FridrichMethod Version Compatibility
Reference examples tested with: numpy 1.26+, scipy 1.12+, statsmodels 0.14+
Before using code patterns, verify installed versions match. If versions differ:
- Python:
pip show <package>thenhelp(module.function)to check signatures
If code throws ImportError, AttributeError, or TypeError, introspect the installed package and adapt the example to match the actual API rather than retrying.
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.poissonfor detection-probability math,scipy.stats.normfor CLSI LoB/LoD,statsmodelsProbit/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; |
| "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