Time-Integrated Micronutrient Adequacy and Chronic Safety: A Tissue-Reservoir Model and a Cited One-Compartment Kinetic Model with Homeostatic Absorption

Manuscript 5 of 8 in the BARC implementation series · Release B-2.1.0 · September 2026

Author. Prasanna M, Growlrr Foods Pvt Ltd, Chennai, India. Correspondence: support@growlrr.com. ORCID: 0009-0003-6812-347X.

DOI. 10.5281/zenodo.22647812 — the concept DOI, which always resolves to the current version of this record · Published. 8 September 2026 · Licence. CC BY-SA 4.0 · Status. Not peer-reviewed as of the date of publication, 8 September 2026.

Competing interests and funding. The author is the founder of Growlrr Foods Pvt Ltd, a pioneer and proponent of the BARC Protocol for pet foods. BARC stands for Biologically Appropriate, Residually Corrected, and is conceptualised as an open protocol under a Creative Commons Attribution–ShareAlike licence, for like-minded collaborators around the world to formulate and develop. Growlrr Foods Pvt Ltd develops and commercialises a BARC-based residual-correction product, BowlBalancer. The work was funded entirely by the company; no external funding was received. No independent laboratory, institution or reviewer has verified the results reported here. Readers should weigh these facts alongside the analysis, which is published in full — including its limitations and open items — precisely so that it can be checked.

Provenance of the results. The tissue-arm figures here were re-run on the current engine build and current formulation for this deposit. They supersede an earlier run of 12 August 2026 that predated the calcium re-architecture [Manuscript 1] and had been carried forward unchanged. Differences are small — copper's reservoir 5th percentile moves 119 → 122% in the dog, and the dog's third-tightest nutrient changes — as expected, since copper is the binding nutrient and the calcium change did not touch it. The superseded figures should not be quoted.

Framework. This work applies BARC — Biologically Appropriate, Residually Corrected — an open protocol for correcting a biologically appropriate fresh base to a published nutrient standard, released under a Creative Commons Attribution–ShareAlike licence (CC BY-SA 4.0) at growlrr.com/barc-protocol (concept DOI 10.5281/zenodo.22648597) so that others may formulate and develop against it. BARC is the protocol; the correction specified and validated here is one implementation of it.

Terms. A glossary of the statistical and nutrition vocabulary used across this series — requirement, floor, safe upper limit, Wilson bound, N-limited, draw, stress multiplier, retention factor, trituration and the rest — is in the series cover note (10.5281/zenodo.22647606), section 4.1.

Computational tooling and AI assistance. The model is implemented in TypeScript under Node.js with no external numerical dependencies. Claude (Anthropic) was used as a coding and analysis assistant in developing this work and preparing these records; it was not used to generate results, all of which were produced by executing the model. See the series cover note (10.5281/zenodo.22647606), section 6.1.

Plain-language material. A non-technical account of hepatic buffering and copper is at growlrr.com/pet-home-cooked-recipes-blog/hepatic-buffering-model. The public reference list behind these claims is at growlrr.com/pet-nutrition-references.

Abstract

A daily adequacy check asks whether one bowl meets a requirement. It cannot answer the two questions that matter over a feeding lifetime: whether adequacy survives the day-to-day variation of a real diet once tissue stores are allowed to buffer it, and whether the nutrients that accumulate stay bounded rather than drifting toward a ceiling.

This paper reports two models addressing the time axis, and gives the parameter set of the second in full.

The reservoir model buffers each nutrient by an exponential moving average parameterised by its biological half-life, with a smoothing constant α = 1 − e^(−1/τ) and τ ranging from 7 days for thiamine to 180 days for cobalamin. Because the correction's contribution is constant day to day while the grocery base varies, integrating over the storage horizon can only raise the effective floor relative to a single-day snapshot. Across 2,000 simulated animals over 365 days per species, the reservoir floor returned a Wilson-C95 lower bound of 99.808% for both species with no sub-90% tail at any body weight, copper the tightest nutrient at a 5th percentile of 122% (dog) and 135% (cat) of requirement.

The kinetic model answers the standard objection that a moving average is a filter with no physiology. It replaces the average with a one-compartment model carrying explicit homeostatic absorption regulation and first-order clearance, and it is a strict generalisation: setting the absorption exponent p = 1 recovers the reservoir exactly, so the kinetic model can add mechanism but cannot contradict the reservoir's floor result. Its parameter set is tabulated in §4: fourteen nutrients, with the source and confidence grade for each half-life and absorption exponent, species extrapolations marked, and values not found in the literature left blank rather than estimated. Nutrients without a citable absorption curve run at p = 1, taking no regulation credit.

The single parameter at which the kinetic model departs from the reservoir is copper's absorption exponent, p = 0.34, derived from a human stable-isotope tracer series. Under it, modelled tissue copper compresses toward its set-point from both directions: the feline median falls from 153% to 116% of requirement and the maximum to 125%, while the 5th percentile remains well above the floor.

Section 7 sets that result against the veterinary literature on copper-associated hepatopathy and states its limits.

Keywords: BARC Protocol · residual correction · homemade dog food · homemade cat food · home-prepared diet · toxicokinetics · tissue reservoir · exponential moving average · one-compartment model · homeostatic absorption · biological half-life · hepatic copper · chronic safety · companion animal nutrition · parameter provenance

1. Introduction — why the time axis needs its own treatment

A stochastic single-day model [Manuscript 2] asks whether a bowl drawn from a realistic distribution clears every floor. It is the right question for manufacturing variance and for a chaotic kitchen, and it is deliberately conservative: it treats each day as if the animal began that day empty.

Animals do not begin each day empty. Tissue reservoirs buffer intake across horizons that differ by an order of magnitude between nutrients — days for the labile water-soluble vitamins, months for cobalamin and iron. Two consequences follow, and they run in opposite directions.

On the floor side, buffering helps. A single low day is absorbed by the store, so a diet that occasionally dips can still be adequate on the horizon that matters physiologically. Any model that ignores this understates adequacy.

On the ceiling side, buffering is the hazard. The same storage that protects against a low day is what allows a nutrient to accumulate. Fat-soluble vitamins and certain trace minerals are the relevant risks, and a diet that is safe on day one can drift over months.

A single-day model cannot see either effect. This paper reports the two models that do.

They sit within a wider validation stack in which each model isolates one axis before the axes are recombined: a smooth-day snapshot establishes the single-day floor; the reservoir model adds noise and storage; rotation models add day-to-day protein alternation. The rotation arm is reported separately, and the reservoir result below is on the smooth-day construction, because prior work on this lineage established that the rotation-versus-smooth reservoir difference is approximately zero — a reservoir averages over its storage horizon, so day order does not matter for a long-τ nutrient.

2. Methods — the reservoir model

2.1 Specification

Each nutrient is buffered by an exponential moving average parameterised by its biological half-life:

tissue_t = (1 − α) · tissue_{t−1} + α · intake_t, with α = 1 − e^(−1/τ)

where intake_t is the delivered amount on day t as a percentage of requirement, and τ is the nutrient's storage time constant in days.

Intake construction. Each daily draw is a full evaluation of the delivered nutrient vector on the dose ladder: the correction's contribution is constant day to day, being a manufactured unit, while the owner's fresh bowl and the retention and assay variance are redrawn each day from the lognormal noise model of Manuscript 2. The reservoir therefore buffers only the grocery-borne variance — which is the physically correct behaviour, and also the conservative one, since it grants the correction no smoothing it does not have.

The gate. The floor is the minimum tissue percentage of requirement across the reservoir nutrients for a given animal. An animal passes if that floor is at or above 90% of requirement, expressed against the NRC recommended allowances [8]. The release statistic is the Wilson score lower bound at z = 1.96 [9].

2.2 The steady-state estimator, and why it is exact rather than a shortcut

The moving average is an unbiased low-pass filter, so its fixed point is the expectation of daily delivered percentage of requirement. That expectation is estimated by averaging J = 64 independent smooth-day draws per animal.

This is not an approximation of the reservoir; it is the reservoir's steady state, for any τ. What the estimator drops is the day-to-day jitter around that mean, whose standard deviation is the daily standard deviation divided by √(2τ) — a factor of 13.4 for the copper binder, and larger for every longer-τ nutrient. The jitter is therefore negligible precisely where the store is deep, which is where the reservoir matters.

2.3 The construction gate

A model of this kind fails silently if it is run on an incorrectly constructed daily vector, so the reservoir carries a built-in check: the day-zero floor from the reservoir loop must equal the smooth-day floor from the single-day model. At day zero the store has had no time to integrate, so the two must agree by construction, and any disagreement indicates the reservoir is running on a different — that is, wrong — daily vector.

Measured: dog 135.5 against 135.3, cat 153.3 against 153.5. The gate passes.

This check earned its place. Evaluating the daily vector outside the engine's own construction path returns approximately 112% rather than 133%, because it misses the runtime sizing the engine applies. The reservoir arm therefore has to run inside that path, and the day-zero identity is what proves it does.

2.4 Result

Full envelope, 2,000 simulated animals × 365 days per species:

The deepest trough never breaches requirement. Beyond the floor test, the model records for each nutrient the minimum the tissue store reaches at any point across the 365 days, and reports the 5th percentile of that minimum across animals. It is 100% of requirement for every nutrient, in both species. For at least 95 of every 100 simulated animals, no tracked nutrient's store dipped below full requirement on any day of the year — a materially stronger statement than clearing a 90% floor, and one a single-day model cannot make.

Copper is the tightest reservoir nutrient in both species, sitting roughly 30 percentage points above the floor. That is by design: copper is the deliberately ceiling-guarded nutrient of this formulation (§6), so it is held closest to the floor from above, and every other nutrient sits higher.

The identical 99.808% for both species is the N = 2,000 ceiling with zero failures, not a measured difference between them. It is also, coincidentally, the value of the feline adult snapshot figure reported at 10,000 draws in Manuscript 2 §4.2; the two are unrelated quantities that agree to three decimal places by arithmetic accident, since n/(n + z²) at n = 2,000 is 99.8083%. With no observed failures the Wilson bound reduces to n/(n + z²), so it reports the animal count and nothing else. The informative results in this table are the zero sub-90% tail and the 5th-percentile margins.

3. Methods — the kinetic model

3.1 Rationale

The standard objection to a reservoir average is fair: it is a linear, symmetric, first-order filter, and it has three physiological gaps. Fill and drain are governed by a single τ, though real tissues fill and clear at different rates. Intake is treated as fully absorbed regardless of how full the store already is. And there is no explicit excretion pathway, so the model cannot speak mechanistically about the ceiling at all.

The reservoir is the right instrument for the floor — a store can only raise an effective floor — which is what §2 uses it for. The ceiling needs a model with a mechanism.

3.2 Specification

Tissue store S is in normalised units, where S = 100 when intake exactly meets requirement at reference absorption and clearance:

S_{t+1} = S_t + k_out · ( netAbs(I_t) − S_t ) netAbs(I) = 100 · (I / 100)^p k_out = ln 2 / t½

• I_t — delivered intake on day t, as a percentage of requirement: the same per-nutrient draw the single-day model produces.
• p — the homeostatic absorption exponent. p < 1 means fractional absorption falls as intake rises and rises as intake falls. It is derived from a cited absorption-versus-intake curve where one exists, and is set to 1 — no regulation credit — wherever one does not.
• k_out — first-order clearance from the cited whole-body biological half-life.

Because the steady state is independent of clearance (§3.4), the implementation evaluates it directly rather than integrating the transient, so the clearance constants do not enter the figures reported here; they would govern a depletion or repletion trajectory, which is not what is reported. The reservoir's smoothing uses the storage time constant τ of §2.1. τ and the biological half-life t½ are related but distinct and are tabulated separately.

3.3 The strict-generalisation property

Setting p = 1 gives netAbs(I) = I, and the recursion becomes

S_{t+1} = (1 − k_out) · S_t + k_out · I_t

which is exactly the reservoir average of §2.1 with α = k_out. The reservoir model is therefore the p = 1 special case of the kinetic model.

This matters for what the paper can claim. The kinetic model can only add the regulation and clearance detail the reservoir omits; it cannot contradict the reservoir's floor result. Any nutrient that cannot be parameterised from the literature degrades gracefully to the reservoir rather than to a guess.

3.4 Steady state, and why regulation is the point

The fixed point solves netAbs(I_mean) = S*, giving

S* = 100 · (I_mean / 100)^p, independent of k_out.

At p = 1 the steady state is the mean intake — the reservoir's answer. At p < 1 a deficient animal is lifted toward requirement and an over-supplied one compressed toward it. Clearance sets the transient speed and the jitter amplitude; it does not move the steady state.

A moving average cannot represent this behaviour. Under the fitted exponent the model has a set-point that bounds both the floor and the ceiling through the same term. Whether dogs and cats regulate copper absorption with this exponent is not established here: it is fitted to three human tracer points and applied cross-species.

4. The parameter set, in full

Every parameter, its source, and its confidence. Every parameter, its source, and its confidence grade.

Confidence legend. ⬤ primary and species-native · ◧ primary or authoritative, but a species extrapolation (human or rodent applied to dog or cat) · ○ secondary or review · ⚠ unverified to a primary source.

Thirteen of the fourteen nutrients run at p = 1. The model takes absorption-regulation credit for copper alone, and nowhere else.

5. What the parameter set does not contain

• Copper's half-life and its absorption curve are human values applied to dogs and cats. They are the two parameters the whole kinetic argument rests on.
• A feline hepatic-copper reference range was not found. The feline copper parameters are therefore extrapolations built on extrapolations.
• Whole-body half-lives for dogs and cats were not found for iron, zinc, manganese, selenium, vitamins A and D, thiamine, riboflavin, pyridoxine and folate. Human or rodent values are used and flagged as such in the table.
• Species-native values exist for only two entries: canine pantothenate, and feline serum cobalamin.
• The manganese half-life is unverified to a primary source. It is used only for a non-binding nutrient and the result is insensitive to it, but it should not be quoted as established.
• Safe upper limits were accessed through secondary summaries of the primary requirement volume and should be verified against it before any label claim.
• Two real protective mechanisms are deliberately uncredited — thyroid autoregulation of iodine, and canine urinary vitamin A clearance — so the reported margins are conservative relative to a model that credited them.

6. Results — copper

Same run as §2.4. The reservoir columns are the p = 1 case; the kinetic columns apply p = 0.34.

Homeostatic absorption compresses copper toward the set-point from both directions. The feline median falls from 153% to 116% of requirement and the modelled maximum to 125%; the 5th percentile stays comfortably above the floor rather than being dragged toward it. Under a deliberate three-fold over-supply the modelled tissue copper plateaus near 155% of requirement.

Two conservatisms are built into that result. All thirteen non-copper nutrients run at p = 1, so the model never lowers a floor anywhere. And the uncredited mechanisms of §5 mean the true ceiling protection is greater than modelled.

7. Discussion — copper in context, and what this does and does not claim

Copper deserves separate treatment, because the veterinary literature has moved on it and a modelling result could easily be read as saying something it does not.

Center et al. [7] argue that copper-associated hepatopathy in dogs has become more common, and link it to increases in bioavailable copper in commercial diets, calling for the guidelines on copper content to be reconsidered. That position is not in tension with this work, and it is worth being exact about why.

What this paper claims. In this formulation copper is deliberately held at a ceiling — it is the nutrient kept closest to its floor from above, so that every other nutrient sits higher. The kinetic model supplies the mechanism for why that cap behaves as intended over a feeding horizon: regulated absorption compresses the modelled tissue concentration toward a set-point rather than letting it track intake linearly upward.

What this paper does not claim. It does not claim that homeostatic absorption makes dietary copper excess safe in general. That would be precisely the inference Center et al. caution against, and the modelling here does not support it: the compression demonstrated in §6 operates on an input that is already capped by formulation, and it is derived from a human tracer series across a modest intake range, not from an animal in which regulation has been overwhelmed.

The narrower reading: a formulation that caps copper by design is responsive to the concern in that literature, and the kinetic model explains why the cap holds rather than asserting that a cap is unnecessary.

8. Scope and limitations

• These are models. No animal was fed, no tissue or serum copper concentration was measured, and no finished lot was assayed. A non-invasive household observation study is planned to begin closing this; see the series cover note, Limitations and future work.
• N = 2,000 animals caps the reservoir statistic at 99.808% with zero failures. That figure reports the animal count, not a difference between species.
• Species extrapolation is pervasive (§5), and it is the reason the ceiling argument rests almost entirely on copper.
• The kinetic model takes regulation credit for one nutrient. For the other thirteen it is the reservoir model with a different name, and should be read as such.
• The reservoir result is on the smooth-day construction. The rotation arm is reported in Manuscript 2; prior work on this lineage established the reservoir difference between them is approximately zero, which is why the smooth-day arm is used here.
• A tissue model is not a clinical endpoint. Reaching a modelled steady state below a safe upper limit is not the same as demonstrating an absence of harm in an animal.
• Healthy animals only. Not a clinical or therapeutic system, and no disease claim is made.
• Not peer-reviewed as of the date of publication (8 September 2026). Should it later be reviewed or independently replicated, that will be recorded in a subsequent version of this record.

9. Conclusion

A daily adequacy check cannot answer whether adequacy holds across months or whether any nutrient drifts toward a toxic reservoir under continuous feeding. Two models are used to address that: a tissue-reservoir model buffering each nutrient by its biological half-life, and a one-compartment kinetic model adding homeostatic absorption regulation and first-order clearance. The second is a strict generalisation of the first — setting the absorption exponent to unity recovers the reservoir exactly — so it can add mechanism but cannot contradict the reservoir's floor result.

Across 2,000 simulated animals over 365 days per species, the fifth percentile of each nutrient's deepest sustained trough was at or above requirement for every nutrient in both species, and no nutrient accumulated toward its safe upper limit. Under the fitted absorption exponent, modelled tissue copper compresses toward a set-point from both directions.

The parameter table is the substance of this paper, and its weakest entries are named rather than smoothed over. The two copper parameters on which the kinetic argument rests are human tracer values applied cross-species, as are several whole-body half-lives. Nutrients without a citable absorption curve are run at an exponent of unity, taking no regulation credit. Whether dogs and cats regulate copper absorption with this exponent is not established here.

References

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How to cite

Prasanna M. (2026). Time-Integrated Micronutrient Adequacy and Chronic Safety: A Tissue-Reservoir Model and a Cited One-Compartment Kinetic Model with Homeostatic Absorption (Manuscript 5 of 8, Release B-2.1.0). Growlrr Foods Pvt Ltd. Zenodo. https://doi.org/10.5281/zenodo.22647812