Mathematical analysis of a model-constrained inverse problem for the reconstruction of early states of prostate cancer growth
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909174507503616 |
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| author | Beretta, Elena Cavaterra, Cecilia Fornoni, Matteo Lorenzo, Guillermo Rocca, Elisabetta |
| author_facet | Beretta, Elena Cavaterra, Cecilia Fornoni, Matteo Lorenzo, Guillermo Rocca, Elisabetta |
| contents | The availability of cancer measurements over time enables the personalised assessment of tumour growth and therapeutic response dynamics. However, many tumours are treated after diagnosis without collecting longitudinal data, and cancer monitoring protocols may include infrequent measurements. To facilitate the estimation of disease dynamics and better guide ensuing clinical decisions, we investigate an inverse problem enabling the reconstruction of earlier tumour states by using a single spatial tumour dataset and a biomathematical model describing disease dynamics. We focus on prostate cancer, since aggressive cases of this disease are usually treated after diagnosis. We describe tumour dynamics with a phase-field model driven by a generic nutrient ruled by reaction-diffusion dynamics. The model is completed with another reaction-diffusion equation for the local production of prostate-specific antigen, which is a key prostate cancer biomarker. We first improve previous well-posedness results by further showing that the solution operator is continuously Fréchet differentiable. We then analyse the backward inverse problem concerning the reconstruction of earlier tumour states starting from measurements of the model variables at the final time. Since this problem is severely ill-posed, only very weak conditional stability of logarithmic type can be recovered from the terminal data. However, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we can derive an optimal Lipschitz stability estimate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_12198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mathematical analysis of a model-constrained inverse problem for the reconstruction of early states of prostate cancer growth Beretta, Elena Cavaterra, Cecilia Fornoni, Matteo Lorenzo, Guillermo Rocca, Elisabetta Analysis of PDEs 35K51, 35K58, 35R30, 35Q92, 92C50 The availability of cancer measurements over time enables the personalised assessment of tumour growth and therapeutic response dynamics. However, many tumours are treated after diagnosis without collecting longitudinal data, and cancer monitoring protocols may include infrequent measurements. To facilitate the estimation of disease dynamics and better guide ensuing clinical decisions, we investigate an inverse problem enabling the reconstruction of earlier tumour states by using a single spatial tumour dataset and a biomathematical model describing disease dynamics. We focus on prostate cancer, since aggressive cases of this disease are usually treated after diagnosis. We describe tumour dynamics with a phase-field model driven by a generic nutrient ruled by reaction-diffusion dynamics. The model is completed with another reaction-diffusion equation for the local production of prostate-specific antigen, which is a key prostate cancer biomarker. We first improve previous well-posedness results by further showing that the solution operator is continuously Fréchet differentiable. We then analyse the backward inverse problem concerning the reconstruction of earlier tumour states starting from measurements of the model variables at the final time. Since this problem is severely ill-posed, only very weak conditional stability of logarithmic type can be recovered from the terminal data. However, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we can derive an optimal Lipschitz stability estimate. |
| title | Mathematical analysis of a model-constrained inverse problem for the reconstruction of early states of prostate cancer growth |
| topic | Analysis of PDEs 35K51, 35K58, 35R30, 35Q92, 92C50 |
| url | https://arxiv.org/abs/2404.12198 |