Page 644 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 644
MATHEMATICS IN SCIENCE AND TECHNOLOGY
A Mathematical Model for Low Grade Gliomas and the effects of
chemotherapy
María Vela Pérez, maria.vela@ucm.es
Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain
Coauthor: Marek Bodnar
The optimal management of low grade gliomas remains an open problem. Although complete
surgical resection is recommended, given the diffuse infiltrative nature of these tumors and the
risk of major surgery in important areas of the brain, they often result in tumors that are in-
completely resected or simply biopsied. For this reason, the use of TMZ combined with other
options is increasingly used [1, 2]. We present a modification of the system appeared in [2] to
understand the late response to the chemotherapy observed in LGG without using an excessive
number of unknown parameters. In our approach we consider separately the proliferation pro-
cess and natural death process [3]. This allows us to obtain a good agreement of model solutions
with medical data which was impossible in the approach presented in [2].
References
[1] Bogdanska M. U. , Bodnar M. , Piotrowska M. J., et al. A mathematical model describes
the malignant transformation of low grade gliomas: Prognostic implications. PLoS One,
12(8):e0179999, (2017).
[2] Bogdanska M., Bodnar M., Belmonte-Beitia J., Murek M., Schucht P., Beck J., Pérez-
García V. M.. A mathematical model of low grade gliomas treated with temozolomide
and its therapeutical implications. Math Biosci., (288):1–13, (2017).
[3] Bodnar M., Vela-Pérez M. Mathematical and Numerical Analysis of Low Grade Gliomas
Model and the Effects of Chemotherapy. Submitted to Commun. Nonlinear Sci. Numer.
Simul.
Image completion with approximate convex hull tensor decomposition
Rafal Zdunek, rafal.zdunek@pwr.edu.pl
Wroclaw University of Science and Technology, Poland
Coauthor: Tomasz Sadowski
Many structural image completion methods are based on a low-rank approximation of the un-
derlying image using matrix or tensor decomposition models. In this study, we assume that
the image to be completed is represented by a multi-way array and can be approximated by
a conical hull of subtensors in the observation space. If an observed tensor is near-separable
along at least one mode, the extreme rays, represented by the selected subtensors, can be found
by analyzing the corresponding convex hull. Following this assumption, we propose a geo-
metric algorithm to address a low-rank image completion problem. The extreme rays are ex-
tracted with a segmented convex-hull algorithm that is suitable for performing noise-resistant
non-negative tensor factorization. The coefficients of a conical combination of such rays are
estimated using Douglas-Rachford splitting combined with the rank-two update least-squares
algorithm. The proposed algorithm was applied to incomplete RGB images and hyperspec-
tral arrays with a large number of randomly missing entries. Experiments confirm its good
642
A Mathematical Model for Low Grade Gliomas and the effects of
chemotherapy
María Vela Pérez, maria.vela@ucm.es
Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain
Coauthor: Marek Bodnar
The optimal management of low grade gliomas remains an open problem. Although complete
surgical resection is recommended, given the diffuse infiltrative nature of these tumors and the
risk of major surgery in important areas of the brain, they often result in tumors that are in-
completely resected or simply biopsied. For this reason, the use of TMZ combined with other
options is increasingly used [1, 2]. We present a modification of the system appeared in [2] to
understand the late response to the chemotherapy observed in LGG without using an excessive
number of unknown parameters. In our approach we consider separately the proliferation pro-
cess and natural death process [3]. This allows us to obtain a good agreement of model solutions
with medical data which was impossible in the approach presented in [2].
References
[1] Bogdanska M. U. , Bodnar M. , Piotrowska M. J., et al. A mathematical model describes
the malignant transformation of low grade gliomas: Prognostic implications. PLoS One,
12(8):e0179999, (2017).
[2] Bogdanska M., Bodnar M., Belmonte-Beitia J., Murek M., Schucht P., Beck J., Pérez-
García V. M.. A mathematical model of low grade gliomas treated with temozolomide
and its therapeutical implications. Math Biosci., (288):1–13, (2017).
[3] Bodnar M., Vela-Pérez M. Mathematical and Numerical Analysis of Low Grade Gliomas
Model and the Effects of Chemotherapy. Submitted to Commun. Nonlinear Sci. Numer.
Simul.
Image completion with approximate convex hull tensor decomposition
Rafal Zdunek, rafal.zdunek@pwr.edu.pl
Wroclaw University of Science and Technology, Poland
Coauthor: Tomasz Sadowski
Many structural image completion methods are based on a low-rank approximation of the un-
derlying image using matrix or tensor decomposition models. In this study, we assume that
the image to be completed is represented by a multi-way array and can be approximated by
a conical hull of subtensors in the observation space. If an observed tensor is near-separable
along at least one mode, the extreme rays, represented by the selected subtensors, can be found
by analyzing the corresponding convex hull. Following this assumption, we propose a geo-
metric algorithm to address a low-rank image completion problem. The extreme rays are ex-
tracted with a segmented convex-hull algorithm that is suitable for performing noise-resistant
non-negative tensor factorization. The coefficients of a conical combination of such rays are
estimated using Douglas-Rachford splitting combined with the rank-two update least-squares
algorithm. The proposed algorithm was applied to incomplete RGB images and hyperspec-
tral arrays with a large number of randomly missing entries. Experiments confirm its good
642
