3399

Comparing L1 and L2 Regularizations for Quantitative Transport Mapping of Tumor: an Image Quality Analysis
Dominick Romano1,2, Qihao Zhang1,2, Ilhami Kovanlikaya2, Pascal Spincemaille2, and Yi Wang2,3
1Biomedical Engineering, Cornell University, New York, NY, United States, 2Radiology, Weill Cornell Medical College, New York, NY, United States, 3Biomedical Engineering, Cornell University, Ithaca, NY, United States

Synopsis

Keywords: Quantitative Imaging, Perfusion, Regularization

This study compared L1 and L2 regularized Quantitative Transport Mapping (QTM1-3) of dynamic contrast enhanced (DCE) MRI in breast and neck tumor using image quality scoring. Improved consistent soft tissue and lesion characterization was observed when using the L1 norm.

Introduction

Advanced stage tumors are known to alter the surrounding vasculature. This causes increased uptake and blood flow, which can be visualized with dynamic contrast enhanced MRI imaging (DCE-MRI), which encodes physiological transport parameters. DCE-MRI can be modeled using the convection-diffusion equation, in a method called Quantitative Transport Mapping (QTM1-3). QTM is ill-posed and requires regularization to achieve consistent results. In this study, we compare $$$L_{1}$$$ and $$$L_{2}$$$ regularized QTM on breast and neck tumor data. The images were evaluated using a scoring system by an experienced radiologist.

Theory

The DCE-MRI signal can be modeled by the simplified convection-diffusion equation3: $$\partial_{t}c(\textbf{r},t)=-\nabla \cdot (c(\textbf{r},t)\textbf{u}(\textbf{r}))$$ The $$$L_{1}$$$ regularized QTM inverse problem is defined as follows: $$\textbf{u}^{*}=\underset{u}{\operatorname{argmin}}\sum_{k=1}^{N-1}\sum_{k=1}^{N-1}\lVert \partial_{t}c(\textbf{u},t) +\nabla \cdot (c(\textbf{r},t)\textbf{u}(\textbf{r}))\rVert_{2}^{2}+\lambda \lVert \nabla \textbf{u}\rVert_{1}$$ And $$$L_{2}$$$ QTM is defined as follows: $$\textbf{u}^{*}=\underset{u}{\operatorname{argmin}}\sum_{k=1}^{N-1}\sum_{k=1}^{N-1}\lVert \partial_{t}c(\textbf{u},t) +\nabla \cdot (c(\textbf{r},t)\textbf{u}(\textbf{r}))\rVert_{2}^{2}+\lambda \lVert \nabla \textbf{u}\rVert_{2}^{2}$$ Once $$$\textbf{u}$$$ is obtained from the QTM reconstruction, we then report the speed map defined by $$u^{*}=\sqrt{\textbf{u}\cdot \textbf{u}}$$ Regularization weights were selected from the L-curve method.

Methods

For the neck tumor dataset, 88 subjects diagnosed with Nasopharyngeal carcinoma (NPC) with a DCE-MRI acquisition were selected for QTM processing. The scanning parameters at 3T (Siemens, Erlangen, Germany): FLASH/vibe sequence; time points = 50; dt = 4.9s; TR=4.9ms; TE=1.47ms; flip angle = 9 degrees; image orientation: axial/transversal; Phase FOV = 75%; bandwidth = 400 Hz; dz = 4mm; slice spacing = 0mm; in-plane FOV = 180x180 mm2; in-plane matrix = 192x144; Acquisition Time = 245s
The breast tumor cohort consisted of 26 subjects. The subjects were scanned at 3T with an 8-channel breast coil with acquisition parameters: time points = 5; dt = 15.4s; TR=3.95 ms; TE=1.7 ms; flip angle = 10; image orientation: axial/transversal; in-plane resolution = 0.71mm; dz = 1.8mm.
The resulting speed maps from $$$L_{1}$$$ and $$$L_{2}$$$ are compared with DCE-MRI as a reference. An experienced radiologist (I.K.) scored the images on the following scale: $$$L_{1}$$$ preferred, $$$L_{2}$$$ preferred, and equal preference. The rationale for each case preference was based on the following: 1) Lesion intensity should not be exaggerated, 2) Soft tissue contrast should be considered, 3) QTM reconstructed lesion should not appear artifactual, 4) QTM reconstructed structures should not be exaggerated.

Results

In NPC cases, $$$L_{1}$$$ regularized maps were preferred in 81.8% of the cases, with $$$L_{2}$$$ regularized maps were preferred in 10.2$ of the cases, and 8.0% were equally preferred (Table 1). This may be explained by the fact that $$$L_{1}$$$ tissue maps present consistent characterization of the normal appearing tissue, while the $$$L_{2}$$$ speed maps provide artifactual normal appearing tissue (Figure 1). Furthermore, $$$L_{1}$$$ speed maps are consistently preserved at and within the lesion boundary when compared with $$$L_{2}$$$ maps.
In the breast tumor dataset, $$$L_{1}$$$ was preferred in 73.1% of the cases, $$$L_{2}$$$ was preferred in 3.8% of the cases, and the remaining 21% of cases were preferred equally (Table 2). It was found that the normal appearing tissue and lesion characterization was exaggerated in the $$$L_{2}$$$ computed speed maps (Figure 2).

Discussion

Previous studies have found $$$L_{1}$$$ regularization of perfusion models perform best in numerical simulations and acquired CT data, especially in lower SNR cases4-7. Furthermore, these works found that $$$L_{1}$$$ regularization localized perfusion quantities to tissue types, in which our work falls into agreement. In MRI, many studies report piecewise constant CBF between white and grey matter8-10, which supports the choice for the $$$L_{1}$$$ norm. Notably, our work fills a gap of comparing regularization methods for convection-diffusion based modeling of time-resolved perfusion imaging signals. For the QTM inverse problem, we find improved consistent soft tissue and lesion characterization when using the L1 norm.

Acknowledgements

No acknowledgement found.

References

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Figures

Table 1: Radiologist L1 vs L2 preference of neck tumor speed maps

Table 2: Radiologist L1 vs L2 preference of breast tumor speed maps

An example of anatomical and quantitative speed maps for neck tumors. (A) DCE-MRI pre-contrast. (B) DCE-MRI post-contrast. (C) QTM L1 regularized speed map $$$|\textbf{u}|$$$. (D) QTM L2 regularized speed map $$$|\textbf{u}|$$$.

Anatomical and quantitative speed maps for breast tumors. (A) DCE-MRI pre-contrast. (B) DCE-MRI post-contrast. (C) QTM L1 regularized speed map $$$|\textbf{u}|$$$ (D) QTM L2 regularized speed map $$$|\textbf{u}|$$$

Proc. Intl. Soc. Mag. Reson. Med. 31 (2023)
3399
DOI: https://doi.org/10.58530/2023/3399