Synopsis

INTRODUCTION

Pattern recognition techniques based on blind source separation (BSS) can be used as a model-free approach to identify perfusion patterns in cancer tumours and form maps that reflect tumour heterogeneity^1,2. BSS can decompose a set of mixed signals into separate source signals without a priori information on the source signals or the mixing process. Non-negative matrix factorization (NMF)³ is a well-known BSS approach that uses dimensionality reduction with visualization capabilities. NMF in DCE-MRI is plagued by non-uniqueness of the matrix factorization for a given dataset due to random initialization4. This work proposes “multi-run” NMF, or multiNMF, to address the non-uniqueness of NMF^4,5 in analysis of DCE-MRI. The performance is demonstrated on data from patients with high grade soft tissue sarcoma⁶.

METHODS

The multiNMF approach performs multiple runs of NMF on a given dataset and uses the mean of all weight maps produced as the most representative weight map of the patient dataset. To determine the number of runs required to obtain a representative mean weight map, a method was devised based on the variation of the mean when a new run is added to the averaging process. The Euclidian distance between successive mean weight maps is computed as a difference metric at each iteration and normalized by the number of pixels in the image. Additional runs of NMF are added to the running mean until the difference metric drops below a pre-set tolerance. The running mean weight map calculated during the process can be affected by solutions far from the mean, driven by the random initialization. A more stable running mean can be obtained by considering the ten most recent outputs (using a window of 10 individual runs of NMF). Finally, once the final mean weight map is computed, the sources that best fit the data can be recomputed using the appropriate decomposition step from the NMF algorithm with the original data.  

In this work, the ANLS-BPP algorithm⁷ was used as the core NMF method, set up a priori to identify two sources. A low number of sources enforces a sparse description of perfusion in tissue, and two sources were selected here as qualitatively enough to ensure a stable factorization.

Data were selected ad hoc from 8 of 18 patients with histologically proven high-grade soft-tissue sarcoma⁶. This dataset featured exams performed at 1.5 T, including DCE-MRI, over a course of neoadjuvant radiotherapy, acquired prospectively under institutional approval with informed consent. Only data from pre-treatment exams were used here. Dynamic T1-weighted time-series images were acquired during intravenous injection of gadobutrol (Gadovist, Bayer, 0.1 mmol/kg), using a 3D fast spoiled gradient echo sequence (number of frames ~ 45 and frame time = 10.2–14.4 s depending on the patient, TR=6.0 ms, TE=4.2 ms, flip angle=25°, matrix =256×128, and field-of-view = 24-28 cm). Manual contours of the sarcoma were produced by a radiation oncologist and used to include only the tumour in the analysis, excluding healthy tissue.  
Test data for multiNMF were generated using the ANLS-BPP algorithm to create 1,000 distinct single runs from random initializations in the patient datasets. multiNMF was then applied retrospectively to these individual runs to generate up to 10 mean weight maps per data set, depending on the number of runs used in each multiNMF. Three tolerance levels were tested (0.01, 0.001, and 0.001). The variability of the NMF and multiNMF outputs were calculated from the standard deviation of the weight maps.

RESULTS

The multiNMF approach resulted in a reduction of the variability of weight maps, as illustrated for one patient dataset in Figure 1. Gains in repeatability were seen at all tolerances for the distance between successive averages, with decreases in the average standard deviation of at least a factor of 3 for the highest tolerance value (Figure 2). Reaching the tolerance of 0.01 typically require up to 10 runs of NMF, or around 1 minute of processing, which is largely justified to stabilize the output of the analysis. A tolerance of 0.0001 requires on the order of 100 runs (around 10 minutes). The source curves generated by multiNMF were identical to the average of curves from 1000 runs, confirming that the multiNMF trends towards the average behaviour (Figure 3).

DISCUSSION AND CONCLUSION

The multiNMF algorithm improves on native NMF (ANLS-BPP here) by decreasing variability in the perfusion weight maps. Decreasing the tolerance in the multiNMF further reduces the variability of the perfusion weight maps but increases the processing time: the optimal tolerance is work in progress. multiNMF can nominally be applied to any NMF algorithm to reduce variability on the perfusion maps.

A bootstrapping method previously proposed to address the non-uniqueness of NMF in DCE-MRI selects the most frequent solution ⁴ and would be interesting to compare. Other initialization techniques have been proposed for NMF algorithms applied DCE-MRI datasets. Data-driven initialization of the Block-coordinate Update NMF (BU-NMF) algorithm with wash-in or K^trans maps, produced interpretable and repeatable results on data from patients with sarcomas^2,8,9. Our approach reduces the variability without requiring a priori information such as the K^trans maps (which would require T₁ and B₁ maps)