Yogurt CFU/g Estimator

Recipe Inputs

Estimate viable culture concentration from starter load, incubation, and refrigerated storage assumptions.

Ready.

Model Scope / Use At Your Own Risk

This model estimates yogurt fermentation trends, viable starter concentration, refrigerated post-acidification, and coarse texture proxies. Real batches vary by strain, milk treatment, inoculum age, solids, thermal history, and refrigerator profile.

Not food safety advice. This tool is not a pathogen-safety model, regulatory compliance tool, sterilization validator, production process authority, or substitute for pH/temperature/plate-count measurement.

No warranty or liability. The tool is provided as-is. The creators and maintainers assume no responsibility or liability for errors, omissions, spoiled batches, unsafe food handling, failed processes, compliance issues, losses, or decisions made from this tool or its output.

Verify independently. Use calibrated thermometers, pH meters, validated procedures, published references, appropriate food-safety guidance, and qualified professionals before using results for safety, production, regulatory, commercial, clinical, or other consequence-bearing decisions.

Milk And Starter

Milk amount Milk unit Starter amount Starter unit Starter mode Substrate mode Additional solids

Starter Viability

Total starter CFU/g S. thermophilus split (%) L. bulgaricus split (%)

Incubation

Incubation time (hr)

Temperature

°C °F

Storage / Post-Acidification

Refrigerated storage (day)

`0` disables refrigerated storage. Storage pH is anchored to published refrigerated pH curves near 4 C with a hybrid absolute/delta fit, but is suppressed when incubation does not reach a viable acidified yogurt state. Storage CFU uses a separate published retention envelope derived from refrigerated viable-count datasets.

Advanced Model Parameters

These inputs are editable model parameters. The optimum-temperature defaults are literature-anchored starting points; lag, maximum slope, and plateau are tunable model values rather than strain-certified constants.

S. thermophilus

Lag (hr) Max slope (log10/hr) Plateau log10 CFU/g

Optimum temp

°C °F

L. bulgaricus

Lag (hr) Max slope (log10/hr) Plateau log10 CFU/g

Optimum temp

°C °F

Initial Total -- --

Final Total -- --

Stored Total -- --

Final Low / High -- --

Final Modeled pH -- --

Stored pH -- --

Storage ΔpH -- --

Storage CFU Shift -- --

Set Proxy -- --

Syneresis Proxy -- --

Batch Mass -- Assumes milk density 1.03 g/mL

Starter Loading -- --

Dominant Endpoint Species -- --

Low/high bands are deterministic scenario bounds. The plotted curves remain the likely scenario.

Growth Curves

Likely deterministic `CFU/g` estimate over incubation time. Use this as a process estimate, not a direct lab substitute. Hover for 15-minute snapshots.

Total S. thermophilus L. bulgaricus

pH / Acidification Proxy

This panel tracks an internal modeled pH state over incubation time. It uses a 42 C incubation backbone built from digitized published modified-Gompertz fits, then applies smaller secondary corrections for temperature, inoculum, substrate mode, and thermal stress. It remains a fermentation-state model rather than a direct pH meter reading or titratable-acidity assay.

Solid magenta line: modeled pH trajectory during incubation. It drops as realized coculture growth accumulates and slows further growth later in fermentation.
Dashed guide at pH 4.6: literature-facing reference threshold. Ge et al. compared time to reach pH 4.6 across starter ratios, so this guide is used in the validation table.
Species response: the model makes S. thermophilus more pH-sensitive than L. bulgaricus at low pH so acidification and final viability are not treated as the same outcome.

Anchor And Direction References

These papers serve two roles in the estimator: direct incubation-phase pH anchors and directional constraints. They support the displayed pH curve and refrigerated storage layer, but they do not yet justify a fully mechanistic acid-production or rheology model.

Linares et al. 2016: supports a 1:1 coculture anchor at 42 C with pH below 5 after 5 hr and about 4.4 by 8 hr.
Popovic et al. 2020: supports a 1:2 mixed-starter anchor at 42 C with pH about 4.74 after 5 hr.
Li et al. 2023: supports faster acidification at 42 C than 37 C even when final viable LAB are higher at 37 C.
Dan et al. 2023: supports starter-ratio effects on pH and titratable acidity during fermentation.
Ge et al. 2024: supports comparing starter ratios by time to pH 4.6 and by later storage/post-acidification behavior.
Salmazo et al. 2023: supports explicitly treating temperature and pH as growth-response dimensions rather than leaving pH out of the model.

pH Modeling Details

Incubation-phase pH is modeled with a modified Gompertz acidification curve rather than a freehand proxy. This follows the common sigmoidal treatment of yogurt acidification in the literature and exposes interpretable parameters instead of only a shape.

pH(t) = pH0 - A * exp(-exp(((mu * e) / A) * (lambda - t) + 1))

pH0: initial milk pH before appreciable fermentation.
A: total modeled pH drop, where A = pH0 - pH_inf.
mu: maximum acidification rate in pH units per hour.
lambda: apparent acidification lag before the sharp pH drop.
pH_inf: fitted terminal incubation pH for the displayed ratio, temperature, substrate mode, and inoculum level.

In the estimator, mu, lambda, and pH_inf are driven by a fitted response surface over temperature, starter ratio, substrate mode, and inoculum strength. The resulting pH curve then feeds back into the species growth step through species-specific low-pH inhibition, so the pH model affects both the chart and the final CFU estimates.

Scope: incubation-phase fitting plus refrigerated post-acidification pH drift. Refrigerated CFU retention is handled by a separate published retention envelope, and the pH-growth feedback is not yet a full Baranyi-Roberts plus Rosso/cardinal coupled model.

Derivation detail: the 42 C ratio effect comes from interpolation between published ratio anchors such as 1:2, 1:1, 2:1, 19:1, and 100:1, rather than from unnamed ratio-basis features. Temperature, inoculum, substrate mode, and thermal suppression are still applied afterward as smaller empirical corrections.

The values below are fixed for the displayed formulation. Only t varies along the plotted curve.

Displayed Gompertz Parameters

Parameter	Reported value	How it is derived
No pH model values available yet.

Displayed Driver Terms

These intermediate terms show which published ratio anchors are being blended for the displayed formulation and which smaller empirical corrections are still applied afterward.

Driver term	Reported value	Meaning
No driver terms available yet.

Modeling references: De Brabandere & De Baerdemaeker 1999 for acidification descriptors and Salmazo et al. 2023 for temperature/pH-dependent starter growth modeling.

Substrate / Driver Proxy

Model-driver view only. These are internal model quantities, not direct laboratory assay outputs.

Orange line: internal substrate reserve proxy (%). It represents the remaining fraction of modeled accessible fermentation headroom. It is conceptually tied to nutrient-limited batch-growth ideas, but it is not a direct lactose concentration measurement.
Green dashed line: internal growth-limitation factor (%). It is a dimensionless multiplier applied to the unconstrained next-step growth increment. `100%` means no substrate-based suppression; `0%` means full suppression of the next-step increment. This is conceptually analogous to substrate-limitation terms used in Monod-type and Logistic-Monod growth models, but it is not itself a direct Monod fit.
Lactose-hydrolyzed milk: changes accessibility and late-stage constraint behavior in a directional, literature-informed way, mainly to reflect the reported increase in L. bulgaricus cell numbers in hydrolyzed milk coculture.

A green dashed value of `65%` means about `65%` of the next-step growth increment is allowed through, not that growth is increased by `65%`.

Concept References

These references support the general nutrient-limited growth framing used here. They do not directly validate the exact orange and green internal proxy curves shown by the estimator.

Growth Kinetics of Suspended Microbial Cells: distinguishes nutrient-limited biomass yield constraints from substrate-controlled growth-rate terms and summarizes Monod-type growth modeling.
Hybrid Logistic-Monod Cell Growth Model: shows how batch-growth limitation can be expressed with a combined substrate and carrying-capacity style formulation.
The Last Generation of Bacterial Growth in Limiting Nutrient: summarizes the Monod-law basis for nutrient-limited bacterial growth.
Yamamoto et al. 2021: supports the directional difference between standard and lactose-hydrolyzed milk in the estimator's substrate mode.

Storage / Post-Acidification

This panel treats refrigerated storage as a separate post-fermentation phase. It does not extend the incubation Gompertz curve. Storage pH uses a hybrid absolute/delta fit to published storage curves and is gated off when the incubation endpoint is not a viable acidified yogurt state. Storage CFU is modeled as a separate published retention envelope because the available literature does not support reducing viability loss to pH alone.

Blue line: modeled refrigerated pH after incubation is complete, plotted against storage day.
Storage delta pH: the net pH change from the modeled fermentation endpoint to the modeled stored pH at the selected day. The internal fit uses published absolute storage pH values and published storage deltas so endpoint pH residuals do not carry through storage one-for-one. If the endpoint remains under-acidified or heat-collapsed, this storage pH drift is suppressed.
CFU retention chart: total and species-specific viable-count retention during refrigerated storage, plotted in log10 CFU/g.
Scope: refrigerated pH drift plus published CFU retention envelopes near 4 C. It is not a user-specific refrigerator temperature model or strain-specific survival fit.

Storage References

Popovic et al. 2020: provides refrigerated pH drift for a mixed 1:2 starter through 28 days.
Ge et al. 2024: provides ratio-sensitive refrigerated pH drift across 1:1 through 100:1 and identifies 19:1 as a low-post-acidification ratio.
Hamann and Marth 1984: provides refrigerated viable-count curves for S. thermophilus, L. bulgaricus, and total yogurt organisms at 5 C and 10 C.
Anbukkarasi et al. 2014: provides short-storage pH, acidity, lactose/galactose, and starter counts at 4 C.

Displayed Storage Model Values

Term	Value	Definition
No storage model values available yet.

Texture / Set Proxy

This panel reports a coarse set and syneresis proxy. It uses the modeled acidification state plus Ramchandran and Shah texture anchors, so it should be read as a screening estimate rather than a direct texture-analyzer or rheometer output.

Set score: normalized proxy driven mainly by endpoint pH, time after pH 4.6, storage maturation, added solids, and over-acidification penalty.
Firmness proxy: mapped against Ramchandran and Shah Figure 2 low-fat yogurt firmness anchors.
Syneresis proxy: mapped against Ramchandran and Shah Figure 3 spontaneous whey separation anchors.
Yield stress proxy: mapped against Ramchandran and Shah Table 4 Herschel-Bulkley yield-stress anchors.

Texture References

Ramchandran and Shah 2009: provides firmness, whey separation, yield stress, EPS content, and relative starter-survival context during 4 C storage.

Displayed Texture Values

Output	Value	Interpretation
No texture proxy available yet.

Texture Proxy Coefficients

Term	Value	Definition
No texture coefficient values available yet.

Estimated Batch Profile

Initial and incubation-endpoint likely concentration estimates plus deterministic low/high endpoint bands for the displayed inputs above. Stored totals are summarized in the metric cards when storage is enabled.

Species	Initial CFU/g	Incubated CFU/g	Stored Endpoint CFU/g	Incubated Low / High	Displayed Endpoint Share

Literature Back-Checks Collapsed by default. Rows compare the model against published anchor points, trend checks, and envelope checks. They show agreement and known gaps, not batch-specific prediction.

Check	Benchmark	Model	Status

Assumptions

CFU/g is viable culturability, not biomass mass.
Growth is capped with a bounded lag-plus-plateau model.
Temperature effects are simplified with separate growth-speed and viability-retention penalties.
Modeled pH is a fitted fermentation-state proxy, not a direct pH-meter reading.
Final mass is treated as the initial mixture mass for v0.1.
Substrate is modeled as a fermentation proxy, not a direct lactose or metabolite assay.
Storage mode applies a hybrid absolute/delta refrigerated pH layer plus published CFU retention envelopes; both are anchored near 4 C, and storage pH drift is gated by endpoint fermentation success.

Warnings

No warnings.

Validation Notes

The inoculum back-calculation is the most constrained output in the current model.
The incubation-phase pH curve uses a 42 C Gompertz backbone fitted to published incubation curves, but fast-acidifying strain systems can still fall outside the current fit.
Storage/post-acidification combines refrigerated pH drift, published viable-count retention envelopes, and a coarse texture/set proxy; these are screening estimates, not strain-specific predictions.
The Literature Back-Checks table summarizes the main published anchors, residual size, and current gaps.

Literature Basis

Reference: Dan et al. 2023 Note: Starter ratio changes viable counts and fermentation outcomes.
Reference: Ge et al. 2024 Note: St:Lb ratio sweeps materially change fermentation behavior, and the optimum need not be 1:1.
Reference: van de Guchte et al. 2006 Note: Describes L. bulgaricus adaptation to the milk environment and explicitly discusses protocooperation with S. thermophilus; included as biological background rather than as a direct kinetic-fit source.
Reference: Yamamoto et al. 2021 Note: Lactose state changes culture behavior, so substrate should not be treated as a fixed generic factor.
Reference: Linares et al. 2016 Note: Gives a concrete 1:1 coculture acidification anchor at 42 C, including pH below 5 after 5 hr and about 4.4 by 8 hr.
Reference: Popovic et al. 2020 Note: Gives a concrete 1:2 mixed-starter pH anchor of about 4.74 after 5 hr at 42 C.
Reference: Li et al. 2023 Note: Temperature and ratio affect viable populations; acidification optimum and viability optimum can differ.
Reference: Salmazo et al. 2023 Note: Temperature and pH jointly shape growth response, so the fermentation model should not leave pH completely external.
Reference: Li et al. 2011 Note: A shift from 42 C to 50 C triggers a heat-shock response in S. thermophilus, so 50 C should be treated as stress territory rather than normal fermentation.
Reference: Gouesbet et al. 2002 Note: Shows severe heat-shock injury in L. bulgaricus at 65 C, including a rapid early drop in viable survivors and non-monotonic recovery behavior; used here as a high-temperature collapse anchor rather than a normal fermentation condition.
Reference: Shu et al. 2016 Note: Goat-yogurt adjunct study showing that temperature, strain ratio, and inoculum size affect viable counts in a probiotic system containing L. acidophilus and B. bifidum; included only as out-of-scope adjunct context.
Reference: Varga et al. 2014 Note: ABT-type fermented camel/cow milk storage study including B. animalis subsp. lactis BB-12; included only as adjunct-culture storage context outside the default model.

Modeling decision informed by the references above:

Scope remains with classical yogurt: S. thermophilus + L. bulgaricus. Additional probiotic species should be a separate adjunct mode, not silently folded into the default model.