Incommensurable Targets in Machine Learning for Nanomaterials: How Dose-Metric Heterogeneity Imposes an Irreducible Error Floor on Pooled Training Data

Abstract

Nanomaterial datasets are assembled from the published literature rather than generated in a single laboratory, and the studies pooled in this way do not agree on what dose means. Exposure is reported as mass concentration, as particle number, as total or specific surface area, or as particles delivered per cell, and conversion between these metrics requires size, density, porosity and aggregation data that source reports frequently omit. This theoretical and analytical paper treats the resulting heterogeneity not as a reporting inconvenience to be settled by better conventions, but as a measurement-model problem that places a quantifiable ceiling on what any machine-learning model trained on such data can achieve. The study is non-empirical. It derives closed-form relations among the dose metrics for spherical particles of uniform density, showing that mass, surface area and particle number scale with the zeroth, inverse first and inverse third powers of radius, so that conversion factors are functions of particle size rather than constants. From these relations the paper obtains a closed-form condition under which two materials compared at equal mass dose are ranked oppositely by particle number and by surface area, and verifies that condition exhaustively against brute-force computation over two hundred thousand randomly drawn pairs. It then derives the irreducible error floor that dose-metric heterogeneity imposes on a pooled training set, MSE_floor = β²p²σ²(ln r) + σ²_ε, which grows with the square of the metric mismatch, with the dispersion of log particle radius across the corpus, and with the steepness of the dose response. The floor is shown to be independent of model class and of hyperparameter optimisation, and not to recede as further data drawn from the same literature are added. The paper converts the derivation into practice through a six-step procedure for bounding the achievable performance of an existing pooled dataset from its own curation records, together with declaration, partitioning and applicability-domain requirements for training-set construction expressed in the context-of-use idiom of current regulatory guidance.