SPF: A spatial and functional data analytic approach to cell imaging data

The tumor microenvironment (TME), which characterizes the tumor and its surroundings, plays a critical role in understanding cancer development and progression. Recent advances in imaging techniques enable researchers to study spatial structure of the TME at a single-cell level. Investigating spatial patterns and interactions of cell subtypes within the TME provides useful insights into how cells with different biological purposes behave, which may consequentially impact a subject’s clinical outcomes. We utilize a class of well-known spatial summary statistics, the K-function and its variants, to explore inter-cell dependence as a function of distances between cells. Using techniques from functional data analysis, we introduce an approach to model the association between these summary spatial functions and subject-level outcomes, while controlling for other clinical scalar predictors such as age and disease stage. In particular, we leverage the additive functional Cox regression model (AFCM) to study the nonlinear impact of spatial interaction between tumor and stromal cells on overall survival in patients with non-small cell lung cancer, using multiplex immunohistochemistry (mIHC) data. The applicability of our approach is further validated using a publicly available multiplexed ion beam imaging (MIBI) triple-negative breast cancer dataset.

Investigating spatial patterns and interactions of cells in the tumor microenvironment (TME) provides useful insights into cancer development and progression. In this work, we proposed a novel approach which combined established spatial summary functions with functional data analysis to flexibly model the cell-cell interactions with overall survival at different inter-cell distances, in conjunction with other clinical predictors such as age, disease stage. By applying the proposed framework to multiplex immunohistochemistry (mIHC) data of patients with non-small cell lung cancer (NSCLC), we studied the nonlinear impact of spatial interactions between tumor and stromal cells on overall survival. The applicability of our proposed method is further validated using a publicly available multiplexed ion beam imaging (MIBI) triple-negative breast cancer (TNBC) dataset.

Attributional & Consequential Life Cycle Assessment: Definitions, Conceptual Characteristics and Modelling Restrictions

To assess the potential environmental impact of human/industrial systems, life cycle assessment (LCA) is a very common method. There are two prominent types of LCA, namely attributional (ALCA) and consequential (CLCA). A lot of literature covers these approaches, but a general consensus on what they represent and an overview of all their differences seems lacking, nor has every prominent feature been fully explored. The two main objectives of this article are: (1) to argue for and select definitions for each concept and (2) specify all conceptual characteristics (including translation into modelling restrictions), re-evaluating and going beyond findings in the state of the art. For the first objective, mainly because the validity of interpretation of a term is also a matter of consensus, we argue the selection of definitions present in the 2011 UNEP-SETAC report. ALCA attributes a share of the potential environmental impact of the world to a product life cycle, while CLCA assesses the environmental consequences of a decision (e.g., increase of product demand). Regarding the second objective, the product system in ALCA constitutes all processes that are linked by physical, energy flows or services. Because of the requirement of additivity for ALCA, a double-counting check needs to be executed, modelling is restricted (e.g., guaranteed through linearity) and partitioning of multifunctional processes is systematically needed (for evaluation per single product). The latter matters also hold in a similar manner for the impact assessment, which is commonly overlooked. CLCA, is completely consequential and there is no limitation regarding what a modelling framework should entail, with the coverage of co-products through substitution being just one approach and not the only one (e.g., additional consumption is possible). Both ALCA and CLCA can be considered over any time span (past, present & future) and either using a reference environment or different scenarios. Furthermore, both ALCA and CLCA could be specific for average or marginal (small) products or decisions, and further datasets. These findings also hold for life cycle sustainability assessment.

Additive Functional Cox Model

We propose the Additive Functional Cox Model to flexibly quantify the association between functional covariates and time to event data. The model extends the linear functional proportional hazards model by allowing the association between the functional covariate and log hazard to vary non-linearly in both the functional domain and the value of the functional covariate. Additionally, we introduce critical transformations of the functional covariate which address the weak model identifiability in areas of information sparsity and discuss their impact on interpretation and inference. We also introduce a novel estimation procedure that accounts for identifiability constraints directly during model fitting. Methods are applied to the National Health and Nutrition Examination Survey (NHANES) 2003-2006 accelerometry data and quantify new and interpretable circadian patterns of physical activity that are associated with all-cause mortality. We also introduce a simple and novel simulation framework for generating survival data with functional predictors which resemble the observed data. The accompanying inferential R software is fast, open source and publicly available. Our data application and simulations are fully reproducible through the accompanying vignette.

Methods for scalar-on-function regression

Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images, etc. are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorizing the basic model types as linear, nonlinear and nonparametric. We discuss publicly available software packages, and illustrate some of the procedures by application to a functional magnetic resonance imaging dataset.

Additive Function-on-Function Regression

We study additive function-on-function regression where the mean response at a particular time point depends on the time point itself, as well as the entire covariate trajectory. We develop a computationally efficient estimation methodology based on a novel combination of spline bases with an eigenbasis to represent the trivariate kernel function. We discuss prediction of a new response trajectory, propose an inference procedure that accounts for total variability in the predicted response curves, and construct pointwise prediction intervals. The estimation/inferential procedure accommodates realistic scenarios, such as correlated error structure as well as sparse and/or irregular designs. We investigate our methodology in finite sample size through simulations and two real data applications. Supplementary Material for this article is available online.

Classical Testing in Functional Linear Models

We extend four tests common in classical regression - Wald, score, likelihood ratio and F tests - to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.