[Paper Review] Adjoint method for a tumour growth PDE-constrained optimization problem

Damián Knopoff, Damián Fernández+2 more|arXiv (Cornell University)|Sep 13, 2012

Numerical methods in inverse problems12 references3 citations

TL;DR

This paper presents an adjoint method for solving a PDE-constrained optimization problem to estimate unknown parameters in a mathematical model of avascular tumour growth. By formulating the inverse problem as a functional minimization using real experimental and imaging data, the adjoint method achieves higher accuracy and efficiency than pattern search, reducing parameter estimation error to approximately 1% even with 5% noisy data.

ABSTRACT

In this paper we present a method for estimating unknown parameters that appear on an avascular, spheric tumour growth model. The model for the tumour is based on nutrient driven growth of a continuum of live cells, whose birth and death generate volume changes described by a velocity field. The model consists on a coupled system of partial differential equations whose spatial domain is the tumour, that changes in size over time. Thus, the situation can be formulated as a free boundary problem. After solving the forward problem properly, we use the model for the estimation of parameters by fitting the numerical solution with real data, obtained via in vitro experiments and medical imaging. We define an appropriate functional to compare both the real data and the numerical solution. We use the adjoint method for the minimization of this functional, getting a better performance than the obtained with the pattern search method.

Motivation & Objective

• To develop a robust PDE-constrained optimization framework for estimating unknown parameters in a spheroid tumour growth model.
• To address the challenge of fitting a nonlinear, free-boundary PDE system to experimental and medical imaging data.
• To improve upon prior work using pattern search by implementing the adjoint method for gradient-based optimization.
• To enable accurate, patient-specific parameter estimation for use in predicting tumour response to therapies.
• To provide a numerically stable method capable of handling nonlinearities and singularities in the adjoint system.

Proposed method

• Formulates a PDE-constrained optimization problem based on the Ward and King tumour growth model, coupling nutrient diffusion and cell population dynamics.
• Defines a least-squares functional to compare numerical solutions with observed data, including tumour radius and cell density distributions.
• Derives the adjoint system to compute the gradient of the functional with respect to unknown parameters, enabling efficient optimization.
• Introduces numerical algorithms to handle singularities and nonlinearities in the adjoint PDEs, particularly at the moving tumour boundary.
• Employs a gradient-based minimization algorithm (e.g., conjugate gradient) using adjoint-derived derivatives to iteratively update parameters.
• Validates the method using synthetic data with and without 5% random noise to assess robustness and convergence.

Experimental results

Research questions

• RQ1 Can the adjoint method outperform derivative-free optimization (e.g., pattern search) in estimating parameters for a tumour growth PDE model?
• RQ2 How accurately can the adjoint method recover unknown parameters when fitted to noisy experimental data?
• RQ3 What is the impact of data noise (e.g., 5%) on the convergence and accuracy of the parameter estimation process?
• RQ4 How does the adjoint-based gradient computation improve computational efficiency and solution accuracy compared to derivative-free methods?
• RQ5 Can the method be extended to include additional biological parameters, such as drug efficacy or nutrient consumption rates?

Key findings

• The adjoint method achieved significantly better accuracy and faster convergence than the pattern search method used in prior work.
• With 5% random noise in synthetic data, the method estimated parameters with an error of approximately 1%.
• The functional value decreased steadily over iterations, and the algorithm converged when changes in the functional became negligible.
• The tumour radius predicted by the optimized model closely matched the observed radius, with a small difference visible only upon zooming in (Figure 12).
• The method successfully recovered the true value of the critical parameter $ c_c $, as shown in Figure 10, where the estimated value converged toward the real value.
• The adjoint-based gradient computation was robust and scalable, allowing extension to multiple unknown parameters in principle.