Formalising Limits of Circulating Tumour DNA Detection: A Signal Detection Framework for Clinical Threshold Specification

Background: Circulating tumour DNA (ctDNA) liquid biopsy is now established across oncology for early cancer detection, minimal residual disease surveillance, and treatment monitoring. Detection thresholds for all current ctDNA assays are derived empirically through receiver operating characteristic analysis on training cohorts - a statistically valid but theoretically uninformed approach that does not specify the minimum detectable tumour fraction given assay technical characteristics, nor identify when increasing sequencing depth ceases to provide additional clinical information. Methods: We model ctDNA detection as a binary hypothesis testing problem with Binomial-distributed mutant allele counts against a sequencing error noise floor. The Neyman-Pearson lemma is applied to derive the uniformly most powerful detector and the minimum detectable tumour fraction in closed form. The sequencing assay is modelled as a binary symmetric channel and Shannon channel capacity is calculated. Empirical validation uses n=61 data points extracted from five published peer-reviewed analytical validation studies across five independent institutions in the US and EU (2018 - 2025): Yu et al. 2022, Stetson et al. 2018, Frydendahl et al. 2023, Northcott et al. 2024, and Cheng et al. 2025. Results: The minimum detectable tumour fraction is derived in closed form as f_min approximately equal to (z_alpha + z_beta) multiplied by the square root of (epsilon divided by N), where N is sequencing depth, epsilon is the platform error rate, and z_alpha, z_beta are standard normal quantiles at the specified false positive and false negative rates. Shannon channel capacity is C = 1 minus H(epsilon) bits per read, where H(epsilon) is binary entropy. Empirical validation yields 84.3% agreement for single-locus assays. Discordance for multi-locus tumour-informed assays (NeXT Personal, duplex WGS) is consistent with the single-locus model scope and identifies the principal theoretical extension required. Conclusions: This framework provides the first formal Neyman-Pearson optimality proof for ctDNA detection, a closed-form detection limit, and a platform-independent efficiency metric for NHS and regulatory standardisation. Keywords: circulating tumour DNA; liquid biopsy; Neyman-Pearson detection; Shannon channel capacity; sequencing depth; limit of detection; minimal residual disease; signal detection theory