Evaluation of Multi-Breed Allometric Energy Formulations Against Published Calorimetric Expenditure Data in Canines

Manuscript 3 of 8 in the BARC Protocol specification 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.22647612 — 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 Not peer-reviewed as of the date of publication, 8 September 2026.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.

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. The resulting feeding quantities, by body weight, are set out at growlrr.com/dog-feeding-chart-by-weight. The public reference list behind these claims is at growlrr.com/pet-nutrition-references.

Abstract

This paper discloses in full the allometric energy model that sizes the daily nutrient target for a residual micronutrient correction, and traces the provenance of every constant in it.

Daily maintenance energy is sized as MER = 98 · BW^0.64 · M for dogs and 90 · BW^0.67 · M for cats, where M is a five-axis additive metabolic multiplier clamped per species. The canine exponent 0.64 is a directly measured value, taken from whole-body calorimetry in 22 dogs of seven breeds spanning 5.8–48.8 kg [1]. It is not derived from, and should not be conflated with, Kleiber's interspecific 0.75 [2]; the Kleiber relation is used only as a consistency check. Only the exponent belongs to the cited source: the coefficient 98 and the entire multiplier model are this framework's own maintenance construct, anchored so that a 20 kg reference dog at M = 1.5 sizes to 1,000 kcal/day.

We report an external triangulation of the multiplier envelope against three independent clinical references, quoting the clinical band rather than a favourable point within it, and marking the rows the product declines to serve.

The paper then sets out the property that makes a single fixed-composition correction viable across a 3–110 kg range. The correction scales at BW^0.667, matched to the energy exponent, so delivery expressed as a percentage of requirement drifts by only BW^0.027 — approximately ±5% across a 37-fold weight range, and ±1.5 percentage points within the reference band. We show what other exponents would have cost, record two alternative scaling schemes that were examined and rejected, and explain the consequence for sedentary and ageing animals: because the correction is dosed by weight class rather than by energy, an animal eating 20% less receives 100% of its correction, and micronutrient density per calorie rises rather than falls.

Keywords: BARC Protocol · residual correction · homemade dog food · homemade cat food · home-prepared diet · allometry · maintenance energy requirement · metabolic scaling · Burger and Johnson · Kleiber · intra-species exponent · micronutrient dilution · weight-class dosing · companion animal nutrition

1. Introduction and methods — the model

For a healthy adult animal at body weight BW in kilograms:

• Dog: MER = 98 · BW^0.64 · M (kcal/day)
• Cat: MER = 90 · BW^0.67 · M (kcal/day)

The coefficient cannot be read without its multiplier band. M is not centred on 1 for both species, and the two conventions are not interchangeable — quoting a coefficient alone is meaningless, since 90 × 1.1 and 100 × 1.0 differ by 1%.

These two anchors are the engine's built-in self-checks: dog 20 kg at M 1.5 gives 1,000 kcal and 12 units; cat 4 kg at M 1.10 gives 250 kcal and 5 units. The cat anchor is set near the reference active adult cat of approximately 250 kcal/day at 4 kg [3] rather than at a sedentary minimum, with the band spanning sedentary to active; the neutered indoor household cat at M = 1.00 sizes to 228 kcal.

Served energy versus the floor. A delivery buffer is applied to the served ration but never to the floor lookup:

served energy = 1.10 × MER

The 10% buffer accommodates bowl residue, incomplete consumption and day-to-day portioning, so the animal reliably receives its target. Nutrient-floor calculations always use the un-buffered MER, so the buffer neither inflates nor deflates the adequacy target.

2. The metabolic multiplier M

M is an additive five-axis model. Rather than multiplying independent factors, it starts at 1.0, sums a signed delta from each axis, and clamps the result:

M = 1.0 + Σ(deltas) over neuter status, season, body condition, housing and activity; clamped to [1.2, 1.8] for dogs and [0.9, 1.3] for cats.

Activity carries the largest deltas, reflecting how strongly real movement drives maintenance energy. The clamp prevents the model producing targets outside the range observed in healthy maintenance, however the inputs combine.

Two floors are deliberate design choices rather than fitted values. The dog floor M = 1.20 is an anti-starvation bound below which the model will not size a maintenance ration: the framework serves healthy maintenance, not sub-maintenance restriction. The cat floor M = 0.90 matches mainstream clinical practice in omitting an aggressive weight-loss mode for cats, because rapid fat mobilisation carries a hepatic-lipidosis risk; the floor lands near 1.0 × resting energy, approximately 205 kcal at 4 kg.

A public-facing calculator presents body condition as a simplified three-point slider. That is a user-interface simplification of the model above, not the model.

3. Exponent provenance — 0.64 is measured

The canine exponent is taken from Burger and Johnson [1], who measured whole-body energy expenditure by calorimetry in 22 dogs of seven breeds spanning 5.8–48.8 kg and reported a resting relationship of 678 · W^0.64 kJ/day (r = 0.96, P < 0.001). The exponent is an empirically fitted metabolic scaling for the domestic dog specifically.

For completeness the identity

98 · BW^0.64 ≡ 75 · BW^0.75 × 1.31 · BW^−0.11

can be written to show the model is consistent with a Kleiber-anchored resting rate modulated by an activity factor that declines with body size. This is a consistency check, not the derivation. Causality runs from the measurement: 0.64 is measured, and 0.75 is the convention the model departs from, with cause. An earlier internal statement had this backwards and has been corrected.

The declining activity factor is a residual of the model identity above rather than an independent measurement: having fixed the coefficient and the exponent, the factor is whatever reconciles them with observed intakes across the range. It is nonetheless consistent with the long-standing observation that energy requirement does not scale within the species as a single flat factor on BW^0.75 [1]. Applied across the range, the implied factor falls from roughly 1.6 at 3 kg to roughly 1.0 at 110 kg.

The direction of that decline is contested, and the contest should be stated rather than glossed. The pooled meta-analysis of adult dogs [4] fits 81.5 · BW^0.93 — a higher exponent than 0.75, which predicts that giant breeds require more energy than a flat factor on BW^0.75 would give, not less. This model takes the opposite view at the giant end. §6 sets out what adopting 0.93 would cost the fixed-composition architecture, and the disagreement is recorded as an open item in §8 rather than resolved here.

Only the exponent belongs to the cited source. Burger and Johnson's coefficient of 678 kJ/day is a resting construct, approximately 162 kcal at 1 kg. The coefficient 98 and the whole multiplier model are this framework's maintenance construct. The two are convertible but are different quantities, so the exponent alone is attributed.

The feline exponent 0.67 is the NRC 2006 feline maintenance scaling, MEm ≈ 100 · BW^0.67 [3]. The coefficient 90 = 0.9 × 100 embeds the modern-household baseline — neutered, indoor, ambient temperature, normal body condition, somewhat active — directly into the allometry rather than applying it as a downstream multiplier. At the ideal M ≈ 1.1 the product recovers the reference exactly: 90 × 1.1 ≈ 99.

4. External triangulation of the multiplier envelope

The exponent and the envelope are separate questions. The measurement fixes the shape of the curve; it does not set the level of M or the width of its clamp. The coefficients and bands were tuned empirically against household pets and cross-referenced against external references only afterwards, so the alignment below indicates convergence rather than curve-fitting.

4.1 Dog, at the 20 kg anchor

The clinical reference at this weight is a band — 110 · 20^0.75 = 1,040 kcal at the upper end, 100 · 20^0.75 = 946 kcal in the middle — and this model's M = 1.5 value of exactly 1,000 kcal sits within it. The band, not a single point within it, is the appropriate comparison.

Rows marked "mapped, not served" lie below the dog clamp of 1.2. The model can express them; the product declines to size sub-maintenance rations for dogs, and presenting them as offered would claim a capability that is deliberately withheld.

4.2 Cat, at the 4 kg anchor

The population spread the multiplier must reproduce is bounded by an adult-dog meta-analysis [4] and a field estimate across activity levels [8]; the weight-management anchor is a lifespan diet-restriction cohort [6]; the neuter-status and body-condition axes are informed by clinical nutritional-assessment guidance [9], weight-management guidance [7,10] and the neuter-associated energy shift documented in cats [11]. These validate the envelope. They are kept distinct from the exponent, which stands on the measurement alone.

5. Results — correction scaling and the exponent-matching rule

5.1 Unit count

The correction is scaled on the same metabolic footing as energy, referenced to a 20 kg dog and a 4 kg cat:

F = anchorUnits · (BW / anchorBW)^0.667

with 11 units at 20 kg for dogs — plus 2 units if BW ≤ 4 kg and 1 unit if 4 < BW ≤ 8 kg — and 4 units at 4 kg for cats. The count is then snapped to whole units. For dogs a small floor-defence term adds units at low metabolic rate, and the result is rounded with a 0.75 bias:

dog:  units = max(1, ⌊F · (1.05 + f) + 0.75⌋)
      where f = 0 for M ≥ 1.5, else f = 0.14 · (1 − u³),  u = clamp((M − 0.8)/0.7, 0, 1)
cat:  units = max(1, ⌊F + 0.75⌋ + 1)

At the anchors these give 12 units (dog) and 5 units (cat). The count is deliberately decoupled from energy — no multiplier term for cats, only the small floor-defence term for dogs — which is what holds the micronutrient floor on a light-eating day.

5.2 Floor scaling and the matching rule

Per-nutrient daily floors are stored at the anchor weight and scaled by (BW / anchorBW) with exponents 0.64 (dog) and 0.67 (cat) — deliberately matched to the energy exponents. Because NRC 2006 states requirements per 1,000 kcal, requirements track energy; the correction scales at 0.667; the difference for dogs is 0.027. Delivery as a percentage of requirement therefore drifts by only BW^0.027:

{{CHART:weight-drift|Why one fixed composition serves the whole weight range. The correction scales at BW^0.667 against an energy exponent of 0.64, so delivery relative to the 20 kg anchor drifts only as BW^0.027 — about five percent across a thirty-seven-fold span of body weight, and within 1.5 percentage points across the reference band.}}

Band endpoints are rounded to whole percentages, so the four band spreads do not sum exactly to the full-range figure; the full-range spread is computed from the endpoints directly. The stochastic validation of Manuscript 2 samples a wider canine envelope than the dose ladder spans, from 2.5 kg, where delivery is 94.5% of the anchor and the spread widens to 10.2 pp. That is ±5% across a 37-fold weight range and ±1.5 percentage points in the reference band. This matching is why a single fixed-composition correction can serve 3–110 kg without per-band reformulation. What other exponents would have cost:

At 0.75 the correction under-delivers 13% at the giant end; at 0.93, 36%. Either would force per-band reformulation.

Design invariant. Any change to the energy exponent breaks the fixed-composition architecture and requires the correction exponent to move with it. They may not be altered independently.

5.3 Two alternatives examined and rejected

Per-nutrient exponents — calcium at BW^1.0 because bone is structural, iron at BW^1.0 because blood volume is linear — were rejected. The error is stock versus flux: those arguments describe the size of a body pool and infer a dietary requirement, but an adult maintenance requirement replaces obligatory losses, which scale with turnover rather than reservoir volume. Canine maintenance calcium is itself published on a metabolic-body-weight basis, and the claim that sub-linear scaling would cause bone-mineral dilution in larger breeds is contradicted by the large-breed growth literature, which restricts calcium [Manuscript 7]. The scheme is also not implementable: the correction is a fixed composition, so one exponent sets the unit count and every nutrient follows in fixed proportion. There is no second degree of freedom to spend.

Per-band exponents are legitimate in principle — the energy exponent is genuinely uncertain at the extremes — but were rejected on magnitude. Within a band the unit count is fixed while continuous need varies 44–179%, so whole-unit quantisation dominates exponent drift by one to two orders of magnitude. Tuning a 2-percentage-point effect inside a 44-point envelope is invisible in practice. The snap rule and the ≥ 90% floor gate absorb quantisation; they are the operative controls.

6. Discussion — the micronutrient-dilution trap

In a fixed-composition food, micronutrients are locked to calories. When an animal becomes sedentary or ages and the owner reduces the ration to prevent weight gain, every trace mineral is diluted by exactly that reduction — at the life stage when absorption is already declining. A dog that eats less to stay lean does not need less copper; organ demand scales with body mass, not with calories.

Weight-class dosing decouples the two. A 20 kg dog receives the same daily unit count whether M is 1.2 or 1.8; M drives the grocery, not the correction. At a constant 60 g/day:

This is the mechanism behind what the framework calls decoupling: because the unit count is a function of body weight and not of how much food is served, an owner may raise or lower the ration — for appetite, for body condition, for a working day — without altering micronutrient delivery at all. A heavy animal can therefore be calorie-restricted without being nutrient-restricted, which is the trade-off a fixed-composition food forces. A senior eating 20% less still receives 100% of its correction, and density per calorie rises automatically — the right direction, since requirements per calorie do not fall with age. The low-multiplier dog case lies inside the validated envelope [Manuscript 2] and needs no separate run.

Two boundaries on the ageing case. First, there is no NRC or AAFCO senior nutrient profile; what this framework offers for older animals is adult floors plus an energy correction, a grocery-composition shift and a monitoring schedule, never a separate floor set. Second, feline energy requirement follows a U-curve, falling through the mature years and rising again in the geriatric years as fat and protein digestibility decline in a substantial minority of older cats [12]. The raised-multiplier senior-cat case may sit at the top of, or outside, the validated cat band; guidance for that population is clinical rather than an engine output, and checking the band bounds against it is an open item.

7. Growth-to-adult continuity

Growth and reproduction are sized with conventional 0.75-family curves — puppy (254.1 − 135·BW/BW_mature) · BW^0.75; lactating bitch 145·BW^0.75 + BW·(24n + 12m)·L; intact queen anchor 100·BW^0.67 × 1.10 × 1.10 — while adult maintenance uses the model of §1. Two exponents at the same body weight look like a discontinuity. They are not.

At maturity the growth factor collapses. The terminal growth forms — puppy 119.1 · BW^0.75, kitten 96.6 · BW^0.75 — equal the adult curve at a crossing multiplier

M = 1.215 · BW^0.11 (dog), M = 1.073 · BW^0.08 (cat)

so the terminal growth curve is the adult curve evaluated at a young-adult activity level, near the top of the healthy band, which is what one expects of a newly mature animal. The handoff is continuous and monotone non-increasing; for giant breeds the mild weight-dependence of M* leaves a bounded taper of approximately 6% at 60 kg, consistent with the documented slower metabolic maturation of large breeds.

This is an energetic continuity result only; it closes no nutrient floor, and the maturity handoff remains a scheduled reconciliation item in the engine rather than a closed one.

8. Scope and limitations

• The measurement envelope is 5.8–48.8 kg [1]. Within it the dog curve is empirically grounded. Below 5.8 kg and above 48.8 kg the exponent is a principled extrapolation beyond the data it was fitted on — the same caveat the meta-analysis raises for the extremes [4]. Confidence is highest near the anchor and degrades outward; that is a property of the evidence, not of the equation.
• The giant end is unconfirmed empirically. At 110 kg the model sizes approximately 2,980 kcal at M 1.5, against 130 · BW^0.75 = 4,416 on a colony-derived convention. The declining activity factor explains the gap and the giant-breed literature supports it, but it has not been checked against a real 60–110 kg dog maintaining condition over time.
• The model is a model of a population. Individual animals vary; body-condition monitoring and ration adjustment remain necessary. The 1.10 served-energy buffer is a population convenience, not an individual prescription.
• The coefficient and multiplier are this framework's construct, not the cited literature's, and are labelled as such throughout.
• Healthy animals only. Not a clinical or therapeutic system.
• 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

The energy model sizes the ration to a maintenance requirement scaled allometrically, with a five-axis multiplier for life stage, neuter status, activity, age and body condition. Its load-bearing property is not the coefficient but the relationship between two exponents: the correction scales at BW^0.667 against an energy exponent of 0.64, so delivery expressed as a percentage of requirement drifts only as BW^0.027 — roughly ±5% across a thirty-seven-fold span of body weight. That is what allows one fixed composition to serve the whole range without per-band reformulation.

The canine exponent of 0.64 rests on a single calorimetric dataset, and the pooled meta-analysis of adult dogs reports a materially higher value of 0.93, which would predict greater rather than lesser energy need at the giant end. This model takes the opposite view there. The disagreement is recorded rather than resolved: a sensitivity analysis across the plausible exponent range would strengthen the position and has not been performed.

References

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How to cite

Prasanna M. (2026). Evaluation of Multi-Breed Allometric Energy Formulations Against Published Calorimetric Expenditure Data in Canines (Manuscript 3 of 8, Release B-2.1.0). Growlrr Foods Pvt Ltd. Zenodo. https://doi.org/10.5281/zenodo.22647612

The series

This manuscript is one of eight deposited together under release B-2.1.0, with a series cover note carrying the concept DOI and the full DOI list. All eight apply the BARC Protocol, published openly and separately at growlrr.com/barc-protocol.