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The remarkable functionality of the brain is made possible by the metabolism (chemical reaction) of oxygen (O₂) and nutrients in the brain. These metabolism components are supplied to the brain by an intricate blood circulatory system (vasculature). The blood brain barrier (BBB), which is the central topic of this book, determines the rate of transfer from the blood to the brain tissue.In particular, mathematical models are developed for mass transfer across the BBB based on partial differential equations (PDEs) applied to the blood capillaries, the endothelial membrane, and the brain tissue. The PDEs derived from mass balances and computer routines in R are presented for the numerical (computer-based) solution of the PDEs. The computed concentration profiles of the transferred components are functions of time and space within the BBB system, i.e., spatiotemporal solutions.The R routines and the associated numerical algorithms for computing the numerical solutions are discussed in detail. The discussion is introductory, without formal mathematics, e.g., theorems and proofs. The general methodology (algorithm) for numerical PDE solutions is the method of lines (MOL).The models are used to study the transfer of oxygen and nutrients, harmful substances that should not enter the brain such as chemicals and pathogens (viruses, bacteria), and therapeutic drugs. The intent of the book is to provide a quantitative approach to the study of BBB dynamics using a computer-based methodology programmed in R, a quality open-source scientific programming system that is easily downloaded from the Internet for execution on modest computers.
This book discusses the computer-based implementation of prototype partial differential equation models for the dynamics of mass transfer across the blood brain barrier. The numerical algorithm for the solution of PDE models is termed the method of lines. Numerical and graphical output from this model is presented with a discussion of possible application to neurodynamics.
Covid-19 is primarily a respiratory disease which results in impaired oxygenation of blood. The O2-deficient blood then moves through the body, and for the study in this book, the focus is on the blood flowing to the brain. The dynamics of blood flow along the brain capillaries and tissue is modeled as systems of ordinary and partial differential equations (ODE/PDEs). The ODE/PDE methodology is presented through a series of examples, 1. A basic one PDE model for O2 concentration in the brain capillary blood. 2. A two PDE model for O2 concentration in the brain capillary blood and in the brain tissue, with O2 transport across the blood brain barrier (BBB). 3. The two model extended to three PDEs to include the brain functional neuron cell density. Cognitive impairment could result from reduced neuron cell density in time and space (in the brain) that follows from lowered O2 concentration (hypoxia). The computer-based implementation of the example models is presented through routines coded (programmed) in R, a quality, open-source scientific computing system that is readily available from the Internet. Formal mathematics is minimized, e.g., no theorems and proofs. Rather, the presentation is through detailed examples that the reader/researcher/analyst can execute on modest computers. The PDE analysis is based on the method of lines (MOL), an established general algorithm for PDEs, implemented with finite differences. The routines are available from a download link so that the example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the blood/brain hypoxia models, such as changes in the ODE/PDE parameters (constants) and form of the model equations.
Covid-19 is primarily a respiratory disease which results in impaired oxygenation of blood. The O2-deficient blood then moves through the body, and for the study in this book, the focus is on the blood flowing to the brain. The dynamics of blood flow along the brain capillaries and tissue is modeled as systems of ordinary and partial differential equations (ODE/PDEs). The ODE/PDE methodology is presented through a series of examples, 1. A basic one PDE model for O2 concentration in the brain capillary blood. 2. A two PDE model for O2 concentration in the brain capillary blood and in the brain tissue, with O2 transport across the blood brain barrier (BBB). 3. The two model extended to three PDEs to include the brain functional neuron cell density. Cognitive impairment could result from reduced neuron cell density in time and space (in the brain) that follows from lowered O2 concentration (hypoxia). The computer-based implementation of the example models is presented through routines coded (programmed) in R, a quality, open-source scientific computing system that is readily available from the Internet. Formal mathematics is minimized, e.g., no theorems and proofs. Rather, the presentation is through detailed examples that the reader/researcher/analyst can execute on modest computers. The PDE analysis is based on the method of lines (MOL), an established general algorithm for PDEs, implemented with finite differences. The routines are available from a download link so that the example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the blood/brain hypoxia models, such as changes in the ODE/PDE parameters (constants) and form of the model equations.
Transcranial stimulation encompasses noninvasive methods that transmit physical fields-such as magnetic, electric, ultrasound, and light-to the brain to modulate its function. The most widespread approach, transcranial magnetic stimulation (TMS), has emerged as an important tool in several areas of neuroscience as well as in clinical applications in psychiatry and neurology. Originally envisioned as a way to measure the responsiveness and conduction speed of neurons and synapses in the brain and spinal cord, TMS has also become an important tool for changing the activity of brain neurons and the functions they subserve as well as an causal adjunct to brain imaging and mapping techniques. Along with transcranial electrical stimulation techniques, TMS has diffused far beyond the borders of clinical neurophysiology and into cognitive, perceptual, behavioural, and therapeutic investigation and attracted a highly diverse group of users and would-be users. Another major success of TMS has been as a treatment in psychiatry, where it is now in routine use worldwide. The field of noninvasive neuromodulation has matured and diversified considerably in the past decade, with an expansion in the number of tools available and our understanding of their mechanisms of action. This second edition of The Oxford Handbook of Transcranial Stimulation brings together the latest developments and important advances in all areas of Transcranial stimulation. The new volume captures the rapid progress made since the first edition, and provides an authoritative and comprehensive review of the state of the art. It also highlights challenges, opportunities, and future directions for this rapidly changing field. The book focuses on the scientific and technical background required to understand transcranial stimulation techniques and a wide-ranging survey of their burgeoning applications in neurophysiology, neuroscience, and therapy. Each of its six sections deals with a major area and is edited by an international authority therein. It will serve researchers, clinicians, students, and others as the definitive text in this area for years to come.
Mathematics of Computing -- Numerical Analysis.
Non-thermal irreversible electroporation is a new minimally invasive surgical p- cedure with unique molecular selectivity attributes – in fact it may be considered the first clinical molecular surgery procedure. Non-thermal irreversible electro- ration is a molecular selective mode of cell ablation that employs brief electrical fields to produce nanoscale defects in the cell membrane, which can lead to cell death, without an effect on any of the other tissue molecules. The electrical fields can be produced through contact by insertion of electrode needles around the undesirable tissue and non-invasively by electromagnetic induction. This new - dition to the medical armamentarium requires the active involvement and is of interest to clinical physicians, medical researchers, mechanical engineers, che- cal engineers, electrical engineers, instrumentation designers, medical companies and many other fields and disciplines that were never exposed in their training to irreversible electroporation or to a similar concept. This edited book is designed to be a comprehensive introduction to the field of irreversible electroporation to those that were not exposed or trained in the field before and can also serve as a reference manual. Irreversible electroporation is broad and interdisciplinary. Therefore, we have made an attempt to cover every one of the various aspects of the field from an introductory basic level to state of the art.