Weir Equation Energy Expenditure Calculator
Weir Equation Energy Expenditure Calculator
What a metabolic cart actually computes: measured energy expenditure from oxygen consumption and carbon dioxide production, with and without the urinary nitrogen term, and the three coefficients different sources print.
Measured energy expenditure (Weir)
Gas exchange, not predictionFull Weir. VO₂ 0.250 L/min, VCO₂ 0.200 L/min, urinary nitrogen 12 g/24 h
Formula
Abbreviated form: the nitrogen term is dropped
- 3.941 and 1.106
- the caloric equivalents of oxygen and carbon dioxide in kcal per litre. VO₂ dominates: at a respiratory quotient of 0.8 it supplies about 82% of the answer, which is why CO₂-only methods are a compromise rather than an equivalent
- 1440, or 1.44
- minutes in a day. Sources print the same relation two ways: the gases in L/min with a 1440 multiplier, or in mL/min with 1.44. They are identical, and a factor of a thousand is the thing to check
- the VCO₂ coefficient, three values
- 1.106 in Weir’s original, 1.11 in one widely cited ICU paper, 1.1 in the bedside form. At a VCO₂ of 0.2 L/min the three span about 3 kcal/day out of roughly 1,700 — under 0.2%, so this is a disagreement worth knowing about and not worth worrying about. The page implements 1.106 and shows 1.11 alongside
- − 2.17 × urinary nitrogen
- the protein correction, 2.17 kcal per gram of TOTAL urinary nitrogen a day. Weir’s own formulation expressed it as a proportion of expenditure instead: EE = VO₂ × (3.941 + 1.106 × RQ) ÷ (1 + 0.082P), where P is the fraction of expenditure from protein, with about 2.3% error reported from leaving it out on a high-protein diet. This is a measurement of this patient over this period, not a prediction and not a target
Worked example
Full Weir. VO₂ 0.250 L/min, VCO₂ 0.200 L/min, urinary nitrogen 12 g/24 h
3.941 × 0.250 = 0.98525 and 1.106 × 0.200 = 0.22120 kcal/min; sum 1.20645
× 1,440 = 1,737.288 kcal/day — the abbreviated Weir answer
− 2.17 × 12 = −26.04, so 1,737.288 − 26.04 = 1,711.248 kcal/day, rounded to 1,711
The nitrogen term is worth 26.04 kcal/day, 1.52% of the answer. Switch to the abbreviated form and the figure rises to 1,737 — omitting it always raises the result
The respiratory quotient implied by these gases is 0.200 ÷ 0.250 = 0.80, inside the 0.67 to 1.3 physiologic range, so the measurement passes its validity check
In mL/min: 1.44 × (3.941 × 250 + 1.106 × 200) = 1,737.288 — the same number, and the check to run if a cart reports in mL/min. With the VCO₂ coefficient at 1.11 the answer becomes 1,712.4, a 1.15 kcal/day difference
The coefficients different sources print
| Source | VO₂ coefficient | VCO₂ coefficient | Nitrogen term | Answer at the worked example |
|---|---|---|---|---|
| Weir’s original, as transcribed in a 2019 validation paper | 3.941 | 1.106 | Expressed as a proportion of expenditure, ÷ (1 + 0.082P) | 1,711.2 with the mass form of the term |
| Oshima and colleagues, Critical Care 2017 | 3.941 | 1.11 | Not included in the form printed | 1,712.4 |
| The bedside form, as taught | 3.94 | 1.1 | Usually omitted | 1,735.2 abbreviated |
What the nitrogen term is worth
| Total urinary nitrogen (g/24 h) | Protein it corresponds to at 6.25 (g/day) | Energy removed (kcal/day) | As a share of 1,737 kcal/day |
|---|---|---|---|
| 0 | 0 | 0 | 0% |
| 6 | 37.5 | 13.0 | 0.75% |
| 12 | 75 | 26.0 | 1.50% |
| 20 | 125 | 43.4 | 2.50% |
| 30 | 187.5 | 65.1 | 3.75% |
The reference method, and the two ways to enter it wrong
Indirect calorimetry does not measure energy. It measures oxygen consumption and carbon dioxide production, and converts them with caloric equivalents that depend only weakly on which substrate is being burnt. That conversion is the Weir equation, and it is what every metabolic cart runs behind its display. Oxygen consumption does most of the work: with coefficients of 3.941 against 1.106, VO₂ supplies roughly 82% of the answer at a respiratory quotient of 0.8 — which is why methods built on CO₂ alone are described as more accurate than predictive equations but less accurate than full calorimetry.
Weir’s own formulation handled protein as a proportion of expenditure, and the clinical form converts that into a mass term: 2.17 kcal per gram of total urinary nitrogen a day. Two things about it are worth stating plainly. It is small — at 12 g/day it removes 26 kcal/day, about 1.5% of the answer, and under 4% even at 30 g/day. And it requires total urinary nitrogen from a complete 24-hour collection, which is not the same measurement as urine urea nitrogen; substituting one for the other is the error to avoid, and the urine urea nitrogen converter and the nitrogen balance calculator cover that distinction. Where total nitrogen is unavailable, use the abbreviated form and say so.
The two input errors that matter are both unit errors. The first is L/min against mL/min: the same relation is published in litres with a 1440 multiplier and in millilitres with 1.44, so entering 250 where 0.250 belongs inflates the answer a thousandfold. The second is the coefficient, and here the news is reassuring — sources print 1.106, 1.11 and 1.1 for the CO₂ term, and at ordinary gas flows the three span about 3 kcal/day out of 1,700. The page implements 1.106 and shows 1.11 beside it.
One validity check comes free. The respiratory quotient implied by the two gases must lie between 0.67 and 1.3 in a human; outside that, the Academy of Nutrition and Dietetics advises suspecting an error and repeating the measurement, and this page flags it. Inside that range an RQ “should not be used alone to reject” a measurement, and it should not be read as a feeding verdict either — see the respiratory quotient calculator. Where a cart is unavailable, the Penn State equations are what the Academy recommends for ventilated adults and Mifflin-St Jeor for everyone else; both predict a population mean and this measures a patient. This page computes a figure and prints the published targets beside it with the guideline each comes from. The prescription is the clinician’s, taken with the patient in front of them.
Frequently asked questions
What is the Weir equation?
The conversion indirect calorimetry uses to turn measured gas exchange into energy expenditure: EE in kcal/day = [3.941 × VO₂ + 1.106 × VCO₂] × 1440 − 2.17 × urinary nitrogen, with the gases in L/min and nitrogen in g/day. The abbreviated form drops the nitrogen term.
Do I need the urinary nitrogen?
Usually not — most clinical carts report the abbreviated form, because a complete 24-hour collection is rarely available. The term is 2.17 kcal per gram of nitrogen, about 26 kcal/day or 1.5% at 12 g/day and under 4% even at 30 g/day, and omitting it always raises the answer. It requires TOTAL urinary nitrogen, not urine urea nitrogen; if only urea nitrogen is available, use the abbreviated form rather than substituting.
Why do sources give different coefficients for VCO₂?
1.106 is Weir’s original, 1.11 appears in a widely cited critical care paper, and 1.1 is the bedside form. At a VCO₂ of 0.2 L/min the three span about 3 kcal/day out of 1,700 — under 0.2%. The page implements 1.106 and prints the 1.11 version alongside. The unit of the gases matters far more than the coefficient.
Should the gases be in L/min or mL/min?
This calculator takes L/min. The same equation is published both ways — L/min with a 1440 multiplier, mL/min with 1.44 — and they are identical. Entering a millilitre figure into a litre field inflates the answer a thousandfold, so check what the cart reports: a resting adult’s VO₂ is about 0.2 to 0.3 L/min, or 200 to 300 mL/min.
Is a measured energy expenditure the same as the feeding target?
No, and ESPEN is explicit: the early acute phase should not exceed 70% of measured expenditure (rec. 17), rising to 80 to 100% after day 3 (rec. 18). The measurement says what the patient is spending over this period; the target is a separate, phase-dependent judgement, and the prescription is the clinician’s.
Related calculators
References
- Oshima T, Graf S, Heidegger CP, Genton L, Pugin J, Pichard C. Can calculation of energy expenditure based on CO₂ measurements replace indirect calorimetry? Crit Care. 2017;21:36. Prints the modified Weir equation with a 1.11 VCO₂ coefficient and the 0.67 to 1.3 window, and states that RQ “is neither a reliable indicator of the feeding status”.
- Kaiyala KJ, Wisse BE, Lighton JRB. Validation of an equation for energy expenditure that does not require the respiratory quotient. PLoS One. 2019;14(2):e0211585. Gives Weir’s original form, EE = VO₂ × (3.941 + 1.106 × RQ) ÷ (1 + 0.082P), and about 2.3% error from omitting the protein term on a high-protein diet.
- Academy of Nutrition and Dietetics Evidence Analysis Library. Energy Expenditure: respiratory quotient recommendations. An RQ within 0.67 to 1.3 “should not be used alone to reject” a measurement (Fair); outside it, “suspect an error and repeat the RMR measurement” (Consensus); “RQ has poor accuracy to evaluate under- and over-feeding” (Strong).
- Singer P, Blaser AR, Berger MM, et al. ESPEN guideline on clinical nutrition in the intensive care unit. Clin Nutr. 2019;38(1):48–79. Recommendations 15 (indirect calorimetry in ventilated patients, Grade B), 17 (not exceeding 70% of energy expenditure in the early phase), 22 (1.3 g/kg protein equivalents), 23 (glucose “should not exceed 5 mg/kg/min”) and 25 (lipids “should not exceed 1.5 g lipids/kg/day”). Predictive equations are “associated with significant inaccuracy (up to 60%)”, with 20–25 kcal/kg/day named as the simplest alternative.
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/