488-Metabolic Equations

Overview and Rationale

Metabolic equations allow estimation of steady-state oxygen consumption (VO₂) when direct VO₂ measurement is impractical.
Purpose: estimate energy expenditure and support more effective exercise prescriptions.
Approach: use a systematic method to read the question, select the correct equation, and perform careful stepwise calculations.
Steps emphasized: read question carefully, write down each step, make all conversions, write out the formula and substitute known values, check if the answer makes sense.

Use and Approach

Read the question carefully and determine which equation applies.
Write the equation in the form VO₂ = R + H + V (resting plus horizontal plus vertical components).
Convert all units as needed before substitution.
Substitute known values into the formula and compute.
Evaluate whether the resulting VO₂ (and any derived kcal) makes sense in the context of the problem.

Guidelines and References

2018 Physical Activity Guidelines (reference): https://health.gov/paguidelines/second-edition/report/
2021 WHO PA report (context for activity levels).
These guidelines underpin the context for metabolic calculations and exercise prescription thresholds.

VO₂ Concepts: Gross vs Net; Absolute vs Relative

VO₂: Gross vs Net
- Gross VO₂ = total amount of oxygen consumed (rest + exercise).
- Net VO₂ = amount needed for exercise only (gross − resting VO₂).
VO₂: Absolute vs Relative
- Absolute VO₂: measured in L·min⁻¹ or mL·min⁻¹.
- Relative VO₂: normalized by body mass to allow comparisons across individuals; expressed as mL·kg⁻¹·min⁻¹.

Interrelationships of VO₂, METS, KCAL & Work

VO₂ relative (ml·kg⁻¹·min⁻¹) increases with exercise intensity.
Resting VO₂ ≈ 3.5 ml·kg⁻¹·min⁻¹ = 1 MET.
Max VO₂ values can be reported per kg or as absolute ml·min⁻¹ or L·min⁻¹.
1 L of O₂ consumed ≈ 5 kcal (rough physiological conversion).
Example: Resting VO₂ ≈ 3.5 ml·kg⁻¹·min⁻¹; 250 mL·min⁻¹ corresponds to a certain MET value: 250 mL·min⁻¹ ÷ 70 kg ≈ 3.5 ml·kg⁻¹·min⁻¹ (1 MET).
1 MET = 3.5 ml·kg⁻¹·min⁻¹.

Common Conversions and Memorization

Power settings and ergometry:
- Power in Watts (W) × 6 = workload/power setting in kg·m·min⁻¹.
Mass conversions:
- Weight in lb ÷ 2.2 = mass in kg.
Caloric content:
- Adipose tissue: 1 lb adipose tissue ≈ 3500 kcal.
Speed conversions:
- mph × 26.8 = m·min⁻¹.
MET conversions:
- 1 MET = 3.5 ml·kg⁻¹·min⁻¹.
Oxygen to kcal:
- 1 L O₂ ≈ 5 kcal.
Additional quick conversions:
- 1 inch = 2.54 cm.

Common Conversions (Ergometry and general)

Distance traveled per revolution on ergometers:
- Monark arm ergometer: 2.4 m/rev
- Monark leg ergometer: 6.0 m/rev
- Tunturi/BodyGuard: 3.0 m/rev
1 L of O₂ = 5 kcal (as above).
1 MET = 3.5 ml·kg⁻¹·min⁻¹.
1 mph = 26.8 m/min.
1 W = 6 kg·m/min.
Power equation (ergometry):
- Power (kg·m/min) = R × D × f, where
- R = resistance setting (kg),
- D = distance the flywheel travels per revolution (m) – values commonly 3, 6, 2.4,
- f = revolutions per minute (rpm).

Ergometry Equations

General form: VO₂ = R + H + V, where
- R = resting component,
- H = horizontal (cost of moving the body mass through space in walking/running ergometry),
- V = vertical (cost of height gain or resistance component).
Walking (overground):
- VO₂ = 0.1(speed) + 1.8(speed)(grade) + 3.5
- Units: speed in m·min⁻¹; grade is fractional (e.g., 0.05 = 5%).
- Interpretation:
- 0.1 ml·kg⁻¹·min⁻¹ per m·min⁻¹ for horizontal motion.
- 1.8 ml·kg⁻¹·min⁻¹ per m·min⁻¹ per unit grade for vertical ascent.
- 3.5 ml·kg⁻¹·min⁻¹ for resting (standby) cost.
- Typical walking speeds: 50–100 m/min (about 1.9–3.7 mph).
Running (overground):
- VO₂ = 0.2(speed) + 0.9(speed)(grade) + 3.5
- 0.2 ml·kg⁻¹·min⁻¹ per m·min⁻¹ for horizontal motion.
- 0.9 ml·kg⁻¹·min⁻¹ per m·min⁻¹ per unit grade for vertical component.
- Typical running speeds produce higher horizontal cost than walking at the same speed.
Leg ergometry (cycling):
- VO₂ = (10.8 × W × M) + 7
- Alternate equivalent form: VO₂ = 1.8 × (Work Rate)/M + 7
- Where
- W = external work rate in watts,
- M = mass in kg.
- Unloaded cycling cost: resting + 3.5; hence the 7 = 3.5 + 3.5.
- Power outputs range typically 300–1200 kg·m/min or 50–200 W.
Arm ergometry (arm crank):
- VO₂ = (18 × W × M) + 3.5
- Equivalent form: VO₂ = 3 × (Work Rate)/M + 3.5
- Note: No horizontal component; the vertical component is minimal and there is no separate unloaded cycling term like “7.”
- Power outputs range approximately 150–750 kg·m/min or 25–125 W.
Stepping (a stepping test or exercise):
- Stepping rate: 12–30 steps per minute.
- Step heights: 0.04–0.4 m (1.6–15.7 inches).
- Involves concentric and eccentric contractions; metabolic cost depends on rate and height.

Stepwise Approach to Using Metabolic Calculations

Step 1: Convert to appropriate units and know common equivalents.
Step 2: Transform VO₂ into the most appropriate units for the problem (relative vs. absolute).
Step 3: Write the appropriate equation in the form VO₂ = R + H + V and substitute values.
Always check units and re-check the final VO₂ against the expected range for the task.

Practice Problems and Worked Solutions

Practice Conversions
- Q1: What is the MET equivalent to 8.75 ml·kg⁻¹·min⁻¹?
- Solution: 8.75 ÷ 3.5 = 2.5 METs.
- Q2: What is the absolute oxygen consumption equivalent to 10 METs for a 155‑pound (lb) male?
- Convert mass: 155 lb ÷ 2.2 = 70.45 kg.
- Relative VO₂ at 10 METs: VO₂,rel = 10 × 3.5 = 35 ml·kg⁻¹·min⁻¹.
- Absolute VO₂: VO₂,abs = VO₂,rel × M = 35 × 70.45 ≈ 2465.75 ml·min⁻¹.
- Q3: What is the equivalent total caloric expenditure of 2.5 pounds of fat?
- 2.5 lb × 3500 kcal/lb = 8750 kcal.
- Q4: Convert 8 METs to relative VO₂.
- VO₂,rel = 8 × 3.5 = 28 ml·kg⁻¹·min⁻¹.
- Q5: Convert 4.5 mph to m·min⁻¹.
- Speed = 4.5 mph × 26.8 ≈ 120.6 m·min⁻¹.
Practice Problem: Walking prescription (Sample from notes)
- Problem: A 30-year-old man, resting HR 60 bpm, max HR 190 bpm, weight 180 lb, VO₂max 48 ml·kg⁻¹·min⁻¹. He wants to walk on a treadmill at 3.5 mph, starting at 70% VO₂max.
- Step A: Target VO₂ = 0.70 × VO₂max = 0.70 × 48 = 33.6 ml·kg⁻¹·min⁻¹.
- Step B: Convert speed to m·min⁻¹: 3.5 mph = 3.5 × 26.8 ≈ 93.8 m·min⁻¹.
- Step C: Solve for grade using walking VO₂ equation: VO₂ = 0.1(speed) + 1.8(speed)(grade) + 3.5.
- 33.6 = 0.1(93.8) + 1.8(93.8)(grade) + 3.5
- 33.6 = 9.38 + 168.84 × grade + 3.5
- 33.6 − 12.88 = 168.84 × grade
- 20.72 ≈ 168.84 × grade
- grade ≈ 0.1226, or about 12.3% grade.
Practice Problem: Running (Sample from notes)
- Problem: A man weighing 176 lb runs at 9 minutes per mile on level ground. Estimate gross VO₂.
- Step A: Convert weight to kg: 176 lb ÷ 2.2 ≈ 80 kg.
- Step B: Determine speed in m·min⁻¹: 9 min per mile → speed = 1 mile / 9 min → ≈ 0.111… miles/min; in m/min: 0.111… × 1609.34 ≈ 178.7 m·min⁻¹.
- Step C: Using walking-to-running VO₂ formulation for running on level ground: VO₂ = 0.2(speed) + 0.9(speed)(grade) + 3.5 with grade = 0.
- VO₂ = 0.2(178.7) + 0.9(178.7)(0) + 3.5 ≈ 35.74 + 0 + 3.5 ≈ 39.24 ml·kg⁻¹·min⁻¹.
- Reported value in notes: ≈ 39.19 ml·kg⁻¹·min⁻¹, which matches within rounding.
Summary of Practice Solutions
- MET conversion, absolute and relative VO₂ conversions, and kcal estimates can be derived directly from the formulas above.
- Always convert units first, select the correct VO₂ equation, substitute, and verify the plausibility of the result.

Practice Conversions and Quick Facts

1 mph = 26.8 m/min.
1 inch = 2.54 cm.
1 kg = 2.2 lb.
1 L O₂ ≈ 5 kcal.
1 MET = 3.5 ml·kg⁻¹·min⁻¹.
1 W = 6 kg·m/min.
Power (kg·m/min) = R × D × f (R in kg, D in m/rev, f in rev/min).
Step rate: 12–30 steps/min; step height: 0.04–0.4 m.

Summary

The metabolic equation framework (VO₂ = R + H + V) provides a structured method to estimate oxygen cost and energy expenditure across activities.
Distinguish between gross vs net VO₂ and absolute vs relative VO₂ to suit the problem context.
Use walking and running VO₂ equations to estimate horizontal and vertical costs, plus a resting component.
For ergometry, use specific equations for leg and arm ergometers, noting the presence/absence of horizontal components and the specific constants (e.g., the extra 7 or 3.5 adjustments).
Apply the stepwise methodology to ensure unit consistency, proper equation selection, and logical checks of results.
Memorize key conversions and formulas to enable quick, accurate problem solving during exams.

Quick Reference Formulas (LaTeX)

VO₂ components: VO_2 = R + H + V
Walking VO₂: VO_2 = 0.1\,(speed) + 1.8\,(speed)\,(grade) + 3.5
Running VO₂: VO_2 = 0.2\,(speed) + 0.9\,(speed)\,(grade) + 3.5
Leg ergometry: VO_2 = 10.8\, W\, M + 7
Leg ergometry (equivalent): VO_2 = 1.8\frac{W}{M} + 7
Arm ergometry: VO_2 = 18\, W\, M + 3.5
Arm ergometry (equivalent): VO_2 = 3\frac{W}{M} + 3.5
Resting cost: R = 3.5\ ext{ml} \cdot \text{kg}^{-1} \cdot \text{min}^{-1}
1 MET: 1\,MET = 3.5\ \text{ml} \cdot \text{kg}^{-1} \cdot \text{min}^{-1}
1 L O₂ to kcal: 1\,\text{L O}_2 \rightarrow \approx 5\ \text{kcal}
1 mph to m/min: 1\ \text{mph} = 26.8\ \text{m/min}
1 inch to cm: 1\,\text{in} = 2.54\ \text{cm}
1 W to kg·m/min: 1\,\text{W} = 6\ \text{kg} \cdot \text{m} \cdot \text{min}^{-1}
Power: \text{Power} = R \times D \times f
Step parameters: step rate 12–30 min⁻¹; height 0.04–0.4 m
Mass and unit conversions: \text{kg} = \text{lb} / 2.2; \text{m} = \text{cm} / 100