bayesian fitting method almost systematically improved the

<footnote>* *P* < 0.05;</footnote>

<footnote>** *P* < 0.01;</footnote>

<footnote>*** *P* < 0.001.</footnote>
<figure><img src="image_9.png" /><figcaption>FIGURE 4: Decision tree structure. The unit of parameters ($D_s$, $\sigma$, and $D^*$) are mm$^2$/s.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIGURE 5: Diagnostic values of diffusion parameters in the classification of mild and severe fibrosis. NLS = nonlinear least squares method; DT = decision tree; QD = quadratic discriminant; SVM = support vector machine; KNN = K-nearest neighbor; AUC = area under the curve; P = precision; Se = sensitivity; Sp = specificity; Ac = accuracy; F1 = F1 score.</figcaption></figure>
accuracy of the parameters compared with the NLS-based
method (Fig. 1a,b). The relative errors in pure diffusion
parameters were less than 10%, and in signal amplitudes even
less than 2%. However, there were greater errors with
advanced diffusion parameters, especially for σ and D*.

In Vivo Study

Figure 2 shows the representative diffusion parameters fitted
by the three methods for patients with severe and mild fibro-
sis. Figure 3 summarizes the statistically significant correlation
coefficients that were found between limited isotropic diffu-
sion parameters and steatosis (Fig. 3a) or fibrosis area
(Fig. 3b). A higher correlation was observed for parameters
fitted by the Bayesian method compared with the other
methods. The highest correlation was found with the Bayes-
ian method for Ds in the Stat_D model that showed the
strongest and statistically significant negative correlation with
steatosis (r = -0.46). For fibrosis area, a weak but still signif-
icant correlation (α: r = 0.23, D*: r = 0.24) was found with
the parameters fitted with the Bayesian method.

Significant differences were found between the groups
with different degrees of fibrosis: Ds, Dapp as well as σ, α, and
D*(Table 5). In addition, statistically significant differences
were only found for diffusion parameter estimations obtained
with the Bayesian inference method (Ds and σ). When con-
sidering inflammation groups, no significant differences
between groups were found for the different fitted parameters
(no inflammation [A0] vs. hepatitis [A1–A2], P-values ranged
from 0.11 to 0.95). Decision tree with Bayesian fitted param-
eters, as shown in Fig. 4, achieved the best AUC. The AUC
was 0.92 (sensitivity = 0.91 and specificity = 0.70), as
shown in the ROC curve in Fig. 5.

Discussion

In this work, we evaluated the effect of different fitting
methods on the quantification of diffusion parameters from
different models. Compared with the NLS-based method, the
Bayesian approach provided greater accuracy in diffusion
parameter estimation and fibrosis classification.

The present numerical experiments conducted on simu-
lated signals demonstrated that the Bayesian approach per-
formed better than the NLS-based methods for most
parameter estimations. Bouhrara et al also reported similar
results for the IVIM and SEM models in a simulation
study.²⁸ The Bayesian method systematically outperformed
the NLS-based method in parameter estimation, probably
because the Bayesian method reduced the influence of noise
on parameters according to the prior and also because it
started from the NLS-based estimations.²⁸ The Bayesian
approach may constrain the signal deviations in a larger solu-
tion space than the NLS method. To reduce the residual, the
NLS solver moves along the gradient descent direction. The
algorithm tends to optimize the parameters that contributed

the most to the model residual reduction and may remain at
a saddle point since the parameter change is not always
linearly continuous with signal residuals. With a combination
of parameter distribution and knowledge regularization,
Bayesian regression self-adaptively updated the parameters.
Despite the lower error obtained by the Bayesian method,
advanced parameters like σ and D* are observed with greater
errors for all estimators. This result is consistent with the
finding given in the literature that the D* is sometimes
associated with inflammation activity.²⁹ Sensitivity of the
parameter estimation to noise and the use of b-values resulted
in inaccurate parameter estimation.

Concerning the segmented method, which shared the
same optimization algorithm as the NLS, the approximation
made for the signals at each step could inevitably introduce
systematic errors. In low-noise conditions, the segmented
methods have lower accuracy than NLS in terms of signal or
parameters. When additional noise becomes important, the
segmented NLS method improved the accuracy of the statisti-
cal diffusion model. This may partly demonstrate that the
segmented NLS method may help obtain quality and correct
estimations using the particular *b*-values at each step. In other
words, the segmented NLS method improved the perfor-
mance in comparison with the NLS method when parameters
were difficult to estimate (i.e., are correlated) and in the pres-
ence of noise.

Since fibrosis can be accompanied by fat accumulation
in CLD, we performed a correlation study to evaluate the
confounding effect of fat on diffusion parameters in the
assessment of fibrosis. As already found by other studies,<sup>12,30</sup>
we found a strong correlation between parameters related to
isotropic limited diffusion and steatosis. This effect can be
explained by the additional restriction of proton motion due
to the fat vesicles. Indeed, all "pure" diffusion parameters
fitted by the three methods decreased gradually with steatosis
severity.

There was a significant correlation between perfusion-
related diffusion parameters and fibrosis. In this in vivo study,
diffusion parameters estimated with the Bayesian method
could lead to differences between groups of patients with
varying degrees of liver fibrosis. Statistically significant differ-
ences were observed for Ds, σ, D*, and f values between
patients with mild vs. severe fibrosis using the Bayesian method.
We also found that Ds values were significantly lower in the
severe fibrosis group than in the mild fibrosis group. This find-
ing may support the notion that both fibrosis and steatosis
decrease the limited isotropic diffusion parameters, and fibrosis
may impact the limited isotropic diffusion, non-Gaussian diffu-
sion, and perfusion parameters in independent ways.

Several previous studies have focused on staging the grade
of fibrosis associated with inflammation and steatosis.12,28–31
The grading of fibrosis is difficult to guarantee when using
only the ADC parameters.32 In the present study, Ds, σ, D*,
and *f* were used to train the variety of classifiers. For the classification between mild and severe fibrosis, the AUC of multiple classifiers used parameters computed with the Bayesian approach, has comparable values to the NLS-based method. In particular, the decision tree classifier achieved the best diagnostic performance with parameters fitted by the Bayesian method. This result may indicate that the accuracy of the fibrosis classification was also influenced by the estimation method. The parameters of each model tested individually were evaluated for their ability to classify fibrosis grade, but none of them provided better results than the combination of parameters from different models in our data. In this regard, many studies have shown that some diffusion parameters can be used to stage liver fibrosis, but they were all conducted on the statistical level (as the first study on in vivo data of this contribution) and did not perform a classification task.<sup>31-34</sup> The decision tree could allow us to get a clearer picture of the relevance of the different fitted parameters. Based on decision tree, *D*<sub>s</sub> could be routinely monitored as a non-invasive screening reference for fibrosis. On the successfully found classifier, *D*<sub>s</sub> have two nodes in 4 depth decision tree and appear at the head of the decision tree. The nodes closer to the root are more representative and understandable as well as the purity of the nodes are also related to the importance of features. For descendant nodes, the decision tree is not straightforwardly readable, given that it used a combination of different parameter level to refine the decision. For example, *f* values were used for classification on both layer 3 and layer 4. However, we have not found a significant correlation between fibrosis severity and *σ* values. This later result may support that some advanced parameters are non-linearly correlated with fibrosis severity. Despite the very good classification results achieved by the decision tree, four patients from the test set were not well-classified. In our data, the etiologies of CLD among patients were numerous and we suspect that for these patients impaired vascularization (for one misclassified patients), high level of steatosis (for two misclassified patient), or focal lesion (for one misclassified patient) may have had an impact on diffusion impedance, which was challenging for the classification.

## Limitations

We did not consider the imaging gradient effect on low *b*-values in the simulation study. Indeed, this could be an additive source of signal fluctuation. Second, the cohort of patients was small including different etiologies of CLDs (such as nonalcoholic steatohepatitis [NASH], viral hepatitis, or alcoholic hepatitis), and it is known that the topography of fibrosis differs according to the etiology (peri-sinusoidal in viral hepatitis vs. centro-lobular in NASH).<sup>35</sup> Consequently, in this work, the etiologies may have introduced some bias in the diffusion parameters for fibrosis evaluation. Additionally, due to the limited dataset size, it was not possible to develop

individual classifications for fibrosis grades using a complex classifier. Third, this study did not discuss the sampling choice of *b*-values. Indeed, the present work focused on comparing different estimation approaches and diffusion function models for liver characterization. However, it can be shown that *b*-value sampling schemes impact the parameter fitting and thus their correlation with liver disease features. In order not to underestimate perfusion-related parameters, low *b*-values (0–50 s/mm²) were used in this study as recommended in previous liver studies.<sup>36,37</sup> In the present study, due to insufficient signal-to-noise ratio (SNR), gradient strength value, and liver short T2 value, the MRI scanner was limited in the ability to obtain images at *b*-values greater than 800 s/mm². To keep the acquisition time short and to keep good SNR, DWI data were thus acquired with averages and with the maximum *b*-value set at 800 s/mm². Historically, the model capturing non-Gaussian behavior has been performed with high *b*-values, even has been reported as 2000 s/mm².<sup>31</sup> However, the Stat_D has never been applied to liver fibrosis and therefore the required *b*-values for this model is not fully established. Finally, spatial resolution may impact IVIM parameter values, particularly *f* and *D*<sup>*</sup>. Indeed, when voxel size increases, the risk to include large vessels or biliary ducts due to the partial volume effect increases. In this context, another faster-decaying diffusion component may corrupt the bi-exponential model and a third component should be added. To address this issue, different approaches have been proposed. Gambarota et al used a non-negative least squares (NLLS) analysis to determine the number of components in the NLSS spectrum and discarded the first *b*-values for the fitting step in voxels where three components were present.<sup>7</sup> Another work has suggested to start IVIM modeling at *b* = 2 s/mm² instead of 0.<sup>38</sup> Li et al demonstrated that compared with IVIM analysis starting from *b* = 2 s/mm², IVIM analysis that included *b* = 0 s/mm² led to higher inter-subject variations among healthy young volunteers.<sup>38</sup> In addition, an increased sensitivity of IVIM in detecting liver fibrosis has been demonstrated when starting from *b* = 2 s/mm².<sup>38</sup> Another group has suggested to acquire additional sets of very small *b*-values (typically between 0 and 20 s/mm²) and performed a tri-exponential analysis.<sup>39</sup> Thus, a future study investigating *b*-value sampling could be used to enhance the findings of the present work.

## Conclusion

In this work, we compared three different fitting methods for the estimation of diffusion parameters in terms of estimation accuracy and relevance for fibrosis evaluation. Based on this analysis, we trained a decision tree combining different advanced diffusion parameters computed with the Bayesian approach to predict severe fibrosis (*F* > 2 based on the ISHAK classification) with an AUC of 0.92. Our results also
demonstrate that the Bayesian approach improves the diagnostic performance of diffusion parameters for assessing liver fibrosis and may confirm the confounding effect of fat in the use of diffusivity (D) for assessing liver fibrosis.

## Acknowledgments

We warmly thank professor Pierre-Jean Valette (radiologist at “Hospices Civils de Lyon”) for his involvement and for facilitating the implementation of this work. This work was supported by the China Scholarship Council (CSC), and LabEx PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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