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

Technical Summary: Formalising Limits of Circulating Tumour DNA Detection

Problem Statement
Despite the clinical establishment of circulating tumour DNA (ctDNA) liquid biopsy for early cancer detection, minimal residual disease (MRD) surveillance, and treatment monitoring, a critical theoretical gap remains in how detection limits are defined. Current clinical assays determine their limits of detection (LOD) empirically through receiver operating characteristic (ROC) analysis on training cohorts. While statistically valid, this approach is "theoretically uninformed." It fails to specify the minimum detectable tumour fraction based on fundamental assay characteristics (sequencing depth and error rates) and does not identify the point at which increasing sequencing depth ceases to yield additional clinical information. Consequently, there is no platform-independent metric for standardisation or a formal proof of optimality for current detection thresholds.

Methodology
The paper models ctDNA detection as a binary hypothesis testing problem where the observed mutant allele counts follow a Binomial distribution against a sequencing error noise floor.

• Hypothesis Formulation:
  • $H_0$ (Null): No ctDNA present ($f=0$); mutant reads arise solely from sequencing errors ($\epsilon$).
  • $H_1$ (Alternative): ctDNA present at fraction $f > 0$; reads arise from true mutant molecules and errors on wild-type molecules.
• Optimal Detector: Applying the Neyman-Pearson lemma, the authors derive the uniformly most powerful (UMP) detector. This establishes that the optimal decision rule is a likelihood ratio test, which simplifies to declaring ctDNA present if the observed mutant count $X$ exceeds a specific integer threshold $k$.
• Information-Theoretic Analysis: The sequencing assay is modelled as a binary symmetric channel to calculate Shannon channel capacity ($C = 1 - H(\epsilon)$), representing the maximum mutual information recoverable per read.
• Empirical Validation: The theoretical framework was validated against $n=61$ data points extracted from five peer-reviewed analytical validation studies (2018–2025) across five independent institutions in the US and EU. These studies covered standard large-panel NGS, UMI error-corrected NGS, and duplex whole-genome sequencing (WGS).

Key Contributions

1. Closed-Form Detection Limit: The paper derives the minimum detectable tumour fraction ($f_{min}$) as a closed-form function of sequencing depth ($N$), error rate ($\epsilon$), and statistical parameters ($\alpha, \beta$):
   $$f_{min}(N, \epsilon, \alpha, \beta) \approx (z_{1-\alpha} + z_{1-\beta}) \sqrt{\frac{\epsilon}{N}}$$
   This formula provides a first-principles derivation of detection limits, replacing empirical calibration with theoretical calculation.
2. Optimality Proof: The work provides the first formal proof that the standard thresholding approach, when derived via the Neyman-Pearson lemma, constitutes the uniformly most powerful detector for this class of problems.
3. Scaling Laws: The analysis identifies three critical properties:
   • Depth Scaling: $f_{min} \propto 1/\sqrt{N}$. Quadrupling depth halves the detectable fraction.
   • Error Correction Scaling: $f_{min} \propto \sqrt{\epsilon}$. Reducing error rates by four orders of magnitude reduces the detectable fraction by 100-fold.
   • Equivalence: Both depth escalation and error correction follow the same square-root scaling law.
4. Information Ceiling: The study calculates that modern platforms (Standard NGS, UMI NGS, Duplex WGS) operate within 2% of the theoretical per-read Shannon information ceiling ($C \approx 1$ bit/read). This suggests that further clinical leverage comes from extending $f_{min}$ via error correction, not from increasing per-read information gain.

Results

• Theoretical vs. Empirical Agreement: For single-locus and small-panel assays, the theoretical framework correctly classified 84.3% of data points (43/51). Agreement was highest for standard large-panel NGS (87%) and hybrid capture assays (83%).
• Multi-Locus Discordance: The model showed lower agreement for multi-locus tumour-informed assays (NeXT Personal: 40%; Duplex WGS: 20%). The authors attribute this not to model failure, but to the model's scope: multi-locus assays aggregate signals across hundreds of loci ($L$), improving the detection limit by a factor of $\sqrt{L}$. The observed discordance aligns with the theoretical extension required for joint multi-locus detection.
• Clinical Benchmarking: The framework confirmed that current clinical practices for three scenarios—early detection (UMI NGS), MRD post-surgery (Duplex WGS), and treatment response (Standard NGS)—meet the theoretical minimum depth requirements for their stated applications.

Significance and Claims
The paper claims to provide the first formal Neyman-Pearson optimality proof for ctDNA detection and a platform-independent efficiency metric. Its primary significance lies in:

• Standardisation: Offering a theoretical basis for regulatory bodies (FDA, MHRA) and the NHS Genomics Medicine Service to specify LODs and compare heterogeneous platforms using a common efficiency metric ($LOD_{empirical} / f_{min}$).
• Assay Design: Replacing empirical trial-and-error with a calculable, economically optimal sequencing depth ($N^*$) for specific clinical targets.
• Future Direction: Identifying the derivation of joint multi-locus detection limits as the principal theoretical extension required to fully model tumour-informed assays.

The authors explicitly state that this framework does not invent new applications but rather formalises the limits of existing ones, providing the mathematical machinery to define "when increasing sequencing depth ceases to provide additional clinical information."