Does the random nature of cell-virus interactions during in vitro infections affect TCID$_{50}$ measurements and parameter estimation by mathematical models?

Catherine A. A. Beauchemin; Christian Quirouette; Kyosuke Adachi; Risavarshni Thevakumaran

REVIEW 3 major objections 4 minor 51 references

Does the random nature of cell-virus interactions during in vitro infections affect TCID$_{50}$ measurements and parameter estimation by mathematical models?

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read TCID50 numbers are not virion counts; a new formula converts between them.

desk verdict Solid methodological core, undermined by an acknowledged but unresolved cell-type mismatch in the empirical application; still worth a peer review with major revisions. read the letter →

arxiv 2412.12960 v1 pith:K5G2WNSJ submitted 2024-12-17 physics.bio-ph q-bio.QM

Christian Quirouette , Risavarshni Thevakumaran , Kyosuke Adachi , Catherine A. A. Beauchemin This is my paper · ORCID

classification physics.bio-phq-bio.QM

keywords TCID50endpointdilutionassayinfectiousvirionconcentrationestablishmentprobabilitystochasticinfectionmodellikelihood-basedparameterestimationinfluenzaAvirusnoise

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the concentration of infection-causing doses measured by a TCID50 endpoint dilution assay (SIN/ml) is not the concentration of infectious virions (IV/ml), but that concentration multiplied by the probability that an infection seeded by one infectious virion establishes rather than goes extinct. Because that establishment probability depends on the very parameters being estimated, the assay's undercounting is not a fixed calibration factor. The paper introduces a likelihood for the raw endpoint-dilution outcome that handles below-detection measurements correctly, and shows that expressing viral titre in IV units, with physical constraints, makes parameters such as gamma and rho individually identifiable. Re-analysis of influenza A infection data suggests that the stochasticity of cell-virus interactions has little effect on the infection time course itself, while the random noise of the TCID50 assay explains the scatter seen between replicate infections. This matters because it provides a route from routine infectivity readings to biologically meaningful virion-level parameters.

What carries the argument

The load-bearing object is the extinction probability $P_{V\to\text{Extinction}}$ in Eqn (1), derived from a stochastic model with Erlang-distributed eclipse and infectious phases, negative-binomial per-cell virion production, and loss of infectivity and cell entry. Its complement, $P_{V\to\text{Establishment}} = 1 - P_{V\to\text{Extinction}}$, converts infectious virions to measured specific infections as in Eqn (2). The new likelihood $L_{ED}$ (Eqn 5) is a product over dilution columns of binomial infection probabilities, with no Gaussian error assumption, and it naturally includes measurements below the detection limit. Parameter estimation uses MCMC, and for the stochastic model the random number seed is added as a parameter so each parameter set maps to a single deterministic trajectory.

What would settle it

Measure the establishment probability directly: inoculate many replicate wells with a dilution expected to contain about one infectious virion, count the fraction that become infected, and compare with Eqn (1) evaluated at the estimated parameters. If the measured fraction differs from the predicted $P_{V\to\text{Establishment}}$ by more than the assay's sampling error, the conversion underlying every IV-unit estimate is wrong; a simpler version is to repeat the authors' parameter estimation with the ED assay performed in the same A549 cells used for the infections and check whether $\gamma$, $\rho$, and $P_{V\to\text{Establishment}}$ shift.

Watch

Extended reading notes

Core claim

The paper's central claim is expressed as Eqn (2): the measured infection-causing dose concentration (SIN/ml) equals the actual infectious virion concentration (IV/ml) multiplied by $P_{V\to\text{Establishment}} = 1 - P_{V\to\text{Extinction}}$, the probability that an infection initiated with a single infectious virion establishes. That probability is parameter-dependent and given by Eqn (1), so the conversion factor between SIN and IV is not a constant but part of what the model must estimate. Building on this, the paper proposes a likelihood, $L_{ED}$ (Eqn 5), that uses the full endpoint-dilution outcome rather than a Gaussian residual assumption, and it re-expresses the model's infectious titre in IV units rather than SIN units. In IV units, physical constraints (one IV cannot infect more than one cell; IV cannot exceed total vRNA) break the degeneracies that otherwise plague parameter estimation. Applied to experimental influenza A infections, the framework yields statistically similar phase durations and production rates to standard approaches, but with the added ability to estimate quantities such as the probability of infection establishment and the number of IV entry events per successful cell infection.

Load-bearing premise

The entire conversion and all IV-unit parameter estimates rest on treating the ED assay and the infection experiments as sharing the same infection parameters, even though the infections used A549 cells and the ED assay used MDCK cells; the paper states explicitly that this assumption was not true.

Editorial extensions

If this is right

Measurements below the assay's detection limit (no infected wells at any dilution) are handled by $L_{ED}$ directly, instead of being set to the limit of detection as in Gaussian-residual likelihoods.
Expressing virus in IV units with the constraints $\gamma \le 1$ cell/IV and IV $\le$ vRNA removes the $(\gamma, \rho)$ degeneracy, giving finite, meaningful posterior intervals for each parameter.
The stochastic model and the ODE give statistically indistinguishable parameter estimates for the in vitro infections considered, so the 15-fold extra computation buys little for such well-inoculated infections.
Simulated ED assay noise alone reproduces the scatter among experimental triplicates, implying that replicate variability in such experiments comes largely from the measurement assay and could be reduced by more wells per dilution or finer dilution steps.
The estimated parameters imply that 1 SIN corresponds to roughly 1.1 to 5.8 infectious virions under the conditions studied, and that about 1 to 5 infectious virion entry events are needed for one successful cell infection.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The authors stop short of saying so, but the same logic implies that comparing viral titres across cell types, or between an assay and an infection done in different cells, carries an unknown conversion factor: equal TCID50 readings can correspond to very different numbers of infectious virions in the two settings.
A testable extension is to run the identical infection experiment with the ED assay performed in the same cell line as the infection; if the shared-parameter assumption is correct, parameter posteriors should match the authors' results, and if not, the discrepancy would quantify cell-type effects on establishment.
A concrete assay-design improvement follows directly from the paper: increasing the number of replicate wells per dilution and using finer dilution steps should narrow the measurement noise identified as the dominant source of inter-replicate scatter, and this can be verified in silico before spending reagents.
The framework should port to plaque and focus-forming assays, which share the 'one dose causes one infection' assumption; expressing those titres in IV units would require the same establishment-probability conversion, possibly with assay-specific parameters.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this one for Eqn. (2) and the LED likelihood. The central identity—measured SIN/ml equals IV/ml times the establishment probability of a single IV—is correct, and the likelihood for raw well counts is a genuine extension of your own midSIN work. The paper is also honest about its biggest problem, right in the Discussion: the A549 infections and MDCK ED assays do not share infection parameters, so the conversion factor used in the likelihood is likely wrong by a cell-type-specific constant. The paper says this is incorrect and quantifies an example (1 SIN in MDCK = 2 IV, in A549 = 10 IV), but then still uses the A549-derived establishment probability to convert MDCK ED outcomes and to impose the vRNA bound on IV. Every IV-unit parameter estimate—γ, ρ, β, P_Est—inherits that bias, and so does the claim that ED assay noise explains inter-replicate variability. That is a genuinely soft spot, not a manufactured one.

What's genuinely new and valuable: the systematic comparison of LNR vs LED and SIN vs IV units is overdue, and the physical constraints (γ ≤ 1 cell/IV, IV ≤ vRNA) are a clever way to partially identify parameters that otherwise are degenerate. The stochastic vs ODE comparison is less interesting in itself—the RNS trick works, but the result that stochasticity doesn't matter under high MOI is expected—though the derivation of the SM infecting time distribution is a nice piece of work.

The p-value table is a problem too. The paper claims LED is better than LNR, but none of the parameter differences reach significance, and the only significant finding (tinf) is about an ODE-vs-SM artifact, not the likelihood choice. For a paper recommending 'should be widely adopted,' the lack of a clear statistical win weakens the sales pitch.

There's no code or data shipped, which makes the 500,000-step MCMC and LED implementation harder to verify. That's fixable.

Overall: this is a paper for modelers who fit viral time courses from TCID50 data. The framework is worth having, but the empirical application is not trustworthy as-is. I'd send it to a serious referee with a request for a same-cell dataset or at least a sensitivity analysis, plus code/data sharing. The theoretical core is sound, so the right outcome is major revision, not rejection.

Referee Report

3 major / 4 minor

Summary. The paper develops a stochastic framework for interpreting TCID50 endpoint dilution (ED) assays in terms of infectious virions (IV) for in vitro virus infection models. The central result is Eqn. (2): the experimentally measured infection-causing dose concentration (SIN/ml) equals the actual infectious virion concentration (IV/ml) multiplied by the establishment probability P_V→Establishment of an infection initiated with a single infectious virion. The authors introduce a new likelihood (LED) based on the raw ED assay outcome (number of infected wells per dilution), compare it to a Gaussian residual likelihood (LNR), express the model virus variable in IV rather than SIN units with physical constraints, and compare ODE versus stochastic model predictions. Using published A549 influenza infection data with MDCK-based ED assays, they estimate parameters under four model variants and argue that ED assay stochasticity explains the inter-replicate variability observed in experimental infections. The paper includes substantial methodological derivations, including the extinction probability, the ED likelihood, and the infecting-time distribution, all validated against stochastic simulations.

Significance. If the results hold, the paper offers an important conceptual advance: it gives a principled, parameter-dependent conversion between a commonly measured infectivity unit (SIN/TCID50) and the biologically meaningful number of infectious virions, and it introduces a likelihood that correctly handles below-detection ED outcomes. The methodological strengths include careful stochastic derivations checked against simulation (e.g., Eqn. (1), the infecting-time distribution in Sec. IV.H), a deliberate comparison of likelihood choices, and physically motivated constraints in IV units that can improve identifiability of parameters such as γ and ρ. The paper is also refreshingly explicit about its limitations. However, the empirical demonstration of the framework rests on an acknowledged incorrect assumption that A549 infection experiments and MDCK ED assays share all infection parameters, which biases the quantitative parameter estimates and the conclusions built on them. The methodological components remain valuable and potentially correct, but the paper's current claims about IV-unit parameter values and the source of inter-replicate variability require revision.

major comments (3)

[§II.D and Discussion (third limitation)] The conversion C_model(t) = [V(t)/s] · P_V→Establishment(π) for the ED assay (Eqn. (2) and its use in the LED likelihood) relies on parameters π estimated from the A549 SC/MC infections being applied to the MDCK ED assay. The manuscript itself states, in the Discussion, that this shared-parameter assumption 'was not the case.' Because P_V→Establishment depends on cell-type-specific parameters (notably γ and β), any difference in establishment probability between A549 and MDCK cells multiplies the IV-to-SIN conversion by a constant. This directly biases the IV-unit estimates of γ, β, ρ, and all derived quantities (B, R0, tinf, P_V→Establishment) reported in Tables I and II, and consequently the identifiability discussion in §II.D and the noise-attribution conclusion in §II.E. To make the empirical claims load-bearing, the authors should either re-analyze the data with ED assays performed in the same cell type as the infections, or explicitly propagate the uncertainty in the cell-type-specific establishment probability and demonstrate that the qualitative conclusions are robust to plausible differences between A549 and MDCK.
[§II.D, Figure 8O and Table IV] The claimed individual identifiability of γ and ρ when [V] = IV relies on the physical constraint that IV cannot exceed vRNA, which in practice is imposed by a single data pair: the post-rinse SC infectious titre (10^7 SIN/ml, measured by MDCK ED) and the post-rinse total vRNA (10^7.76 vRNA/ml), giving P_V→Establishment ≥ 0.174. This constraint mixes MDCK-based SIN measurements with A549-based vRNA measurements, so it is exactly the type of quantity affected by the A549/MDCK parameter mismatch. If the true MDCK establishment probability differs from the A549-derived value, the inferred IV scale and hence the posterior bounds on γ and ρ change, weakening the conclusion that IV units resolve the γ–ρ degeneracy. The authors should show how the γ and ρ posteriors change under a range of plausible P_V→Establishment values in the ED assay, or obtain a same-cell-type measurement to anchor this constraint.
[§II.E, Figure 13] The conclusion that ED assay stochasticity alone explains the experimentally observed inter-replicate variability is based on simulating ED assays from SM-predicted IV time courses using the MAP parameter set (Table I, SM,IV,LED). Since that parameter set is obtained under the acknowledged A549/MDCK sharing assumption, the simulated ED noise band in Figure 13B is not an unbiased representation of the actual ED assay noise for the experimental samples. If P_V→Establishment in the MDCK ED assay differs from the A549-based value, the simulated SIN values and their dispersion shift, and the visual match with the experimental triplicates in Figure 13C is no longer meaningful. The authors should either verify this conclusion with same-cell-type data or temper the claim to state that the observed variability is consistent with ED noise under the (unverified) assumption of shared parameters.

minor comments (4)

[§II.C, Eqns. (4)–(5)] The notation C_model(t|π) is used both for the model prediction in SIN/ml (when [V]=SIN) and for the model prediction converted to SIN/ml via the establishment probability (when [V]=IV). The text explains the distinction, but the equations would be clearer if the two quantities were written as C_model^SIN(t) and C_model^IV(t)·P_V→Establishment.
[Table II] The p-values are described as one-tailed fractions of pairwise comparisons, which is not a standard hypothesis test. This should be stated in the caption so that readers do not interpret them as conventional two-sided p-values.
[§IV.F, Eqn. (17)] The derivation of the ED likelihood uses the small-x approximation ln(1−x) ≈ −x. The approximation is well justified because p = C_actual·V_vir with V_vir ≈ 5.2×10^−16 ml, but the paper should state the numerical magnitude explicitly so that the approximation's validity is transparent.
[§II.E, Figure 13] The grey band in panel (B) represents the 95% range of 10,000 simulated ED outcomes at each time point for the SM-predicted IV concentration, while the experimental points are three independent infection replicates. The text should note that this comparison does not include infection-to-infection variability beyond the mean IV trajectory, only ED sampling noise.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-consistency in the Eqn. (2) demonstration; the central likelihood and parameter-estimation framework is not circular.

self definitional [Section II.A, Eqn. (2) and Figure 2B/Figure 5]
"The random outcome for each well, i.e. whether infection took place or not (1 or 0, respectively), is determined based on the number of IV received by the well (V i,j 0) and the SM-predicted likelihood that infection will fail (PV → Extinction). ... Cmeasured = Cactual · PV → Establishment."

In the simulated ED assay, a well receiving V0 IVs is scored infected with probability 1 − (PV→Extinction)^V0, so for V0 = 1 the per-IV establishment probability is exactly PV→Establishment by construction. The 'demonstration' in Figure 5 divides the midSIN-estimated titres by PV→Establishment and recovers Cactual; this recovery is guaranteed by the simulation's own infection rule, not by independent data. Eqn. (2) is therefore a built-in identity of the simulation rather than an empirically established relation. This does not undermine the LED likelihood, which uses Eqn. (2) as a deliberate modeling conversion, but the paper's wording 'demonstrated' overstates the independence of this step.

full rationale

The central parameter-estimation methodology is otherwise self-contained. The LED likelihood is derived analytically from the binomial/Poisson sampling structure of the ED assay (Section IV.F), and the establishment probability PV→Extinction comes from the authors' prior peer-reviewed derivation (Quirouette et al. 2023), whose stated model assumptions do not include the present target results; it is a mathematical consequence of the stochastic model, not an unverified ansatz or fitted parameter. The acknowledged A549-versus-MDCK mismatch is explicitly stated by the authors as an incorrect simplifying assumption; it is a validity threat to the reported numerical estimates, but it is not circularity. The comparisons among likelihoods, units, and stochastic versus ODE models are in-sample methodological evaluations rather than fitted parameters renamed as predictions. The only notable circular aspect is the self-consistency check used to present Eqn. (2), which is minor because the relation is subsequently used as a modeling conversion rather than as an externally validated prediction.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The framework's central claim is the relationship between IV and SIN via PV→Establishment. That relationship depends on the stochastic infection model and its fitted parameters, and on several assumptions about the measurement process. The most fragile are the cross-cell-line parameter sharing and the vRNA-to-virion conversion, both acknowledged by the authors.

free parameters (6)

τE (mean eclipse phase duration) = 7 h (mode, SM-IV-LED)
Estimated via MCMC from SC/MC infectious and total titre data; enters the time course and the extinction probability.
τI (mean infectious phase duration) = 25 h (mode, SM-IV-LED)
Estimated via MCMC; affects burst size B = ρτI and the extinction probability.
ρRNA (total virus production rate) = 10^3.1 vRNA/(cell·h)
Estimated via MCMC; controls total virus time course and the V_IV ≤ V_RNA constraint.
ρ (infectious virion production rate) = 10^1.7 IV/(cell·h)
Estimated via MCMC when [V]=IV; correlated with γ and constrained by the vRNA inequality.
γ (cells infected per IV entry) = 10^-0.64 cell/IV
Estimated via MCMC; enters PV→I and the IV-to-SIN conversion.
β (IV loss rate due to cell entry) = 10^-6.8 ml/(cell·h)
Estimated via MCMC; enters the extinction probability and the infection time course.

assumptions (4)

domain assumption PV→Extinction (Eqn. 1), derived in Quirouette et al. 2023, correctly describes infection establishment in an ED well.
Invoked to convert IV to SIN in the likelihood and in simulations; if this model is wrong, the core conversion Eqn. (2) fails.
ad hoc to paper Infections (A549 cells) and ED assays (MDCK cells) share all infection parameters except cell number and volume.
Acknowledged as incorrect in the Discussion; required because the ED assay alone cannot identify its own parameters.
domain assumption One viral RNA measured by qRT-PCR corresponds to one physical virion, and sampled virions retain the infectivity they had during the infection (no freeze-thaw loss).
Used for the constraint V_IV ≤ V_RNA and for relating sample IV to infection IV; both are listed as limitations.
domain assumption The ED well infection process can be modeled by independent Bernoulli trials with per-virion success probability PV→Establishment.
Underlies the derivation of LED from the binomial likelihood in Cresta et al. 2021 (Methods IV F).

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Pith. "Pith review of Does the random nature of cell-virus interactions during in vitro infections affect TCID$_{50}$ measurements and parameter estimation by mathematical models?." pith.science (2026). https://pith.science/paper/K5G2WNSJ

@misc{pith2026241212960,
  author       = {Pith},
  title        = {Pith review of: Does the random nature of cell-virus interactions during in vitro infections affect TCID$_50$ measurements and parameter estimation by mathematical models?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5G2WNSJ}},
  note         = {Machine review of arXiv:2412.12960}
}

read the original abstract

Endpoint dilution (TCID50) assays cannot count the number of infectious virions (IVs), and instead are limited to counting the number of Specific INfections caused by the sample (SIN). The latter depends not only on whether virions are infectious, but also on the cells and the experimental conditions under which they interact. These interactions are random and controlled by parameters such as the rates at which IVs lose infectivity, enter cells, or fail to replicate following cell entry. Here, stochastic TCID50 assays are simulated to determine how the random number of infected wells relates to the parameters and the number of IVs in a sample. We introduce a new parameter estimation method based on the likelihood of observing a given TCID50 assay outcome given the model-predicted number of IVs in the sample. We then successively evaluate how parameter estimates are affected by the use of: 1) the new likelihood function vs the typical assumption of Gaussian-distributed measurement errors; 2) IV vs SIN units to express virus in the model; and 3) a stochastic vs an ODE model to simulate the course of a virus infection. Unlike previous methods, the new likelihood correctly handles measurements beyond the detection limits, and results in non-Gaussian distributions. Expressing virus using IV units makes it possible to impose physical constraints (e.g. one IV cannot infect more than one cell), and yields more biologically useful parameters (e.g. mutation emergence likelihood depends on the number of IVs, not SIN, produced). Using a stochastic rather than an ODE model we show that the variability observed between replicate in vitro virus infections is consistent with the level of stochasticity introduced by the TCID50 assay, which can be reduced through better assay design. The framework introduced herein offers several important improvements over current methods and should be widely adopted.

Figures

Figures reproduced from arXiv: 2412.12960 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (17 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: shows the probability (y-axis), computed using either LNR or LED, that the log10 SIN concentration measured at a particular time t takes on a certain value (x axis), given the three experimental measurements from replicate MC infections taken at different times. Figur…

**Figure 7.** Figure 7: A–D shows the ODE-predicted infectious and total viral titres over the course of the SC and MC infections, and Figure 7E–J the corresponding marginalized posterior distributions (MPDs) for the 6 parameters estimated based on either LNR or LED. All parameters have simil…

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

**Figure 13.** Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

**Figure 14.** Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]

**Figure 15.** Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]

**Figure 16.** Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]

**Figure 17.** Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p028_17.png]

**Figure 18.** Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]

**Figure 19.** Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]

**Figure 20.** Figure 20: shows that the normalized histogram of the infecting time, generated from 105 SM simulations, is in agreement with Eqn. (28), over a wide range of infection parameters. ACKNOWLEDGMENTS This work was supported in part by Discovery Grants RGPIN/3774-2022 (C.A.A.B.) from…

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