Bivariate deconvolution for cancer detection after surgery

1. Problem Statement

Context: Following curative-intent cancer surgery, undetected residual cancer cells (Minimal Residual Disease, or MRD) can lead to recurrence. Early detection of MRD via liquid biopsies (measuring circulating tumor DNA, ctDNA) is crucial for guiding adjuvant treatment.
Challenges:

• Low Signal-to-Noise Ratio: Post-surgery, the tumor burden is extremely low, making the tumor signal weak or absent against the background of healthy cell-free DNA (cfDNA).
• Limitations of Current Methods:
  • Tumor-informed approaches (mutation-based Variant Allele Frequency, VAF) require patient-specific mutation panels, which are expensive, time-consuming, and require tumor tissue.
  • Tumor-agnostic approaches (e.g., copy number variations, fragmentation) often lack sensitivity or tumor specificity.
  • Existing Deconvolution Models: Most current deconvolution methods treat pre- and post-surgery measurements as independent and are univariate. They fail to leverage the richer tumor signal present in pre-surgery samples to inform the estimation of the weak post-surgery signal.
• Goal: Develop a method to accurately estimate tumor proportion ($\pi$) from genome-wide cfDNA methylation profiles by linking pre- and post-surgery data, improving sensitivity for MRD detection without requiring patient-specific mutation panels.

2. Methodology

The authors propose a Bivariate Deconvolution Model that jointly models pre- and post-surgery methylation data.

A. Statistical Model

• Observation Model: For $N$ individuals and $p$ molecular features (CpG sites), the observed expression levels ($Y$) are modeled as a convolution of two latent components: a tumor-specific component ($T$) and a background component ($B$), weighted by the tumor proportion ($\pi$).
  $$Y_{ij}^k = \pi_i^k T_{ij}^k + (1 - \pi_i^k) B_{ij}^k$$
  Where $k \in \{0, 1\}$ represents pre-surgery and post-surgery, respectively.
• Distributional Assumptions:
  • Both latent components ($T$ and $B$) are assumed to follow bivariate log-normal distributions. This accounts for the strictly positive and right-skewed nature of methylation read counts.
  • Correlation Structure: The model introduces a feature-wise correlation parameter ($\rho$) between the pre- and post-surgery measurements for both tumor and background signals. This links the two time points, allowing the pre-surgery signal to inform the post-surgery estimate.
• Independence Assumption: To reduce complexity, features ($j$) are treated as independent (pseudo-likelihood approach), though the bivariate nature captures the correlation between time points for each feature.

B. Estimation Strategy

• Likelihood Approximation: The convolution of two log-normal distributions has no analytical solution. The authors propose a discrete approximation:
  1. The bivariate density is discretized onto a 2D grid based on the observed values.
  2. A hybrid approximation is used:
     • Midpoint approximation: Used for most bins for computational speed.
     • Cumulative distribution approximation: Used for "extreme bins" where the density is concentrated in a single cell (identified via a specific criterion) to ensure accuracy.
• Optimization: Parameters are estimated via Block Coordinate Ascent maximizing the discrete log-pseudo-likelihood in three stages:
  1. Estimate tumor proportions ($\pi$) given distribution parameters.
  2. Estimate correlation coefficients ($\rho$).
  3. Update remaining distribution parameters (means and variances).
• Constraints: To ensure identifiability, the model imposes $\pi_i^0 > \pi_i^1$ (tumor proportion must decrease or stay same after surgery).

3. Key Contributions

1. Bivariate Framework: First application of a bivariate deconvolution model for cfDNA methylation that explicitly links pre- and post-surgery measurements via correlation parameters, rather than treating them as independent univariate problems.
2. Computational Efficiency: Development of a fast, discrete approximation algorithm for the convolution of bivariate log-normal variables, enabling maximum likelihood estimation on large genomic datasets.
3. Clinical Utility without Tumor Tissue: The method is "tumor-agnostic" (genome-wide) yet accounts for patient-specific characteristics, removing the need for expensive, patient-specific mutation panels.
4. Robustness: The model performs well even when the correlation between time points is weak or non-existent, avoiding the loss of power seen in other complex models.

4. Results

A. Simulation Studies

• Setup: Simulated data for 100 patients and 600 features based on real methylation profiles.
• Performance:
  • The bivariate model (incorporating correlations $\rho^t$ and $\rho^b$) significantly outperformed univariate models and models assuming zero correlation.
  • Post-surgery accuracy: The bivariate approach showed lower Root Mean Squared Error (RMSE) and higher association with true tumor proportions specifically for the post-surgery estimates ($\pi^1$), where the signal is weakest.
  • Robustness: The model remained accurate even when initial parameter values were incorrect or when the simulated data was uncorrelated.

B. Real Data Analysis

• Dataset: 112 patients with colorectal liver metastases (CRLM) undergoing liver resection. Samples taken pre-surgery and 3 weeks post-surgery.
• Comparison: Estimated tumor proportions ($\hat{\pi}^1$) were compared against mutation-based VAF values ($\hat{\pi}^1_{VAF}$) and clinical outcomes (1-year Recurrence-Free Survival - RFS).
• Key Findings:
  • Differentiation: $\hat{\pi}^1_{VAF}$ values were heavily concentrated near zero with little distinction between relapsed and non-relapsed patients. In contrast, the bivariate estimates showed clear separation between groups.
  • Predictive Power:
    • Logistic regression showed $\hat{\pi}^1_{VAF}$ was not significantly associated with 1-year RFS ($p=0.150$).
    • The bivariate estimate ($\hat{\pi}^1$) was significantly associated with RFS ($p=0.017$). Patients with above-median estimates had 3.4 times higher odds of relapse (OR = 3.429).
    • The Area Under the Curve (AUC) for the bivariate model (0.693) was superior to VAF (0.618).
  • Survival Analysis: Kaplan-Meier curves demonstrated that the bivariate estimates better stratified patients' survival probabilities over time compared to VAF values.

5. Significance

• Clinical Impact: This method offers a more sensitive and accurate tool for detecting MRD in the critical post-surgical window. By leveraging the pre-surgery signal to "anchor" the post-surgery estimate, it overcomes the limitations of low tumor burden.
• Cost-Effectiveness: It eliminates the need for tumor tissue sequencing to create personalized panels, making MRD monitoring more accessible and scalable.
• Methodological Advancement: The proposed discrete approximation for bivariate log-normal convolution provides a robust statistical framework for similar deconvolution problems in genomics and signal processing.
• Future Directions: The authors suggest extending the model to four-variate (incorporating unmethylated reads from WGBS) and integrating cell-type heterogeneity to further refine tumor and background composition estimates.

In conclusion, the paper demonstrates that a bivariate deconvolution approach significantly improves the accuracy of tumor burden estimation from liquid biopsies, leading to better prediction of cancer recurrence compared to standard mutation-based methods or univariate models.