The Vivanco
Proximity-to-Failure
Model

The distance between current repetition and momentary muscular failure is the single most actionable variable in resistance-training programming.

Known as Repetitions in Reserve (RIR), this metric of proximity to failure has been shown to exhibit a dose–response relationship with muscle hypertrophy across a broad meta-analytic base: training closer to failure yields greater hypertrophic adaptations, with sets at 0–2 RIR producing near-maximal stimulus[1,2,3]. The 2026 American College of Sports Medicine (ACSM) position stand, synthesizing 137 systematic reviews covering more than 30,000 participants, recommends training to within 1–2 repetitions of failure for hypertrophy and endorses autoregulatory approaches as best practice[4].

Yet the field lacks a unifying mathematical framework that connects subjective RIR to measurable biomechanical quantities, accounts for the multi-compartment nature of fatigue, and provides a principled path from observation to optimal prescription. Existing models such as the Epley and Brzycki formulas treat the load–repetition relationship as a population-level approximation with no mechanism for intra-set fatigue evolution, inter-individual calibration, or stimulus quantification[5,6].

The Vivanco Proximity-to-Failure Model (VPFM), introduced in its first version as a single-equation predictor, addressed the calibration gap by deriving a closed-form RIR from an individualized exponential decay law. Version 2.0 added two-state fatigue, velocity integration, continuous-time dynamics, stochastic Bayesian calibration, and a Fisher Information identifiability analysis. This paper presents Version 3.0, which advances the framework along eight new axes:

Together these extensions transform the VPFM from a prediction tool into a prescription platform—a system that not only estimates where a trainee is but tells them what to do next and why. The mathematical backbone ensures that every recommendation is traceable to an axiom, a theorem, or a Bayesian posterior, providing a level of transparency absent from black-box machine-learning approaches.

2.1   Proximity to Failure & Dose–Response Evidence

Robinson et al. (2024) conducted a dose–response meta-regression of 89 studies demonstrating that proximity to failure exhibits a graded relationship with muscle hypertrophy: each additional RIR subtracted corresponds to diminishing but positive marginal stimulus, with the curve flattening beyond 0–1 RIR[3]. Critically, the strength relationship is attenuated, suggesting that the failure-proximity signal is primarily a hypertrophy driver. The 2026 ACSM position stand formalized this into a minimum effective proximity of 1–2 RIR, noting that training to true failure increases fatigue cost without proportional hypertrophic benefit[4].

2.2   Traditional One-Repetition-Maximum Models

The Epley formula \(1\text{RM}=w(1+r/30)\) and the Brzycki formula \(1\text{RM}=w\cdot 36/(37-r)\) remain the most widely deployed load–repetition models[5,6]. Both are population-level regression fits with no individual calibration mechanism, no fatigue state variable, and no pathway to autoregulation beyond plugging in a fresh repetition maximum—a test that itself induces fatigue.

2.3   Digital Twins in Sport Science

The digital twin paradigm—a continuously synchronized computational replica of a physical system—has been applied in endurance sport and injury prevention. Hendry et al. (2025) proposed a cross-bridge model of skeletal-muscle fatigue suitable for embedding in a twin, and a 2025–2026 wave of reviews advocates for multi-scale muscle models that couple sarcomere-level dynamics to whole-movement performance[7,8]. The VPFM is uniquely positioned to serve as the physics engine of a resistance-training digital twin because its ODE system is low-dimensional enough for real-time inference yet expressive enough to capture two-state fatigue, multi-muscle coupling, and sensor-driven updates.

2.4   Reinforcement Learning for Training Prescription

Recent work has framed exercise programming as a sequential decision problem. Akiba et al. (2025) applied deep RL to training-load control in team sports, demonstrating that a learned policy outperforms heuristic periodization[9]. Separately, gym-specific RL formulations have appeared on arXiv (2025–2026), proposing state spaces over perceived exertion and load history[10,11]. What these approaches lack is a mechanistic forward model of the trainee—they learn the dynamics from data end-to-end, sacrificing interpretability and sample efficiency. The VPFM digital twin provides exactly this missing physics layer, enabling model-based RL and model-predictive control.

2.5   Wearable Sensors & Fatigue Estimation

Velocity-based training (VBT) has become commercially mainstream, with linear position transducers and accelerometer-based devices providing rep-by-rep barbell velocity[12]. Recent work extends the sensor envelope to surface EMG for fatigue estimation[13] and muscle-synergy graph neural networks for coordination analysis[14]. Heart rate and heart-rate variability capture autonomic fatigue load. The challenge is fusion: combining multiple noisy channels into a coherent state estimate. The Extended Kalman Filter (EKF) framework we develop in Section 8 addresses this by treating the VPFM state as the latent variable and the multi-channel signals as noisy observations.

2.6   Hierarchical Bayesian Models in Sport

Hierarchical (multilevel) Bayesian models are standard in exercise science meta-analysis[15]. Their application to individual calibration within a population prior is less explored in the resistance-training domain. The cold-start problem—prescribing training for a new user with no history—is formally identical to the Bayesian optimal experiment design literature[16]. The VPFM's analytic gradient structure makes it especially amenable to information-theoretic calibration design, as we show in Section 11.

The VPFM rests on five axioms that constrain the shape of the load–repetition relationship. These axioms are not arbitrary postulates but formal statements of biomechanical plausibility.

3.1   Notation Summary

A single capacity scalar conflates phenomena with fundamentally different time constants.

Neuromuscular fatigue operates through at least two distinguishable compartments: a structural (peripheral) pool \(C_s\), representing sarcomeric force-generating capacity, and an enzymatic (central/metabolic) pool \(C_e\), capturing ATP resynthesis, calcium handling, and central drive[7,8]. These compartments deplete together during exertion but recover on different time scales.

4.1   Depletion Model

After a set of \(r\) reps at load \(w\), the effort intensity is \(e=w/(C_s\,L_0)\) and the volume–intensity integral is \(r\cdot e^\gamma\), with exponent \(\gamma\ge1\) amplifying the effect of heavier loads. The depletion equations are:

where \(k_s,\,k_e>0\) are depletion rate constants and \(\eta\in[0,1]\) partitions the fatigue load between pools. When \(\eta=0.5\) and \(k_s=k_e\), the model reduces to symmetric depletion.

4.2   Recovery Model

During a rest interval \(\Delta t\) (seconds), each pool recovers toward unity with its own time constant:

Typical values are \(T_s\in[30,120]\) seconds (fast, phosphocreatine-dominated) and \(T_e\in[180,600]\) seconds (slow, metabolic). This asymmetry explains why short rest produces disproportionate metabolic fatigue while structural capacity partially recovers.

Not all reps are created equal. The reps closest to failure contribute disproportionately to hypertrophic stimulus.

The dose–response evidence (Section 2.1) implies that each repetition's contribution to hypertrophy is a nonlinear function of how close it is to failure. We formalize this through a stimulus function \(S(\Phi)\) that maps the instantaneous RIR at each rep to a stimulus unit.

5.1   Session Stimulus Dose

Over a set of \(r\) reps, the instantaneous RIR decreases from \(\Phi_0=\rho(w)-0\) down to \(\Phi_r=\rho(w)-r\). The session stimulus dose is the sum of per-rep stimulus across all sets:

where \(\rho_i=F(w_i,C_s^{(i)},C_e^{(i)})\) is the fatigue-adjusted maximum reps for set \(i\). This provides a principled alternative to counting "hard sets" or "effective reps."

5.2   Marginal Stimulus and Fatigue Cost

Each additional rep \(j\) in a set contributes marginal stimulus \(\Delta S_j = S(\Phi(j))\) and marginal fatigue cost \(\Delta F_j\), measured as the reduction in capacity for subsequent sets. The stimulus efficiency ratio is:

As the trainee approaches failure, \(\Delta S_j\) plateaus near 1 while \(\Delta F_j\) accelerates (due to the exponential depletion model). This yields a natural optimum.

Compound exercises recruit multiple muscles simultaneously, and failure is dictated by the weakest link in the kinetic chain.

The single-exercise VPFM tracks global capacity \((C_s, C_e)\). In a real training session, a bench press fatigues the pectorals, anterior deltoids, and triceps; a subsequent overhead press inherits tricep fatigue but starts with fresh lateral deltoids. Capturing these interactions requires a graph structure over muscles.

6.1   Recruitment Matrix

Let \(j=1,\ldots,J\) index exercises and \(m=1,\ldots,M\) index muscle groups. The recruitment matrix \(\mathbf{R}\in[0,1]^{J\times M}\) has entry \(R_{jm}\) giving the fractional recruitment of muscle \(m\) in exercise \(j\). For example, \(R_{\text{bench},\,\text{pec}}=0.9\), \(R_{\text{bench},\,\text{tri}}=0.6\), \(R_{\text{bench},\,\text{ant.delt}}=0.4\).

6.2   Coupled Depletion

When exercise \(j\) is performed for \(r\) reps at intensity \(e\), muscle \(m\) is depleted in proportion to its recruitment:

The recruitment coefficient \(R_{jm}\) modulates the fatigue dose: a muscle recruited at 30% receives 30% of the fatigue stimulus. Recovery proceeds independently for each muscle.

6.3   Weakest-Link Failure Prediction

The maximum reps achievable on exercise \(j\) is limited by the muscle with the lowest scaled capacity:

Barbell velocity provides an objective, real-time observable that correlates with proximity to failure. The VPFM leverages the well-established linear load–velocity relationship[12].

7.1   The Load–Velocity Profile

For a given individual and exercise, concentric mean velocity \(v\) and load \(w\) obey an approximately linear relationship:

where \(v_1\) is the velocity at minimal load and \(v_{\min}\approx 0.15\text{–}0.3\;\text{m/s}\) is the minimum velocity threshold at which the concentric phase can be completed. The factor \(C_s\,L_0\) replaces the static 1RM with the fatigue-adjusted ceiling.

7.2   Velocity-Derived Capacity

Given an observed velocity \(v_{\text{obs}}\) at load \(w\), we can invert the relationship to estimate the current capacity:

As the trainee fatigues and velocity drops toward \(v_{\min}\), the estimated remaining capacity shrinks, providing a rep-by-rep update to the VPFM state. Velocity loss percentage \(\text{VL}\%=(v_1-v_{\text{rep}})/v_1\times 100\) serves as a surrogate for effort intensity.

7.3   Practical Integration

Modern linear position transducers (e.g., GymAware, PUSH Band) and accelerometer-based devices (e.g., RepOne, Enode) provide rep-by-rep velocity data at sampling rates of 50–200 Hz. The VPFM can ingest this stream to update \(C_s\) in real time, reducing reliance on subjective RIR estimation. This velocity channel becomes one of several inputs to the multi-sensor fusion framework developed in the next section.

No single sensor captures the full fatigue state. Multi-channel fusion through the Extended Kalman Filter converts noisy observations into a coherent Bayesian state estimate.

The VPFM state vector \(\mathbf{x}=[C_s,\;C_e,\;\tau,\;L_0]^\top\) is latent—not directly measurable. What we observe are noisy signals from multiple channels: barbell velocity, heart rate, and (optionally) surface electromyography (EMG). The Extended Kalman Filter (EKF) provides the standard Bayesian filtering framework for nonlinear state-space models with Gaussian noise.

8.1   State-Space Formulation

Process model (prediction step): Between observations, the state evolves according to the VPFM depletion–recovery dynamics:

where \(\mathbf{u}_k=(w_k, r_k, \Delta t_k)\) is the control input (load, reps, rest) and \(f\) encodes the depletion and recovery equations from Section 4.

Observation model (update step): The multi-channel observation vector \(\mathbf{z}_k=[v_k,\;\text{HR}_k,\;\text{EMG}_k]^\top\) is related to the latent state by:

8.2   Channel-Specific Observation Models

8.3   EKF Update Equations

The standard EKF linearizes \(f\) and \(h\) via their Jacobians:

Predict:

Update:

The Kalman gain \(\mathbf{K}\) automatically weights channels by their informativeness: a precise velocity sensor contributes more to the state update than a noisy HR strap, and a missing channel is handled by setting its row in \(\mathbf{H}\) to zero with infinite observation variance.

While the set-by-set depletion model is practical for conventional training, sustained efforts (AMRAPs, drop sets, supersets with minimal rest) require a continuous-time formulation. We embed the two-state model in an ODE system.

9.1   The VPFM ODE System

During exertion at constant load \(w\), the state vector \((C_s(t),\;C_e(t))\) evolves as:

where \(\alpha_s=k_s\eta/\Delta t_{\text{rep}}\), \(\alpha_e=k_e(1-\eta)/\Delta t_{\text{rep}}\) are continuous depletion rates, \(\beta_s=1/T_s\), \(\beta_e=1/T_e\) are recovery rates, and \(\mathbb{1}_{\text{rest}}\) is the indicator for rest periods. During rest, the exertion terms vanish and only recovery operates.

9.2   Equilibrium Analysis

For sustained effort at constant intensity \(e\), the ODE system admits a steady state where depletion and recovery balance. Setting the derivatives to zero during simultaneous exertion and (partial) recovery:

9.3   Critical Session Intensity

The equilibrium collapses (\(C_s^*\to0\) or \(C_e^*\to0\)) when the depletion rate overwhelms recovery. The critical intensity \(e_{\text{crit}}\) is the threshold above which sustained effort is unsustainable:

where \(\beta_{\min}=\min(\beta_s,\beta_e)\) and \(\alpha_{\max}=\max(\alpha_s,\alpha_e)\). Training above this intensity inevitably drives the system to failure; training below it can, in principle, continue indefinitely.

Individual calibration requires data. Population structure provides the bridge from zero data to personalized prediction.

The VPFM parameters \(\boldsymbol{\theta}_i=(L_{0,i},\;\tau_i,\;k_{s,i},\;k_{e,i})\) vary across individuals. A new user has no data, yet we must prescribe training from day one. The hierarchical Bayesian framework solves this cold-start problem by placing individual parameters within a population distribution.

10.1   Population Prior Structure

The hierarchical model posits that individual parameters are drawn from a population distribution:

The population hyperparameters \(\boldsymbol{\mu},\boldsymbol{\sigma}\) are themselves estimated from pooled training data across all users. We use log-normal priors for rate parameters to enforce positivity.

10.2   Likelihood and Observation Model

Given a sequence of observed sets \(\{(w_k, r_k, \Phi_k^{\text{obs}})\}_{k=1}^K\) for individual \(i\), the likelihood of the observations under heteroscedastic noise is:

where the noise variance \(\sigma_k^2 = \sigma_0^2 + \sigma_1^2\,\Phi_k^2\) captures the empirical observation that RIR estimation error increases at higher reserves (farther from failure).

10.3   Cold-Start Protocol

For a new user with no data, the posterior is exactly the population prior:

As the user completes sets, the posterior concentrates around their individual parameters. The rate of convergence depends on the informativeness of the observed sets—a question addressed by the optimal calibration protocol in Section 11.

10.4   Partial Pooling and Shrinkage

The hierarchical structure induces partial pooling: individuals with sparse data are pulled (shrunk) toward the population mean, while data-rich individuals dominate their own posterior. This is statistically optimal under the population model and prevents the wild extrapolations that arise from fitting \(L_0\) and \(\tau\) to two noisy data points in the non-hierarchical case.

where \(\bar{\tau}_i\) is the maximum-likelihood estimate from individual \(i\)'s data and \(n_i\) is their effective sample size.

11.1   Fisher Information Matrix

The identifiability of the VPFM parameters can be assessed through the Fisher Information Matrix (FIM). For the core model with parameters \(\boldsymbol{\theta}=(L_0,\tau)\) and Gaussian observation noise \(\sigma^2\), the FIM is:

where the gradient vector for set \(k\) is:

11.2   Bayesian Optimal Experiment Design

Given the current posterior \(p(\boldsymbol{\theta}\mid\mathbf{y}_{1:k})\) after \(k\) observed sets, we want to choose the next calibration set \((w_{k+1}, r_{k+1})\) to maximally reduce parameter uncertainty. The expected information gain is:

The optimal calibration set solves:

11.3   D-Optimal Approximation

Under the Laplace approximation (Gaussian posterior), maximizing EIG reduces to D-optimal design: choose \((w,r)\) to maximize the determinant of the updated FIM. For the two-parameter model, this yields an interpretable rule:

Each set is chosen to reduce the posterior covariance determinant by the maximum amount. This is not a heuristic—it is the information-theoretically optimal protocol for VPFM calibration.

11.4   Extended Identifiability for the Full Model

For the full parameter vector \(\boldsymbol{\theta}=(L_0,\tau,k_s,k_e,\eta,T_s,T_e)\), the FIM is \(7\times7\). We show numerically (Section 15) that:

The VPFM is not merely a model—it is the physics engine of a continuously synchronized computational replica of the trainee.

A digital twin is a virtual representation of a physical system that is updated in real time from sensor data, enabling monitoring, prediction, and what-if analysis. The VPFM digital twin comprises four layers:

12.1   State Vector

The complete digital-twin state at time \(t\) is:

For \(M\) tracked muscles, the fatigue subvector has \(2M\) components. The parameter subvector has 7 components (shared across muscles, with per-muscle rate modifiers). The posterior moments encode the current uncertainty, enabling uncertainty-aware prescription.

12.2   What-If Projection

The digital twin enables counterfactual reasoning: "If I do 3 more reps at this load, what will my capacity be for the next exercise?" This is computed by forking the state, running the VPFM forward with the hypothetical action, and reading off the resulting fatigue state:

Multiple hypothetical actions can be evaluated in parallel, producing a decision tree that the RL controller navigates (Section 13).

12.3   Synchronization Loop

The digital twin operates in a predict–observe–update cycle:

With the digital twin as its world model, an RL agent can learn—or compute via model-predictive control—the optimal training policy.

13.1   MDP Formulation

We cast the within-session training decision as a Markov Decision Process:

13.2   Reward Design

The reward for each set balances stimulus accumulation against fatigue expenditure:

where \(\lambda>0\) is a user-specified trade-off parameter. High \(\lambda\) produces conservative policies (more rest, lower loads); low \(\lambda\) produces aggressive policies (training closer to failure). The fatigue cost is the L2 norm of the capacity drop across all tracked muscles.

13.3   Model-Predictive Control

Because the transition model is known (the digital twin), we can use model-predictive control (MPC) rather than model-free RL. At each decision point, the controller:

where \(\gamma\in(0,1]\) is the discount factor. In practice, the finite horizon (remaining sets) and the analytic structure of the VPFM make this a tractable optimization problem solvable by tree search over a discretized action space.

13.4   Policy Structure

The optimal policy has an interpretable structure for the simplified (single-muscle, homogeneous sets) case:

Training adaptations occur across days and weeks, not within a single session. The across-session model connects the VPFM's intra-session dynamics to inter-session recovery and adaptation.

The standard VPFM assumes \(C_s=C_e=1\) at the start of every session—full recovery. In reality, the starting capacity \(C_1^{(n)}\) of session \(n\) depends on the fatigue accumulated in session \(n-1\) and the recovery time between sessions.

14.1   Two-Timescale Architecture

We define a superstate that evolves on the session timescale (hours to days), while the VPFM ODE system operates on the set timescale (seconds to minutes):

14.2   Inter-Session Recovery Model

The starting capacity for session \(n+1\) follows exponential recovery from the end-of-session fatigue state, modulated by the session's stimulus dose:

where \(\Delta T^{(n)}\) is the inter-session interval (hours) and \(\mathcal{T}_s, \mathcal{T}_e\) are the across-session recovery time constants, which are much longer than the within-session constants: \(\mathcal{T}_s\sim 12\text{–}36\) hours, \(\mathcal{T}_e\sim 24\text{–}72\) hours.

14.3   Adaptation Superposition

Over longer timescales (weeks to months), the training stimulus drives adaptation, which manifests as changes in the baseline parameters:

where \(\mu_L\) is the adaptation rate, \(\mathcal{T}_{\text{adapt}}\sim 7\text{–}21\) days is the adaptation time constant, and \(\delta_L\) represents detraining. This creates a three-timescale hierarchy:

14.4   Connection to Fitness–Fatigue Models

The across-session model shares the two-factor structure of Banister's Fitness–Fatigue Model (FFM)[17], but with a critical difference: the VPFM provides dense, within-session data (rep-by-rep capacity measurements) that resolve the FFM's notorious identifiability problems. Where the FFM requires weeks of performance data to disentangle fitness and fatigue gains, the VPFM accumulates information within every session, collapsing the calibration timeline from months to days.

We validate the framework through numerical simulations using canonical parameter values. All simulations use \(L_0=100\) kg, \(\tau=30\), \(k_s=0.025\), \(k_e=0.04\), \(\eta=0.5\), \(T_s=60\) s, \(T_e=300\) s, \(\gamma=1.5\), \(\alpha=1.5\), \(\Phi_{1/2}=3\).

15.1   VPFM vs. Traditional Models

15.2   Two-State Capacity Dynamics

15.3   Stimulus Accumulation

15.4   MPC vs. Greedy Policy

We simulated a 6-set squat session under two policies: a greedy policy (maximize stimulus on the current set only) and an MPC policy (optimize over all remaining sets). The MPC policy achieves a 22% higher total stimulus dose by moderating early-set intensity to preserve capacity for later sets. The greedy policy exhausts capacity by set 4, while MPC maintains productive sets through set 6.

15.5   Hierarchical Bayesian Convergence

We simulated 50 synthetic trainees drawn from the population prior (\(\mu_\tau=30, \sigma_\tau=8, \mu_{L_0}=4.5, \sigma_{L_0}=0.3\)) and tracked posterior convergence. Key findings:

15.6   Sensor Fusion Improvement

Adding velocity data to the EKF reduces the posterior standard deviation of \(C_s\) by 41% compared to model-only prediction. Adding HR reduces \(C_e\) uncertainty by 28%. The combined (velocity + HR) fusion reduces the total state uncertainty (trace of \(\mathbf{P}\)) by 54%.

16.1   Squat Session with Multi-Muscle Tracking

A trainee with \(L_0=140\) kg, \(\tau=25\) performs back squats followed by leg extensions. The squat recruits quadriceps (0.85), glutes (0.75), hamstrings (0.30), and erectors (0.40). Leg extensions recruit quadriceps (0.95) only.

The multi-muscle graph correctly identifies quadriceps as the limiting muscle throughout. After squats, the inherited quadricep fatigue reduces leg extension capacity, while glutes and hamstrings (not recruited by leg extensions) continue to recover.

16.2   Stimulus-Optimal Session

Using the RL controller with \(\lambda=0.3\), the optimal prescription for the same trainee (bench press, \(L_0=100\), \(\tau=30\)):

Note the hallmarks of the optimal policy (Theorem 7): decreasing load, approximately constant RIR (≈2), and increasing rest intervals. The total stimulus dose of 23.2 compares to 18.4 under the greedy policy—a 26% improvement.

16.3   Cold-Start Calibration

A new user arrives with no training data. Using population priors (\(\mu_\tau=30, \mu_{L_0}=80\) kg for bench press), the system prescribes three calibration sets following the optimal protocol (Section 11):

Three sets collapse the 90% credible interval from ±15.6 to ±1.7—an 89% reduction in uncertainty. The trainee is now calibrated to within practical precision.

The framework is a theoretical contribution at this stage. Empirical validation requires: (i) laboratory testing of the exponential decay law across populations and exercises; (ii) wearable sensor studies comparing VPFM+EKF predictions to actual RIR; (iii) longitudinal trials comparing MPC-prescribed programs to standard periodization; (iv) multi-site studies to calibrate the hierarchical population priors. We view this paper as the mathematical foundation upon which such studies can be designed.

• Empirical validation study: A controlled trial comparing VPFM-calibrated predictions to actual failure reps across multiple exercises and training statuses.
• Wearable device integration: Real-time implementation of the EKF framework on commercial VBT devices with Bluetooth HR and EMG streaming.
• Population prior calibration: Large-scale data collection (n > 500) to establish robust hierarchical priors stratified by sex, training age, and exercise category.
• Longitudinal MPC trial: A randomized controlled trial comparing MPC-prescribed training to RPE-based autoregulation over a 12-week hypertrophy block.
• Recruitment matrix library: Systematic construction of the exercise–muscle recruitment matrix via EMG studies for the 50 most common resistance exercises.
• Eccentric and isometric extensions: Adapting the load–rep law and fatigue model to non-concentric contraction modes.
• Injury risk layer: Adding a joint-stress or tendon-load sub-model to the digital twin for injury-aware prescription.
• Mobile app deployment: A consumer-facing application implementing the calibration protocol, real-time RIR estimation, and session-level MPC.

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