1 Introduction
Diagnosis of renal dysfunction based on blood and urine tests often produces inaccurate results as the creatinine levels in blood are detectable only after 60% of the renal dysfunction has taken place zollner2007assessment . Therefore, to address this limitation Dynamic Contrast Enhanced Magnetic Resonance Imaging (DCEMRI) has been proposed tofts2010t1 ; sourbron2013classic . DCEMRI is a nonionising alternative to conventional radioisotope renography. It has particular attraction in cases of Chronic Kidney Disease (CKD), which requires repeated functional assessment, as well as in paediatric cases where exposure to repeated radiation doses is of greater concern than in adults khrichenko2010functional ; michaely2007functional ; prasad2006functional . A further benefit of DCEMRI is that anatomical images can also be obtained during the same imaging session, providing a direct comparisons to the observed physiological abnormalities.
Absolute quantification of kidney (renal) function in DCEMRI (see Figure 1) is often obfuscated as it requires manual segmentation zollner2009assessment ; zollner2012assessment of the kidney ROI, (i.e., a region of kidney is selected as a template by human experts by manually delineating the kidney ROI). Semiautomatic methods, however, work by specifying the target ROI apriori to an automated segmentation algorithm huang2004functional . Although these approaches are potentially correct, the major issue is the need for human intervention in the segmentation process of the target region. In addition, this can be labour intensive, timeconsuming and inefficient as the human expert has to examine the whole sequence of images to find the most suitable frame. This is typically also inconvenient since human experts require proprietary software for delineating the ROI, and also errorprone as the selection of the template (ROI) is subjected to observer variations de2000mr .
Automated methods johnson2011determinants ; rusinek2007performance have the potential to overcome these limitations and moreover offer a more reproducible approach. In order to achieve a complete automatic approach reliably, we propose to use Dynamic Mode Decomposition (DMD), which has been used extensively in modelling fluid dynamics. Despite the complexity of the dynamics of an image sequence containing anatomical structures, the information dynamics in our approach are represented in an extremely efficient manner within individual “modes”. These experiments show, for the first time, that DMD can capture distinctive features that clearly distinguish various functional segments within a dynamic MRI image sequence. In this paper, we thus report in detail a framework utilising DMD in conjunction with a simple thresholding technique to obtain functional segmentation of the kidney ROI.
Previously, automated segmentation methods utilising clustering and classification methods such as kmeans clustering
zollner2009assessment and knearest neighbour classification hodneland2011vivohave been suggested. These methods work by considering the signal intensity values across the images in time, thus obtaining a highdimensional feature vector in each voxel based on the actual tissue response before and after injecting the contrast agent. Methods based on active contours
abdelmunim2008kidney and related methods have also offered solutions that consider the region boundary coupled with shape constraints ali2007graph . A recent work by Hodneland et al.in 2014 applied the temporal tissue response and minimal boundary length as shape information for obtaining the kidney segmentation hodneland2014segmentation in 4D DCEMRI videos.Zollner et al zollner2007assessment
introduced the Independent Component Analysis (ICA) technique for functional segmentation of human kidney ROI in DCEMRI recordings. An approach based on spatiotemporal ICA (STICA)
kiani2012linewas also developed recently that offers a fully datadriven approach exploiting the distribution of the properties of the spatial data incrementally in the direction of the time axis. A major limitation for this ICA based approach is finding an optimal filter that maximises the statistical independence of the observed signals. The ICA method is typically also approached with a substantial number of assumptions/heuristics and is computationally expensive.
In our previous studies Tirunagari2017 , we presented a novel automated, registrationfree movement correction approach based on windowed and reconstruction variants of Dynamic Mode Decomposition (WRDMD) to suppress unwanted complex organ motion in DCEMRI image sequences caused due to respiration.
Our methodological framework in Tirunagari2017 consisted of the following steps:1) DCEMRI sequence consisting of images was processed using the windowed DMD (WDMD) algorithm in order to output each WDMD components C1 and C2. At this stage the WDMD(C1) produced the low rank images and WDMD(C2) produced sparse images. 2) WDMD(C1) was then given as an input to DMD which produced DMD modes. The first DMD modes were then selected for reconstructing the motion stabilised image sequence. Out of the first three dynamic modes, the first mode revealed a lowrank model and the remaining modes captured the sparse representations. The contrast changes were captured in the most significant modes, in particular, mode2 was captured kidney region and mode3 and 4, captured spleen and the liver regions respectively. Noise and residuals including the motion components were captured in the remaining of the modes. Therefore, this study investigates whether we could perform segmentation of the kidney region of interest from dynamic mode2 for automatic quantification the kidney function.
Dynamic mode decomposition, due to its ability to identify regions of dominant motion in an image sequence in a completely datadriven manner without relying on any prior assumptions about the patterns of behaviour within the data, has gained significant applications in various fields tirunagari2015windowed ; tirunagari2016can ; grosek2014dynamic ; 6926317 ; 2015arXiv150804487M ; brunton2016extracting ; tirunagari2015detection . Therefore, it is thus potentially be wellsuited to analyse a wide variation of blood flow and filtration patterns seen in renography pathology Tirunagari2017 .
The novelty of our proposal thus lies in utilising DMD to carry out functional segmentation from medical image sequences in a manner that is both extremely efficient and completely data driven (and thus heuristicfree). Extracting relevant key organs from the DCEMRI images can potentially provide clinicians with a better way to manage clinical time and cost.
The remainder of this paper is organised as follows: Section 2 considers the theory for DMD as well as presenting evaluation criteria based on Jaccard Similarity Coefficient. Section 3 describes datasets that are based on synthetic data and a set of 10 healthy volunteers’ MRI datasets. Section 4 derives our experimental objectives and presents the results and, finally, conclusions are drawn and are discussed in Section 5.
2 Methodology
In this section, we present our methodological framework which consecutively consists of DMD, thresholding, selection of kidney regions using connected component analysis and finally modelling of the kidney function. The overall process pipeline is shown in Figure 2. At the end of this section, we also present our evaluation criteria i.e., Jaccard similarity coefficient for comparing segmentation results.
2.1 Dynamic Mode Decomposition (DMD)
Let be the dynamic image frame in a DCEMRI sequence, whose size is . This image frame is converted to column vector, resulting in the construction of a data matrix X of size for image frames.
(1) 
The images in the DCEMRI data are collected over regularly spaced time intervals and hence each pair of consecutive images are correlated. It can be justified that a mapping exists between them forming a span of krylov subspace krylov1931numerical ; saad1981krylov ; ruhe1984rational :
(2) 
Here, is the vector of residuals that accounts for behaviours that cannot be described completely by , , and . The system
is unknown and it captures the overall dynamics within the dynamic image sequence in terms of the eigenvalues and eigenvectors of
, which are referred to as the DMD eigenvalues and DMD modes respectively.The sizes of the matrices and are both each. Therefore, the size of unknown matrix would be . Since, matrix captures the dynamical information within the image sequence, solving it provides us with the dynamics captured in the image sequence in terms of modes. Unfortunately, solving for A is computationally very expensive due to it size. For instance, if an image has a size of i.e., and , the size of is then .
From the literature there are two approaches for obtaining these eigenvalues and modes. The first is Arnoldi based approach, which is useful for theoretical analysis due to its connection with Krylov subspace methods krylov1931numerical ; saad1981krylov ; ruhe1984rational
. The second is a singular value decomposition (SVD) based approach that is more robust to noise in the data and to numerical errors
Schmid3 .Therefore, SVD can be directly applied on in Eq. (2), to obtain , and matrices.
(3) 
,
here, is the singular value of , and are the corresponding left and right singular vectors respectively; and is the total number of singular values.
Rearranging Eq. (3), we obtain the the fullrank matrix ,
(4) 
Since the eigenvalue analysis is agnostic to any linear projection; so solving the eigen problem of is easier than that of solving for directly. Moreover, the associated eigenvectors of provide the coefficients for the linear combination that is necessary to express the dynamics within the time series basis.
(5) 
where, are the eigenvectors and a diagonal matrix containing the corresponding eigenvalues of matrix. The eigenvalues of approximate some of the eigenvalues of the full system DBLP:journals/corr/GrosekK14 , we then have:
(6)  
Therefore is determined on the subspace spanned by the orthogonal singular basis vectors obtained via ,
(7)  
which can be rewritten as:
(8) 
Here and are the conjugate transpose of and , respectively; and denotes the inverse of the singular values . By replacing in Eq. (6) i.e., , we obtain the dynamic modes . , therefore, we have:
(9) 
The complex eigenvalues contain growth/decay rates and frequencies of the corresponding DMD modes Schmid2 ; Schmid1 . If are the diagonal elements of from Eq. (5), the temporal behaviour of the DMD modes is then formed via Vandermonde matrix , which raises its column vector to the appropriate power. with elements will be defined by the following:
(10) 
is a standard Vandermonde matrix for reconstruction but if , this is used for forecasting. DMD modes with frequencies is defined by:
(11) 
where is the lag between the images. The real part of regulates the growth or decay of the DMD modes, while the imaginary part of drives oscillations in the DMD modes.
2.2 Ordering Dynamic modes
In order to select the most significant dynamic modes, the method suggested in kutz2016multiresolution ; grosek2014dynamic is to calculate the logarithmic values of the . The frequencies which are near origin are the most significant modes. The other way we propose here is by calculating the phaseangles for the complex eigenvalues.
The absolute value for the phaseangles are calculated and modes with unique phaseangles are selected. Doing this will remove one of the conjugate pairs in the dynamic modes. These conjugate modes have same phaseangles but with different signs and look and capture similar information sayadi2013dynamic . After discarding one of the conjugate pairs, the dynamic modes are then sorted in ascending order of their phaseangles. The resultant dynamic modes are thus sorted according to their significance. In this study we have considered the first three significant dynamic modes when reconstructing the original sequence.
The effectiveness of DMD, as a preliminary analysis, is demonstrated in Figure 3. The top five figures show random DCEMRI images in temporal order for a healthy volunteer. Since renal perfusion is taking place inside the kidneys region, one can observe that the corresponding DMD mode in (b) is highlighting the dynamic changes inside the kidney region. A simple thresholding and binarization of the DMD mode thus highlights the kidney region as shown in (c). Selecting the region, or the largest area of connected pixels, can thus give us the kidney template in an exemplar 4D dynamic medical imaging application.
2.3 Performance Measure
To evaluate the performance of our segmentation we use Jaccard similarity coefficient, a standard measure of similarity between finite sample sets levandowsky1971distance (here, in particular, the similarity between the two segmented sets). It is defined as the size of the intersection (of pixels values) between the segmented sets divided by the size of the union of the segmented sets:
(12) 
Here A and B are two segmented images. Values of the Jaccard index range from 0 to 1. (0 if the intersection of the two sets is empty; 1 if the two sets are equal; the more similar the sets are, the closer to 1 is the metric).
2.4 Evaluation Criteria
Let and be the segmented results from the three human experts and DMD framework respectively. For an unbiased evaluation criterion, let us calculate three different groundtruths based on the segmentation results obtained from the human experts.
(13) 
The groundtruth for the human expert1 is the mutual agreement of segmentation results from expert2 and expert3. Similarly the groundtruth for expert2 can be calculated as the mutual agreement of expert1 and expert3 and vice versa. The evaluation criteria for the human experts can be calculated as follows:
(14) 
The evaluation for the DMD framework is given by the average of :
(15) 
where,
(16) 
3 Dataset
In this section we briefly describe the datasets that we have used in this study i.e., DCEMRI data and synthetically generated data.
3.1 DCEMRI data
(1)  (2)  (3)  (4)  (5) 
(6)  (7)  (8)  (9)  (10) 
The functional kidney DCEMRI datasets used in this study have been provided by collaborators at the Great Ormond Street Hospital using 1.5T Siemens Avanto scanner with 32channel body phasedarray coil. The datasets obtained are from ten healthy volunteers as shown in Figure 4. The acquired DCEMRI datasets cover the abdominal region, enclosing left and right kidneys and abdominal aorta. The anatomical images can be used to produce organ templates. The dataset consists of 120 MR images taken in sequence for seconds showing the central kidney slice i.e., the largest portion of kidney region.
3.2 Synthetic data
In order to validate our experiments, and to demonstrate the capacity of our framework for functional separation, we artificially generate synthetic data corresponding to broad DCEMRI events in terms of simple mathematical functions.
A series of 100 images are produced representing the kidney, liver and background functions as labelled in Figure 6 with temporal evolution shown in Figure 5(a). Kidney function is hence modelled as a linear combination of Poisson and log distributions; liver function is modelled as a sigmoid distribution and the background function is derived from a Gaussian noise distribution.
4 Experiments
In this section we outline the experimental procedure as well as the corresponding results.
4.1 Evaluating the DMD framework on synthetically generated data.
We firstly investigate the performance of the proposed framework on the syntheticallygenerated data embodying a coarsegrained simulation of DCEMRI composite dynamics. This evaluation will provide the proposed DMD framework with a ground truth and establish the reproducibility and repeatability of our experimental results. Since DMD is completely datadriven, we conjecture that DMD will capture the kidney, liver as well as the background functions within its modes.
Following the outlined methodology, synthetic data consisting of a image sequence is given as an input to the modified DMD algorithm producing DMD modes (In theory, for a image sequence with images, we obtain DMD modes). Recall that, in principle, each dynamic mode captures one of the Key dynamic axes of the image sequence. Modes that show predominating liver, kidney and background functions are manually selected, as shown in Figure 7 (top). These modes are then thresholded to obtain the segmented versions of respective functions as shown in Figure 7 (middle). As the DMD is completely datadriven, aside from mode selection (in this case), we have thus demonstrated our conjecture that DMD can capture intact the distinct kidney, liver as well as the background functions from the synthetic data, irrespective of their intrinsically varied morphological kinds. More generally, we have demonstrated the intrinsic capacity of DMD for functional separation as distinct from ICA’s stochastic component separation. We further characterise this separability by projecting the segmented versions of Dynamic modes 1, 2 and 7 for quantifying kidney, liver and background functions as shown in Figure 7(bottom). The Jaccard Similarity Coefficient achieved for synthetic dataset is and Mean Square Error between the DMD result and the ground truth is .
DMD Mode1  DMD Mode2  DMD Mode7 
Thresholded DMD Mode1  Thresholded DMD Mode2  Thresholded DMD Mode7 
(a)  (b)  (c) 
4.2 Implementation of the DMD framework on DCEMRI data
Due to the injection of contrast agent, large scale intensity fluctuations can be observed inside the kidney region which also has minor impact on the liver region. We therefore wish to examine whether these features are indeed picked up by DMD.
DCEMRI image sequences from the healthy volunteers’ data are given as input to DMD algorithm. The output of DMD is thus modally separated large and small scale fluctuations in voxel intensity. These modes are obtained by suppressing the stable information present within the DCEMRI image sequences (with the exception of mode 1); DMD is thus able to suppress the background information from the image sequence insofar as this is stationary. Renal perfusion (due to the injection of contrast agent) inside the kidney region exhibits large scale voxel intensity fluctuations, followed, in order of magnitude, by liver and spleen region fluctuations respectively. DMD thus captures key kidney, liver and spleen regions in distinct modes as shown in Figure 8. Dynamic mode1 captures the lowrank image of the DCEMRI dataset1 as shown in Figure 8(a) revealing the background function of the dataset. The kidney function is empirically observed to be captured in dynamic mode2 (b) and the spleen and liver functions captured in dynamic mode4 (c) and 5 (d) respectively. The lower order modes reveal the noise components inside the dataset1 as seen in Figure 8(e). These results hence further provide visual evidence for our conjecture that DMD is capable of isolating key functional regions such as kidney and liver. These rankings however are not given intrinsically by DMD (as discussed previously); we now look at how this can be accomplished.
(a)  (b)  (c)  (d)  (e) 
The main aim in this study is to segment the kidney region so as to enable quantification of the kidney function. We thus select the second dynamic mode for this functional characterisation as this configuration is observed to produce kidney region segmentation, consistent with the above argument, across all of the datasets as shown in Figure 9 (top). These dynamic mode2s are first normalised in the range of . Binarized images of dynamic mode2s are then obtained by replacing all pixels in the image with intensity values greater than an adaptive threshold with the value 1 (white), and replacing all other pixels with the value 0 (black). These result in the binarized versions of dynamic mode 2 shown in Figure 9 (middle). Kidney templates are then automatically selected as the largest area of connected components of binarized dynamic mode2 images across all the datasets as shown in Figure 9 (bottom). The connected components are selected by scanning the binarized image from toptobottom. All the pixels in a connected component are given a greedy label. Thus, all the pixels in the first connected component are labelled as and those in the second as and so on. These components are then ordered by decreasing area (i.e. the sum of all the pixels present in that particular component) and the label with the largest area is then selected as the kidney template. The produced kidney templates are then projected onto DCEMRI images to obtain the kidney function (as shown in Figure 2 from Section 2) by calculating the mean intensity pixel values inside region specified by the template. The kidney functions across all the datasets obtained through the DMD framework are shown in Figure 10.
Dynamic mode2 across 10 datasets of DCEMRI data 
Binarization across 10 datasets of DCEMRI data 
Kidney template across 10 datasets of DCEMRI data 
4.3 Evaluation of the DMD framework on DCEMRI.
Evaluation of our framework is twofold:

Evaluation of the performance of the segmented kidney template.

Evaluation of the performance of the quantified kidney function.
After obtaining the kidney ROIs (i.e the templates), the Jaccard Similarity Measure is used to evaluate the performance of the proposed DMD framework with respect to the groundtruth data. The aim of this experiment is to benchmark the generalisation performance of our framework against expert annotations.
For an unbiased evaluation, three different groundtruths are calculated based on the three different segmentation results obtained from the human experts. The groundtruth for the human expert1 is calculated as the mutual agreement of the segmentation between expert2 and expert3. Similarly the groundtruth for expert2 is calculated as the mutual agreement of segmentation between expert1 and expert3 and vice versa. We also have considered an additional baseline given by the minimum bounding box region around the human annotated regions (to simulate ’blind’ a priori segmentation). The DMD framework is evaluated as the mean of JSC values from all three groundtruths. Table 1 shows the JSC values for the DMD framework against the three groundtruths . The last column of the table presents the mean of the JSC values obtained from the three groundtruths across all the 10 datasets.
G_1  G_2  G_3  avg(DMD)  
Dataset1  0.8789  0.8672  0.8727  0.8729 
Dataset2  0.8291  0.8525  0.8758  0.8525 
Dataset3  0.8426  0.8628  0.8411  0.8488 
Dataset4  0.9038  0.9133  0.9279  0.9150 
Dataset5  0.8583  0.9027  0.8854  0.8821 
Dataset6  0.8651  0.8418  0.8386  0.8485 
Dataset7  0.8578  0.8239  0.8065  0.8294 
Dataset8  0.8927  0.9075  0.8861  0.8954 
Dataset9  0.8949  0.8702  0.8828  0.8826 
Dataset10  0.8356  0.8169  0.8799  0.8441 
The performance of the kidney segmentation achieved by the DMD framework, the blind boundingbox region and three domain experts with respect to their groundtruths are reported in Table 2. The JSC values achieved by expert2 are generally high in comparison to the other experts. The DMD framework has achieved higher JSC values for the datasets 4 and 8 and 9. The Overall average JSC values for the experts are and for the DMD framework is around while for minimum boundingbox region the JSC is below
DMD  E1  E2  E3  Bbox  
D1  0.8729  0.8642  0.8994  0.8974  0.5891 
D2  0.8525  0.8718  0.8925  0.8906  0.4563 
D3  0.8488  0.8642  0.8781  0.8719  0.4298 
D4  0.9150  0.8880  0.8894  0.8894  0.4073 
D5  0.8821  0.8668  0.9064  0.8934  0.3162 
D6  0.8485  0.8589  0.8679  0.8536  0.3016 
D7  0.8294  0.8548  0.8833  0.8529  0.2692 
D8  0.8954  0.8155  0.8726  0.8708  0.2984 
D9  0.8826  0.8221  0.8711  0.8391  0.2624 
D10  0.8441  0.8426  0.8292  0.8792  0.2455 
avg  0.8671  0.8548  0.8789  0.8738  0.3575 
Figure 10 shows the kidney functions produced by the DMD framework, groundtruth ( ), three domain experts as well as kidney function produced by blind boundingbox over the kidney region. The results in these figures show that kidney function quantified by DMD is closely aligned with the experts annotation.
(1)  (2)  (3)  (4)  (5) 
(6)  (7)  (8)  (9)  (10) 
Evaluation for the kidney functions are calculated based on the Mean Square Error (MSE) criteria as shown in Table 3.
avg (DMD)  E1  E2  E3  avg (Bbox)  
D1  0.0001  0.0001  0.0001  0.0000  0.0029 
D2  0.0001  0.0001  0.0001  0.0000  0.0126 
D3  0.0002  0.0001  0.0001  0.0000  0.0082 
D4  0.0000  0.0000  0.0001  0.0001  0.0085 
D5  0.0001  0.0000  0.0000  0.0000  0.0166 
D6  0.0000  0.0000  0.0000  0.0000  0.0188 
D7  0.0001  0.0001  0.0001  0.0000  0.0162 
D8  0.0001  0.0002  0.0000  0.0000  0.0431 
D9  0.0000  0.0002  0.0001  0.0000  0.0176 
D10  0.0000  0.0000  0.0002  0.0001  0.0213 
drvdk144@gmail.com
5 Discussions & Conclusions
This study aims to demonstrate the significance of the proposed DMD framework as a viable functional segmentation algorithm when coupled with simple thresholdingbinarization and selection of the largest area of connected pixels to effectively quantify kidney function in DCEMRI data. We applied the DMD framework to10 sets of DCEMRI data collected from healthy volunteers. We also applied the DMD framework to synthetically generated data mimicking the DCEMRI data. The results demonstrate that the proposed framework is extremely promising in obtaining functional segmentation in general and thus to quantifying segmented functions, in particular kidney functionality.
DMD can be demonstrated to extract local variations as a low rank representation within an image sequence, as well as capturing dominating regions of causallyconnected intensity fluctuations. In our context, perfusion inside the kidney region is the most dominating region of intensity fluctuations due to the injection of contrast agent. DMD was thus able to naturally capture the kidney region as mode2. Using a simple thresholding technique and selecting the largest area with connected pixels then automatically generates the proposed kidney region. The results for the proposed DMD framework when compared with expert annotations and groundtruth clearly shows the strength of the framework suggesting that manual selection of kidney region is no longer needed and that the entire process can be automated.
Acknowledgements.
The funding for this work has been provided by the Department of Computer Science and the Centre for Vision, Speech and Signal Processing (CVSSP)  University of Surrey. ‘I.G’ would like to express gratitude towards Kidney Research UK for funding the DCEMRI data acquisition as part of a reproducibility study. ‘S.T’ and ‘N.P’ have benefited from the Medical Research Council (MRC) funded project “Modelling the Progression of Chronic Kidney Disease” under the grant number R/M023281/1. The details of the project are available at www.modellingCKD.org. ‘D.W’ acknowledges the financial support from the Horizon 2020 European Research project “DREAMS4CARS” (#731593).References

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