Methods: Optimization Strategies

OptiConn employs two distinct strategies to solve the problem of parameter selection in structural connectomics.

1. Bayesian Optimization (Gaussian Processes)

This is the primary and recommended method. It treats the connectome quality as a "black box" function \(f(x)\) where \(x\) is the vector of tracking parameters (FA threshold, turning angle, etc.) and \(f(x)\) is the composite quality score.

Algorithm

We use Gaussian Process (GP) Regression to model \(f(x)\). 1. Prior: A GP defines a prior distribution over functions. 2. Acquisition Function: We use Expected Improvement (EI) to decide which point \(x_{next}\) to evaluate next. EI balances exploitation (sampling where the model predicts high quality) and exploration (sampling where uncertainty is high). 3. Update: After evaluating \(f(x_{next})\) (by running tracking and scoring the network), the GP is updated to produce a posterior distribution.

Subject Sampling

To avoid overfitting to a single subject's anatomy, we implement Stochastic Bayesian Optimization. - In standard optimization, \(f(x)\) is deterministic. - In our case, \(f(x, s)\) depends on the subject \(s\). - By sampling a new random subject \(s_i\) at each iteration \(i\), the optimizer learns parameters that maximize the expected quality \(E_s[f(x, s)]\) across the population.

This method serves as a rigorous baseline and validation tool.

Design

  • Split-Half Validation: The cohort is split into two "waves" (Wave 1 and Wave 2).
  • Exhaustive Search: A grid of parameters is defined (e.g., FA \(\in \\{0.1, 0.2\\}\), Angle \(\in \\{30, 60\\}\)).
  • Evaluation: Every combination is run on both waves.
  • Selection: We select parameters that:
    1. Score highly in Wave 1.
    2. Score highly in Wave 2.
    3. Show low variance between waves (high stability).

This method is computationally expensive (\(O(N^k)\) where \(k\) is the number of parameters) but provides a complete landscape of the parameter space.

Scoring Function

The objective function maximizes a composite score derived from graph-theoretic metrics:

\[ \mathrm{Score} = w_1 \cdot \mathrm{Density}_{\mathrm{score}} + w_2 \cdot \mathrm{Efficiency}_{\mathrm{score}} + w_3 \cdot \mathrm{SmallWorld}_{\mathrm{score}} + \cdots \]
  • Density: Penalizes unconnected or fully connected graphs.
  • Global Efficiency: Measures integration.
  • Small-Worldness: Measures the balance of segregation and integration.
  • Rich Club: Measures the connectivity of high-degree nodes.

All metrics are normalized to a [0, 1] scale before combination.