Noname manuscript No. (will be inserted by the editor) Functional Segmentation through Dynamic Mode Decomposition: Automatic Quantification of Kidney Function in DCE-MRI Images Santosh Tirunagari · Norman Poh · Kevin Wells · Miroslaw Bober · Isky Gorden · David Windridge Received: date / Accepted: date Abstract Quantification of kidney function in Dynamic We find that the segmentation result obtained from Contrast Enhanced Magnetic Resonance imaging (DCE- our proposed DMD framework is comparable to that MRI) requires careful segmentation of the renal region of expert observers and very significantly better than of interest (ROI). Traditionally, human experts are re- that of an a-priori bounding box segmentation. Our re- quired to manually delineate the kidney ROI across sult gives a mean Jaccard coefficient of 0.87, compared multiple images in the dynamic sequence. This approach to mean scores of 0.85, 0.88 and 0.87 produced from is costly, time-consuming and labour intensive, and there- three independent manual annotations. This represents fore acts to limit patient throughout and acts as one of the first use of DMD as a robust automatic data driven the factors limiting the wider adoption of DCR-MRI in segmentation approach without requiring any human clinical practice. intervention. This is a viable, efficient alternative ap- Therefore, to address this issue, we present the first proach to current manual methods of isolation of kidney use of Dynamic Mode Decomposition (DMD) as a the function in DCE-MRI. basis for automatic segmentation of a dynamic sequence, Keywords DMD · W-DMD · R-DMD · WR-DMD · in this case, kidney ROIs in DCE-MRI. Using DMD DCE-MRI · Movement correction coupled combined with thresholding and connected com- ponent analysis is first validated on synthetically gener- ated data with known ground-truth, and then applied 1 Introduction to ten healthy volunteers DCE-MRI datasets. Diagnosis of renal dysfunction based on blood and urine Santosh Tirunagari and Norman Poh have benefited from the tests often produces inaccurate results as the creatinine Medical Research Council (MRC) funded project \Modelling the Progression of Chronic Kidney Disease" under the grant levels in blood are detectable only after 60% of the renal number R/M023281/1. dysfunction has taken place [34]. Therefore, to address S. Tirunagari · N. Poh this limitation Dynamic Contrast Enhanced Magnetic Department of Computer Science, University of Surrey, Resonance Imaging (DCE-MRI) has been proposed [32, Guildford, Surrey GU2 7XH, UK. 27]. DCE-MRI is a non-ionising alternative to conven- arXiv:1905.10218v1 [eess.IV] 24 May 2019 Tel.: +44-(0)1483-6179. tional radioisotope renography. It has particular attrac- S. Tirunagari · K. Wells · M. Bober tion in cases of Chronic Kidney Disease (CKD), which Center for Vision, Speech and Signal Processing (CVSSP), requires repeated functional assessment, as well as in University of Surrey, Guildford, Surrey GU2 7XH, UK. paediatric cases where exposure to repeated radiation I. Gorden doses is of greater concern than in adults [11,17,18]. A University College London (UCL) Institute of Child Health, 30 Guildford Street, London WCIN lEH, UK. further benefit of DCE-MRI is that anatomical images can also be obtained during the same imaging session, D. Windridge Department of Computer Science, Middlesex University, The providing a direct comparisons to the observed physio- Burroughs, Hendon, London NW4 4BT, UK. logical abnormalities. Absolute quantification of kidney (renal) function E-mail: fs.tirunagari, n.poh, k.wells, [email protected] in DCE-MRI (see Figure 1) is often obfuscated as it re- 2 Tirunagari et al. quires manual segmentation [35,36] of the kidney ROI, Previously, automated segmentation methods util- (i.e., a region of kidney is selected as a template by hu- ising clustering and classification methods such as k- man experts by manually delineating the kidney ROI). means clustering [35] and k-nearest neighbour classifi- Semi-automatic methods, however, work by specifying cation [8] have been suggested. These methods work by the target ROI a-priori to an automated segmentation considering the signal intensity values across the im- algorithm [9]. Although these approaches are poten- ages in time, thus obtaining a high-dimensional feature tially correct, the major issue is the need for human vector in each voxel based on the actual tissue response intervention in the segmentation process of the target before and after injecting the contrast agent. Methods region. In addition, this can be labour intensive, time- based on active contours [1] and related methods have consuming and inefficient as the human expert has to also offered solutions that consider the region bound- examine the whole sequence of images to find the most ary coupled with shape constraints [2]. A recent work suitable frame. This is typically also inconvenient since by Hodneland et al. in 2014 applied the temporal tissue human experts require proprietary software for delin- response and minimal boundary length as shape infor- eating the ROI, and also error-prone as the selection mation for obtaining the kidney segmentation [7] in 4D of the template (ROI) is subjected to observer varia- DCE-MRI videos. tions [19]. Zollner et al [34] introduced the Independent Com- ponent Analysis (ICA) technique for functional segmen- Automated methods [10,21] have the potential to tation of human kidney ROI in DCE-MRI recordings. overcome these limitations and moreover offer a more An approach based on spatio-temporal ICA (STICA) [12] reproducible approach. In order to achieve a complete was also developed recently that offers a fully data- automatic approach reliably, we propose to use Dy- driven approach exploiting the distribution of the prop- namic Mode Decomposition (DMD), which has been erties of the spatial data incrementally in the direction used extensively in modelling fluid dynamics. Despite of the time axis. A major limitation for this ICA based the complexity of the dynamics of an image sequence approach is finding an optimal filter that maximises the containing anatomical structures, the information dy- statistical independence of the observed signals. The namics in our approach are represented in an extremely ICA method is typically also approached with a sub- efficient manner within individual \modes". These ex- stantial number of assumptions/heuristics and is com- periments show, for the first time, that DMD can cap- putationally expensive. ture distinctive features that clearly distinguish var- ious functional segments within a dynamic MRI im- In our previous studies [30], we presented a novel age sequence. In this paper, we thus report in detail a automated, registration-free movement correction ap- framework utilising DMD in conjunction with a simple proach based on windowed and reconstruction variants thresholding technique to obtain functional segmenta- of Dynamic Mode Decomposition (WR-DMD) to sup- tion of the kidney ROI. press unwanted complex organ motion in DCE-MRI im- age sequences caused due to respiration. Our methodological framework in [30] consisted of the following steps:1) DCE-MRI sequence consisting of N images was processed using the windowed DMD (W- Liver Spleen DMD) algorithm in order to output each N2 W-DMD components C1 and C2. At this stage the W-DMD(C1) produced the low rank images and W-DMD(C2) pro- duced sparse images. 2) W-DMD(C1) was then given as an input to DMD which produced N3 DMD modes. The first 3 DMD modes were then selected for recon- Right Left structing the motion stabilised image sequence. Out of Kidney Kidney the first three dynamic modes, the first mode revealed a low-rank model and the remaining N − 4 modes cap- tured the sparse representations. The contrast changes Fig. 1 DCE-MRI image with anatomical parts: kidney, liver were captured in the most significant modes, in particu- and spleen. Renal perfusion takes place inside the kidney re- lar, mode-2 was captured kidney region and mode-3 and gions after the injection the contrast agent. The kidney func- 4, captured spleen and the liver regions respectively. tion is then quantified by calculating the mean pixel intensity values inside the kidney region (in practice using a tracer- Noise and residuals including the motion components kinetic method [33]). For this purpose, a proper segmentation were captured in the remaining of the modes. There- of kidney region is required. fore, this study investigates whether we could perform Movement Correction in DCE-MRI via WR-DMD 3 segmentation of the kidney region of interest from dy- The images in the DCE-MRI data are collected over namic mode-2 for automatic quantification the kidney regularly spaced time intervals and hence each pair of function. consecutive images are correlated. It can be justified Dynamic mode decomposition, due to its ability to that a mapping A exists between them forming a span identify regions of dominant motion in an image se- of krylov subspace [13,22,20]: quence in a completely data-driven manner without re- lying on any prior assumptions about the patterns of X = [x¯ ;Ax¯ ;A2x¯ ;A3x¯ ; ··· ;AN−1x¯ ]: behaviour within the data, has gained significant appli- 1 1 1 1 1 cations in various fields [28,29,5,3,16,4,31]. Therefore, [x¯2; x¯3; ··· ; x¯N ] = A[x¯1; x¯2; ··· ; x¯N−1]: (2) T it is thus potentially be well-suited to analyse a wide P2 = AP1 + reN−1: variation of blood flow and filtration patterns seen in renography pathology [30]. Here, r is the vector of residuals that accounts for behaviours that cannot be described completely by A, The novelty of our proposal thus lies in utilising e = f0; 0;; 1g 2 RN−1, P = [x¯ ; x¯ ; ··· ; x¯ ] and DMD to carry out functional segmentation from medi- N−1 2 2 3 N P = [x¯ ; x¯ ; ··· ; x¯ ]. The system A is unknown and cal image sequences in a manner that is both extremely 1 1 2 N−1 it captures the overall dynamics within the dynamic im- efficient and completely data driven (and thus heuristic- age sequence in terms of the eigenvalues and eigenvec- free).