Calculation Methods & Formulas
Complete documentation of all threshold calculations used, with exact formulas, scientific sources, and validation data.
Threshold Calculations (LT1/LT2)
Log-Poly-ModDmax
StandardDefault method. Like Modified Dmax, but the reference line starts at the log-log breakpoint (onset of the lactate rise after Beaver 1985) instead of the first 0.4 mmol/L rise. Curve: 3rd-order polynomial through all data points. If there is no log-log breakpoint (no baseline plateau), the app falls back to Modified Dmax and labels the result.
1. Start point S = log-log breakpoint (see Log-Log)
2. Reference line from S to the last data point E
3. Curve: 3rd-order polynomial y = a + b·x + c·x² + d·x³ (least squares)
4. LT2 = curve point between S and E with the maximum perpendicular distance to the lineModified Dmax
Maximum perpendicular distance of the polynomial curve (3rd order) to the line from the point preceding the first lactate rise ≥ 0.4 mmol/L to the final data point (Bishop et al. 1998).
1. Find start point S:
If La[i] - La[i-1] ≥ 0.4 mmol/L → S = (P[i-1], La[i-1])
No such rise before the last stage → not determinable
2. Curve: 3rd-order polynomial through all data points (least squares)
3. Distance of every curve point (x,y) to the line S→E:
d(x,y) = |A·x + B·y + C| / √(A² + B²)
where: A = -(La_E - La_S) / (P_E - P_S), B = 1, C = A·P_S - La_S
4. LT2 = curve point with max(d)Dmax (Original)
LegacyOriginal Dmax method by Cheng et al. 1992: line from the first to the last data point, 3rd-order polynomial as the curve.
Identical to Modified Dmax, but:
Start point S = (P[0], La[0]) ← First data pointThe result depends strongly on the first stage: elevated starting lactate shifts the reference line.
Dickhuth (IAT)
Individual Anaerobic Threshold based on the lactate equivalent. The minimum of the equivalent indicates maximum metabolic efficiency.
1. Calculate lactate equivalent:
LE(i) = La(i) / P(i)
2. Find minimum:
i_min = argmin[LE(i)]
La_min = La(i_min)
3. Thresholds:
LT1 = P(i_min) ← the minimum itself
LT2 (IAT) = Power at (La_min + 1.5 mmol/L)
If the minimum falls on the first or last stage, LT1 is not determinable.Delta (+1.5 mmol/L)
Fixed offset above the starting lactate (minimum of the first three stages). Simple, but dependent on the starting value.
1. Determine baseline:
La_baseline = min(La[0], La[1], La[2])
2. Thresholds:
LT1 = Power at (La_baseline + 0.5 mmol/L)
LT2 = Power at (La_baseline + 1.5 mmol/L)
3. Interpolate power:
LT2_P = P[j] + (LT2_La - La[j])/(La[j+1] - La[j]) × (P[j+1] - P[j])Mader (4.0 mmol/L)
LegacyFixed threshold at 4.0 mmol/L (OBLA). Historically widely used, but not individual.
LT2_Lactate = 4.0 mmol/L (fixed value)
LT2_P = P[j] + (4.0 - La[j])/(La[j+1] - La[j]) × (P[j+1] - P[j])
⚠️ The individual MLSS varies between 2.0-7.0 mmol/L!This method is NOT individual and systematically overestimates the anaerobic threshold in many athletes.
Log-Log (LT1)
Onset of the lactate rise after Beaver et al. 1985: on a double-logarithmic plot the blood lactate curve consists of a phase of slow and a phase of rapid increase; the intersection of the two regression lines is the first lactate threshold. LactateThreshold uses it as LT1 of the default method and as the starting point of Log-Poly-ModDmax.
1. x' = log₁₀(P), y' = log₁₀(La)
2. For every split point two regression lines (≥ 2 points per segment);
split with the lowest residual sum of squares
3. LT1 = intersection of the two lines, back-transformed: P = 10^x'
Valid only if the upper line is steeper, the intersection lies inside the measured range
and La there ≤ baseline + 1.0 mmol/L — otherwise the curve has no breakpoint.Baseline + 0.5 mmol/L (LT1)
LT1 for Modified Dmax, Dmax, Delta and Mader: the power at which lactate is 0.5 mmol/L above the starting value (minimum of the first three stages), linearly interpolated.
La_baseline = min(La[0], La[1], La[2])
LT1 = Power at (La_baseline + 0.5 mmol/L)Where to find the comparison
LABThe interactive method comparison runs on your own measurements, so it lives with the data: open a saved step test and you will find it directly below the analysis.
Go to my tests →Uncertainty & Validation
★ V4The core innovation: honest measurement uncertainty instead of false precision. Every threshold receives a CI, every curve gets coverage, and every progression comparison is checked against measurement tolerance.
Bootstrap confidence intervals
New in V4Parametric bootstrap: all lactate values are perturbed 500 times with device-typical noise and the thresholds recomputed; the 68% interval of the results gives the ±. The noise model grows with the measured value (σ = 0.10 + 0.03·y) and lies within the repeatability of hand-held analysers (< 0.5 mmol/L between 1 and 10 mmol/L). Deterministically seeded, the same procedure for every number of stages.
σ(y) = σ_base + σ_cv · y
σ_base = 0.10 mmol/L (device floor)
σ_cv = 0.03 (3% CV)
CI₆₈ = [P₁₆, P₈₄] → ±Half-widthGoodness of fit (R²)
New in V4Polynomial fit (cubic for n ≥ 5, quadratic for n = 4) used to quantify lactate-curve noise. Low R² can indicate noisy measurements, stress lactate or a non-monotone curve.
R² = 1 − (Σ(y_i − ŷ_i)²) / (Σ(y_i − ȳ)²)
Categories:
≥ 95% → Excellent
≥ 90% → Good
≥ 80% → Acceptable
< 80% → NoisyCoverage score
New in V4Checks whether the test actually reached LT2. R² only measures noise: a 4-stage test at 40-60% of LT2 can have R² ≈ 1.0 and still miss LT2 completely. Coverage answers whether the curve truly brackets LT2.
Complete ⟺ all 3 criteria are met:
1. Peak lactate ≥ 4.0 mmol/L
2. ≥ 2 stages > 2.5 mmol/L
3. Δ(last 2 stages) ≥ 0.8 mmol/LProgression significance (error propagation)
New in V4Compares two tests with error propagation: is the improvement significant relative to measurement uncertainty? This prevents athletes from interpreting noise as progress. Three levels are used: 95%, 68% and not significant.
σ_combined = √(CI_a² + CI_b²)
|Δ| > 1.96·σ → sig. 95%
|Δ| > 1.00·σ → sig. 68%
otherwise → n.s.LT1-anchored training zones
New in V4Z2 upper bound = measured LT1 instead of a fixed percentage of LT2. San Millán & Brooks 2018 show that blood lactate and fat oxidation are inversely correlated across all fitness levels — a Z2 bound at LT1 keeps Z2 in the range where fat oxidation still dominates. The Z1 upper bound (0.85 × LT1) and the Z3 upper bound (midpoint LT1–LT2) are conventions of this app, not published values.
lt1Frac = lt1 / lt2 (bike)
lt1Frac = lt2 / lt1 (run, speed ratio)
Z1_upper = 0.85 · lt1Frac ← convention
Z2_upper = lt1Frac ← measured LT1
Z3_upper = (lt1Frac + 1) / 2 ← convention
Z4_upper = 1.05 ← +5% LT2MLSS calibration & method ranking
New in V4Gold-standard validation: the user uploads MLSS constant-test values, the app compares all 6 LT2 methods against this anchor, and ranks them by absolute error. Across multiple comparison pairs this creates a per-athlete method recommendation. Lab tier.
For each method M:
error_M = |LT2_M − MLSS|
Ranking: ascending by mean(error_M)
across all (stepTest, mlssTest) pairs.VLamax Calculation
VLamax Sprint Test
StandardMaximal lactate accumulation rate from an all-out sprint of 10–12 s (Quittmann 2025), lactate before the sprint and after it until the peak.
VLamax = ΔLa / t_eff
where:
ΔLa = La_peak - La_rest [mmol/L]
t_eff = t_sprint - t_alac [s]
t_alac = 3.0 s (10 s sprint) to 3.5 s (15 s), linear (Quittmann 2025)
Example:
Sprint: 15 s, Rest-La: 1.2, Peak-La: 12.8
t_eff = 15 - 3.5 = 11.5 s
VLamax = (12.8 - 1.2) / 11.5 = 1.01 mmol/L/sMathematical Foundations
Linear Interpolation
Simple interpolation between two known points.
y = y₁ + (x - x₁)/(x₂ - x₁) × (y₂ - y₁)
For threshold calculation (Y → X):
x = x₁ + (y - y₁)/(y₂ - y₁) × (x₂ - x₁)Fritsch-Carlson Monotone Spline
New in V4Monotone cubic Hermite interpolation. It guarantees that no new extrema appear between data points, unlike Catmull-Rom splines, which can oscillate on steep or non-monotone curves. This matters for Dmax geometry because overshoot can create false inflection points.
1. δ_k = (y_{k+1} − y_k) / (x_{k+1} − x_k)
2. m_k = (δ_{k-1} + δ_k) / 2
3. if δ_{k-1}·δ_k ≤ 0 → m_k = 0
4. if α² + β² > 9 → scale m_k, m_{k+1}
5. Hermite basis: h₀₀, h₁₀, h₀₁, h₁₁Point-to-Line Distance (Dmax)
Core formula for all Dmax-based methods.
Perpendicular distance of a point (x,y) to the line:
|(y₂-y₁)·x - (x₂-x₁)·y + x₂·y₁ - y₂·x₁|
d(x,y) = ─────────────────────────────────────────
√[(y₂-y₁)² + (x₂-x₁)²]
where (x₁,y₁) = Start point, (x₂,y₂) = End pointReferences
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