brimstone 🜏

brimstone is a variational inverse planning algorithm for radiotherapy treatment planning. It is related to variational bayes methods, though free energy is implicitly represented.

Variational Bayes Connection

The dH algorithm implements a simplistic variational Bayes approach for treatment planning optimization. The connection manifests through several key mechanisms:

Core Variational Elements

1. KL Divergence Minimization (RtModel/KLDivTerm.cpp)

   • Minimizes KL(P_calc || P_target) between calculated and target dose-volume histograms (DVHs)
   • This is the fundamental operation in variational inference, seeking the best approximation to a target distribution

2. Implicit Free Energy

   • The objective function implicitly minimizes variational free energy: F = KL(q||p) + Expected log likelihood
   • Implemented as weighted sum of KL divergence terms across anatomical structures
   • Unlike explicit variational Bayes, free energy is not directly computed but emerges from the optimization

3. Gaussian Approximation (RtModel/include/HistogramGradient.h)

   • Dose histograms are convolved with adaptive Gaussian kernels
   • Similar to mean-field approximation in variational Bayes
   • Variance parameters represent posterior uncertainty in the dose calculation

4. Adaptive Variance (RtModel/Prescription.cpp)

   • Dynamic covariance optimization adjusts uncertainty representation during optimization
   • Acts as variational parameter analogous to posterior variance in Bayesian inference
   • Variance scaling uses sigmoid derivatives: actVar = baseVar * dSigmoid(input)² * varWeight²

5. Hierarchical Structure (RtModel/include/PlanPyramid.h)

   • Multi-scale pyramid (4 levels) provides coarse-to-fine optimization
   • Similar to hierarchical variational models without full hierarchical Bayes

Simplifications

The algorithm is "simplistic" in that it:

• Does not explicitly model posterior distributions
• Uses sigmoid-transformed parameters rather than full probabilistic representation
• By default computes implicit rather than explicit free energy (though explicit calculation is available as an option)
• Focuses on point estimates rather than full posterior inference

Explicit Free Energy Calculation (Optional)

An optional explicit free energy calculation has been implemented (RtModel/ConjGradOptimizer.cpp:220-254):

Enable via: optimizer.SetComputeFreeEnergy(true)

Calculation Method:

1. Entropy from Covariance: Computes differential entropy from the dynamically-built covariance matrix

   H = 0.5 * (n * log(2πe) + log(det(Σ)))
   

   • Uses eigenvalue decomposition for numerical stability
   • No Hessian approximation required - uses existing search-direction-based covariance

2. Free Energy: Combines KL divergence objective with entropy

   F = KL_divergence - Entropy
   

   • KL divergence represents expected log likelihood term
   • Entropy term accounts for posterior uncertainty
   • Both terms logged during optimization iterations

This implementation leverages the existing DynamicCovarianceOptimizer covariance approximation (built from orthogonalized conjugate gradient search directions) rather than requiring expensive Hessian computation.

Mathematical Formulation

The optimization problem solved is:

minimize: Σ_structures [ w_i * KL(P_calc_i || P_target_i) ]
subject to: 0 ≤ beamlet_weight_j ≤ max_weight (via sigmoid transform)

This is fundamentally a variational inference problem where the algorithm seeks optimal beamlet weights that produce dose distributions matching target specifications in an information-theoretic sense.

-MIND THE LICENSE-

U. S. Patent 7,369,645

Copyright (c) 2007-2021, Derek G. Lane All rights reserved.

• documents
  • see zebrastack
  • frame notes
  • autoencoder, mdl, free energy
  • em free energy
  • free energy and the brain
  • variational bayes inverse planning
• notebook_zoo
  • entropy_max
• diy_ml
  • pytorch tutorial
  • CMatrixNxM