Synopsis

Keywords: Quantitative Imaging, Modelling

Joint fitting of ADC and R₂ improves the noise characteristics of whole-body R₂ maps when using an iterative weighted least squares (IWLS) estimation that accounts for the signal offset, ρ, between diffusion-weighted and multi-echo-time imaging sequences. This is demonstrated using simulation experiments using relevant values of R₂, ADC, and signal-to-noise ratio (SNR), where we find an improvement in R₂ precision using the joint-fitting approach. Initial validation was further performed in clinical imaging examples to demonstrate the improvement in R₂ map detail.

Introduction

The utility of the apparent diffusion coefficient (ADC) and R₂ has been examined in many clinical applications[1–6]. Although previous studies have investigated joint fitting of ADC and R₂ using identical EPI protocols in pulmonary disease[7], to our knowledge no studies have investigated whole-body estimation for evaluation of metastatic disease. Furthermore, previous attempts at joint fitting of ADC and R₂ do not account for differences in signal intensities between DWI and multi-echo-time sequences acquired as separate acquisitions.

In this study we (i) propose an updated signal model for joint ADC/R₂ fitting, (ii) compare iterative weighted least-squares (IWLS) versus ordinary least-squares (OLS) for joint ADC/R₂ fitting using simulation experiments, and (iii) demonstrate the utility of whole-body ADC/R₂ estimation in a patient with metastatic melanoma.

Methods

i) Updated signal model for joint ADC/R₂ fitting
Implementing joint ADC/R₂ fitting on a clinical MRI scanner may require acquisition of the sequences for the estimation of R₂ (multi-TE data) and DWI data separately. Variation in receiver gain between separate acquisitions will affect signal intensity, which impairs joint fitting. We model this offset by the variable factor ρ such that the models for DWI and multi-TE data become:
$$\begin{gathered}S_{\mathrm{DWI}}\left(b_i,\mathrm{TE}_{\mathrm{DWI}}\right)=\rho S_0\exp\left(-b_i * \mathrm{ADC}\right)\exp\left(-\mathrm{TE}_{\mathrm{DWI}}*\mathrm{R}_2\right)+\varepsilon_i \\S_{\mathrm{R}_2}\left(\mathrm{TE}_i\right)=S_0\exp\left(-\mathrm{TE}_i*\mathrm{R}_2\right)+\varepsilon_i\end{gathered}$$ (Equation 1).
$$$\varepsilon_i$$$ is assumed to be zero-mean Gaussian-distributed noise with standard deviation, σ.

ii) Comparing OLS and IWLS
Following linearisation of data using log-transformation of Equation 1, OLS fitting can estimate the values of ρ, S₀, ADC and R₂ for known values of $$$b_i$$$, $$$\mathrm{TE}_{\mathrm{DWI}}$$$ and $$$\mathrm{TE}_i$$$. A drawback is that it incorrectly assumes homoscedastic noise variance in the data after log-transformation, which is known to be false. IWLS optimisation can improve this for ADC fitting[8] and so we compared OLS and IWLS strategies.

To match our clinical protocol, DWI data were generated according to Equation 1 using b-values of 50, 600, and 900 s/mm² (repeated 12 times) with $$$\mathrm{TE}_{\mathrm{DWI}}$$$=73 ms; multi-echo-time data were generated with TE=39/118 ms (repeated twice); ADC^true=1.5x10^-3mm²/s and R₂^true=0.02 s^-1. S₀=1 and Rician-distributed noise was added to the generated data with variance=1/SNR² where SNR={5,10,20,50,100,200,300} and $$$\rho\in(0,0.25, \ldots,2)$$$ for each value of SNR. ADC and R₂ were estimated using IWLS and OLS using three approaches: (a) separate fitting of DWI and multi-TE data, (b) joint-fitting, including estimation of ρ, and (c) joint fitting assuming ρ=1. The bias in the estimate of parameter X was defined as:
$$\operatorname{bias}(\widehat{\mathrm{X}})=\frac{1}{N}\sum_{j=1}^N\frac{\widehat{\mathrm{X}}_j-\mathrm{X}^{\text{true}}}{\mathrm{X}^{\text{true}}}$$
and the variance was:
$$\operatorname{var}(\widehat{\mathrm{X}})=\frac{1}{N}\sum_{j=1}^N\frac{\left(\widehat{\mathrm{X}}_j-\mathrm{X}^{\text{true}}\right)^2}{\mathrm{X}^{\text{true}}}$$
for N=1000 simulation samples, where $$$\widehat{\mathrm{X}}$$$ is the estimated parameter.

iii) Whole-body ADC/R₂ estimation
Data were acquired in a single patient with metastatic melanoma according to the protocol defined in the simulation study, with DWI acquired using EPI with bipolar diffusion-encoding gradients and using the same sequence with b=0 and monopolar diffusion-encoding at matched slice locations and resolution. Joint fitting with the IWLS algorithm was used to estimate a map of ρ_est which was smoothed to ρ_G using a gaussian filter (Figure 1). Maps of the ADC and R₂ were recalculated using IWLS following rescaling of the DWI data using ρ_G.

Results

Figure 2 shows the changes in bias and variance of $$$\widehat{\mathrm{ADC}}$$$ and $$$\widehat{\mathrm{R_2}}$$$. For SNR>50, var($$$\widehat{\mathrm{ADC}}$$$) and var($$$\widehat{\mathrm{R_2}}$$$) are both reduced using IWLS (values for high SNR simulations shown in Figure 3). However, when ρ is assumed to be 1, bias is observed in estimation of these parameters.

Figures 4 shows maps of ADC and R₂, estimated separately or jointly using IWLS and smoothed ρ_G (Figure 5). In a single square ROI of 16 pixels drawn in healthy liver the variance of the pixel values was 2.6x10^-5s^-1 in the separately fitted R₂ map and 9.0x10^-6s^-1 in the map estimated using IWLS and ρ_G.

Discussion

The separate fittings to the equations in Equation 1 show the same changes in the bias and variance for $$$\widehat{\mathrm{ADC}}$$$ and $$$\widehat{\mathrm{R_2}}$$$ as the joint fitting estimating ρ_est. When ρ_est=1, for ρ<1 bias($$$\widehat{\mathrm{ADC}}$$$) is reduced when using the IWLS method, however bias($$$\widehat{\mathrm{R_2}}$$$) is larger than with OLS for the same values. When ρ_est=ρ=1 and SNR>20 bias($$$\widehat{\mathrm{ADC}}$$$) and bias($$$\widehat{\mathrm{R_2}}$$$) are the same with both IWLS and OLS. Var($$$\widehat{\mathrm{ADC}}$$$) and var($$$\widehat{\mathrm{R_2}}$$$) are both reduced with IWLS for SNR>50 because the fitting has fewer degrees of freedom: the value of ρ is known.

In the clinical case, values of ρ_est are shown to vary spatially across the image. Areas of flow, motion, small structures, or other mechanisms leading to localised signal loss result in overestimations of the ADC or R₂ and extreme values of ρ_est. Smoothing to ρ_G magnifies this effect resulting in large differences in the ADC or R₂ as seen in the spinal CSF and the aorta respectively. Further work is necessary to investigate the impact of smoothing ρ_est on the ADC and R₂ maps.

While a significant change in the noise characteristics is not observed in the ADC map corrected with ρ_G, the corrected R₂ map displays better signal homogeneity in central regions of the liver, spleen, and muscle.

Conclusion

Joint fitting of DWI and multi-TE data improves noise characteristics of the R₂ map when using IWLS with a smoothed estimate of the signal gain factor, ρ_G. Further validation is necessary using quantitative measurements of the SNR and estimation of ρ using phantom and clinical experiments.

Acknowledgements

This work represents independent research funded by the National Institute for Health and Care Research (NIHR) Biomedical Research Centre and the Clinical Research Facility in Imaging at The Royal Marsden NHS Foundation Trust and the Institute of Cancer Research, London. The views expressed are those of the authors and not necessarily those of the NIHR or the Department of Health and Social Care.