Model‐based reconstructions for intravoxel incoherent motion and diffusion tensor imaging parameter map estimations

Abstract

Intravoxel incoherent motion (IVIM) imaging and diffusion tensor imaging (DTI) facilitate noninvasive quantification of tissue perfusion and diffusion. Both are promising biomarkers in various diseases and a combined acquisition is therefore desirable. This comes with challenges, including noisy parameter maps and long scan times, especially for the perfusion fraction f and pseudo‐diffusion coefficient D*. A model‐based reconstruction has the potential to overcome these challenges. As a first step, our goal was to develop a model‐based reconstruction framework for IVIM and combined IVIM‐DTI parameter estimation. The IVIM and IVIM‐DTI models were implemented in the PyQMRI model‐based reconstruction framework and validated with simulations and in vivo data. Commonly used voxel‐wise nonlinear least‐squares fitting was used as the reference. Simulations with the IVIM and IVIM‐DTI models were performed with 100 noise realizations to assess accuracy and precision. Diffusion‐weighted data were acquired for IVIM reconstruction in the liver (n = 5), as well as for IVIM‐DTI in the kidneys (n = 5) and lower‐leg muscles (n = 6) of healthy volunteers. The median and interquartile range (IQR) values of the IVIM and IVIM‐DTI parameters were compared to assess bias and precision. With model‐based reconstruction, the parameter maps exhibited less noise, which was most pronounced in the f and D* maps, both in the simulations and in vivo. The bias values in the simulations were comparable between model‐based reconstruction and the reference method. The IQR was lower with model‐based reconstruction compared with the reference for all parameters. In conclusion, model‐based reconstruction is feasible for IVIM and IVIM‐DTI and improves the precision of the parameter estimates, particularly for f and D* maps.

We propose a model‐based reconstruction framework for intravoxel incoherent motion (IVIM) and combined IVIM and diffusion tensor imaging (IVIM‐DTI) parameter estimation. The framework was validated with simulations and in vivo data. With model‐based reconstruction, the parameter maps exhibit less noise, which was most pronounced in the f and D* maps, both in the simulations and in vivo. We found that model‐based reconstruction is feasible for IVIM and IVIM‐DTI and improves the precision of the parameter estimates, particularly for f and D* maps.

Abbreviations used

DTI

diffusion tensor imaging

FA

fractional anisotropy

IQR

interquartile range

IRGN

iteratively regularized Gauss–Newton

IVIM

intravoxel incoherent motion

MD

mean diffusivity

ROI

region of interest

SENSE

sensitivity encoding

SNR

signal‐to‐noise ratio

SVD

singular value decomposition

TGV

total generalized variation

1. INTRODUCTION

Diffusion tensor imaging (DTI) is an important noninvasive tool to investigate the diffusion properties of tissue. The introduction of acceleration methods and new fitting algorithms made DTI feasible for applications outside the brain, such as in the kidneys ¹ , ² and skeletal muscle. ³ , ⁴ , ⁵ However, the signal decay observed in DTI is not purely caused by diffusion, but is also influenced by perfusion originating from water molecules in the vascular system. This causes an additional signal decay, which may result in biased DTI parameter estimates. ⁶ This effect is taken into account in the intravoxel incoherent motion (IVIM) model. IVIM has been shown to be useful for tumor classification, ⁷ , ⁸ , ⁹ in stroke assessment, ¹⁰ and for measuring muscle perfusion after exercise. ¹¹ , ¹² , ¹³ While in most applications either DTI or IVIM is used, the combination of both permits more stable parameter estimations and is highly desirable in organs with anisotropic diffusion. ¹⁴

However, combining DTI with IVIM is challenging because:

1. The scan time increases significantly compared with DTI or IVIM only because of the need for the acquisition of additional b‐values and diffusion directions.

2. The DTI and IVIM models are conventionally fitted to the magnitude data in image space. The noise in MRI magnitude images follows a Rician distribution. ¹⁵ In low signal‐to‐noise ratio (SNR) regions, this introduces a noise‐dependent bias in parameter estimates. ¹⁶

3. Diffusion data are often noisy (low SNR), in part because of the long echo time and the additional signal decay induced by diffusion‐weighting the signal. Fitting those noisy images causes bias in the parameters and results in an overall poor quality of the parameter maps. This effect is amplified in tissues with low T2 values found outside the brain, such as for some abdominal organs and skeletal muscle. ¹⁷

4. Some organs like the brain and skeletal muscle, have a low perfusion fraction, typically in the range of 2%–12%, ¹⁸ , ¹⁹ making fitting of the IVIM parameters f and D* challenging.

A suitable approach to overcome these shortcomings is model‐based reconstruction. Instead of reconstructing the image series via an inverse Fourier transformation and fitting the model in a voxel‐wise fashion to the magnitude images, model‐based reconstruction aims to obtain the parameter maps directly from the raw k‐space data using an iterative optimization algorithm (Figure 1).

FIGURE 1.

Workflow of the (A) conventional and (B) model‐based parameter estimations. (A) In conventional fitting, the data are first reconstructed to image space, typically with an inverse Fourier transformation (iFFT). Then the model is fitted to the data series. (B) In model‐based reconstruction, the parameter maps are estimated using an iterative algorithm starting from an initial guess of the unknown parameters u. The signal is calculated with the signal model from u and compared with the acquired k‐space data d. A regularization term (R_TGV) with weight γ is added as constraint. MD, mean diffusivity.

Model‐based reconstruction offers distinct advantages over conventional image‐based model fitting, which could help to tackle the challenges of IVIM‐DTI fitting, specifically:

1. Conventionally, quantitative imaging requires many repeated acquisitions with changing parameters, resulting in high data redundancy between the scans. By skipping the reconstruction of each individual image, model‐based reconstruction can exploit all available (image) information from all k‐spaces. ²⁰ , ²¹

2. The noise in the complex k‐space is Gaussian distributed, ¹⁵ which should prevent a Rician noise bias.

3. The loss function is calculated on the entire dataset. In combination with spatial regularization, individual “outlier” voxels have less weight in the total loss function, which suppresses voxel‐wise high overestimation or underestimation of parameters. This can improve the quality of the perfusion‐related parameter maps (f and D*), particularly when SNR and/or the perfusion fraction is low.

Model‐based reconstructions were first proposed for relaxation mapping ²² , ²³ and have been successfully applied to T1 and T2 mapping, ²⁰ , ²¹ , ²⁴ , ²⁵ diffusion MRI, ²⁶ and DTI. ²⁷ , ²⁸ However, currently no model‐based reconstruction framework exists for IVIM or combined IVIM‐DTI.

The aim of this study was to develop and validate a model‐based reconstruction framework for IVIM and combined IVIM‐DTI parameter map estimations and to compare the parameter estimates with conventional image‐based fitting. We considered IVIM and IVIM‐DTI fits in different tissues to investigate the frameworks' performance in situations of high and low perfusion fractions in combination with high and low SNR due to T2 decay. Specifically, IVIM fitting was performed for the liver, a highly perfused organ with low T2 and isotropic diffusion. ²⁹ Additionally, IVIM‐DTI fitting was assessed for the kidney, an organ with high perfusion and high T2, as well as for skeletal muscle, which has low perfusion and low T2.

2. THEORY

2.1. DTI and IVIM

The signal of a DTI scan can be described as a monoexponentially decaying function:

$$S\left(b\right)=S_0·e^{-b{\mathit{g}}^T\mathit{Dg}},$$ (1)

with b (b‐value) the measure for the diffusion weighting in s/mm², g the diffusion gradient direction vector, D the symmetric rank‐2 diffusion tensor, and S ₀ the signal for b = 0 s/mm². The elements of the diffusion tensor are given by [D_xx, D_yy, D_zz, D_xy, D_xz, D_yz] with the diagonal elements D_diag = [D_xx, D_yy, D_zz] and the off‐diagonal elements D_offdiag = [D_xy, D_xz, D_yz]. To determine all elements of the diffusion tensor, diffusion‐weighting gradients need to be applied in at least six independent directions.

With the addition of perfusion effects the signal decay becomes bi‐exponential and can be described by the IVIM signal equation ³⁰ :

$$S\left(b\right)=S_0·\left(f·e^{-b{D}^{*}}+\left(1-f\right)·e^{-\mathit{bD}}\right),$$ (2)

with f the perfusion fraction, D the tissue diffusivity, and D* the pseudo‐diffusion coefficient related to capillary perfusion. Because the water molecules in the vascular system move faster than diffusing molecules in tissue, D* is an order of magnitude larger than D and thus the IVIM effect is most pronounced at low b‐values.

Combining IVIM and DTI results in the IVIM‐DTI signal model:

$$S\left(b\right)=S_0·\left(f·e^{-b{D}^{*}}+\left(1-f\right)·e^{-b{\mathit{g}}^T\mathit{Dg}}\right).$$ (3)

2.2. Model‐based reconstruction

Model‐based reconstruction aims to solve the following nonlinear optimization problem:

$$\underset{u}{u=\mathit{arg}\mathit{min}}{\left|\left|\mathrm{A}\left(u\right)-d\right|\right|}_2^2+\mathit{\gamma R}\left(u\right),$$ (4)

with u the unknown (quantitative) parameters, A the forward operator (the nonlinear MRI signal model consisting of the Fourier operator, the coil sensitivity profiles, and the sequence‐specific signal equation), d the acquired raw k‐space data and R an optional regularization term, reflecting a priori knowledge on the unknown parameters. The regularization weight γ can be used to balance between the data term and the regularization term.

In this work, the PyQMRI ³¹ toolbox written in Python was used for the model‐based reconstructions. A detailed description of the PyQMRI algorithm can be found in the work of Maier et al. ²¹ , ³¹ In short, PyQMRI uses an iteratively regularized Gauss–Newton (IRGN) approach to solve the optimization problem in Equation (4). This means that the regularization weight γ is decreased each Gauss–Newton linearization step with a factor γ_dec until a minimum value γ_min is reached. A second‐order total generalized variation (TGV) regularization ³² with a joint Frobenius norm is used to maximize information between individual parameter maps. The regularization weight γ can be adjusted using a customized configuration file to balance data consistency and regularization. The subproblems in each Gauss–Newton step are solved with a Primal‐Dual algorithm combined with a line search. ³³ The Primal‐Dual iterations are doubled each Gauss–Newton step until a maximum number of iterations is reached.

Preconditioning of the optimization problem was implemented in PyQMRI to improve and speed up convergence. The algorithm is given in the Appendix. Because the unknown parameters can vary in fit stability, a weighting of the parameters can be useful. If preconditioning is used, this is done automatically based on the singular values of the Jacobian of A(u).

3. METHODS

3.1. Parameter estimations

We implemented the IVIM and the combined IVIM‐DTI model (Equations 2 and 3) in the PyQMRI framework with the unknown parameters u_IVIM = {S₀, D, f, D*} and u_IVIM‐DTI = {S₀, D_xx, D_yy, D_zz, D_xy, D_xz, D_yz, f, D*}. The image series was reconstructed by PyQMRI using a conjugate‐gradient sensitivity encoding ³⁴ (CG‐SENSE) algorithm. The first b = 0 s/mm² image of these images was used as the initial guess for S₀. To account for possible phase errors arising from the diffusion gradients, the image phase for each b‐value and diffusion direction was estimated from the reconstructed image series relative to S₀ and multiplied to the forward signal and gradient in the model. The physical properties of the diffusion tensor require D to be symmetric and positive (semi‐) definite, which was enforced by fitting the elements of the Cholesky decomposition of $\mathbf{D}=\mathbf{L}·{\mathbf{L}}^{\mathrm{T}}$. The fit constraints were set to $\mathrm{D}=\left[0,5\right]·{10}^{-3}$ mm²/s (IVIM model), L = [−10, 10] $·$ 10⁻³ mm²/s (elements of the Cholesky decomposition for DTI), f = [0, 1], and D* = [5.001, 300] $·$ 10⁻³ mm²/s.

For all reconstructions, the following configuration parameters were used: 20 Gauss‐Newton iterations, 100 Primal‐Dual start iterations and 1000 maximum iterations, regularization (start) weight γ = 0.1, decreasing factor γ_dec = 0.5, and γ_min = 0.006. The full PyQMRI configuration file can be found in the supporting information (Text S1). All reconstructions were performed offline on a GPU (NVIDIA A100‐SXM4‐80GB).

The full PyQMRI code, including preconditioning and the IVIM and IVIM‐DTI models, is available from Gitlab (https://gitlab.tugraz.at/F23B736137140D66/PyQMRI).

3.2. Reference methods

A conventional voxel‐wise fit to the data in image space was performed in Matlab Release 2021a (MathWorks, Natick, MA, USA) as a comparison to the proposed model‐based reconstruction. The image series reconstructed with PyQMRI's implementation of the CG‐SENSE algorithm was used for parameter fitting.

Two fit algorithms were used for conventional fitting:

1. Free fit, where all unknown parameters u_IVIM or u_IVIM‐DTI were fitted at once. This was considered the fairest comparison with the model‐based reconstruction because with model‐based reconstruction all unknown parameters are estimated in one step.

2. Two‐step fit, which is probably the most widely used algorithm for IVIM fitting. In this approach, the diffusion component was fitted first to the data with b‐values ≥ b_thr. Then the tissue diffusion component was fixed to fit f and D* using all b‐values. b_thr was set to 200 s/mm² for both IVIM and IVIM‐DTI fitting.

A nonlinear least‐squares algorithm was used for fitting (Matlab's lsqcurvefit). The fit boundaries were kept the same as for the model‐based reconstruction. Similar to the model‐based reconstruction, the Cholesky decomposition of the diffusion tensor D was used.

3.3. Synthetic data

To evaluate the IVIM and IVIM‐DTI model‐based reconstructions, a numerical phantom was generated in Matlab with a matrix size of 128 x 128 pixels. Three regions were simulated with values representative for liver (region of interest [ROI] 1), kidney (ROI 2), and skeletal muscle tissue (ROI 3; see Figure 2). The signal was generated with the IVIM and IVIM‐DTI signal equations (Equations 2 and 3) with the b‐values and diffusion directions similar to those used for in vivo acquisition of the liver (IVIM signal) and muscle/kidneys (IVIM‐DTI signal) (Table 1).

FIGURE 2.

Parameter maps of the IVIM and IVIM‐DTI simulations. The noise‐free ground‐truth parameters are shown in the left column and the ground‐truth values of the three ROIs and the background are written in those regions. D*, pseudo‐diffusion coefficient; DTI, diffusion tensor imaging; f, perfusion fraction; FA, fractional anisotropy; IVIM, intravoxel incoherent motion; MD, mean diffusivity; ROI, region of interest.

TABLE 1.

MRI scan parameters for the IVIM and IVIM‐DTI acquisitions of the liver, kidney, and calf muscle. Because the liver and kidney scans were respiratory triggered, the given TR and scan time are the minimum values.

	Liver		Kidney		Calf muscle
Field of view (mm3) (RL x AP x FH)	450 x 247.5 x 188		432 x 157 x 80		320 x 171 x 60
Voxel size (mm3)	3 x 3 x 6		2.5 x 2.5 x 5		3 x 3 x 6
Slices	27		15–18		10
Slice orientation	Axial		Axial		Axial
TR (ms)	2050 (minimum)		1470 (minimum)		2500
TE (ms)	59		61		58
Bandwidth (Hz)	3283		2900		3429
Breathing navigator	External trigger		Navigator		—
b‐value (s/mm2) and gradient directions (dirs)	0	9 x	0	16 x	0	16 x
	1, 2, 5, 10, 20, 30, 40, 50, 75, 100, 150, 200, 300, 400, 500, 600, 700	3 dirs	10, 20, 35, 50, 75, 100	6 dirs	10, 20, 35, 50, 75, 100	6 dirs
			200, 400, 600	32 dirs	200, 400, 600	32 dirs
Fat suppression	SPAIR + Grad. Rev.		SPAIR + Grad. Rev.		SPAIR + Grad. Rev.
SENSE factor	2		2		2
Scan time	6 min 6 s (minimum)		7 min 30 s (minimum)		6 min 15 s

Abbreviations: AP, anterior–posterior; DTI, diffusion tensor imaging; FH, foot‐head; Grad. Rev., gradient reversal fat suppression; IVIM, intravoxel incoherent motion; RL, right–left; SENSE, sensitivity encoding; SPAIR, spectral attenuated inversion recovery; TE, echo time; TR, repetition time.

Coil sensitivity maps were computed using Biot–Savart's law and the 12 coil profiles were multiplied by the signal. The images were then transformed into k‐space via Fourier‐transformation. Gaussian noise with zero mean was added in k‐space to mimic SNR values found in the in vivo images. Effective SNR values of approximately 30 in the liver, 52 in the kidneys, and 36 in skeletal muscle were used. The k‐space was undersampled with SENSE factor 2. The IVIM and IVIM‐DTI parameter maps were computed with PyQMRI and the reference methods. As an initial guess, the mean value of the ground‐truth parameters from all three regions was used (IVIM: D = 1.6 $·$ 10⁻³ mm²/s, f = 0.14, D* = 50 $·$ 10⁻³ mm²/s; IVIM‐DTI: D_diag = 1.5 $·$ 10⁻³ mm²/s, D_offdiag = 0 mm²/s, f = 0.14, D* = 50 $·$ 10⁻³ mm²/s). To assess a possible bias due to the regularization, the model‐based reconstruction was repeated without regularization by setting γ = γ_min = 1e⁻²⁰.

One hundred different noise realizations were simulated to assess the accuracy and precision of the parameter estimations. As a measure of accuracy, the bias was calculated by subtracting the ground‐truth parameter from the median of each region. The precision was assessed by the interquartile range (IQR) between the 25th and 75th quartile for each region and parameter. A repeated measures Friedman test followed by a post hoc Dunn's test with correction for multiple comparisons was used to test for significant differences between the methods (GraphPad PRISM 9.1, San Diego, CA, USA).

The results from the individual regions were evaluated based on a summarizing score. The lowest bias and IQR per parameter and ROI received a score of 1 and the highest a score of 4. For an overall comparison, the scores for bias and IQR were averaged for the IVIM and IVIM‐DTI simulations separately.

3.4. In vivo data

To evaluate the model‐based reconstructions of IVIM and IVIM‐DTI in vivo, data were acquired in healthy volunteers on a 3‐T Philips Ingenia MRI (Philips Healthcare, Best, The Netherlands). The study was approved by the institutional medical ethical committee and the subjects provided written informed consent.

Data of the liver were acquired in five subjects (two females/three males, age 23–42 years, mean 30.0 years) for IVIM fitting. Data for the IVIM‐DTI model were acquired in the kidneys (five subjects, two females/three males, age 23–43 years, mean 31.4 years) and calf muscles (six subjects, three females/three males, age 26–33 years, mean 30.2 years). A 16‐channel anterior body receiver coil in combination with a 12‐channel posterior table‐top coil was used for all scans. Volunteers were positioned in the feet‐first supine position. For IVIM and IVIM‐DTI acquisitions, a diffusion‐weighted single‐shot spin‐echo EPI sequence was used. For the kidney scan, saturation slabs were placed above and below the kidneys. The detailed scan parameters are listed in Table 1.

The raw k‐space data were preprocessed using ReconFrame (Gyrotools LLC, Zurich, Switzerland). Basic corrections were applied (random phase and measurement phase correction, profile‐dependent amplification and DC offset correction, EPI phase correction). The data were sorted, gridded, and zero‐filled. Coil sensitivity maps from the scanner were obtained and normalized to unfold the SENSE undersampled data. Coil compression was applied using singular value decomposition (SVD). The k‐space data were sorted for b‐values and stored together with b‐values, diffusion gradient directions, and coil sensitivity maps as HDF5 files. For parameter estimations, initial guess values for all three methods were identical and based on literature values (liver ³⁵ : D = 1.0 $·$ 10⁻³ mm²/s, f = 0.2, D* = 130 $·$ 10⁻³ mm²/s; kidneys ³⁶ : D_diag = 2.3 $·$ 10⁻³ mm²/s, D_offdiag = 0 mm²/s, f = 0.2, D* = 25 $·$ 10⁻³ mm²/s; muscle ¹⁸ : D_diag = 1.5 $·$ 10⁻³ mm²/s, D_offdiag = 0 mm²/s, f = 0.1, D* = 30 $·$ 10⁻³ mm²/s). The IVIM and IVIM‐DTI parameter maps were estimated with PyQMRI and the reference methods.

For quantitative analysis, ROIs were drawn in the liver, kidney cortex and medulla, and gastrocnemius medialis (GM) and tibialis anterior (TA) muscles. The outcome parameters for the IVIM model were D, f, and D*, and for the IVIM‐DTI model the diffusion tensor eigenvalues λ₁, λ₂, and λ₃, mean diffusivity (MD), fractional anisotropy (FA), f, and D*. The median values of the outcome parameters in the ROIs and the IQR were calculated and compared between model‐based reconstruction and the reference methods as measures for bias and stability, respectively. A repeated measures Friedman test followed by a post hoc Dunn's test with correction for multiple comparisons was used to compare the free fit, two‐step fit, and model‐based reconstruction.

4. RESULTS

4.1. Synthetic data

The model‐based reconstruction with regularization was the best performing method (lowest overall score) for the simulations (Table 2). Visually, the D‐ and MD‐maps obtained with the proposed method showed closest similarity to the noise‐free ground truth (Figure 2). Noise amplification was observed for f and D* with all methods. However, the three simulated ROIs were only resolved using model‐based reconstruction, especially in the D* maps. In the FA maps, the background was pushed to a lower value with model‐based reconstruction with regularization, while higher values were favored with the other methods. In the MD maps of the IVIM‐DTI simulation, a residual SENSE undersampling aliasing artifact was observed in the reference methods. This was not visible in the model‐based reconstructions.

TABLE 2.

Averaged scoring of the bias and IQR in the IVIM and IVIM‐DTI simulations. Scoring for individual parameters was performed on a scale from 1 (best) to 4 (worst) and averaged. The lowest value per row is marked in bold. In the overall score, given in the bottom row, the values from the individual comparisons are added. Overall, the model‐based reconstruction with regularization resulted in the lowest score.

	Free fit	Two‐step fit	Model‐based reconstruction with regularization	Model‐based reconstruction without regularization
Bias IVIM	2.4	2.2	3.4	1.9
IQR IVIM	2.5	3.6	1.0	2.7
Bias IVIM‐DTI	2.9	2.0	2.3	2.7
IQR IVIM‐DTI	3.0	3.3	1.0	2.7
Overall	2.7	2.8	1.9	2.5

Abbreviations: DTI, diffusion tensor imaging; IQR, interquartile range; IVIM, intravoxel incoherent motion.

The above visual evaluation was supported by the quantitative analysis of the bias and IQR, summarized in Table 2 and detailed for the IVIM simulations in Figure  S1 and the IVIM‐DTI simulations in Figure 3. Overall, the model‐based reconstruction with regularization yielded the highest precision, reflected by the lowest IQR for all parameters compared with the reference methods. This quantitative assessment can also be visually appreciated in Figure 2, in which the parameter maps obtained with model‐based reconstruction were clearly the least noisy. For the IVIM model, the model‐based reconstruction resulted in an overall higher bias compared with the other methods. However, the differences in absolute values were small (Figure  S1 ). For the IVIM‐DTI model, the biases between all fitting methods were within the same range. Interestingly, model‐based reconstruction yielded the smallest bias for FA and λ₃ for all ROIs. For all fitting methods and both models, a small negative bias for the tissue diffusivity and D* and a positive bias for f were observed. The reconstruction times for one IVIM and IVIM‐DTI simulation dataset were 30 and 45 min, respectively.

FIGURE 3.

Bias (difference of median to reference value) and IQR of the IVIM‐DTI simulations. Nonsignificant changes are marked (ns = not significant). D*, pseudo‐diffusion coefficient; DTI, diffusion tensor imaging; f, perfusion fraction; FA, fractional anisotropy; IQR, interquartile range; IVIM, intravoxel incoherent motion; λ₃, third eigenvalue of the diffusion tensor; MD, mean diffusivity; recon, reconstruction.

4.2. In vivo data

Model‐based reconstruction results of in vivo‐acquired IVIM and IVIM‐DTI data are presented in Figures 4, 5, 6, 7, 8. Model‐based reconstructions resulted in parameter maps with more anatomical details, less noise, and fewer outliers compared with traditional fitting, particularly for f and D* (Figures 4, 6, and 8) and are in line with the observations in the synthetic images. The reconstruction of a full dataset took approximately 1 h for the muscle and 5 h for the liver and kidney data.

FIGURE 4.

IVIM parameter maps of the liver of one example volunteer obtained with the two‐step fit as the reference method and the model‐based reconstruction. The liver contour is overlaid in red. D, tissue diffusion coefficient; D*, pseudo‐diffusion coefficient; f, perfusion fraction; IVIM, intravoxel incoherent motion; recon, reconstruction.

FIGURE 5.

Median and interquartile range (IQR) values of the IVIM parameters D, f, and D* in the liver for all three fitting methods. The individual subjects are shown as dots and the mean ± standard deviation over the subjects is shown in bold bars. Significant differences are indicated by an asterisk (p < 0.05). D, tissue diffusion coefficient; D*, pseudo‐diffusion coefficient; f, perfusion fraction; IVIM, intravoxel incoherent motion; recon, reconstruction.

FIGURE 6.

IVIM‐DTI parameter maps of the muscle of an example volunteer. The two‐step fit as the reference method and the model‐based reconstruction (recon) are shown. The ROIs in the tibialis anterior and gastrocnemius medialis are overlaid in red and white, respectively. D*, pseudo‐diffusion coefficient; DTI, diffusion tensor imaging; f, perfusion fraction; FA, fractional anisotropy; IVIM, intravoxel incoherent motion; MD, mean diffusivity; recon, reconstruction; ROI, region of interest.

FIGURE 7.

Median and interquartile range (IQR) values of the IVIM‐DTI parameters MD, λ₃, f, and D* in the tibialis anterior and gastrocnemius medialis muscle for all three fitting methods. The individual subjects are shown as dots and the mean ± standard deviation over the subjects is shown in bold bars. Significant differences are indicated by an asterisk (p < 0.05). D*, pseudo‐diffusion coefficient; DTI, diffusion tensor imaging; f, perfusion fraction; IVIM, intravoxel incoherent motion; λ₃, third eigenvalue of the diffusion tensor; MD, mean diffusivity; recon, reconstruction.

FIGURE 8.

IVIM‐DTI parameter maps of both kidneys in one example volunteer, showing the two‐step fit as the reference and the model‐based reconstruction (recon) results. The ROIs in the kidney's cortex are overlaid in red and the medulla segmentation in white. D*, pseudo‐diffusion coefficient; DTI, diffusion tensor imaging; f, perfusion fraction; FA, fractional anisotropy; IVIM, intravoxel incoherent motion; MD, mean diffusivity; recon, reconstruction; ROI, region of interest.

In the liver, the model‐based reconstructed parameter maps exhibited less noise compared with the reference methods, which was especially pronounced in the D* map (Figure 4). While with two‐step fitting no anatomical details could be identified in the D* map, large vessels and structures such as the spine became visible with model‐based reconstruction. The visually observed noise reduction was also reflected in the IQR (Figure 5), with the lowest values for model‐based reconstruction for all IVIM parameters. The differences in IQR values between two‐step fitting and model‐based reconstruction were significant, while no significant differences were found between free fit and two‐step fit and between free fit and model‐based reconstruction. In the median values, significant differences between the two‐step fit and model‐based reconstruction were found for D and f, but not between the free fit and model‐based reconstructions.

In the calf muscles, the model‐based IVIM‐DTI reconstructions yielded smoother parameter maps with less noise amplification for f and D* (Figure 6). The differences in parameter maps between two‐step fitting and model‐based reconstruction for MD and FA were visually minimal. The quantitative analysis showed that the median values of the tissue diffusion‐related parameters were only slightly different between the reference methods and model‐based reconstruction (Figure 7). For the perfusion‐related parameters, lower median values were found for f and D* with model‐based reconstruction. However, these differences were only significant for f (free fit vs. model‐based reconstruction in the TA muscle) and D* (GM muscle). It was noticeable that the median D* values with the reference methods remained at the initial guess value of 30 $·$ 10⁻³ mm²/s. No significant differences in IQR between the model‐based reconstruction and the reference methods were found for all IVIM‐DTI parameters (except for MD and D* in the TA muscle). However, the IQR values of the diffusion‐related parameters tended to be lower than both reference methods.

In the kidneys, the visual differences between the parameter maps obtained with the two‐step fit and model‐based reconstruction were minimal for all parameters (Figure 8). Only in the D* map some outlier pixels were present with two‐step fitting. This is also reflected by the quantitative analysis (Figure 9), where almost no significant differences in median values between the model‐based reconstruction and the reference methods were found (with the exception of D* in the medulla). The IQR was significantly lower with model‐based reconstruction compared with the free fit for all IVIM‐DTI parameters in the medulla and no significant differences between model‐based reconstruction and the two‐step fit were found in the medulla and the cortex.

FIGURE 9.

Median and interquartile range (IQR) values of the IVIM‐DTI parameters MD, FA, f, and D* in the kidney cortex and medulla for all three fitting methods. The individual subjects are shown as points and the mean ± standard deviation over the subjects is shown in bold bars. Significant differences are indicated by an asterisk (p < 0.05). D*, pseudo‐diffusion coefficient; DTI, diffusion tensor imaging; f, perfusion fraction;FA, fractional anisotropy; IVIM, intravoxel incoherent motion; MD, mean diffusivity; recon, reconstruction.

5. DISCUSSION

In this work, we introduced a model‐based reconstruction method for IVIM and combined IVIM‐DTI parameter estimations. To the best of our knowledge, we are the first to use model‐based reconstruction for IVIM and combined IVIM‐DTI. In simulations, we showed that model‐based reconstruction generally outperforms conventional fitting, particularly for the f and D* maps. In vivo, model‐based reconstructions substantially reduce noise in the parameter maps. Moreover, model‐based reconstruction resulted in anatomically detailed f and D* parameter maps.

In the simulations, we found a decreased IQR with the proposed methods for all parameters and simulated regions. This is in agreement with previous work using PyQRMI ²¹ , ³⁷ for model‐based reconstruction of relaxation time parameter maps. It indicates that model‐based reconstruction with TGV regularization provides improved noise suppression in the parameter maps and results in higher precision. However, it is known that regularization may introduce a bias. ³⁸ Overall, we found a slightly higher bias with model‐based reconstruction for the IVIM parameters and a comparable bias for all IVIM‐DTI parameters. The largest bias with the proposed method was found for the region with low ground‐truth f and D* values, such as found in skeletal muscle (ROI 3). Regarding all parameters and regions, no clear trend towards a higher bias with the proposed method compared with the reference methods was observed, indicating that the biases found are not a result of regularization. This suggests that the accuracy of the parameter estimates is preserved with model‐based reconstruction.

The in vivo IVIM parameter maps of the liver with model‐based reconstruction appeared smoother and revealed more anatomical detail in the D* maps compared with the reference methods. The IQR was smallest with model‐based reconstruction for all parameters while no major differences in the median values were found, indicating that the noise suppression in the parameter maps is improved with model‐based reconstruction without introducing a bias. The same holds for the IVIM‐DTI parameter maps in skeletal muscle, where the greatest improvement was seen in the f and D* maps. It is noteworthy that the median values of D* for the reference methods stayed at the initial guess value of 30 $·$ 10⁻³ mm²/s, which indicates that D* values of the muscles could not be retrieved from the data with voxel‐wise fitting to the magnitude image data. Model‐based reconstruction can thus overcome this main challenge of IVIM‐DTI fitting. Minor differences between fitting methods were found in the kidneys. This is not surprising because the higher T2 values, and thus high SNR, of the kidneys compared with skeletal muscle and liver and the relatively high perfusion fraction of roughly 17%, facilitate the IVIM‐DTI parameter estimation for both the reference methods and model‐based reconstruction.

A limitation of the study is that we used only five subjects per organ (and six in skeletal muscle), so the statistical findings should be treated with care. However, we performed measurements and reconstructions of liver, kidney, and skeletal muscle and thus successfully demonstrated that the model‐based reconstructions are applicable to tissues with very different SNR and perfusion characteristics. In this study no motion correction was applied to the abdominal scans. Although breathing triggering was applied for the liver and kidney scans, motion from the cardiac cycle, peristaltic, or respiration might have influenced the image quality. Because model‐based reconstruction estimates the quantitative parameter maps directly from k‐space, image registration techniques are not directly applicable. Implementing motion correction in the model‐based reconstruction framework would be highly desirable to make the technique more robust to motion.

Model‐based reconstruction has been applied to DTI before and has been shown to outperform conventional methods for the acceleration of undersampled Cartesian, ³⁹ , ⁴⁰ radial, ²⁷ and multishot EPI DTI data. ²⁸ All those works used first‐order total variation (TV) regularization. PyQMRI uses a second‐order total generalized variation regularization, which is known to outperform TV ³² in most cases and is suitable for DTI fitting. ⁴¹ Compared with pure DTI, the complexity for the IVIM‐DTI model is increased by adding two unknown parameters, namely f and D*. Moreover, in regions with low perfusion (low f) and low SNR, like skeletal muscle, estimation of the perfusion‐related parameters f and D* remains challenging, because the signal fraction available to fit f and D* depends strongly on f. Our results in the liver and skeletal muscle suggest that model‐based IVIM and IVIM‐DTI reconstructions might be especially useful for organs with low SNR and/or perfusion fraction to improve the quality of f and D* parameter maps. In the kidneys, however, differences between the model‐based reconstruction and the reference methods were less pronounced. For organs with good SNR, like the kidneys, model‐based IVIM and IVIM‐DTI reconstruction could facilitate higher undersampling rates and a reduction in scanning time.

When using the complex k‐space data for model‐based reconstruction of diffusion‐weighted scans, the phase errors arising from the diffusion gradients need to be taken into account. In previous works, the phase has been estimated from low‐resolution data from a fully sampled k‐space center for each diffusion‐weighting. ²⁸ , ³⁹ , ⁴⁰ In this work, we used SENSE as a parallel imaging technique, which also undersamples the k‐space center. Therefore, we estimated the phase of each diffusion weighting relative to the first b = 0 s/mm² scan from the reconstructed image series. Acquiring a fully sampled k‐space center region, for example, by using generalized autocalibrating partial parallel acquisition (GRAPPA ⁴² ) instead of SENSE, might improve the phase estimation and thus the results obtained with model‐based reconstruction. Generally, a random undersampling pattern is favorable for compressed‐sensing iterative reconstructions. ⁴³ Because this is not widely available yet on clinical scanners for diffusion imaging, we decided to use parallel imaging in this work to keep the scan and echo time within an acceptable range. However, as seen in the IVIM‐DTI simulations, this might, in some cases, result in some residual aliasing and/or unfolding artifacts. The use of advanced sampling schemes in combination with model‐based reconstruction might facilitate acceleration of the IVIM and IVIM‐DTI acquisition in future research.

A disadvantage of model‐based reconstruction methods is the typically long computation timeassociated with the technique. ⁴⁴ In this work, we did not optimize the configuration parameters for speed. However, we found that the loss function already converged after 5–10 Gauss‐Newton iterations (Figure S2). We used 20 Gauss‐Newton iterations in this work to ensure convergence in all cases. Reducing the number of Gauss–Newton iterations and/or the number of maximum Primal‐Dual iterations could be a first step towards speeding up the reconstruction.

6. CONCLUSIONS

Model‐based reconstruction with TGV regularization and preconditioning is demonstrated for IVIM and combined IVIM‐DTI parameter estimations in regions with relatively high and low SNR and perfusion fractions. Furthermore, the precision of the parameter maps is improved compared with conventional fitting, especially for the perfusion fraction f and pseudo‐diffusion D*. We found that model‐based reconstruction does not introduce a regularization bias. Taken together, model‐based reconstruction yields high‐quality IVIM and combined IVIM‐DTI parameter maps, particularly for tissues exhibiting low SNR and low perfusion.

Supporting information

Text S1: Configuration file used for model‐based reconstruction with comments.

Figure S1: Bias and IQR of the IVIM simulations.

Figure S2: Cost function of the model‐based reconstruction.

APPENDIX A.

Preconditioned IRGN algorithm

Starting from the following optimization problem (also given in Equation (4):

$$u=\underset{u}{\mathit{arg}\mathit{min}}\frac{1}{2}{‖A\left(u\right)-d‖}_2^2+\gamma R_{\mathit{TGV}}\left(u\right).$$ (5)

In each Gauss–Newton step k, the forward operator A is linearized around u _k by Taylor expansion:

$$\mathrm{A}\left(\mathrm{u}\right)\approx \mathrm{A}\left({\mathrm{u}}_{\mathrm{k}}\right)+{\left.\frac{\mathrm{\partial A}\left(\mathrm{u}\right)}{\mathrm{\partial u}}\right|}_{\mathrm{u}={\mathrm{u}}_{\mathrm{k}}}\left(\mathrm{u}-{\mathrm{u}}_{\mathrm{k}}\right).$$ (6)

Plugging Equation (6) into Equation (5) yields the inner problem:

$$u_k=\underset{u}{\mathit{arg}\mathit{min}}\frac{1}{2}{‖A\left(u_k\right)+{\left.\frac{\partial A\left(u\right)}{\partial u}\right|}_{u=u_k}u-{\left.\frac{\partial A\left(u\right)}{\partial u}\right|}_{u=u_k}{u}_k-d‖}_2^2+\gamma R_{\mathit{TGV}}\left(u\right)=\underset{u}{\mathit{arg}\mathit{min}}\frac{1}{2}{‖\mathit{DA}u-\left[d-A\left(u_k\right)+\mathit{DA}{u}_k\right]‖}_2^2+\gamma R_{\mathit{TGV}}\left(u\right),$$ (7)

with DA the Jacobian of A.

Summarizing the constant terms [d – A (u _k ) +  DAu _k ] to d _k the optimization problem to solve in each Gauss–Newton step simplifies to:

$$\underset{u}{\mathit{min}}\frac{1}{2}{‖\mathit{DA}u-d_k‖}_2^2+\gamma R_{\mathit{TGV}}\left(u\right).$$ (8)

Preconditioning of the above equation can be achieved by finding a suitable preconditioner for the Jacobian DA . This is done by computing the SVD along the unknown dimension in each pixel, leading to:

$$\mathit{DA}u={\mathbf{V}}^{\mathbf{T}}\mathbf{\Sigma U}u={\mathbf{V}}^{\mathbf{T}}{\mathbf{\Sigma UU}}^{\mathbf{T}}{\mathbf{\Sigma }}^{\mathbf{-}\mathbf{1}}\mathbf{\Sigma U}u.$$ (9)

Depending on the eigenvalues of Σ (sorted in descending order) it is necessary to clip them based on the maximum Σ₀₀ in order to avoid stability problems due to exploding values at the inversion, that is:

$${\tilde{\Sigma }}_{\mathit{ii}}=\max \left\{{\Sigma }_{\mathit{ii}},{\mathit{\epsilon \Sigma }}_{00}\right\},$$ (10)

with ε = 0.01.

Inserting the truncated singular values $\tilde{\Sigma }$ into Equation (9) gives:

$$\mathit{DA}u=\left({\mathbf{V}}^{\mathrm{T}}\mathbf{\Sigma U}{\mathbf{U}}^{\mathbf{T}}{\tilde{\mathbf{\Sigma }}}^{-\mathbf{1}}\right)\left(\tilde{\mathbf{\Sigma }}\mathbf{U}\right)\mathrm{u}.$$ (11)

Defining $\widetilde{\mathbf{DA}}={\mathbf{V}}^{\mathrm{T}}\mathbf{\Sigma U}{\mathbf{U}}^{\mathbf{T}}{\tilde{\mathbf{\Sigma }}}^{\mathbf{-}\mathbf{1}}$ and y = $\tilde{\mathbf{\Sigma }}\mathbf{U}\mathrm{u}$ and plugging into Equation (8) yields the preconditioned inner problem:

$${\widehat{y}}_k=\underset{y}{\mathit{arg}\mathit{min}}\frac{1}{2}{‖\widetilde{\mathit{DA}}y-d_k‖}_2^2+\gamma R_{\mathit{TGV}}\left({\mathit{U}}^{\mathit{T}}{\tilde{\mathit{\Sigma }}}^{-1}y\right).$$ (12)

Comparing Equation (12) with Equation (5) one sees that preconditioning comes at the additional cost of a matrix–vector product prior to computing the regularization.

Rauh SS, Maier O, Gurney‐Champion OJ, et al. Model‐based reconstructions for intravoxel incoherent motion and diffusion tensor imaging parameter map estimations. NMR in Biomedicine. 2023;36(8):e4927. doi: 10.1002/nbm.4927