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Analysis of the Parametric Correlation in Mathematical Modeling of In Vitro Glioblastoma Evolution Using Copulas. MATHEMATICS 2020. [DOI: 10.3390/math9010027] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Abstract
Modeling and simulation are essential tools for better understanding complex biological processes, such as cancer evolution. However, the resulting mathematical models are often highly non-linear and include many parameters, which, in many cases, are difficult to estimate and present strong correlations. Therefore, a proper parametric analysis is mandatory. Following a previous work in which we modeled the in vitro evolution of Glioblastoma Multiforme (GBM) under hypoxic conditions, we analyze and solve here the problem found of parametric correlation. With this aim, we develop a methodology based on copulas to approximate the multidimensional probability density function of the correlated parameters. Once the model is defined, we analyze the experimental setting to optimize the utility of each configuration in terms of gathered information. We prove that experimental configurations with oxygen gradient and high cell concentration have the highest utility when we want to separate correlated effects in our experimental design. We demonstrate that copulas are an adequate tool to analyze highly-correlated multiparametric mathematical models such as those appearing in Biology, with the added value of providing key information for the optimal design of experiments, reducing time and cost in in vivo and in vitro experimental campaigns, like those required in microfluidic models of GBM evolution.
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Ayensa-Jiménez J, Pérez-Aliacar M, Randelovic T, Oliván S, Fernández L, Sanz-Herrera JA, Ochoa I, Doweidar MH, Doblaré M. Mathematical formulation and parametric analysis of in vitro cell models in microfluidic devices: application to different stages of glioblastoma evolution. Sci Rep 2020; 10:21193. [PMID: 33273574 PMCID: PMC7713081 DOI: 10.1038/s41598-020-78215-3] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2020] [Accepted: 10/26/2020] [Indexed: 12/31/2022] Open
Abstract
In silico models and computer simulation are invaluable tools to better understand complex biological processes such as cancer evolution. However, the complexity of the biological environment, with many cell mechanisms in response to changing physical and chemical external stimuli, makes the associated mathematical models highly non-linear and multiparametric. One of the main problems of these models is the determination of the parameters’ values, which are usually fitted for specific conditions, making the conclusions drawn difficult to generalise. We analyse here an important biological problem: the evolution of hypoxia-driven migratory structures in Glioblastoma Multiforme (GBM), the most aggressive and lethal primary brain tumour. We establish a mathematical model considering the interaction of the tumour cells with oxygen concentration in what is called the go or grow paradigm. We reproduce in this work three different experiments, showing the main GBM structures (pseudopalisade and necrotic core formation), only changing the initial and boundary conditions. We prove that it is possible to obtain versatile mathematical tools which, together with a sound parametric analysis, allow to explain complex biological phenomena. We show the utility of this hybrid “biomimetic in vitro-in silico” platform to help to elucidate the mechanisms involved in cancer processes, to better understand the role of the different phenomena, to test new scientific hypotheses and to design new data-driven experiments.
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Affiliation(s)
- Jacobo Ayensa-Jiménez
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain.,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain
| | - Marina Pérez-Aliacar
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain.,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain
| | - Teodora Randelovic
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain.,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain
| | - Sara Oliván
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain.,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain
| | - Luis Fernández
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain.,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain.,Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), C/ Monforte de Lemos 3-5, Pabellón 11. Planta 0, 28029, Madrid, Spain
| | - José Antonio Sanz-Herrera
- School of Engineering, Department of Mechanics of Continuous Media and Theory of Structures, University of Seville, Camino de los descubrimientos, s/n, 41092, Sevilla, Spain
| | - Ignacio Ochoa
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain.,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain.,Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), C/ Monforte de Lemos 3-5, Pabellón 11. Planta 0, 28029, Madrid, Spain
| | - Mohamed H Doweidar
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain.,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain.,Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), C/ Monforte de Lemos 3-5, Pabellón 11. Planta 0, 28029, Madrid, Spain
| | - Manuel Doblaré
- Aragón Institute of Engineering Research (I3A), University of Zaragoza, Mariano Esquillor s/n, 50018, Zaragoza, Spain. .,Institute for Health Research Aragón (IIS Aragón), Avda. San Juan Bosco, 13, 50009, Zaragoza, Spain. .,Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), C/ Monforte de Lemos 3-5, Pabellón 11. Planta 0, 28029, Madrid, Spain.
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