Analytical validation

Version Compatibility

Reference examples tested with: numpy 1.26+, scipy 1.12+, statsmodels 0.14+

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• Python: pip show <package> then help(module.function) to check signatures

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Analytical Validation and Detection Limits

"What is the real limit of detection of my ctDNA assay, and can I trust the number I am about to report?" -> Quantify the Poisson sampling ceiling, the error-suppression floor, and the LoB/LoD/LoQ that together define a defensible sensitivity claim.

• Python: scipy.stats.poisson for detection-probability math, scipy.stats.norm for CLSI LoB/LoD, statsmodels Probit/Logit for a dilution-series LoD95 fit.

The Single Most Important Modern Insight -- LoD Is Set by Genome Equivalents Sampled and Error Suppression, NOT by Sequencing Depth or the Caller

A ctDNA assay is a molecule-counting experiment at the Poisson edge. The mutant signal is a fixed, tiny number of physical template molecules in the tube, and the limit of detection is governed by two ceilings: how many genome equivalents were sampled (Poisson), and how low the background error floor was driven (error suppression). 1 ng of human DNA is ~330 haploid genome equivalents; the expected mutant-molecule count is lambda = input_GE x VAF. A 0.1% variant on 1,000 GE (~3.0 ng) has lambda = 1, so e^-1 ~= 37% of the time the mutant template was never in the tube and a perfect sequencer detects nothing. Past the point where every input molecule has been read once (sampling saturation, visible as a deduplication plateau in UMI families), additional read depth re-sequences the same physical molecules and adds zero information. Reporting an LoD as a bare VAF -- with no input mass, no unique-molecule (consensus) depth, no replicate detection rate -- is reporting an undefined quantity.

The second ceiling is the per-base background error rate, which sets the VAF floor independently: a 0.1% variant cannot be distinguished from noise if the assay manufactures that base at 0.1%. Error suppression is a ladder (raw NGS ~1e-3 -> single-strand UMI consensus ~1e-4/1e-5 -> duplex <1e-7), and single-strand consensus does NOT remove template-resident damage (C->T deamination, G->T 8-oxoG) because every PCR copy of that strand inherits the lesion -- only duplex strand-concordance catches it. The achieved LoD is the worse of the two ceilings: error dominates above ~0.1% VAF for tumor-naive single-locus calling, sampling dominates below it. The escape hatch is integration -- a bespoke panel summing mutant molecules across 16-50 loci against summed background reaches single-ppm even though each locus alone is ~1e-3 to 1e-4 (per-locus vs panel-integrated LoD).

Methods Landscape

Concept	Definition	Source
LoB (Limit of Blank)	Highest signal expected from an analyte-free blank (95th pct): LoB = mean_blank + 1.645*SD_blank; the false-positive anchor on true negatives	CLSI EP17-A2
LoD (Limit of Detection)	Lowest level reliably distinguishable from LoB: LoD = LoB + 1.645*SD_low; a sample at LoD is detected ~95% of the time	CLSI EP17-A2
LoD95	The concentration/VAF where detection probability = 95%; a point on a probit/logistic detection curve, not a separate definition	CLSI EP17-A2; Newman 2016
LoQ (Limit of Quantitation)	Lowest level measurable with stated precision (e.g. CV<=20%); LoQ >= LoD always, so a "VAF" near the floor is detectable but not trustworthy	CLSI EP17-A2
Per-locus LoD	Single-variant LoD; sampling- and error-limited (~0.05-0.1% VAF typical)	Newman 2014/2016
Panel-integrated LoD	Evidence summed across N tracked variants via a >=k-of-N positivity rule (binomial over per-locus Poisson detection), reaching single-ppm at 16-50 loci; ~sqrt(N) variance-averaging is only a loose lower bound	Reinert 2019
Reference standards	Contrived defined-VAF cell-line admixtures fragmented to ~160 bp into normal cfDNA; SEQC2 Sample A / HCC1395 truth sets	Fang 2021 (SEQC2)

Decision Tree by Scenario

Scenario	Recommended	Why
"How many GE for 95% detection at VAF X?"	Solve lambda = input_GE x VAF >= 3, so input_GE >= 3/VAF	1 - e^-3 = 0.95; ~30,000 GE (~91 ng at 330 GE/ng) for a single 1e-4 variant -- often more than one tube provides
"Why is more depth not helping?"	Report unique (consensus) molecular coverage, not raw depth; check the dedup plateau	Past sampling saturation, depth re-reads the same molecules; the ceiling is GE in the tube
Single hotspot vs bespoke panel for low VAF	Single locus: error/sampling-limited ~0.1%; need ppm -> integrate across 16-50 clonal loci	Per-locus Poisson/error floor is escaped only by summing independent detections (panel-integrated LoD)
Reporting an LoD	Condition on input mass (GE) + consensus depth + replicate detection rate (e.g. "LoD95 0.1% VAF at 30 ng / 2x duplex / 95% of 20 replicates")	A bare VAF omits the input mass, the unique depth, and per-locus vs integrated -- it is undefined
Estimating LoD95 from a dilution series	Probit (or logistic) regression of detection (0/1) on VAF; read off the 95% point with a CI	CLSI EP17-A2 detection-curve method; binary detection is a clean GLM target
Distinguishing detection from quantitation	Set LoD for yes/no calls; set LoQ (CV<=20%) separately for any reported VAF/TF	MRD calls are binary and can sit far below LoQ; a near-floor VAF number is not quantitative
Validating against truth	Contrived SEQC2 Sample A / HCC1395 admixtures, fragmented to cfDNA-like ~160 bp	Real low-VAF patient material is scarce/unverifiable; commutability with plasma is the caveat

Genome-Equivalent and Poisson Detection Calculator

Goal: Convert an input mass and target VAF into an expected mutant-molecule count and a detection probability, so a sensitivity claim is anchored to molecules rather than to a VAF alone.

Approach: Convert ng to haploid genome equivalents (~330/ng), set lambda = input_GE x VAF, and read the detection probability as a Poisson tail P(X >= k) = 1 - cdf(k-1, lambda); invert for the minimum GE that puts lambda at the >=3 sampling-detection threshold.

import numpy as np
from scipy.stats import poisson

GE_PER_NG = 330  # haploid ~3.3 pg -> strict 1 ng / 3.3 pg = 303; 330 is the common diploid-6.6 pg/rounding convention

def genome_equivalents(input_ng):
    return input_ng * GE_PER_NG

def detection_probability(input_ng, vaf, min_mutant_molecules=1):
    '''P(at least min_mutant_molecules present) under Poisson(lambda = GE * VAF).'''
    lam = genome_equivalents(input_ng) * vaf
    return float(poisson.sf(min_mutant_molecules - 1, lam))

def ge_for_sampling_detection(vaf, target_lambda=3.0):
    '''GE needed so lambda >= 3 -> ~95% chance the mutant molecule is present at all.'''
    return target_lambda / vaf

# A 0.1% variant on 3.0 ng (~990 GE) has lambda ~= 1 -> ~63% detected, ~37% missed by sampling alone.
detection_probability(3.0, 0.001)          # ~0.63
ge_for_sampling_detection(1e-4)            # 30000 GE (~91 ng) for a single 0.01% variant

LoB and LoD95 from a Dilution Series (CLSI EP17 style)

Goal: Estimate the VAF at which the assay detects 95% of the time, from a contrived dilution series, and anchor it to the blank-derived false-positive floor.

Approach: Compute LoB from blank replicates (mean + 1.645*SD, one-sided 95th pct), then fit a probit GLM of binary detection on log10(VAF) (CLSI EP17 fits on log concentration) across the dilution series and invert it for the 95% detection point. The series must bracket the 0.95 crossing — all-detected upper levels cause near-complete separation and an unstable slope.

import numpy as np
import statsmodels.api as sm
from scipy.stats import norm

def limit_of_blank(blank_signals):
    '''LoB = mean + 1.645*SD; one-sided 95th percentile of analyte-free blanks.'''
    blank_signals = np.asarray(blank_signals, dtype=float)
    return blank_signals.mean() + 1.645 * blank_signals.std(ddof=1)

def lod95_probit(vaf_levels, detected):
    '''Probit fit of detection (0/1) on log10(VAF); returns the VAF where P(detect) = 0.95.'''
    log_vaf = np.log10(np.asarray(vaf_levels, dtype=float))
    y = np.asarray(detected, dtype=float)
    X = sm.add_constant(log_vaf)
    fit = sm.GLM(y, X, family=sm.families.Binomial(link=sm.families.links.Probit())).fit()
    intercept, slope = fit.params
    return 10 ** ((norm.ppf(0.95) - intercept) / slope)

Per-Locus to Panel-Integrated LoD

Goal: Combine independent per-locus detection probabilities into the panel-level detection probability that a bespoke MRD assay actually achieves, and find the integrated LoD.

Approach: Treat each tracked locus as an independent Poisson sampler at the same tumor VAF; a panel positive call requires at least k loci detected, so the panel detection probability is the binomial-tail over the per-locus probabilities -- this is why summing 16-50 loci reaches ppm.

import numpy as np
from scipy.stats import poisson, binom

def panel_detection_probability(input_ng, vaf, n_loci, min_loci_positive=2):
    '''P(>= min_loci_positive of n_loci detected); >=2-of-N is the Signatera-style positivity rule.'''
    per_locus = float(poisson.sf(0, input_ng * 330 * vaf))
    return float(binom.sf(min_loci_positive - 1, n_loci, per_locus))

def panel_integrated_lod95(input_ng, n_loci, min_loci_positive=2, grid=None):
    '''Lowest VAF on a log grid where the >=k-of-N panel call hits 95%.'''
    grid = np.logspace(-6, -2, 400) if grid is None else np.asarray(grid)
    probs = [panel_detection_probability(input_ng, v, n_loci, min_loci_positive) for v in grid]
    hits = grid[np.asarray(probs) >= 0.95]
    return float(hits.min()) if hits.size else float('nan')

Quantitative Thresholds

Threshold	Source	Rationale
~330 haploid genome equivalents per ng cfDNA	Standard (haploid ~3.3 pg)	Converts input mass to the molecule count that actually sets sensitivity; strict 1 ng / 3.3 pg = 303, with 330 the common diploid-6.6 pg/rounding convention
lambda = input_GE x VAF; lambda >= 3 for ~95% sampling-detection	Poisson, 1 - e^-3 = 0.95	Below lambda~3 the mutant template is often simply absent from the tube regardless of sequencing
Raw NGS error floor ~1e-3	Schmitt 2012 context; field consensus	Sets the per-base VAF floor before any consensus; a global VAF cutoff above this is noise-limited
Single-strand UMI consensus ~1e-4 to 1e-5	Newman 2014/2016 (CAPP-Seq/iDES)	Majority-vote within a UMI family erases PCR/sequencing error not shared across the family
Duplex sequencing <1e-7 (theory <1/1e9 nt)	Schmitt 2012 PNAS 109:14508	Requires the variant on BOTH original strands; independent strand errors cannot agree
iDES adds ~3-15x over baseline; ctDNA to ~4e-5	Newman 2016 Nat Biotechnol 34:547	Position/trinucleotide background model subtracts stereotyped artifacts per locus
ichorCNA tumor-fraction floor ~3%	Adalsteinsson 2017 Nat Commun 8:1324	Copy-number-based TF estimation; sWGS/ULP-WGS cannot resolve TF below ~3% -- an LoD, not a VAF
Bespoke panel reaches single-ppm by integrating 16-50 loci	Reinert 2019 JAMA Oncol 5:1124	Per-locus ~1e-4 floor escaped by summing independent detections; >=2-of-N positivity rule
LoQ >= LoD (e.g. CV<=20% for quantitation)	CLSI EP17-A2	Detection (binary) is easier than quantitation (continuous); near-floor VAFs are not trustworthy numbers

Common Errors

Error / symptom	Cause	Solution
"Assay detects 0.1% VAF" with no input mass	VAF reported as a standalone sensitivity spec	Condition the LoD on input GE + consensus depth + replicate detection rate; 0.1% on 100 GE is noise
Buying more sequencing depth to improve sensitivity	Conflating read depth with molecule count	Past the dedup plateau the assay is sampling-saturated; add plasma volume / conversion efficiency, not depth
Per-locus LoD quoted as the panel LoD (or vice versa)	Ignoring integration across tracked loci	State which is reported; a 50-variant panel's integrated LoD is orders of magnitude below any single locus
VAF used as the sensitivity unit	Omitting the molecule count behind the fraction	Pair every VAF with input GE; lambda = GE x VAF is the quantity that determines detection
Single-strand UMI assumed to remove damage artifacts	Template-resident C->T/G->T inherited by every copy	Use duplex strand-concordance for sub-1e-5 claims; single-strand votes unanimously for the lesion
Reporting a near-floor VAF as a measured value	Confusing LoD (detect) with LoQ (quantify)	Quantitative VAF/TF only at/above LoQ (CV<=20%); below it report detected/not-detected
Global VAF cutoff across all loci	Background error is position/context-dependent	Use a per-locus background model (iDES-style); a flat threshold loses sensitivity and specificity

References

• Diehl F, Schmidt K, Choti MA, et al. 2008. Circulating mutant DNA to assess tumor dynamics. Nat Med 14:985-990. -- ctDNA half-life ~114 min; molecule-counting framing of tumor dynamics.
• Schmitt MW, Kennedy SR, Salk JJ, et al. 2012. Detection of ultra-rare mutations by next-generation sequencing. PNAS 109:14508-14513. -- Duplex sequencing; theoretical error floor <1 per 1e9 nt.
• Newman AM, Bratman SV, To J, et al. 2014. An ultrasensitive method for quantitating circulating tumor DNA with broad patient coverage. Nat Med 20:548-554. -- CAPP-Seq; UMI-consensus error suppression.
• Newman AM, Lovejoy AF, Klass DM, et al. 2016. Integrated digital error suppression for improved detection of circulating tumor DNA. Nat Biotechnol 34:547-555. -- iDES; ~3-15x gain; ctDNA to ~4e-5.
• Razavi P, Li BT, Brown DN, et al. 2019. High-intensity sequencing reveals the sources of plasma circulating cell-free DNA variants. Nat Med 25:1928-1937. -- CHIP as the dominant non-tumor signal in the LoB blank.
• Adalsteinsson VA, Ha G, Freeman SS, et al. 2017. Scalable whole-exome sequencing of cell-free DNA reveals high concordance with metastatic tumors. Nat Commun 8:1324. -- ichorCNA; copy-number tumor-fraction floor ~3%.
• Reinert T, Henriksen TV, Christensen E, et al. 2019. Analysis of plasma cell-free DNA by ultradeep sequencing in patients with stages I to III colorectal cancer. JAMA Oncol 5:1124-1131. -- Signatera; 16-variant integration; >=2-of-N positivity.
• Fang LT, Zhu B, Zhao Y, et al.; SEQC2 Consortium. 2021. Establishing community reference samples, data and call sets for benchmarking cancer mutation detection using whole-genome sequencing. Nat Biotechnol 39:1151-1160. -- SEQC2 Sample A / HCC1395 contrived reference standards.
• CLSI EP17-A2. 2012. Evaluation of Detection Capability for Clinical Laboratory Measurement Procedures; Approved Guideline -- Second Edition. Clinical and Laboratory Standards Institute. -- Governing LoB/LoD/LoQ definitions.

Related Skills

• ctdna-mutation-detection - applies these limits to low-VAF somatic calls
• longitudinal-monitoring - per-timepoint LoD and left-censoring of undetectable samples
• tumor-fraction-estimation - the ~3% CNA-based detection floor as an LoD
• experimental-design/multiple-testing - repeated-surveillance specificity and FDR
• clinical-biostatistics/power-and-sample-size - validation-study design