Corina Drapaca - Profile on Academia.edu (original) (raw)

Papers by Corina Drapaca

Bulletin of Mathematical Biology

Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but... more Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but its use introduces complexities that may not be warranted. Mathematical modelling with differential equations is a standard approach to study and predict treatment outcomes for population-level and patient-specific responses. Here, we use patient data of radiation-treated tumours to discuss the benefits and limitations of introducing fractional derivatives into three standard models of tumour growth. The fractional derivative introduces a history-dependence into the growth function, which requires a continuous death-rate term for radiation treatment. This newly proposed radiation-induced death-rate term improves computational efficiency in both ordinary and fractional derivative models. This computational speed-up will benefit common simulation tasks such as model parameterization and the construction and running of virtual clinical trials.

Fractal and Fractional

Action potentials in myelinated neurons happen only at specialized locations of the axons known a... more Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases...

Frontiers in Physiology

Nitric oxide (NO) is a small gaseous molecule that is involved in some critical biochemical proce... more Nitric oxide (NO) is a small gaseous molecule that is involved in some critical biochemical processes in the body such as the regulation of cerebral blood flow and pressure. Infection and inflammatory processes such as those caused by COVID-19 produce a disequilibrium in the NO bioavailability and/or a delay in the interactions of NO with other molecules contributing to the onset and evolution of cardiocerebrovascular diseases. A link between the SARS-CoV-2 virus and NO is introduced. Recent experimental observations of intracellular transport of metabolites in the brain and the NO trapping inside endothelial microparticles (EMPs) suggest the possibility of anomalous diffusion of NO, which may be enhanced by disease processes. A novel space-fractional reaction-diffusion equation to model NO biotransport in the brain is further proposed. The model incorporates the production of NO by synthesis in neurons and by mechanotransduction in the endothelial cells, and the loss of NO due to i...

Fractal and Fractional

Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vasc... more Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than in aged-matched male patients. Currently, there is no cure for AD. Mathematical models can help us to understand mechanisms of AD onset, progression, and therapies. However, existing models of AD do not account for sex differences. In this paper a mathematical model of AD is proposed that uses variable-order fractional temporal derivatives to describe the temporal evolutions of some relevant cells’ populations and aggregation-prone amyloid-β fibrils. The approach generalizes the model of Puri and Li. The variable fractional order describes variable fading memory due to neuroprotection loss caused by AD progression with age which, in the case of post-menopausal females, is more aggressive because of fast estr...

arXiv (Cornell University), Aug 24, 2018

Mathematical models that accurately predict the mechanical behavior of blood can contribute to th... more Mathematical models that accurately predict the mechanical behavior of blood can contribute to the development of biomedical devices and medications which are relevant in clinical applications. The models existing in the literature are complex enough in order to agree with various in vitro experimental observations. Latest technological advancements opened the possibility of studying blood flow in vivo which could play an important role in the creation of biocompatible implants for health monitoring and treatment purposes. However, most of existing models may fail to predict blood behavior in vivo because they require many parameters which are difficult to find in vivo and they do not incorporate pertinent coupled chemo-mechanical dynamics specific to blood flow in a living body. Recently, we used the fractional model of continuum mechanics proposed by Drapaca and Sivaloganathan [J. Elast.,107: 105-123, 2012] to study blood circulation. In this mathematical formulation the spatial derivatives of the rate of deformation tensor are expressed using Caputo fractional derivatives. The aim of this paper is to compare the Poiseuille flows of blood through an axi-symmetric circular rigid and impermeable pipe where the blood is described by the above mentioned fractional model, the Casson's model, and the power law model. Although the velocity profiles of these three models look similar, the fractional model provides a better fitting to published experimental data.

It is well known that the mechanical behavior of arterial walls plays an important role in the pa... more It is well known that the mechanical behavior of arterial walls plays an important role in the pathogenesis of vascular diseases. Most studies existing in the literature focus on the mechanical interactions between the blood flow and wall’s deformations. However, in the brain, the smaller vessels experience not only oscillatory forces due to the pulsatile blood flow but also structural and morphological changes controlled by the surrounding brain cells. In this study, the mechanical deformation of the cerebral arterial wall caused by the pulsatile blood flow and the dynamics of the neuronal nitric oxide (NO) is investigated. NO is a small diffusive gaseous molecule produced by the endothelial cells and neurons, which is involved in the regulation of cerebral blood flow and pressure. The cerebral vessel is assumed to be a hollow axial symmetric cylinder whose wall thickness is much smaller than the cylinder’s radius and longitudinal length is much less than the propagating wavelength...

Emerging Science Journal, 2020

Myelinated neurons are characterized by the presence of myelin, a multilaminated wrapping around ... more Myelinated neurons are characterized by the presence of myelin, a multilaminated wrapping around the axons formed by specialized neuroglial cells. Myelin acts as an electrical insulator and therefore, in myelinated neurons, the action potentials do not propagate within the axons but happen only at the nodes of Ranvier which are gaps in the axonal myelination. Recent advancements in brain science have shown that the shapes, timings, and propagation speeds of these so-called saltatory action potentials are controlled by various biochemical interactions among neurons, glial cells and the extracellular space. Given the complexity of brain’s structure and processes, the work hypothesis made in this paper is that non-local effects are involved in the optimal propagation of action potentials. A non- local model of the action potentials propagation in myelinated neurons is proposed that involves spatial derivatives of fractional order. The effects of non- locality on the distribution of the...

Science Advances, 2020

Aspiration-assisted bioprinting enables precise positioning of viscoelastic spheroids in both sca... more Aspiration-assisted bioprinting enables precise positioning of viscoelastic spheroids in both scaffold-based and free manner.

Fractal and Fractional, 2018

Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture... more Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common brain diseases. Mathematical models could highlight possible mechanisms of aneurysmal development and suggest specialized biomarkers for aneurysms. Existing mathematical models of intracranial aneurysms focus on mechanical interactions between blood flow and arteries. However, these models cannot be applied to microaneurysms since the anatomy and physiology at the length scale of cerebral microcirculation are different. In this paper, we propose a mechanism for the formation of microaneurysms that involves the chemo-mechanical coupling of blood and endothelial and neuroglial cells. We model the blood as a non-local non-Newtonian incompressible fluid and solve analytically the Poiseuille flow of such a fluid through an axi-symmetric circular rigid and impermeable pipe in the presence of wall slip. The spatial derivatives of the proposed generalization of the rate of deformation tensor are expressed using Caputo fractional derivatives. The wall slip is represented by the classic Navier law and a generalization of this law involving fractional derivatives. Numerical simulations suggest that hypertension could contribute to microaneurysmal formation.

International Journal of Non-Linear Mechanics, 2019

The multiple timescale evolution of polymers' microstructure due to an applied load is a well-kno... more The multiple timescale evolution of polymers' microstructure due to an applied load is a well-known challenge in building models that accurately predict its mechanical behavior during deformation. In the presented work, a constitutive model involving a variable order fractional derivative with piecewise definition is presented to describe the viscoelasticity of polymers under the condition of uniaxial loading at constant strain rates. It is shown that our model requires three parameters for small strains while five parameters are defined for large deformations. By comparing the predictions made by the proposed model with published experimental data and an existing model for polymers, we demonstrate that our model has higher accuracy while it benefits from its simple form of linearly decreasing order function to predict large deformations. An illustration based on the mechanism of molecular chain resistance indicates that the hardening process and the rate dependence of polymers are captured by the variation of fractional order. We conclude that the evolution of microstructure and mechanical properties of polymers during deformation is well represented by the variable order fractional constitutive model.

Emerging Science Journal, 2018

Brain tissue is a complex material made of interconnected neural, glial, and vascular networks. W... more Brain tissue is a complex material made of interconnected neural, glial, and vascular networks. While the physics and biochemistry of brain’s cell types and their interactions within their networks have been studied extensively, only recently the interactions of and feedback among the networks have started to capture the attention of the research community. Thus, a good understanding of the coupled mechano-electrochemical processes that either provide or diminish brain’s functions is still lacking. One way to increase the knowledge on how the brain yields its functions is by developing a robust controlled feedback engineering system that uses fundamental science concepts to guide and interpret experiments investigating brain’s response to various stimuli, aging, trauma, diseases, treatment and recovery processes. Recently, a mathematical model for an implantable neuro-glial-vascular unit, named brain-on-a-chip, was proposed that can be optimized to perform some fundamental cellular ...

Theoretical Biology and Medical Modelling, 2017

Background: One recent area of cancer research is irreversible electroporation (IRE). Irreversibl... more Background: One recent area of cancer research is irreversible electroporation (IRE). Irreversible electroporation is a minimally invasive procedure where needle electrodes are inserted into the body to ablate tumor cells with electricity. The aim of this paper is to propose a mathematical model that incorporates a tissue's conductivity increasing more in the direction of the electrical field as this has been shown to occur in experiments. Method: It was necessary to mathematically derive a valid form of the conductivity tensor such that it is dependent on the electrical field direction and can be easily implemented into numerical software. The derivation of a conductivity tensor that can take arbitrary functions for the conductivity in the directions tangent and normal to the electrical field is the main contribution of this paper. Numerical simulations were performed for isotropic-varying and anisotropic-varying conductivities to evaluate the importance of including the electrical field's direction in the formulation for conductivity. Results: By starting from previously published experimental results, this paper derived a general formulation for an anistropic-varying tensor for implementation into irreversible electroporation modeling software. The anistropic-varying tensor formulation allows the conductivity to take into consideration both electrical field direction and magnitude, as opposed to previous published works that only took into account electrical field magnitude. The anisotropic formulation predicts roughly a five percent decrease in ablation size for the monopolar simulation and approximately a ten percent decrease in ablation size for the bipolar simulations. This is a positive result as previously reported results found the isotropic formulation to overpredict ablation size for both monopolar and bipolar simulations. Furthermore, it was also reported that the isotropic formulation overpredicts the ablation size more for the bipolar case than the monopolar case. Thus, our results are following the experimental trend by having a larger percentage change in volume for the bipolar case than the monopolar case. Conclusions: The predicted volume of ablated cells decreased, and could be a possible explanation for the slight over-prediction seen by isotropic-varying formulations.

A Multiscale Triphasic Biomechanical Model for Tumors’ Classification

Conference Proceedings of the Society for Experimental Mechanics Series, 2011

The aim of this paper is to formulate a novel mathematical model that will be able to differentia... more The aim of this paper is to formulate a novel mathematical model that will be able to differentiate not only between normal and abnormal tissues, but, more importantly, between benign and malignant tumors. We present some very promising preliminary results of a multiscale triphasic model for biological tissues that couple the electro-chemical processes that take place in tissue’s microstructure and

Lecture Notes in Computational Vision and Biomechanics, 2013

Hydrocephalus is a neurological disease which occurs when normal cerebrospinal fluid (CSF) circul... more Hydrocephalus is a neurological disease which occurs when normal cerebrospinal fluid (CSF) circulation is impeded within the cranial cavity. As a result, the brain ventricles enlarge, and the tissue compresses, causing physical and mental problems. Treatment has been mainly through CSF flow diversion by surgically implanting a CSF shunt in the brain ventricles or by performing an endoscopic third ventriculostomy (ETV). However, the patient response to either treatment continues to be poor. Therefore, there is an urgent need to design better therapy protocols for hydrocephalus. An important step in this direction is the development of predictive computational models of the mechanics of hydrocephalic brains. In this paper, we propose a combined level set/mesh warping algorithm to track the evolution of the ventricles in the hydrocephalic brain. Our combined level set/mesh warping method is successfully used to track the evolution of the brain ventricles in two hydrocephalic patients.

Lecture Notes in Computer Science, 2012

Hydrocephalus is a neurological disease which causes ventricular dilation due to abnormalities in... more Hydrocephalus is a neurological disease which causes ventricular dilation due to abnormalities in the cerebrospinal fluid (CSF) circulation. Although treatment via a CSF shunt in the brain ventricles has been performed, poor rates of patient responses continue. Thus, to aid surgeons in hydrocephalus treatment planning, we propose a geometric computational approach for tracking hydrocephalus ventricular boundary evolution via the level set method and a mesh warping technique. In our previous work [1], we evolved the ventricular boundary in 2D CT images which required a backtracking line search for obtaining valid intermediate meshes. In this paper, we automatically detect the ventricular boundary evolution for 2D CT images. To help surgeons determine where to implant the shunt, we also compute the brain ventricle volume evolution for 3D MR images using our approach. Level-set methods (LSM) (e.g., ) are computational techniques for tracking evolving curves or surfaces and have been used extensively in medical imaging and in other fields. The level set approach delineates region boundaries using closed parametric curves (or surfaces, etc.) that deform according to motion prescribed by a partial differential equation (PDE). The problem of how to move the curves is formulated as a front evolution problem. The final contour position is influenced by the speed of the deformation, which may be controlled by local curvature of the contour, the intensity gradient in an image, shape, the initial position of the contour [9], and the intrinsic physics of the problem. One important advantage of LSM is that deforming shapes undergoing topological changes can easily be tracked. This makes the LSM ideal for tracking the evolution of hydrocephalic brain ventricles. Persson et al. developed a moving mesh technique for image-based problems which is based on the incorporation of level sets into an adaptive mesh refinement technique which uses a Cartesian or octree background mesh to determine the mesh motion. Alternatively, mesh warping algorithms compute the mesh deformation from the source domain to the target domain based upon interpolation and/or extrapolation of the vertex coordinates. Several mesh warping techniques for biomedical applications have been developed (e.g., ). However, none of these techniques were designed to handle the large deformations the ventricles undergo due to hydrocephalus.

Mechanics of Biological Systems and Materials, Volume 4, 2013

A Multiscale Pressure-Volume Model of Celebrospinal Fluid Dynamics: Application to Hydrocephalus

ASME 2013 2nd Global Congress on NanoEngineering for Medicine and Biology, 2013

Hydrocephalus is a brain disease characterized by abnormalities in the cerebrospinal fluid (CSF) ... more Hydrocephalus is a brain disease characterized by abnormalities in the cerebrospinal fluid (CSF) circulation. The treatment is surgical in nature and continues to suffer of poor outcomes. The first mathematical model for studying the CSF pressure-volume relationship in hydrocephalus was proposed by Marmarou in the 1970s. However, the model fails to fully capture the complex CSF dynamics controlled by CSF-brain tissue interactions. In this paper we use fractional calculus to introduce multiscaling effects in Marmarou’s model. We solve our fractional order non-linear differential equation analytically using a modified Adomian decomposition method.

Frontiers in Cellular Neuroscience, 2015

Damage of the brain may be caused by mechanical loads such as penetration, blunt force, shock loa... more Damage of the brain may be caused by mechanical loads such as penetration, blunt force, shock loading from blast, and by chemical imbalances due to neurological diseases and aging that trigger not only neuronal degeneration but also changes in the mechanical properties of brain tissue. An understanding of the interconnected nature of the electro-chemo-mechanical processes that result in brain damage and ultimately loss of functionality is currently lacking. While modern mathematical models that focus on how to link brain mechanics to its biochemistry are essential in enhancing our understanding of brain science, the lack of experimental data required by these models as well as the complexity of the corresponding computations render these models hard to use in clinical applications. In this paper we propose a unified variational framework for the modeling of neuronal electromechanics. We introduce a constrained Lagrangian formulation that takes into account Newton's law of motion of a linear viscoelastic Kelvin-Voigt solid-state neuron as well as the classic Hodgkin-Huxley equations of the electronic neuron. The system of differential equations describing neuronal electromechanics is obtained by applying Hamilton's principle. Numerical simulations of possible damage dynamics in neurons will be presented.

Journal of neurosurgery. Pediatrics, 2010

Hydrocephalus has traditionally been quantified by linear measures of ventricular size, with adju... more Hydrocephalus has traditionally been quantified by linear measures of ventricular size, with adjunct use of cortical mantle thickness. However, clinical outcome depends on cognitive function, which is more directly related to brain volume than these previous measures. The authors sought to quantify the dynamics of brain and ventricular volume growth in normal compared with hydrocephalic mice. Hydrocephalus was induced in 14-day-old C57BL/6 mice by percutaneous injection of kaolin into the cisterna magna. Nine hydrocephalic and 6 normal mice were serially imaged from age 2-12 weeks with a 14.1-T MR imaging unit. Total brain and ventricle volumes were calculated, and linear discriminant analysis was applied. Two very different patterns of response were seen in hydrocephalic mice compared with mice with normative growth. In one pattern (3 mice) brain growth was normal despite accumulation of CSF, and in the second pattern (6 mice) abnormal brain enlargement was accompanied by increased...

Robust space-time extraction of ventricular surface evolution using multiphase level sets

SPIE Proceedings, 2004

This paper focuses on the problem of accurately extracting the CSF-tissue boundary, particularly ... more This paper focuses on the problem of accurately extracting the CSF-tissue boundary, particularly around the ventricular surface, from serial structural MRI of the brain acquired in imaging studies of aging and dementia. This is a challenging problem because of the common occurrence ...

Bulletin of Mathematical Biology

Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but... more Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but its use introduces complexities that may not be warranted. Mathematical modelling with differential equations is a standard approach to study and predict treatment outcomes for population-level and patient-specific responses. Here, we use patient data of radiation-treated tumours to discuss the benefits and limitations of introducing fractional derivatives into three standard models of tumour growth. The fractional derivative introduces a history-dependence into the growth function, which requires a continuous death-rate term for radiation treatment. This newly proposed radiation-induced death-rate term improves computational efficiency in both ordinary and fractional derivative models. This computational speed-up will benefit common simulation tasks such as model parameterization and the construction and running of virtual clinical trials.

Fractal and Fractional

Action potentials in myelinated neurons happen only at specialized locations of the axons known a... more Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases...

Frontiers in Physiology

Nitric oxide (NO) is a small gaseous molecule that is involved in some critical biochemical proce... more Nitric oxide (NO) is a small gaseous molecule that is involved in some critical biochemical processes in the body such as the regulation of cerebral blood flow and pressure. Infection and inflammatory processes such as those caused by COVID-19 produce a disequilibrium in the NO bioavailability and/or a delay in the interactions of NO with other molecules contributing to the onset and evolution of cardiocerebrovascular diseases. A link between the SARS-CoV-2 virus and NO is introduced. Recent experimental observations of intracellular transport of metabolites in the brain and the NO trapping inside endothelial microparticles (EMPs) suggest the possibility of anomalous diffusion of NO, which may be enhanced by disease processes. A novel space-fractional reaction-diffusion equation to model NO biotransport in the brain is further proposed. The model incorporates the production of NO by synthesis in neurons and by mechanotransduction in the endothelial cells, and the loss of NO due to i...

Fractal and Fractional

Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vasc... more Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than in aged-matched male patients. Currently, there is no cure for AD. Mathematical models can help us to understand mechanisms of AD onset, progression, and therapies. However, existing models of AD do not account for sex differences. In this paper a mathematical model of AD is proposed that uses variable-order fractional temporal derivatives to describe the temporal evolutions of some relevant cells’ populations and aggregation-prone amyloid-β fibrils. The approach generalizes the model of Puri and Li. The variable fractional order describes variable fading memory due to neuroprotection loss caused by AD progression with age which, in the case of post-menopausal females, is more aggressive because of fast estr...

arXiv (Cornell University), Aug 24, 2018

Mathematical models that accurately predict the mechanical behavior of blood can contribute to th... more Mathematical models that accurately predict the mechanical behavior of blood can contribute to the development of biomedical devices and medications which are relevant in clinical applications. The models existing in the literature are complex enough in order to agree with various in vitro experimental observations. Latest technological advancements opened the possibility of studying blood flow in vivo which could play an important role in the creation of biocompatible implants for health monitoring and treatment purposes. However, most of existing models may fail to predict blood behavior in vivo because they require many parameters which are difficult to find in vivo and they do not incorporate pertinent coupled chemo-mechanical dynamics specific to blood flow in a living body. Recently, we used the fractional model of continuum mechanics proposed by Drapaca and Sivaloganathan [J. Elast.,107: 105-123, 2012] to study blood circulation. In this mathematical formulation the spatial derivatives of the rate of deformation tensor are expressed using Caputo fractional derivatives. The aim of this paper is to compare the Poiseuille flows of blood through an axi-symmetric circular rigid and impermeable pipe where the blood is described by the above mentioned fractional model, the Casson's model, and the power law model. Although the velocity profiles of these three models look similar, the fractional model provides a better fitting to published experimental data.

It is well known that the mechanical behavior of arterial walls plays an important role in the pa... more It is well known that the mechanical behavior of arterial walls plays an important role in the pathogenesis of vascular diseases. Most studies existing in the literature focus on the mechanical interactions between the blood flow and wall’s deformations. However, in the brain, the smaller vessels experience not only oscillatory forces due to the pulsatile blood flow but also structural and morphological changes controlled by the surrounding brain cells. In this study, the mechanical deformation of the cerebral arterial wall caused by the pulsatile blood flow and the dynamics of the neuronal nitric oxide (NO) is investigated. NO is a small diffusive gaseous molecule produced by the endothelial cells and neurons, which is involved in the regulation of cerebral blood flow and pressure. The cerebral vessel is assumed to be a hollow axial symmetric cylinder whose wall thickness is much smaller than the cylinder’s radius and longitudinal length is much less than the propagating wavelength...

Emerging Science Journal, 2020

Myelinated neurons are characterized by the presence of myelin, a multilaminated wrapping around ... more Myelinated neurons are characterized by the presence of myelin, a multilaminated wrapping around the axons formed by specialized neuroglial cells. Myelin acts as an electrical insulator and therefore, in myelinated neurons, the action potentials do not propagate within the axons but happen only at the nodes of Ranvier which are gaps in the axonal myelination. Recent advancements in brain science have shown that the shapes, timings, and propagation speeds of these so-called saltatory action potentials are controlled by various biochemical interactions among neurons, glial cells and the extracellular space. Given the complexity of brain’s structure and processes, the work hypothesis made in this paper is that non-local effects are involved in the optimal propagation of action potentials. A non- local model of the action potentials propagation in myelinated neurons is proposed that involves spatial derivatives of fractional order. The effects of non- locality on the distribution of the...

Science Advances, 2020

Aspiration-assisted bioprinting enables precise positioning of viscoelastic spheroids in both sca... more Aspiration-assisted bioprinting enables precise positioning of viscoelastic spheroids in both scaffold-based and free manner.

Fractal and Fractional, 2018

Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture... more Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common brain diseases. Mathematical models could highlight possible mechanisms of aneurysmal development and suggest specialized biomarkers for aneurysms. Existing mathematical models of intracranial aneurysms focus on mechanical interactions between blood flow and arteries. However, these models cannot be applied to microaneurysms since the anatomy and physiology at the length scale of cerebral microcirculation are different. In this paper, we propose a mechanism for the formation of microaneurysms that involves the chemo-mechanical coupling of blood and endothelial and neuroglial cells. We model the blood as a non-local non-Newtonian incompressible fluid and solve analytically the Poiseuille flow of such a fluid through an axi-symmetric circular rigid and impermeable pipe in the presence of wall slip. The spatial derivatives of the proposed generalization of the rate of deformation tensor are expressed using Caputo fractional derivatives. The wall slip is represented by the classic Navier law and a generalization of this law involving fractional derivatives. Numerical simulations suggest that hypertension could contribute to microaneurysmal formation.

International Journal of Non-Linear Mechanics, 2019

The multiple timescale evolution of polymers' microstructure due to an applied load is a well-kno... more The multiple timescale evolution of polymers' microstructure due to an applied load is a well-known challenge in building models that accurately predict its mechanical behavior during deformation. In the presented work, a constitutive model involving a variable order fractional derivative with piecewise definition is presented to describe the viscoelasticity of polymers under the condition of uniaxial loading at constant strain rates. It is shown that our model requires three parameters for small strains while five parameters are defined for large deformations. By comparing the predictions made by the proposed model with published experimental data and an existing model for polymers, we demonstrate that our model has higher accuracy while it benefits from its simple form of linearly decreasing order function to predict large deformations. An illustration based on the mechanism of molecular chain resistance indicates that the hardening process and the rate dependence of polymers are captured by the variation of fractional order. We conclude that the evolution of microstructure and mechanical properties of polymers during deformation is well represented by the variable order fractional constitutive model.

Emerging Science Journal, 2018

Brain tissue is a complex material made of interconnected neural, glial, and vascular networks. W... more Brain tissue is a complex material made of interconnected neural, glial, and vascular networks. While the physics and biochemistry of brain’s cell types and their interactions within their networks have been studied extensively, only recently the interactions of and feedback among the networks have started to capture the attention of the research community. Thus, a good understanding of the coupled mechano-electrochemical processes that either provide or diminish brain’s functions is still lacking. One way to increase the knowledge on how the brain yields its functions is by developing a robust controlled feedback engineering system that uses fundamental science concepts to guide and interpret experiments investigating brain’s response to various stimuli, aging, trauma, diseases, treatment and recovery processes. Recently, a mathematical model for an implantable neuro-glial-vascular unit, named brain-on-a-chip, was proposed that can be optimized to perform some fundamental cellular ...

Theoretical Biology and Medical Modelling, 2017

Background: One recent area of cancer research is irreversible electroporation (IRE). Irreversibl... more Background: One recent area of cancer research is irreversible electroporation (IRE). Irreversible electroporation is a minimally invasive procedure where needle electrodes are inserted into the body to ablate tumor cells with electricity. The aim of this paper is to propose a mathematical model that incorporates a tissue's conductivity increasing more in the direction of the electrical field as this has been shown to occur in experiments. Method: It was necessary to mathematically derive a valid form of the conductivity tensor such that it is dependent on the electrical field direction and can be easily implemented into numerical software. The derivation of a conductivity tensor that can take arbitrary functions for the conductivity in the directions tangent and normal to the electrical field is the main contribution of this paper. Numerical simulations were performed for isotropic-varying and anisotropic-varying conductivities to evaluate the importance of including the electrical field's direction in the formulation for conductivity. Results: By starting from previously published experimental results, this paper derived a general formulation for an anistropic-varying tensor for implementation into irreversible electroporation modeling software. The anistropic-varying tensor formulation allows the conductivity to take into consideration both electrical field direction and magnitude, as opposed to previous published works that only took into account electrical field magnitude. The anisotropic formulation predicts roughly a five percent decrease in ablation size for the monopolar simulation and approximately a ten percent decrease in ablation size for the bipolar simulations. This is a positive result as previously reported results found the isotropic formulation to overpredict ablation size for both monopolar and bipolar simulations. Furthermore, it was also reported that the isotropic formulation overpredicts the ablation size more for the bipolar case than the monopolar case. Thus, our results are following the experimental trend by having a larger percentage change in volume for the bipolar case than the monopolar case. Conclusions: The predicted volume of ablated cells decreased, and could be a possible explanation for the slight over-prediction seen by isotropic-varying formulations.

A Multiscale Triphasic Biomechanical Model for Tumors’ Classification

Conference Proceedings of the Society for Experimental Mechanics Series, 2011

The aim of this paper is to formulate a novel mathematical model that will be able to differentia... more The aim of this paper is to formulate a novel mathematical model that will be able to differentiate not only between normal and abnormal tissues, but, more importantly, between benign and malignant tumors. We present some very promising preliminary results of a multiscale triphasic model for biological tissues that couple the electro-chemical processes that take place in tissue’s microstructure and

Lecture Notes in Computational Vision and Biomechanics, 2013

Hydrocephalus is a neurological disease which occurs when normal cerebrospinal fluid (CSF) circul... more Hydrocephalus is a neurological disease which occurs when normal cerebrospinal fluid (CSF) circulation is impeded within the cranial cavity. As a result, the brain ventricles enlarge, and the tissue compresses, causing physical and mental problems. Treatment has been mainly through CSF flow diversion by surgically implanting a CSF shunt in the brain ventricles or by performing an endoscopic third ventriculostomy (ETV). However, the patient response to either treatment continues to be poor. Therefore, there is an urgent need to design better therapy protocols for hydrocephalus. An important step in this direction is the development of predictive computational models of the mechanics of hydrocephalic brains. In this paper, we propose a combined level set/mesh warping algorithm to track the evolution of the ventricles in the hydrocephalic brain. Our combined level set/mesh warping method is successfully used to track the evolution of the brain ventricles in two hydrocephalic patients.

Lecture Notes in Computer Science, 2012

Hydrocephalus is a neurological disease which causes ventricular dilation due to abnormalities in... more Hydrocephalus is a neurological disease which causes ventricular dilation due to abnormalities in the cerebrospinal fluid (CSF) circulation. Although treatment via a CSF shunt in the brain ventricles has been performed, poor rates of patient responses continue. Thus, to aid surgeons in hydrocephalus treatment planning, we propose a geometric computational approach for tracking hydrocephalus ventricular boundary evolution via the level set method and a mesh warping technique. In our previous work [1], we evolved the ventricular boundary in 2D CT images which required a backtracking line search for obtaining valid intermediate meshes. In this paper, we automatically detect the ventricular boundary evolution for 2D CT images. To help surgeons determine where to implant the shunt, we also compute the brain ventricle volume evolution for 3D MR images using our approach. Level-set methods (LSM) (e.g., ) are computational techniques for tracking evolving curves or surfaces and have been used extensively in medical imaging and in other fields. The level set approach delineates region boundaries using closed parametric curves (or surfaces, etc.) that deform according to motion prescribed by a partial differential equation (PDE). The problem of how to move the curves is formulated as a front evolution problem. The final contour position is influenced by the speed of the deformation, which may be controlled by local curvature of the contour, the intensity gradient in an image, shape, the initial position of the contour [9], and the intrinsic physics of the problem. One important advantage of LSM is that deforming shapes undergoing topological changes can easily be tracked. This makes the LSM ideal for tracking the evolution of hydrocephalic brain ventricles. Persson et al. developed a moving mesh technique for image-based problems which is based on the incorporation of level sets into an adaptive mesh refinement technique which uses a Cartesian or octree background mesh to determine the mesh motion. Alternatively, mesh warping algorithms compute the mesh deformation from the source domain to the target domain based upon interpolation and/or extrapolation of the vertex coordinates. Several mesh warping techniques for biomedical applications have been developed (e.g., ). However, none of these techniques were designed to handle the large deformations the ventricles undergo due to hydrocephalus.

Mechanics of Biological Systems and Materials, Volume 4, 2013

A Multiscale Pressure-Volume Model of Celebrospinal Fluid Dynamics: Application to Hydrocephalus

ASME 2013 2nd Global Congress on NanoEngineering for Medicine and Biology, 2013

Hydrocephalus is a brain disease characterized by abnormalities in the cerebrospinal fluid (CSF) ... more Hydrocephalus is a brain disease characterized by abnormalities in the cerebrospinal fluid (CSF) circulation. The treatment is surgical in nature and continues to suffer of poor outcomes. The first mathematical model for studying the CSF pressure-volume relationship in hydrocephalus was proposed by Marmarou in the 1970s. However, the model fails to fully capture the complex CSF dynamics controlled by CSF-brain tissue interactions. In this paper we use fractional calculus to introduce multiscaling effects in Marmarou’s model. We solve our fractional order non-linear differential equation analytically using a modified Adomian decomposition method.

Frontiers in Cellular Neuroscience, 2015

Damage of the brain may be caused by mechanical loads such as penetration, blunt force, shock loa... more Damage of the brain may be caused by mechanical loads such as penetration, blunt force, shock loading from blast, and by chemical imbalances due to neurological diseases and aging that trigger not only neuronal degeneration but also changes in the mechanical properties of brain tissue. An understanding of the interconnected nature of the electro-chemo-mechanical processes that result in brain damage and ultimately loss of functionality is currently lacking. While modern mathematical models that focus on how to link brain mechanics to its biochemistry are essential in enhancing our understanding of brain science, the lack of experimental data required by these models as well as the complexity of the corresponding computations render these models hard to use in clinical applications. In this paper we propose a unified variational framework for the modeling of neuronal electromechanics. We introduce a constrained Lagrangian formulation that takes into account Newton's law of motion of a linear viscoelastic Kelvin-Voigt solid-state neuron as well as the classic Hodgkin-Huxley equations of the electronic neuron. The system of differential equations describing neuronal electromechanics is obtained by applying Hamilton's principle. Numerical simulations of possible damage dynamics in neurons will be presented.

Journal of neurosurgery. Pediatrics, 2010

Hydrocephalus has traditionally been quantified by linear measures of ventricular size, with adju... more Hydrocephalus has traditionally been quantified by linear measures of ventricular size, with adjunct use of cortical mantle thickness. However, clinical outcome depends on cognitive function, which is more directly related to brain volume than these previous measures. The authors sought to quantify the dynamics of brain and ventricular volume growth in normal compared with hydrocephalic mice. Hydrocephalus was induced in 14-day-old C57BL/6 mice by percutaneous injection of kaolin into the cisterna magna. Nine hydrocephalic and 6 normal mice were serially imaged from age 2-12 weeks with a 14.1-T MR imaging unit. Total brain and ventricle volumes were calculated, and linear discriminant analysis was applied. Two very different patterns of response were seen in hydrocephalic mice compared with mice with normative growth. In one pattern (3 mice) brain growth was normal despite accumulation of CSF, and in the second pattern (6 mice) abnormal brain enlargement was accompanied by increased...

Robust space-time extraction of ventricular surface evolution using multiphase level sets

SPIE Proceedings, 2004

This paper focuses on the problem of accurately extracting the CSF-tissue boundary, particularly ... more This paper focuses on the problem of accurately extracting the CSF-tissue boundary, particularly around the ventricular surface, from serial structural MRI of the brain acquired in imaging studies of aging and dementia. This is a challenging problem because of the common occurrence ...