Spatial Shape Analysis of Cell Nuclei

In a histopathology image, to assess if the shapes of tumor cell nuclei influence tumor infiltration in surrounding tissues it is important to first quantify and visualize variability in the shape of cell nuclei whilst accounting for their spatial correlations. To avoid loss of shape information when using summaries, we represent the nucleus boundary as a planar curve and develop a geometric framework to compute local spatially-informed summaries of shapes of planar curves. The framework is based on the definition of a novel shape trace-variogram that captures dependence between shapes of curves. The variogram is used to compute a local spatially-weighted shape average and covariance, using which, shape variation is quantified via principal component analysis applied to the weighted covariance operator. Primary directions of spatial shape variation are subsequently visualized as vector fields (and their pointwise magnitudes) on the spatially-weighted average shape. Our approach serves as a practical exploratory tool for clinicians who routinely assess morphological and spatial heterogeneity of cell nuclei and also represents the first step toward probabilistic modeling of spatially-varying cell shapes. We demonstrate utility of the framework on synthetic data, and on data pertaining to cell nuclei from whole slide images of breast cancer tissues.

Relevance to Life Sciences.

Pathologists routinely examine tumor regions in histopathology images to assess severity and progression of cancer. Nuclear shape heterogeneity plays a key role in this process and provides valuable information that guides clinical treatment decisions. To aid in such assessments, there is need for tools that enable statistical quantification, summarization, and visualization of cell nucleus shape variation. While such tools exist for independent shape data, they are not applicable in the current context due to the inherent spatial correlation among cell nuclei. Thus, we define local spatially-informed shape summaries which, when computed based on cell nuclei within histopathology image regions, reveal their shape heterogeneity. The proposed framework is applied to whole slide images of breast cancer tissues.

Mathematical Content.

Cell nucleus boundaries are represented using planar closed curves. Their shape is a property that is unaffected by translation, rescaling, rotation, and reparameterization. To ensure that our framework is invariant to all shape-preserving transformations, we build on the Riemannian geometric elastic shape analysis framework. We first define a shape trace-variogram, which models spatial dependence among the shapes of planar curves. The trace-variogram is then used to define a spatially-weighted shape average and covariance, which enable exploration of shape variability in a local spatial region. Finally, we provide algorithms for estimation of the proposed spatially-informed summaries.