CalQuant logoCalQuant
Loading...

Methods & Equations

This page outlines the main equations, assumptions, and source references used across CalQuant. Each section shows the formula used in the app, defines the variables, and indicates whether a result comes from a published method or from CalQuant's own implementation logic.

1.

Basal Metabolic Rate

Basal metabolic rate is the starting point for the energy model, so CalQuant shows up to four standard BMR equations side by side. Weight-and-height methods are always available, while lean-mass methods appear only when body-fat percentage has been entered or estimated. The active method is used for downstream TDEE calculations.

1.1.

Mifflin-St Jeor

A widely used weight-height-age equation. It is useful when body-fat data is not available, and it provides a simple baseline estimate from standard body measurements.

Formula

BMRkcal = 10 · Wkg + 6.25 · Hcm - 5 · A + S

Variables

Wkg
Body weight, in kilograms.
Hcm
Height, in centimetres.
A
Age, in years.
S
Sex-specific constant: 5 for male, -161 for female.

1.2.

Harris-Benedict (revised)

CalQuant uses the revised Roza-Shizgal form. It is included as a familiar comparison estimate based on age, body weight, height, and sex.

Formula

Male

BMRkcal = 88.362 + 13.397 · Wkg + 4.799 · Hcm - 5.677 · A

Female

BMRkcal = 447.593 + 9.247 · Wkg + 3.098 · Hcm - 4.330 · A

Variables

Wkg
Body weight, in kilograms.
Hcm
Height, in centimetres.
A
Age, in years.

1.3.

Katch-McArdle

A lean-mass-based estimate that is only available when body-fat percentage is known or estimated. It can be more informative when lean mass is likely to matter more than scale weight alone.

Formula

Lkg = Wkg · ⟮1 - F% / 100⟯

BMRkcal = 370 + 21.6 · Lkg

Variables

Lkg
Lean body mass, in kilograms.
Wkg
Body weight, in kilograms.
F%
Body-fat percentage on a 0-100 scale.

1.4.

Cunningham

Another lean-mass-based estimate, available only when body-fat percentage is known or estimated. As with Katch-McArdle, its usefulness depends on the quality of the body-fat input used to derive lean mass.

Formula

BMRkcal = 500 + 22 · Lkg

Variables

Lkg
Lean body mass, in kilograms, as defined in 1.3.

2.

Activity Expenditure And Total Daily Energy Expenditure

This section shows how CalQuant turns the selected BMR and the user's reported day structure into a daily expenditure estimate. The app models sleep, sedentary time, light activity, moderate activity, and vigorous activity, then adds a nutrition-profile-based TEF estimate to report final TDEE and a PAL-style activity ratio.

2.1.

Activity Energy Expenditure

CalQuant converts broad time blocks into energy using representative MET-style anchors for sleep, sedentary time, light activity, moderate activity, and vigorous activity. This keeps the model simple while preserving the main intensity differences that drive daily expenditure.

Formula

Ekcal = ⟮M - 1⟯ · 1.05 · Wkg · Th

Variables

M
Representative MET-style anchor assigned to the time block.
Ekcal
Energy from a given time block, in kilocalories per day.
Wkg
Body weight, in kilograms.
Th
Time, in hours.

Notes

  • Sedentary time is inferred as the remaining minutes in the day after sleep, light, moderate, and vigorous time have been entered.
  • The 2024 Adult Compendium classifies sedentary behaviour as 1.0-1.5 METs, light activity as 1.6-2.9 METs, moderate activity as 3.0-5.9 METs, and vigorous activity as 6.0 or more METs.
  • CalQuant uses fixed anchors within those published ranges rather than an activity-by-activity lookup.

2.2.

Non-Exercise Activity Thermogenesis

In CalQuant, NEAT is approximated as sedentary plus light activity, with the sleep offset carried alongside those terms so the full daily activity stack stays internally consistent against the resting baseline used elsewhere in the calculator.

Formula

NEATkcal = Δ + Ese + El

Variables

Δ (Esl)
Sleep offset, in kilocalories per day.
Ese
Energy from inferred sedentary time, in kilocalories per day.
El
Energy from light activity time, in kilocalories per day.

Notes

  • Including the sleep offset inside NEAT is an app-specific bookkeeping choice so the component totals sum cleanly inside the TDEE breakdown; it is not a claim that sleep itself is part of classical NEAT definitions.
  • The sedentary and light components are estimated from simplified Compendium-informed intensity anchors.

2.3.

Exercise Activity Thermogenesis

EAT is the explicit exercise portion of the activity model. In CalQuant it is the energy attributed to the user's moderate and vigorous time blocks.

Formula

EATkcal = Em + Ev

Variables

Em
Energy from moderate activity time, in kilocalories per day.
Ev
Energy from vigorous activity time, in kilocalories per day.

Notes

  • The Energy Availability section uses this EAT value as exercise energy expenditure inside the EA formula.
  • These exercise blocks are modeled with representative intensity anchors rather than a single exact activity code.

2.4.

Total Energy Expenditure

TEE is the app's pre-TEF daily total. It combines the selected BMR with the sleep offset and the modeled activity components.

Formula

TEEkcal = Bsel + Δ + Ese + El + Em + Ev

Variables

Bsel
Selected BMR estimate, in kilocalories per day.
Δ (Esl)
Sleep offset, in kilocalories per day.
TEEkcal
Total energy expenditure before TEF is added back in, in kilocalories per day.
Ese
Energy from inferred sedentary time, in kilocalories per day.
El
Energy from light activity time, in kilocalories per day.
Em
Energy from moderate activity time, in kilocalories per day.
Ev
Energy from vigorous activity time, in kilocalories per day.
  • Source: CalQuant implementation

2.5.

Thermic Effect Of Food

TEF is not measured directly in the app. Instead, CalQuant uses nutrition-profile presets to estimate how much food-related thermogenesis sits on top of TEE.

Formula

TEFkcal = TEEkcal / f - TEEkcal

Variables

TEEkcal
Total energy expenditure before TEF is added back in, in kilocalories per day.
f
Nutrition-profile factor selected in the UI.

Notes

  • Review literature describes mixed diets as producing roughly 5-15% diet-induced energy expenditure overall, with higher values at relatively high protein intake and lower values at high fat intake.
  • The preset factors are designed to preserve the expected ordering across profiles rather than to claim one exact TEF percentage for every real-world diet.

2.6.

Total Daily Energy Expenditure

This is the final presentation step used by the energy calculator. TDEE is shown once TEF has been added to TEE, and the activity ratio is expressed as Personal Activity Level (PAL) relative to the active BMR estimate.

Formula

TDEEkcal = TEEkcal + TEFkcal

PAL = TEEkcal / Bsel

Variables

TDEEkcal
Total daily energy expenditure, in kilocalories per day.
TEEkcal
Total energy expenditure before TEF is added back in, in kilocalories per day.
TEFkcal
Thermic effect of food, in kilocalories per day.
Bsel
Selected BMR estimate, in kilocalories per day.
PAL
Physical activity level, expressed here as TEE divided by selected BMR.

3.

Body Composition

This section covers the two circumference-based methods CalQuant uses to estimate body-fat percentage when a direct value has not been entered. The estimate chosen here can affect lean-mass-based BMR methods and the body-fat-dependent metrics shown elsewhere in the app.

3.1.

Relative Fat Mass (RFM)

RFM is a simple waist-and-height body-fat estimate. It is quick to use, but the result depends on measuring the waist at a consistent site.

Formula

Male

F% = 64 - 20 · ⟮H / C⟯

Female

F% = 76 - 20 · ⟮H / C⟯

Variables

F%
Body-fat percentage on a 0-100 scale.
H
Height.
C
Waist circumference measured just above the hip bone.

3.2.

Taylor-McClung

Taylor-McClung is a circumference-based body-fat equation developed in an active population. It uses abdominal circumference and body weight, and depends on consistent measurement quality.

Formula

Male

F% = -26.97 + 1.99 · Cabd,in - 0.12 · Wlb

Female

F% = -9.15 + 1.27 · Cabd,in - 0.015 · Wlb

Variables

F%
Body-fat percentage on a 0-100 scale.
Wlb
Body weight, in pounds.
Cabd,in
Abdominal circumference at the belly button, in inches.

4.

Energy Availability

Energy availability is the dietary energy left over after exercise expenditure, scaled to fat-free mass. CalQuant applies the same core equation across maintenance, deficit, and surplus intake scenarios.

Formula

EA = ⟮I - EATkcal⟯ / Lkg

Variables

EA
Energy availability, in kilocalories per kilogram of fat-free mass per day.
I
Daily energy intake, in kilocalories per day.
EATkcal
Exercise energy expenditure, in kilocalories per day.
Lkg
Fat-free mass, in kilograms.

Notes

  • In the literature, Energy Availability is usually expressed using exercise energy expenditure. In CalQuant, that term is approximated with EAT from the moderate and vigorous activity blocks.
  • At maintenance, daily intake is set to TDEE. In deficit and surplus views, intake reflects the goal setting selected in the interface.
  • If fat-free mass is unavailable, the app can still show the intake scenarios, but it leaves energy availability blank.

5.

Derived Physique Metrics

These metrics sit one step downstream from the body-composition estimate. They use lean mass, fat mass, or body-fat percentage to provide physique-oriented context rather than direct measures of health or performance.

5.1.

Normalised Fat-Free Mass Index

Normalised fat-free mass index adjusts fat-free mass for height so muscularity can be compared more fairly across people of different statures. It is best used as a physique reference metric.

Formula

NFFMI = Lkg / ⟮Hm2⟯ + 6.3 · ⟮1.8 - Hm⟯

Variables

NFFMI
Normalised fat-free mass index, in kilograms per square metre.
Lkg
Fat-free mass, in kilograms.
Hm
Height, in metres.

5.2.

Fat Oxidation Limit

Fat oxidation limit is used in CalQuant as an estimate of how much daily energy can be supplied from stored fat. In practice, it acts as a planning guardrail for how aggressive a calorie deficit can be.

Formula

FOL = 31 · Flb

Variables

FOL
Estimated daily fat oxidation limit, in kilocalories per day.
Flb
Fat mass, in pounds.

Notes

  • This is an applied estimate derived from Alpert's fat-store transfer-rate work rather than a directly measured app output.

5.3.

MMP Heuristics

These MMP values are heuristic reference metrics rather than validated personal limits. They are intended as rough physique context, not as a precise prediction of an individual's muscular ceiling.

Formula

MMP = Hcm - 100

MMPBF% = ⟮MMP · 0.95⟯ / ⟮1 - φ⟯

Variables

MMP
Maximum muscular potential, in kilograms.
MMPBF%
Maximum muscular potential adjusted to the current body-fat percentage, in kilograms.
Hcm
Height, in centimetres.
φ
Body-fat proportion on a 0-1 scale.

Notes

  • The base MMP formula is a height-based heuristic and should be read more cautiously than the published equations elsewhere on this page.
  • MMPBF% is a CalQuant presentation layer built on top of that heuristic rather than a separately published equation.
  • These values are shown only when the required inputs are available, and no female-specific base MMP formula is currently generated in the app.

6.

Goal Deficit And Gain Models

The final section covers the app's cutting, bodyweight-gain, and muscle-gain planning logic. These formulas are implementation models built to support planning rather than direct clinical measurement.

6.1.

Goal Deficit Model

CalQuant's cutting model uses a Hall-style weekly simulation. Starting from current body weight and fat-free mass, it estimates how the effective daily deficit changes across the week and converts that into predicted bodyweight loss.

Formula

ΔWd = ⟮δ0 - ⟮TDEE0 - TDEEa⟯⟯ / ⟮9439 · p + 1816 · ⟮1 - p⟯⟯

Variables

ΔWd
Predicted bodyweight loss, in kilograms per day.
δ0
Starting planned daily calorie deficit, in kilocalories per day.
p
Current Forbes-style fat-share term used inside the weekly simulation.
TDEE0
Starting total daily energy expenditure, in kilocalories per day.
TDEEa
Adaptive expenditure estimate used during the weekly simulation.

Notes

  • The Forbes-style term p is recalculated from current fat mass during the simulation, so the model can shift how much predicted loss is attributed to fat versus fat-free tissue.
  • TDEEa is the model's adaptive expenditure estimate. It moves away from the starting TDEE as the simulated body state changes across the week.
  • The denominator blends fat and fat-free tissue energy costs into a single effective energy-per-kilogram term.
  • This is a simplified planning model informed by Hall/Forbes work rather than a paper-exact reproduction of the full Hall system.
  • If the body-composition inputs needed for this Hall-style simulation are unavailable, the dashboard falls back to a simpler single-step estimate in which p is treated as constant rather than being updated across a weekly simulation.

6.2.

Bodyweight Gain Simulation

CalQuant's gain model begins with a Hall-style monthly bodyweight simulation. Starting from current body weight and body-fat percentage, it estimates how the effective daily surplus changes across the month, then converts that into predicted bodyweight gain and energy-supported lean gain.

Formula

ΔWd = ⟮s0 - ⟮TDEEa - TDEE0⟯⟯ / ⟮1816 · ph + 9439 · ⟮1 - ph⟯⟯

ΔLe = ΔWd · ph

Variables

ΔWd
Predicted bodyweight gain, in kilograms per day.
ph
Current Forbes-style lean-share term used inside the monthly simulation.
s0
Starting planned daily surplus, in kilocalories per day, before adaptive expenditure is subtracted.
TDEE0
Starting total daily energy expenditure, in kilocalories per day.
TDEEa
Adaptive expenditure estimate used during the monthly simulation.
ΔLe
Lean gain that the modeled daily surplus could energetically support.

Notes

  • The Forbes-style term ph is recalculated from current fat mass during the simulation, so the model can shift how much predicted gain is attributed to lean versus fat tissue.
  • TDEEa is the model's adaptive expenditure estimate. It moves away from the starting TDEE as the simulated body state changes across the month.
  • The denominator blends fat and fat-free tissue energy costs into a single effective energy-per-kilogram term.
  • ΔLe is an energy-supported lean-gain quantity, not yet a final muscle-gain prediction. The next section applies the separate hypertrophy-capacity constraint.
  • This is a simplified planning model informed by Hall/Forbes work rather than a paper-exact reproduction of the full Hall system.
  • If body-fat percentage is unavailable, the dashboard falls back to a simpler non-simulation estimate rather than using this monthly Hall-style model.

6.3.

Muscle Gain Constraint

After the bodyweight simulation estimates how much lean tissue the surplus could support, CalQuant applies a separate hypertrophy-capacity term. Realized muscle gain is then set to the lower of the energy-supported lean gain and that capacity estimate.

Formula

m = r · Lα / ⟮Lα + H1α⟯ · 1 / ⟮1 + ⟮L / H2⟯β⟯

ΔMr = min{ΔLe, m}

Variables

L
Current fat-free mass used in the hypertrophy drive term.
H1
Lean-mass level at which the Hill-type factor equals 1/2.
H2
Lean-mass level at which the multiplicative inhibition factor equals 1/2.
r
Scaling parameter for the hypertrophy term.
α
Exponent controlling the steepness of the ascent of the curve.
β
Exponent controlling the steepness of the descent of the curve.
ΔLe
Lean gain that the bodyweight simulation says the modeled surplus could energetically support.
m
Daily hypertrophy-capacity estimate, in kilograms per day.
ΔMr
Realized muscle gain after capping energy-supported lean gain by the hypertrophy-capacity term.

Notes

  • H1 is scaled to the individual from starting lean mass using a training-age-specific activation target, so the activation side of the curve is personalized rather than fixed at one absolute value.
  • H2 is scaled to the individual from lean mass and nFFMI headroom, so current muscularity helps shape the inhibitory side of the curve.
  • In the current implementation, α and β are fixed globally, while r varies with training age.
  • Realized muscle gain is set to the lower of the energy-supported lean gain and the hypertrophy-capacity term. In other words, the model must satisfy both an energy constraint and a muscle-growth-capacity constraint.
  • The model was calibrated against a merged NHANES-derived body-composition dataset of 2,442 male and female profiles using in-house developed software tools. That dataset was used as a broad calibration and sanity-check surface rather than as direct longitudinal muscle-gain evidence.
  • A separate verification pass then compared group-level outputs against Hatamoto et al. using achieved energy balance and the reported mean body-weight and fat-free-mass changes.

7.

References & Notes

Published equations are linked below. Throughout this page, "CalQuant implementation" refers to formulas, heuristics, or assumptions defined directly in the current application code rather than taken from a single external source paper.

  1. Mifflin MD, St Jeor ST, Hill LA, Scott BJ, Daugherty SA, Koh YO. A new predictive equation for resting energy expenditure in healthy individuals. American Journal of Clinical Nutrition. 1990;51(2):241-247.
  2. Roza AM, Shizgal HM. The Harris-Benedict equation reevaluated: resting energy requirements and the body cell mass. American Journal of Clinical Nutrition. 1984;40(1):168-182.
  3. McArdle WD, Katch FI, Katch VL. Exercise Physiology: Energy, Nutrition, and Human Performance. 6th ed. Baltimore, MD: Lippincott Williams & Wilkins; 2006.
  4. Cunningham JJ. A reanalysis of the factors influencing basal metabolic rate in normal adults. American Journal of Clinical Nutrition. 1980;33(11):2372-2374.
  5. Cunningham JJ. Body composition and resting metabolic rate: the myth of feminine metabolism. American Journal of Clinical Nutrition. 1982;36(4):721-726.
  6. Ainsworth BE, Haskell WL, Herrmann SD, et al. 2011 Compendium of Physical Activities: a second update of codes and MET values. Medicine and Science in Sports and Exercise. 2011;43(8):1575-1581.
  7. Herrmann SD, Li Y, Sidman CL, et al. 2024 Adult Compendium of Physical Activities: A third update of the energy costs of human activities. Journal of Sport and Health Science. 2024;13(1):6-12.
  8. Compendium of Physical Activities. 2024 Adult Compendium Tracking Guide. Official activity examples and representative MET values drawn from the 2024 Adult Compendium.
  9. Levine JA, Kotz CM. NEAT--non-exercise activity thermogenesis--egocentric and geocentric environmental factors versus biological regulation. Acta Physiologica Scandinavica. 2005;184(4):309-318.
  10. Westerterp KR. Diet induced thermogenesis. Nutrition & Metabolism (Lond). 2004;1(1):5.
  11. Westerterp KR, Wilson SA, Rolland V. Diet induced thermogenesis measured over 24h in a respiration chamber: effect of diet composition. International Journal of Obesity and Related Metabolic Disorders. 1999;23(3):287-292.
  12. FAO/WHO/UNU. Human energy requirements: report of a Joint FAO/WHO/UNU Expert Consultation, Rome, 17-24 October 2001. FAO Food and Nutrition Technical Report Series No. 1. Rome: Food and Agriculture Organization of the United Nations; 2004.
  13. Woolcott OO, Bergman RN. Relative fat mass (RFM) as a new estimator of whole-body fat percentage: a cross-sectional study in American adult individuals. Scientific Reports. 2018;8:10980.
  14. Taylor KM, Castellani MP, Bartlett PM, Oliver TE, McClung HL. Development and cross-validation of a circumference-based predictive equation to estimate body fat in an active population. Obesity Science & Practice. 2024;10(2):e747.
  15. Areta JL, Taylor HL, Koehler K. Low energy availability: history, definition and evidence of its endocrine, metabolic and physiological effects in prospective studies in females and males. European Journal of Applied Physiology. 2021;121(1):1-21.
  16. Kouri EM, Pope HG Jr, Katz DL, Oliva P. Fat-free mass index in users and nonusers of anabolic-androgenic steroids. Clinical Journal of Sport Medicine. 1995;5(4):223-228.
  17. Alpert SS. A limit on the energy transfer rate from the human fat store in hypophagia. Journal of Theoretical Biology. 2005;233(1):1-13.
  18. Berkhan M. Maximum Muscular Potential of Drug-Free Athletes. Leangains. Published December 31, 2010.
  19. Forbes GB. Lean body mass-body fat interrelationships in humans. Nutrition Reviews. 1987;45(8):225-231.
  20. Hall KD. Body fat and fat-free mass inter-relationships: Forbes's theory revisited. British Journal of Nutrition. 2007;97(6):1059-1063.
  21. Hall KD. Mathematical modelling of energy expenditure during tissue deposition. British Journal of Nutrition. 2010;104(1).
  22. Torres M, Trexler ET, Smith-Ryan AE, Reynolds A. A mathematical model of the effects of resistance exercise-induced muscle hypertrophy on body composition. European Journal of Applied Physiology. 2018;118(2):449-460.
  23. Centers for Disease Control and Prevention, National Center for Health Statistics. About NHANES. National Health and Nutrition Examination Survey overview and public data documentation.
  24. Hatamoto Y, Tanoue Y, Tagawa R, et al. Greater energy surplus promotes body protein accretion in healthy young men: A randomized clinical trial. Clinical Nutrition. 2024;43:48-60.