Dynamic Metabolic Model
This simulator integrates the nonlinear ODE system proposed by Mader (2003) for cellular energy metabolism, extended by the two-compartment lactate model of Heck, Bartmus & Grabow (2022). The state vector comprises four coupled variables — phosphocreatine [PCr], pulmonary oxygen uptake V̇O2, muscle lactate [Lam], and blood lactate [Lab] — whose dynamics are governed by the creatine-kinase / adenylate-kinase equilibrium (Mader 2003, Eq. 1–6) and Michaelis–Menten-type rate equations for oxidative phosphorylation and anaerobic glycolysis.
State equations (simplified):
d[PCr]/dt = V̇ox + V̇gly − V̇ATP,demand
dV̇O2/dt = kVO₂ · (V̇O2,ss(ADP) − V̇O2)
d[Lam]/dt = v̇La − vox,La − Kdif · ΔLa
d[Lab]/dt = Vrel · (Kdif · ΔLa − vox,b − vres,b)
The phosphorylation state ([ATP], [ADP], [AMP], Pi) is recomputed at every time step from the CHEP equilibrium (creatine kinase KCK = 1.66 · 10⁹, adenylate kinase KAK = 1.05, total adenine nucleotide pool TAN = 8.2 mmol/kg; Mader 2003). [ADP] serves as the central feedback signal driving oxidative phosphorylation (sigmoidal Hill kinetics with nox) and glycolytic flux (via phosphofructokinase activation). Integration uses an explicit fourth-order Runge–Kutta scheme (dt typically 0.1 s).
Two-Compartment Lactate Model
Muscle and blood lactate are coupled by a concentration-dependent diffusion term Kdif(Lab) = Kdif,0 · Lab−K₁, representing monocarboxylate transporter (MCT1/MCT4) saturation kinetics (Heck et al. 2022, Eq. 27). Lactate elimination occurs via oxidation (vox = KLaO₂ · [La]) in both compartments and gluconeogenic resynthesis (vres, ADP-driven, primarily hepatic/renal). The volume ratio Vrel = mactive / (BWeff − mactive) scales inter-compartmental fluxes, where BWeff represents the effective distribution volume (default 45 kg for 75 kg body mass).
pH and Glycolytic Inhibition
Intramuscular pH is derived from lactate accumulation and non-bicarbonate buffer capacity β (slyke; Mader & Heck 1994): pH = pHrest − Δ[Lam] / β. Rising [H⁺] inhibits phosphofructokinase (PFK) and thereby glycolytic rate via:
V̇Lamax,pH = V̇Lamax / (1 + [H⁺]n / Ks3)
(Mader & Heck 1994, Eq. 18). This pH-mediated feedback explains the self-limiting nature of supra-maximal exercise and the emergence of a maximal lactate steady state (MLSS) where lactate production equals elimination.
Glycogen Depletion
Muscle glycogen is tracked as a fifth state variable, depleted by glycolytic flux and partially replenished by exogenous CHO intake. Depletion reduces V̇Lamax via a Hill function: f(Glyc) = GlycnH / (KmnH + GlycnH) (Km = 0.25, nH = 3; Neubig 2021, in Heck et al. 2022, Eq. 34), representing substrate limitation of glycogen phosphorylase. V̇O2max is attenuated via a cube-root relationship (minimum factor ≈ 0.50; Hargreaves & Spriet 2020), reflecting reduced pyruvate flux to the TCA cycle when glycogen stores are critically low.
Dynamic Gross Efficiency
Constant η — A fixed gross efficiency (η ≈ 25 %, corresponding to Ks4 = 12.322; Mader 2003) converts metabolic ATP turnover to mechanical power. This is the default mode and appropriate for most aerobic simulations.
Dynamic η(t) — Following Dunst et al. (2023), instantaneous efficiency is computed as the ATP-flux-weighted mean of pathway-specific efficiencies: ηox ≈ 24.9 % (= 60 / (Ks4 · CEfat) with CEfat = 19.584 J/mL O₂ at RER = 0.70), ηgly ≈ 15 %, and ηpcr ≈ 10–13 %. This yields η ≈ 13 % during all-out sprints (PCr-dominated) and η ≈ 23–25 % at aerobic steady state. When a measured 1 s peak power Pmax is provided, ηpcr is calibrated so the model reproduces the observed sprint power.
Dynamic η(t) + SC — Adds the V̇O₂ slow component. Above MLSS, effective efficiency decreases progressively: SC = 1 + ksc · f(I) · tsupra, where f(I) = clamp((P − PMLSS) / (PV̇O₂max − PMLSS), 0, 3) is the normalised supra-threshold intensity and tsupra the accumulated time above MLSS. This increases ATP cost per watt by ~3–8 %/min in the severe domain, shortening Tlim by ≈ 20–40 % near PV̇O₂max. The mechanism reflects progressive type-II fibre recruitment, Pi-mediated impairment of cross-bridge cycling, and mitochondrial inefficiency (Jones et al. 2011; Grassi et al. 2015). Auto-activates for Sprint and Max Effort modes.
Simulation Modes
Abort on Exhaustion terminates when any metabolic threshold is reached (PCr < 1.0 mmol/kg, pH < 6.4, or ATP < 2.0 mmol/kg), then continues at recovery power until tsimulation. Thresholds represent the physiological limits of the Lohmann reaction, intracellular acidosis tolerance, and adenine nucleotide depletion, respectively.
Energy Limited allows the athlete to continue beyond metabolic thresholds but reduces realisable power to the instantaneously available metabolic capacity (Preal = min(Pdemand, Pmax,metabolic)).
Max Effort computes the maximal sustainable power at each time step from instantaneous V̇max and bWatt, requiring dynamic efficiency and Pmax calibration.
Environmental Conditions
Altitude. V̇O2max decreases linearly by 6.3 % per 1 000 m above 580 m due to reduced arterial O₂ saturation (Wehrlin & Hallén 2006). V̇Lamax is unaffected as glycolysis is O₂-independent.
Temperature. Above the thermoneutral zone (≤ 22 °C), V̇O2max decreases by 1.5 %/°C due to thermoregulatory competition for cardiac output and peripheral vasodilation (Périard, Racinais & Sawka 2015). V̇Lamax increases by 1.0 %/°C due to the Q10 effect on phosphofructokinase and catecholamine-driven glycogenolysis (Febbraio et al. 1994). Altitude and temperature effects on V̇O2max are applied multiplicatively.
References
Mader A (2003) Eur J Appl Physiol 88:317–338 · Mader A, Heck H (1994) BSW 8(2):124–162 · Heck H, Bartmus U, Grabow V (2022) Laktat — Das Buch, Kap. 4, Feldhaus · Dunst AK, Hesse C, Ueberschär O, Holmberg HC (2023) Sports 11(2):29 · Jones AM, Grassi B, Christensen PM et al. (2011) Med Sci Sports Exerc 43:2046–2062 · Grassi B, Rossiter HB, Zoladz JA (2015) J Physiol 593:3413–3434 · Wehrlin JP, Hallén J (2006) Eur J Appl Physiol 96:404–412 · Périard JD, Racinais S, Sawka MN (2015) Compr Physiol 5:255–281 · Febbraio MA et al. (1994) J Appl Physiol 77:2827–2831 · Hargreaves M, Spriet LL (2020) Nat Metab 2:817–828
This plot shows the relative contribution of the three energy systems (oxidative, glycolytic, and PCr) to overall ATP turnover.
The PCr contribution reflects the instantaneous deficit between ATP demand and aerobic + glycolytic supply. At exercise onset or intensity increases, PCr spikes as it buffers the VO₂ kinetics delay.