Abramo Agosti | Politecnico di Milano (original) (raw)
Papers by Abramo Agosti
arXiv (Cornell University), Jan 19, 2023
In this work, we develop a computational tool to predict the patient-specific evolution of a high... more In this work, we develop a computational tool to predict the patient-specific evolution of a highly malignant brain tumour, the glioblastoma multiforme (GBM), and its response to therapy. A diffuse-interface mathematical model based on mixture theory is fed by clinical neuroimaging data that provide the anatomical and microstructural characteristics of the patient brain. The model is numerically solved using the finite element method, on the basis of suitable numerical techniques to deal with the resulting Cahn-Hilliard type equation with degenerate mobility and single-well potential. We report the results of simulations performed on the real geometry of a patient brain, proving how the tumour expansion is actually dependent on the local tissue structure. We also report a sensitivity analysis concerning the effects of the different therapeutic strategies employed in the clinical Stupp protocol. The simulated results are in qualitative agreement with the observed evolution of GBM dur...
Mathematical medicine and biology : a journal of the IMA, 2021
Interfaces play a key role on diseases development because they dictate the energy inflow of nutr... more Interfaces play a key role on diseases development because they dictate the energy inflow of nutrients from the surrounding tissues. What is underestimated by existing mathematical models is the biological fact that cells are able to use different resources through nonlinear mechanisms. Among all nutrients, lactate appears to be a sensitive metabolic when talking about brain tumours or neurodegenerative diseases. Here we present a partial differential model to investigate the lactate exchanges between cells and the vascular network in the brain. By extending an existing kinetic model for lactate neuro-energetics, we first provide analytical proofs of the uniqueness and the derivation of precise bounds on the solutions of the problem including diffusion of lactate in a representative volume element comprising the interface between a capillary and cells. We further perform finite element simulations of the model in two test cases, discussing the relevant physical parameters governing ...
arXiv: Cell Behavior, 2019
Tissue self-organization into defined and well-controlled three-dimensional structures is essenti... more Tissue self-organization into defined and well-controlled three-dimensional structures is essential during development for the generation of organs. A similar, but highly deranged process might also occur during the aberrant growth of cancers, which frequently display a loss of the orderly structures of the tissue of origin, but retain a multicellular organization in the form of spheroids, strands, and buds. The latter structures are often seen when tumors masses switch to an invasive behavior into surrounding tissues. However, the general physical principles governing the self-organized architectures of tumor cell populations remain by and large unclear. In this work, we perform in-vitro experiments to characterize the growth properties of glioblastoma budding emerging from monolayers. Using a theoretical model and numerical tools here we find that such a topological transition is a self-organised, non-equilibrium phenomenon driven by the trade--off of mechanical forces and physica...
Journal of Clinical Medicine, 2021
Glioblastoma extensively infiltrates the brain; despite surgery and aggressive therapies, the pro... more Glioblastoma extensively infiltrates the brain; despite surgery and aggressive therapies, the prognosis is poor. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of glioblastoma evolution in every single patient, with the aim of tailoring therapeutic weapons. In particular, the ultimate goal of biomathematics for cancer is the identification of the most suitable theoretical models and simulation tools, both to describe the biological complexity of carcinogenesis and to predict tumor evolution. In this report, we describe the results of a critical review about different mathematical models in neuro-oncology with their clinical implications. A comprehensive literature search and review for English-language articles concerning mathematical modelling in glioblastoma has been conducted. The review explored the different proposed models, classifying them and indicating the significative advances of each one. ...
We consider a Cahn-Hilliard type equation with degenerate mobility and single-well potential of L... more We consider a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution together with the convergence to the weak solution. We present simulation results in one and two space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case we find similar r...
Magnetic Resonance Materials in Physics, Biology and Medicine, 2021
In this study we address the automatic segmentation of selected muscles of the thigh and leg thro... more In this study we address the automatic segmentation of selected muscles of the thigh and leg through a supervised deep learning approach. The application of quantitative imaging in neuromuscular diseases requires the availability of regions of interest (ROI) drawn on muscles to extract quantitative parameters. Up to now, manual drawing of ROIs has been considered the gold standard in clinical studies, with no clear and universally accepted standardized procedure for segmentation. Several automatic methods, based mainly on machine learning and deep learning algorithms, have recently been proposed to discriminate between skeletal muscle, bone, subcutaneous and intermuscular adipose tissue. We develop a supervised deep learning approach based on a unified framework for ROI segmentation. The proposed network generates segmentation maps with high accuracy, consisting in Dice Scores ranging from 0.89 to 0.95, with respect to “ground truth” manually segmented labelled images, also showing ...
International Journal of Computer Assisted Radiology and Surgery
Purpose This study aims at exploiting artificial intelligence (AI) for the identification, segmen... more Purpose This study aims at exploiting artificial intelligence (AI) for the identification, segmentation and quantification of COVID-19 pulmonary lesions. The limited data availability and the annotation quality are relevant factors in training AI-methods. We investigated the effects of using multiple datasets, heterogeneously populated and annotated according to different criteria. Methods We developed an automated analysis pipeline, the LungQuant system, based on a cascade of two U-nets. The first one (U-net$$_1$$ 1 ) is devoted to the identification of the lung parenchyma; the second one (U-net$$_2$$ 2 ) acts on a bounding box enclosing the segmented lungs to identify the areas affected by COVID-19 lesions. Different public datasets were used to train the U-nets and to evaluate their segmentation performances, which have been quantified in terms of the Dice Similarity Coefficients. The accuracy in predicting the CT-Severity Score (CT-SS) of the LungQuant system has been also evalu...
Mathematical Methods in the Applied Sciences
Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly pati... more Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly patient specific. Here we aim to estimate parameters in a Cahn-Hilliard type diffuse interface model in an optimised way using model order reduction (MOR) based on proper orthogonal decomposition (POD). Based on snapshots derived from finite element simulations for the full order model (FOM) we use POD for dimension reduction and solve the parameter estimation for the reduced order model (ROM). Neuroimaging data are used to define the highly inhomogeneous diffusion tensors as well as to define a target functional in a patient specific manner. The reduced order model heavily relies on the discrete empirical interpolation method (DEIM) which has to be appropriately adapted in order to deal with the highly nonlinear and degenerate parabolic PDEs. A feature of the approach is that we iterate between full order solves with new parameters to compute a POD basis function and sensitivity based parameter estimation for the ROM problems. The algorithm is applied using neuroimaging data for two clinical test cases and we can demonstrate that the reduced order approach drastically decreases the computational effort.
ESAIM: Mathematical Modelling and Numerical Analysis
This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potent... more This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate...
International Journal of Non-Linear Mechanics
This work evaluates the predictive ability of a novel personalised computational tool for simulat... more This work evaluates the predictive ability of a novel personalised computational tool for simulating the growth of brain tumours using the neuroimaging data collected during one clinical case study. The mathematical model consists in an evolutionary fourth-order partial differential equation with degenerate motility, in which the spreading dynamics of the multiphase tumour is coupled through a growth term with a parabolic equation determining the diffusing oxygen within the brain. The model also includes a reaction term describing the effects of radiotherapy, that is simulated in accordance to the clinical schedule. We collect Magnetic Resonance (MRI) and Diffusion Tensor (DTI) imaging data for one patient at given times of key clinical interest, from the first diagnosis of a giant glioblastoma to its surgical removal and the subsequent radiation therapies. These neuroimaging data allow reconstructing the patient-specific brain geometry in a finite element virtual environment, that is used for simulating the tumour recurrence pattern after the surgical resection. In particular, we characterize the different brain tissues and the tumour location from MRI data, whilst we extrapolate the heterogeneous nutrient diffusion parameters and cellular mobility from DTI data. The numerical results of the simulated tumour are found in good qualitative and quantitative agreement with the volume and the boundaries observed in MRI data. Moreover, the simulations point out a consistent regression of the tumour mass in correspondence to the application of radiotherapy, with an average growth rate which is of the same order as the one calculated from the neuroimaging data. Remarkably, our results display the highest Jaccard index of the tumour region reported in the biomathematical literature. In conclusion, this work represents an important proof-of-concept of the ability of this mathematical framework to predict the tumour recurrence and its response to therapies in a patient-specific manner.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
A soft solid is said to be initially stressed if it is subjected to a state of internal stress in... more A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors sa...
Calcolo
This work concerns the construction and the convergence analysis of a Discontinuous Galerkin Fini... more This work concerns the construction and the convergence analysis of a Discontinuous Galerkin Finite Element approximation of a Cahn-Hilliard type equation with degenerate mobility and single-well singular potential of Lennard-Jones type. This equation has been introduced in literature as a diffuse interface model for the evolution of solid tumors. Differently from the Cahn-Hilliard equation analyzed in the literature, in this model the singularity of the potential does not compensate the degeneracy of the mobility at zero by constraining the solution to be strictly positive. In previous works a finite element approximation with continuous elements of the problem has been developed by the author and coauthors. In the latter case, the positivity of the solution is enforced through a discrete variational inequality, which is solved only on active nodes of the triangulation where the degenerate operator can be inverted. Moreover, a lumping approximation of the L 2 scalar product is introduced in the formulation in order to select the solutions with a moving support with finite speed of velocity from the unphysical solutions with fixed support. As a consequence of this approximation, the order of convergence of the method is lowered down with respect to the case of the classical Cahn-Hilliard equation with constant mobility. In the present discretization with discontinuous elements, the concept of active nodes is delocalized to the concept of active elements of the triangulation and no lumping approximation of the mass products is needed to select the physical solutions. The well posedness of the discrete formulation is shown, together with the convergence to the weak solution. Different algorithms to solve the discrete variational inequality, based on iterative solvers of the associated complementarity system, are derived and implemented. Simulation results in two space dimensions are reported in order to test the validity of the proposed algorithms, in which the dynamics of the spinodal decomposition and the evolution behaviour in the coarsening regime are studied. Similar results to the ones obtained in standard phase ordering dynamics are found, which highlight nucleation and pattern formation phenomena and the evolution of single domains to steady state with constant curvature. Since the present formulation does not depend on the particular form of the potential, but it's based on the fact that the singularity set of the potential and the degeneracy set of the mobility do not coincide, it can be applied also to the degenerate CH equation with smooth potential.
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
In this work, we develop a computational tool to predict the patient-specific evolution of a high... more In this work, we develop a computational tool to predict the patient-specific evolution of a highly malignant brain tumour, the glioblastoma multiforme (GBM), and its response to therapy. A diffuse-interface mathematical model based on mixture theory is fed by clinical neuroimaging data that provide the anatomical and microstructural characteristics of the patient brain. The model is numerically solved using the finite element method, on the basis of suitable numerical techniques to deal with the resulting Cahn-Hilliard type equation with degenerate mobility and single-well potential. The results of simulations performed on the real geometry of a patient brain quantitatively show how the tumour expansion dependens on the local tissue structure. We also report the results of a sensitivity analysis concerning the effects of the different therapeutic strategies employed in the clinical Stupp protocol. The simulated results are in qualitative agreement with the observed evolution of GBM during growth, recurrence and response to treatment. Taken as a proof-of-concept, these results open the way to a novel personalized approach of mathematical tools in clinical oncology.
Mechanics Research Communications
Initial stresses originate in soft materials by the occurrence of misfits in the undeformed micro... more Initial stresses originate in soft materials by the occurrence of misfits in the undeformed microstructure. Since the reference configuration is not stress-free, the effects of initial stresses on the hyperelastic behavior must be constitutively addressed. Notably, the free energy of an initially stressed material may not possess the same symmetry group as the one of the same material deforming from a naturally unstressed configuration. This work assumes an explicit dependence of the hyperelastic strain energy density on both the deformation gradient and the initial stress tensor, taking into account for their independent invariants. Using this theoretical framework, a constitutive equation is derived for an initially stressed body that naturally behaves as an incompressible Mooney-Rivlin material. The strain energy densities for initially stressed neo-Hookean and Mooney materials are derived as special sub-cases. By assuming the existence of a virtual state that is naturally stress-free, the resulting strain energy functions are proved to fulfill the required frame-independence constraints. In the case of plane strain condition, great simplifications arise in the expression of the constitutive relations. Finally, the resulting constitutive relations prove useful guidelines for designing non-destructive methods for the quantification of the underlying initial stresses in naturally isotropic materials.
Mathematical Methods in the Applied Sciences
We consider a Cahn-Hilliard–type equation with degenerate mobility and single-well potential of L... more We consider a Cahn-Hilliard–type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.
Advances in Water Resources, 2016
Abstract The process by which rocks are formed from the burial of a fresh sediment involves the c... more Abstract The process by which rocks are formed from the burial of a fresh sediment involves the coupled effects of mechanical compaction and geochemical reactions. Both of them affect the porosity and permeability of the rock and, in particular, geochemical reactions can significantly alter them, since dissolution and precipitation processes may cause a structural transformation of the solid matrix. Often, the differential problems that arise from the modeling of these chemical reactions may present a discontinuous right hand side, where the discontinuity depends on the solution itself. In this work we have developed a numerical model to simulate this complex multi-physics problem by treating the discontinuous right hand side with a specially tailored event-driven numerical scheme. We show the performance of this strategy in terms of positivity and mass conservation, also in comparison with a more traditional approach that relies on a regularization of the discontinuous terms.
arXiv (Cornell University), Jan 19, 2023
In this work, we develop a computational tool to predict the patient-specific evolution of a high... more In this work, we develop a computational tool to predict the patient-specific evolution of a highly malignant brain tumour, the glioblastoma multiforme (GBM), and its response to therapy. A diffuse-interface mathematical model based on mixture theory is fed by clinical neuroimaging data that provide the anatomical and microstructural characteristics of the patient brain. The model is numerically solved using the finite element method, on the basis of suitable numerical techniques to deal with the resulting Cahn-Hilliard type equation with degenerate mobility and single-well potential. We report the results of simulations performed on the real geometry of a patient brain, proving how the tumour expansion is actually dependent on the local tissue structure. We also report a sensitivity analysis concerning the effects of the different therapeutic strategies employed in the clinical Stupp protocol. The simulated results are in qualitative agreement with the observed evolution of GBM dur...
Mathematical medicine and biology : a journal of the IMA, 2021
Interfaces play a key role on diseases development because they dictate the energy inflow of nutr... more Interfaces play a key role on diseases development because they dictate the energy inflow of nutrients from the surrounding tissues. What is underestimated by existing mathematical models is the biological fact that cells are able to use different resources through nonlinear mechanisms. Among all nutrients, lactate appears to be a sensitive metabolic when talking about brain tumours or neurodegenerative diseases. Here we present a partial differential model to investigate the lactate exchanges between cells and the vascular network in the brain. By extending an existing kinetic model for lactate neuro-energetics, we first provide analytical proofs of the uniqueness and the derivation of precise bounds on the solutions of the problem including diffusion of lactate in a representative volume element comprising the interface between a capillary and cells. We further perform finite element simulations of the model in two test cases, discussing the relevant physical parameters governing ...
arXiv: Cell Behavior, 2019
Tissue self-organization into defined and well-controlled three-dimensional structures is essenti... more Tissue self-organization into defined and well-controlled three-dimensional structures is essential during development for the generation of organs. A similar, but highly deranged process might also occur during the aberrant growth of cancers, which frequently display a loss of the orderly structures of the tissue of origin, but retain a multicellular organization in the form of spheroids, strands, and buds. The latter structures are often seen when tumors masses switch to an invasive behavior into surrounding tissues. However, the general physical principles governing the self-organized architectures of tumor cell populations remain by and large unclear. In this work, we perform in-vitro experiments to characterize the growth properties of glioblastoma budding emerging from monolayers. Using a theoretical model and numerical tools here we find that such a topological transition is a self-organised, non-equilibrium phenomenon driven by the trade--off of mechanical forces and physica...
Journal of Clinical Medicine, 2021
Glioblastoma extensively infiltrates the brain; despite surgery and aggressive therapies, the pro... more Glioblastoma extensively infiltrates the brain; despite surgery and aggressive therapies, the prognosis is poor. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of glioblastoma evolution in every single patient, with the aim of tailoring therapeutic weapons. In particular, the ultimate goal of biomathematics for cancer is the identification of the most suitable theoretical models and simulation tools, both to describe the biological complexity of carcinogenesis and to predict tumor evolution. In this report, we describe the results of a critical review about different mathematical models in neuro-oncology with their clinical implications. A comprehensive literature search and review for English-language articles concerning mathematical modelling in glioblastoma has been conducted. The review explored the different proposed models, classifying them and indicating the significative advances of each one. ...
We consider a Cahn-Hilliard type equation with degenerate mobility and single-well potential of L... more We consider a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution together with the convergence to the weak solution. We present simulation results in one and two space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case we find similar r...
Magnetic Resonance Materials in Physics, Biology and Medicine, 2021
In this study we address the automatic segmentation of selected muscles of the thigh and leg thro... more In this study we address the automatic segmentation of selected muscles of the thigh and leg through a supervised deep learning approach. The application of quantitative imaging in neuromuscular diseases requires the availability of regions of interest (ROI) drawn on muscles to extract quantitative parameters. Up to now, manual drawing of ROIs has been considered the gold standard in clinical studies, with no clear and universally accepted standardized procedure for segmentation. Several automatic methods, based mainly on machine learning and deep learning algorithms, have recently been proposed to discriminate between skeletal muscle, bone, subcutaneous and intermuscular adipose tissue. We develop a supervised deep learning approach based on a unified framework for ROI segmentation. The proposed network generates segmentation maps with high accuracy, consisting in Dice Scores ranging from 0.89 to 0.95, with respect to “ground truth” manually segmented labelled images, also showing ...
International Journal of Computer Assisted Radiology and Surgery
Purpose This study aims at exploiting artificial intelligence (AI) for the identification, segmen... more Purpose This study aims at exploiting artificial intelligence (AI) for the identification, segmentation and quantification of COVID-19 pulmonary lesions. The limited data availability and the annotation quality are relevant factors in training AI-methods. We investigated the effects of using multiple datasets, heterogeneously populated and annotated according to different criteria. Methods We developed an automated analysis pipeline, the LungQuant system, based on a cascade of two U-nets. The first one (U-net$$_1$$ 1 ) is devoted to the identification of the lung parenchyma; the second one (U-net$$_2$$ 2 ) acts on a bounding box enclosing the segmented lungs to identify the areas affected by COVID-19 lesions. Different public datasets were used to train the U-nets and to evaluate their segmentation performances, which have been quantified in terms of the Dice Similarity Coefficients. The accuracy in predicting the CT-Severity Score (CT-SS) of the LungQuant system has been also evalu...
Mathematical Methods in the Applied Sciences
Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly pati... more Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly patient specific. Here we aim to estimate parameters in a Cahn-Hilliard type diffuse interface model in an optimised way using model order reduction (MOR) based on proper orthogonal decomposition (POD). Based on snapshots derived from finite element simulations for the full order model (FOM) we use POD for dimension reduction and solve the parameter estimation for the reduced order model (ROM). Neuroimaging data are used to define the highly inhomogeneous diffusion tensors as well as to define a target functional in a patient specific manner. The reduced order model heavily relies on the discrete empirical interpolation method (DEIM) which has to be appropriately adapted in order to deal with the highly nonlinear and degenerate parabolic PDEs. A feature of the approach is that we iterate between full order solves with new parameters to compute a POD basis function and sensitivity based parameter estimation for the ROM problems. The algorithm is applied using neuroimaging data for two clinical test cases and we can demonstrate that the reduced order approach drastically decreases the computational effort.
ESAIM: Mathematical Modelling and Numerical Analysis
This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potent... more This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate...
International Journal of Non-Linear Mechanics
This work evaluates the predictive ability of a novel personalised computational tool for simulat... more This work evaluates the predictive ability of a novel personalised computational tool for simulating the growth of brain tumours using the neuroimaging data collected during one clinical case study. The mathematical model consists in an evolutionary fourth-order partial differential equation with degenerate motility, in which the spreading dynamics of the multiphase tumour is coupled through a growth term with a parabolic equation determining the diffusing oxygen within the brain. The model also includes a reaction term describing the effects of radiotherapy, that is simulated in accordance to the clinical schedule. We collect Magnetic Resonance (MRI) and Diffusion Tensor (DTI) imaging data for one patient at given times of key clinical interest, from the first diagnosis of a giant glioblastoma to its surgical removal and the subsequent radiation therapies. These neuroimaging data allow reconstructing the patient-specific brain geometry in a finite element virtual environment, that is used for simulating the tumour recurrence pattern after the surgical resection. In particular, we characterize the different brain tissues and the tumour location from MRI data, whilst we extrapolate the heterogeneous nutrient diffusion parameters and cellular mobility from DTI data. The numerical results of the simulated tumour are found in good qualitative and quantitative agreement with the volume and the boundaries observed in MRI data. Moreover, the simulations point out a consistent regression of the tumour mass in correspondence to the application of radiotherapy, with an average growth rate which is of the same order as the one calculated from the neuroimaging data. Remarkably, our results display the highest Jaccard index of the tumour region reported in the biomathematical literature. In conclusion, this work represents an important proof-of-concept of the ability of this mathematical framework to predict the tumour recurrence and its response to therapies in a patient-specific manner.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
A soft solid is said to be initially stressed if it is subjected to a state of internal stress in... more A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors sa...
Calcolo
This work concerns the construction and the convergence analysis of a Discontinuous Galerkin Fini... more This work concerns the construction and the convergence analysis of a Discontinuous Galerkin Finite Element approximation of a Cahn-Hilliard type equation with degenerate mobility and single-well singular potential of Lennard-Jones type. This equation has been introduced in literature as a diffuse interface model for the evolution of solid tumors. Differently from the Cahn-Hilliard equation analyzed in the literature, in this model the singularity of the potential does not compensate the degeneracy of the mobility at zero by constraining the solution to be strictly positive. In previous works a finite element approximation with continuous elements of the problem has been developed by the author and coauthors. In the latter case, the positivity of the solution is enforced through a discrete variational inequality, which is solved only on active nodes of the triangulation where the degenerate operator can be inverted. Moreover, a lumping approximation of the L 2 scalar product is introduced in the formulation in order to select the solutions with a moving support with finite speed of velocity from the unphysical solutions with fixed support. As a consequence of this approximation, the order of convergence of the method is lowered down with respect to the case of the classical Cahn-Hilliard equation with constant mobility. In the present discretization with discontinuous elements, the concept of active nodes is delocalized to the concept of active elements of the triangulation and no lumping approximation of the mass products is needed to select the physical solutions. The well posedness of the discrete formulation is shown, together with the convergence to the weak solution. Different algorithms to solve the discrete variational inequality, based on iterative solvers of the associated complementarity system, are derived and implemented. Simulation results in two space dimensions are reported in order to test the validity of the proposed algorithms, in which the dynamics of the spinodal decomposition and the evolution behaviour in the coarsening regime are studied. Similar results to the ones obtained in standard phase ordering dynamics are found, which highlight nucleation and pattern formation phenomena and the evolution of single domains to steady state with constant curvature. Since the present formulation does not depend on the particular form of the potential, but it's based on the fact that the singularity set of the potential and the degeneracy set of the mobility do not coincide, it can be applied also to the degenerate CH equation with smooth potential.
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
In this work, we develop a computational tool to predict the patient-specific evolution of a high... more In this work, we develop a computational tool to predict the patient-specific evolution of a highly malignant brain tumour, the glioblastoma multiforme (GBM), and its response to therapy. A diffuse-interface mathematical model based on mixture theory is fed by clinical neuroimaging data that provide the anatomical and microstructural characteristics of the patient brain. The model is numerically solved using the finite element method, on the basis of suitable numerical techniques to deal with the resulting Cahn-Hilliard type equation with degenerate mobility and single-well potential. The results of simulations performed on the real geometry of a patient brain quantitatively show how the tumour expansion dependens on the local tissue structure. We also report the results of a sensitivity analysis concerning the effects of the different therapeutic strategies employed in the clinical Stupp protocol. The simulated results are in qualitative agreement with the observed evolution of GBM during growth, recurrence and response to treatment. Taken as a proof-of-concept, these results open the way to a novel personalized approach of mathematical tools in clinical oncology.
Mechanics Research Communications
Initial stresses originate in soft materials by the occurrence of misfits in the undeformed micro... more Initial stresses originate in soft materials by the occurrence of misfits in the undeformed microstructure. Since the reference configuration is not stress-free, the effects of initial stresses on the hyperelastic behavior must be constitutively addressed. Notably, the free energy of an initially stressed material may not possess the same symmetry group as the one of the same material deforming from a naturally unstressed configuration. This work assumes an explicit dependence of the hyperelastic strain energy density on both the deformation gradient and the initial stress tensor, taking into account for their independent invariants. Using this theoretical framework, a constitutive equation is derived for an initially stressed body that naturally behaves as an incompressible Mooney-Rivlin material. The strain energy densities for initially stressed neo-Hookean and Mooney materials are derived as special sub-cases. By assuming the existence of a virtual state that is naturally stress-free, the resulting strain energy functions are proved to fulfill the required frame-independence constraints. In the case of plane strain condition, great simplifications arise in the expression of the constitutive relations. Finally, the resulting constitutive relations prove useful guidelines for designing non-destructive methods for the quantification of the underlying initial stresses in naturally isotropic materials.
Mathematical Methods in the Applied Sciences
We consider a Cahn-Hilliard–type equation with degenerate mobility and single-well potential of L... more We consider a Cahn-Hilliard–type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.
Advances in Water Resources, 2016
Abstract The process by which rocks are formed from the burial of a fresh sediment involves the c... more Abstract The process by which rocks are formed from the burial of a fresh sediment involves the coupled effects of mechanical compaction and geochemical reactions. Both of them affect the porosity and permeability of the rock and, in particular, geochemical reactions can significantly alter them, since dissolution and precipitation processes may cause a structural transformation of the solid matrix. Often, the differential problems that arise from the modeling of these chemical reactions may present a discontinuous right hand side, where the discontinuity depends on the solution itself. In this work we have developed a numerical model to simulate this complex multi-physics problem by treating the discontinuous right hand side with a specially tailored event-driven numerical scheme. We show the performance of this strategy in terms of positivity and mass conservation, also in comparison with a more traditional approach that relies on a regularization of the discontinuous terms.