Magnetic resonance imaging of tissue morphology has been widely used in oncology to detect the presence of disease and to detect treatment response by measuring decreases in tumor size . Since for some therapies treatment-induced tumor cell death can be correlated with patient survival [2-4], a more general method for detecting treatment response would be to image tumor cell death. This could be used to detect response to those therapies that have little or no effect on tumor size and provide earlier detection of response to those therapies that do eventually induce tumor shrinkage .
Even relatively low resolution images, such as those produced by computed tomography (CT) or MRI, can be a sensitive indicator of underlying tissue biology  and therefore could potentially be used to interrogate more subtle features of tumor physiology and changes in this physiology in response to treatment. Such an approach, however, requires the development of image metrics that give objective and quantitative assessments of tissue morphology and that capture the underlying biological information in a routine and automated fashion.
A characteristic of tumors is their heterogeneous appearance in MR images, a consequence of their irregular and uncoordinated growth and a chaotic and intermittent blood supply that leads to periods of transient ischemia and hypoxia. The resulting areas of necrosis and hemorrhage can lead to hyper- and hypointensity, respectively, in T2-weighted images  and may also be influenced by treatment, where successful treatment can result in a change in the size and distribution of these areas .
Various approaches have been adopted for analyzing heterogeneity in MR images of tumors, including k-means clustering [9, 10], texture analysis [11-13], and fractal dimensions [14-16]. However, texture analysis and the k-means clustering algorithm, have some important limitations. Texture analysis requires a large set of image parameters to be calculated with a subset of these parameters then chosen and used to differentiate between tissues of different types. In a recent study, where texture analysis was used to distinguish between benign and malignant soft tissue masses on MR images, only small differences were identified between the two and it was concluded that more data were required to confirm the value of this approach . A number of variations of the k-means clustering algorithm have been used in MRI; however, in most cases, the algorithm requires prior knowledge of the number of expected regions or features and, furthermore, due to the fact that the algorithm has a randomly assigned starting point, the resulting clusters may not always converge to the same point.
More recent approaches to image analysis have focused on the use of shape-orientated descriptors, such as fractal dimension analysis and Minkowski functionals (MFs), which do not require prior assumptions about the number of regions or features in an image. For example, Rose et al.  used fractal dimensions to describe the heterogeneity found in dynamic contrast enhanced (DCE)-MRI parameter maps and showed that the measured heterogeneity could distinguish between low-grade and high-grade gliomas, a distinction that could not be made using distribution-based summary statistics. MFs have been widely employed in cosmology as precise morphological and structural descriptors, and used in the study of the evolution and morphology of galaxies and clusters of galaxies [18, 19]. More recently they have been used as shape functionals in neuromorphometric characterization , for classifying normal and pathological pulmonary tissue [21, 22] and as parameters for the analysis of mineral distribution in hip fractures . We demonstrated recently that MFs can be used to parameterize the heterogeneous distribution of a targeted MRI contrast agent for detecting tumor cell death and showed that this increased the sensitivity of cell death detection in a drug-treated tumor . We show here that MFs can be used with T2-weighted images to detect the morphological changes that accompany tumor cell death following drug treatment in the absence of any exogenous contrast agent.
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Tumor responses to treatment in the clinic are conventionally assessed from CT or MR imaging measurements of decreases in tumor size . We have shown previously that cell death in EL4 lymphomas induced by etoposide treatment could be imaged using 2D MFs at 24 h after treatment using a targeted MR imaging agent that binds to the phosphatidylserine exposed by dying cells . We have also shown that the cell death induced by CA4P treatment could be detected at 6 h post-treatment using 13C-MRS measurements of the increased rates of conversion of hyperpolarized [1,4-13C2]fumarate to malate and at 24 h using diffusion-weighted imaging . Since the cellular necrosis resulting from etoposide treatment was expected to lead to regions of increased intensity in T2-weighted images  whereas the hemorrhagic necrosis resulting from CA4P treatment was expected to lead to regions of hypo- and hyper-intensity . We have examined here the possibility of detecting treatment response by quantitatively assessing, using 2D MFs, the changes in heterogeneity of T2-weighted images following drug treatment.
The ability of MFs to detect treatment response is based on changes in the MF values with treatment, particularly changes in perimeter and genus values. At low threshold values the “holes” in the thresholded images (Fig. 2), produced by eliminating dark pixels, result in negative genus values (Fig. 5c), which is also seen as a corresponding increase in the perimeter MF (Fig. 5b). As the threshold is increased the genus is sensitive to the number of isolated regions of bright pixels in the original image and the genus values become positive. At higher thresholds the perimeter tends to decrease as the number of remaining pixels decreases.
Changes in the 2D MF values were observed in control tumors in the 24 h between scans (Fig. 5), in particular the area MF at the high thresholds was increased in the 24 h scan, indicating that more bright pixels are present. This may represent changes in tumor morphology, particularly an increase in isolated necrotic areas associated with continued tumor growth. The volume of the segmented tumor was increased in the control group between scans, although this increase was not statistically significant. To account for these changes in the control untreated tumors we subsequently used changes in MF values (treated minus control) for each tumor in the construction of the SVM classifiers. In this way each tumor acted as its own control.
In the case of etoposide treatment of EL4 tumors, there was a significant reduction in the segmented tumor volume 24 h post-treatment, which could be explained by the increase in cell death observed in histological sections. In addition, there were changes in the 2D MF values post-treatment relative to the pre-treatment scans, specifically the perimeter and genus functionals were decreased in the post-treatment scans, indicating changes in the image heterogeneity post-treatment (Fig. 6). These changes in MFs were similar to those seen previously with contrast enhanced T1-weighted images of EL4 tumors post etoposide treatment . The decrease in both genus and perimeter is possibly due the presence of fewer but larger hyperintense regions due to necrosis in the post-treatment images.
There was no significant change in tumor volume either 6 or 24 h post CA4P treatment, however changes in MF values were observed. The pattern of changes in MF values was different for CA4P treated tumors compared to etoposide treatment. The change in both perimeter and genus functionals at low thresholds in the 6 h post-treatment data (Fig. 7) suggests that there was an increase in the number of regions of dark pixels, possibly due to the presence of regions of hemorrhage. At higher thresholds for the 6 and 24 h treatment data there was a reduction in both perimeter and genus functionals, which may indicate an increase in the size of the regions of necrosis similar to that seen with etoposide treatment.
The potential of 2D MFs for detecting treatment response from T2-weighted images was evaluated using ROC analysis, where the curves were generated after SVM classification based on the MF parameters for control tumors and for tumors 6 and 24 h after drug-treatment. The SVM classifier for etoposide performed well, with an AUC of 86% and high specificity and sensitivity (Table 2). While these tumors also showed a significant reduction in tumor volume post-treatment, these results demonstrate that changes in image heterogeneity can be used to detect treatment response in the absence of exogenous contrast. For CA4P at 6 h following treatment the SVM classifier gave an AUC of 82%, and again scored well on specificity and sensitivity. Unlike etoposide treatment, however, there was no observable change in tumor volume; therefore 2D MFs may be applicable clinically to detect treatment response before any changes in tumor volume occur. The 24-h post-treatment group performed poorly in the ROC analysis with an AUC of 64%, a low sensitivity of 67%, but a good specificity. Increasing the number of features by using the criteria of P<0.1 as a feature selector did not improve the classification performance (data not shown). Since CA4P-treated tumors show a loss of tumor perfusion using DCE-MRI at 6 h post-treatment but restored perfusion by 24 h post-treatment , the levels of perfusion may be more important than changes in the levels of necrosis in determining whether the 2D MF analysis of image heterogeneity detects treatment response in this case.
Manual segmentation of the tumors, as performed here, does introduce an element of operator-dependent subjectivity into the analysis. The use of automatic segmentation algorithms would be preferable. While the present study has focused on the application of 2D MFs in the analysis of treatment response, 3-dimensional equivalents of the 2D MFs used in this study are available . 3D MFs may be expected to perform as well or better than 2D MFs as in principle they capture more information about the heterogeneity of the tumor as a whole. However, an important consideration is the large difference in resolution between the 2D slices (0.14 × 0.28 mm) when compared to the thickness of the imaging slice, which was 1.5 mm in the dataset used here. This may reduce the capability of the 3D MFs to capture additional heterogeneity when compared to that reported on by the 2D MFs.
Demonstrating that specific aspects of tumor morphology are indeed responsible for the observed features in the T2-weighted MR images requires careful registration of the MR and histological images; a process that is made difficult by the very different slice thicknesses involved; 1.5 mm for the MR slice versus 5 µm for the histological slice. It may be possible to use MF analysis to assign features in the MR image to specific features in the corresponding histological section by demonstrating that they have the same MFs. For example, showing that lightly stained regions in an H&E stained section, which represent regions of necrosis, have the same MFs as hyperintense regions in the corresponding T2-weighted image. However, this analysis was not attempted here.
In conclusion, we have demonstrated that we can detect early responses of tumors to drug treatment by parameterizing, using MFs, changes in heterogeneity in T2-weighted images after drug treatment. This was possible in the absence of observable changes in tumor size in the case of treatment with CA4P. Such an approach provides a quantitative and therefore objective assessment of treatment response. In principle, this method could be used to enhance response detection in any type of MR image, e.g. diffusion-weighted, contrast agent-enhanced and, indeed, in images acquired using other imaging modalities, such as positron emission tomography (PET) and single-photon emission computed tomography (SPECT).