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Iterative algorithms for the reconstruction of early states of prostate cancer growth

Elena Beretta, Cecilia Cavaterra, Matteo Fornoni, Guillermo Lorenzo, Elisabetta Rocca

TL;DR

The paper tackles the inverse problem of recovering early prostate cancer states from a single late-time spatial measurement using a phase-field model coupled to nutrient and PSA dynamics. It develops a Landweber-type reconstruction framework, proving local convergence and Lipschitz stability on finite-dimensional subspaces, and demonstrates adaptive step-size variants to handle long time horizons. Numerical experiments in 2D with Isogeometric space-time discretisation show high-quality reconstructions for short horizons and robust performance for longer horizons (up to one year) when using adaptive step sizes and geometry-informed initial guesses, even with noisy data. These findings support model-constrained, data-efficient forecasting of tumour evolution from limited clinical measurements, contributing to patient-specific decision support and digital-twin development in oncology.

Abstract

The development of mathematical models of cancer informed by time-resolved measurements has enabled personalised predictions of tumour growth and treatment response. However, frequent cancer monitoring is rare, and many tumours are treated soon after diagnosis with limited data. To improve the predictive capabilities of cancer models, we investigate the problem of recovering earlier tumour states from a single spatial measurement at a later time. Focusing on prostate cancer, we describe tumour dynamics using a phase-field model coupled with two reaction-diffusion equations for a nutrient and the local prostate-specific antigen. We generate synthetic data using a discretisation based on Isogeometric Analysis. Then, building on our previous analytical work (Beretta et al., SIAP (2024)), we propose an iterative reconstruction algorithm based on the Landweber scheme, showing local convergence with quantitative rates and exploring an adaptive step size that leads to faster reconstruction algorithms. Finally, we run simulations demonstrating high-quality reconstructions even with long time horizons and noisy data.

Iterative algorithms for the reconstruction of early states of prostate cancer growth

TL;DR

The paper tackles the inverse problem of recovering early prostate cancer states from a single late-time spatial measurement using a phase-field model coupled to nutrient and PSA dynamics. It develops a Landweber-type reconstruction framework, proving local convergence and Lipschitz stability on finite-dimensional subspaces, and demonstrates adaptive step-size variants to handle long time horizons. Numerical experiments in 2D with Isogeometric space-time discretisation show high-quality reconstructions for short horizons and robust performance for longer horizons (up to one year) when using adaptive step sizes and geometry-informed initial guesses, even with noisy data. These findings support model-constrained, data-efficient forecasting of tumour evolution from limited clinical measurements, contributing to patient-specific decision support and digital-twin development in oncology.

Abstract

The development of mathematical models of cancer informed by time-resolved measurements has enabled personalised predictions of tumour growth and treatment response. However, frequent cancer monitoring is rare, and many tumours are treated soon after diagnosis with limited data. To improve the predictive capabilities of cancer models, we investigate the problem of recovering earlier tumour states from a single spatial measurement at a later time. Focusing on prostate cancer, we describe tumour dynamics using a phase-field model coupled with two reaction-diffusion equations for a nutrient and the local prostate-specific antigen. We generate synthetic data using a discretisation based on Isogeometric Analysis. Then, building on our previous analytical work (Beretta et al., SIAP (2024)), we propose an iterative reconstruction algorithm based on the Landweber scheme, showing local convergence with quantitative rates and exploring an adaptive step size that leads to faster reconstruction algorithms. Finally, we run simulations demonstrating high-quality reconstructions even with long time horizons and noisy data.
Paper Structure (21 sections, 5 theorems, 71 equations, 13 figures)

This paper contains 21 sections, 5 theorems, 71 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Analytical results
  4. Landweber iteration scheme
  5. Numerical methods
  6. Spatial discretisation
  7. Time discretisation
  8. Iterative algorithms for the identification of initial data
  9. Short time horizon
  10. Long time horizon
  11. Simulation study
  12. Computational scenario and setup
  13. Model parameters
  14. Numerical implementation details
  15. Ground truth
  16. ...and 6 more sections

Key Result

Theorem 3.1

Assume hypotheses ass:coeff--ass:m. Let $(\varphi_1, \sigma_1, p_1)$ and $(\varphi_2, \sigma_2, p_2)$ be two solutions of eq:phi--ic corresponding to two triples of initial data $(\varphi_0^i, \sigma_0^i, p_0^i) \in \mathcal{I}_{\text{ad}}$ for $i=1,2$. Let $M > 0$, $M_1 > 0$ be as above. Moreover, and assume that where $C_1 = C_1(T) > 0$ is the constant appearing in the Hölder stability estimat

Figures (13)

  • Figure 1: Reconstruction of the initial tumour phase field from a measurement at $T=15$ days using the Landweber iteration scheme. The first two columns compare the tumour phase field from the reference simulation and the corresponding reconstruction at $t=0$ and $t=T$. The plot in the last column provides the values of the four metrics used to assess the reconstruction of the tumour phase field in each iteration of the Landweber algorithm.
  • Figure 2: Convergence of the Landweber algorithm for $T= 15$ days. This figure provides the changes of the relative $L^2$ error of the reconstructed tumour phase field at $t=0$ and $t=T$. For $e_{L^2,0}$, we also plot a straight dotted line with the same slope as the linear portion of the trajectory of the relative $L^2$ error. The value of this slope is -9.47$\cdot 10^{-2}$, which corresponds to a value of parameter $c$ of 1.96$\cdot 10^{-1}$ in Theorem \ref{['thm:landweber']}. Thus, the convergence in all scenarios is infralinear, as it was found theoretically.
  • Figure 3: Reconstruction of the initial tumour phase field from a measurement at $T=15$ days using the Landweber iteration scheme. In each panel, the first two rows represent the tumour phase field and its corresponding adjoint variable at $t=0$ (i.e, $\varphi_0$ and $q_0$), while the last two rows provide the same quantities at $t=T$ (i.e, $\varphi_T$ and $q_T$).
  • Figure 4: Reconstruction of the initial tumour phase field from a measurement at $T=90$ days using the adaptive gradient descent algorithm. The first two columns compare the tumour phase field from the reference simulation and the corresponding reconstruction at $t=0$ and $t=T$. The plot in the last column provides the values of the four metrics used to assess the reconstruction of the tumour phase field in each iteration of the adaptive gradient descent algorithm.
  • Figure 5: Reconstruction of the initial tumour phase field from a measurement at $T=365$ days using the adaptive gradient descent algorithm. The first two columns compare the tumour phase field from the reference simulation and the corresponding reconstruction at $t=0$ and $t=T$. The plot in the last column provides the values of the four metrics used to assess the reconstruction of the tumour phase field in each iteration of the adaptive gradient descent algorithm.
  • ...and 8 more figures

Theorems & Definitions (12)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • Remark 4.4
  • Theorem 4.5
  • ...and 2 more