This work combined two different data analysis tools, octree decomposition and variograms, to study tissue heterogeneity in lung disease. We showed that this merged approach was better able to differentiate rats with mild emphysematous disease from the healthy control group than methods that relied on absolute HU values. The main criterion for octree decomposition was based on the standard deviation of HU values within an octree box. An advantage to this approach is that it avoids thresholding according to HU values, although sophisticated thresholding algorithms may be useful [35–37]; rather, it focuses only on heterogeneity-based signatures that may characterize disease [2]. We propose that a heterogeneity score Δ, the average distance of a rat’s variance from that of the control group average, may be useful to classify disease severity. Furthermore, to visualize the regions of the lung with the greatest heterogeneity, one could determine which boxes had the highest semi-variance within *d*
_{
max
} and map them back to the original image. This would provide 3D information about the spatial distribution of lung tissue heterogeneity and, potentially, disease distribution.

Another approach to a disease metric might be that of fitting the data to an established variogram model, most of which describe an asymptotic rise in variance (i.e. variance becomes independent of distance indicating that spatial relationships become random) [23]. This is seen to some degree in our data (see Figure 6). However, within the range of *d* ≤ *d*
_{
max
}, we found that a power law model generally fit our data better than other models. This model typically describes fractal behavior [38]. Our initial investigations into this model showed potential promise at using fit parameters to distinguish dose groups; however, results lacked statistical significance, and the full-lung dose group tended to not fit this model as well as the other two groups.

There was no single variogram model that satisfactorily fit the all the data over the entire range of distances, because the complex geometries found in the lung result in some problems for variograms. In particular, the direct linear path between two regions of the lung separated by large distances often crosses non-lung tissue, such as the heart, which are essentially treated as holes or voids in the geometry. Furthermore, neighboring lobes generally do not interact physiologically except through the vascular and airway trees, which may only connect regions through many orders of branching. As pointed out by Keil et al. [25], one could go to extraordinary measures to take into account structural distances (the physiological distance at which different regions interact) versus Euclidian (straight line) distances used herein. To limit the problem in this study, we constrained the distance of variogram analysis to approximately half the characteristic diameter of the largest lobe. However, to better understand the inter-and intra-lobe variance relationships, the lung could be segmented into lobes (if the image is of sufficient resolution to discern lobar boundaries), and the decomposition/variogram process repeated on the segmented images. Though, by ignoring inter-lobar variances, this approach would likely not capture information about disease that was confined to a single lobe, particularly if the entire lobe was affected homogeneously.

We employed octree decomposition prior to generating the variograms. In doing this, we assumed that the emphysematous disease is generally slowly varying over space, and that the disease causes changes to homogeneity on the order of or less than the lobar length scale but greater than the octree box length scale [4, 34]. This is consistent with what we previously observed using ^{3}He MRI in the same disease model [26]. Without this assumption, variograms would have to be made directly from the raw images. This is possible but impractical, because the semi-variance computation time (and resulting file size) is proportional to the number of voxel pairs, which rises approximately as *n*
^{2} (see Eq. 2). We measured the variogram computation time for 1078 octree-decomposed boxes from one rat to be 6.6 seconds (on a MacPro model 3.1), and we verified experimentally that the computation time indeed rose in proportion to the square of the number of voxels. Therefore, to create a variogram on the entire masked 3D image, which consisted of 1.48 × 10^{6} voxels (of lung tissue only), it would take us ≈ 1.25 × 10^{7} seconds, or about 5 months–with a resulting file size on the order of 20 GB. Therefore, octree decomposition dramatically reduces the computation time while focusing non-subjectively on regions of the lung that are of greatest interest. We note that the octree decomposition itself was performed in ~2 minutes.

Image noise can confound octree decomposition and affect resulting variograms. Results of image filtering tests indicated that noise reduction using an edge-preserving filter resulted in more 8 × 8 × 8 octree blocks without significantly affecting variogram results. An alternative to octree decomposition is downsampling, which is a quick and straightforward approach to reducing image noise and size. However, we verified that the octree decomposition approach performed much better at separating dose groups than simply downsampling the image and then calculating the semi-variance using every voxel. The octree decomposition assures that only the homogeneous regions of the lungs are singled out for comparison, whereas downsampling the image blurs together proximal voxels, including vasculature, airways, and lung boundaries irrespective of signal intensity or tissue type. Thus, the downsampling approach apparently causes a loss of information. The radius = 4 Gaussian filter had a result similar to downsampling.

One limitation of this pilot study was the small number of animals in each group, which did not allow the statistical evaluation of specificity and sensitivity. Therefore, follow-on work will be required to validate these results and establish specificity and sensitivity [2]. This might be accomplished in conjunction with pulmonary function tests, conventional morphometric measurements [39], and histological techniques particularly sensitive to early emphysematous changes [40]. Additional future work should evaluate the performance of this method on clinical CT images as well as test the effectiveness for distinguishing different diseases and disease models.