Biomedical Physics with Applications in Disease (BPAD)

Magdalena I. Ivanova | Ivo D. Dinov | Contact us |
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BPAD Mathematical Foundations »

Mathematical Foundations

Calculus of Differentiation and Integration

Scalars, Vectors, Matrices, and Tensors

Displacement, Velocity, and Acceleration

Polynomials, Exponents, and Logarithms

Taylor’s Series Expansions

Complex Numbers

Ordinary Differential Equations

Probability and Statistics

Moments: Mean, Standard Deviation, Skewness, Kurtosis

Binomial Probability Distribution

Normal (Gaussian) Probability Distribution

Poisson Distribution

Joint Probability Distributions

Discrete and Continuous Variables

Polar, Spherical and Cylindrical Coordinates

Partial Derivatives and PDEs

Linear Algebra (linear modeling is covered later in Chapter XII)

Dimensionality Reduction (PCA/ICA, t-SNE, UMAP)

Mathematical Foundations

Calculus of Differentiation and Integration

Scalars, Vectors, Matrices, and Tensors

Displacement, Velocity, and Acceleration

Polynomials, Exponents, and Logarithms

Taylor’s Series Expansions

Complex Numbers

Ordinary Differential Equations

Probability and Statistics

Moments: Mean, Standard Deviation, Skewness, Kurtosis

Binomial Probability Distribution

Normal (Gaussian) Probability Distribution

Poisson Distribution

Joint Probability Distributions

Discrete and Continuous Variables

Polar, Spherical and Cylindrical Coordinates

Partial Derivatives and PDEs

Linear Algebra (linear modeling is covered later in Chapter XII)

Dimensionality Reduction (PCA/ICA, t-SNE, UMAP)

Physics & Mechanics »

Distances and Sizes

Models

Forces (Achilles Tendon, Hip)

Translational and Rotational Equilibria

Vector Product

Work

Stress, Strain, Shear

Hydrostatics and Buoyancy

Compressibility and Viscosity

Pressure

Human Circulatory System

Turbulent Flow and Reynolds Number

Distances and Sizes

Models

Forces (Achilles Tendon, Hip)

Translational and Rotational Equilibria

Vector Product

Work

Stress, Strain, Shear

Hydrostatics and Buoyancy

Compressibility and Viscosity

Pressure

Human Circulatory System

Turbulent Flow and Reynolds Number

Exponential Growth & Decay »

Exponential model

Exponential Growth and Decay

Examples Simulation and SARS-CoV-2 (Coronavirus modeling)

Continuously compounded interest (growth) distribution spending (decay)

Half-life and radioactive decay

Logistic equation

Mixture Distribution modeling

Exponential model

Exponential Growth and Decay

Examples Simulation and SARS-CoV-2 (Coronavirus modeling)

Continuously compounded interest (growth) distribution spending (decay)

Half-life and radioactive decay

Logistic equation

Mixture Distribution modeling

Complex Particle Systems »

Gas Molecules in a Box

Microstates and Macrostates

Energy and the First Law of Thermodynamics

Thermal Equilibrium

Entropy and the Second Law of Thermodynamics

Boltzmann Factor

Nernst Equation

Pressure Variation in the Atmosphere

Equipartition of Energy and Brownian Motion

Heat Capacity

Concentration Dependence of the Chemical Potential

Thermodynamic Relationships

Gibbs Free Energy

Gas Molecules in a Box

Microstates and Macrostates

Energy and the First Law of Thermodynamics

Thermal Equilibrium

Entropy and the Second Law of Thermodynamics

Boltzmann Factor

Nernst Equation

Pressure Variation in the Atmosphere

Equipartition of Energy and Brownian Motion

Heat Capacity

Concentration Dependence of the Chemical Potential

Thermodynamic Relationships

Gibbs Free Energy

Transport, Motion, Diffusion »

Reaction-Diffusion Dynamics on Torus Surface »

Flux, Curl and Divergence

Continuity Equation in 1D, 2D and 3D

Brownian Motion

Gas Motion, Mean Free Path, and Collision Time

Liquid Motion and Viscosity

Diffusion: Fick’s First and Second Laws of Diffusion

Time-Independent Solutions

Steady-State Diffusion to a Spherical Cell and End Effects

Diffusion Through a Collection of Pores

Diffusion from a Sphere

Reaction-Diffusion Dynamics on Torus Surface »

Flux, Curl and Divergence

Continuity Equation in 1D, 2D and 3D

Brownian Motion

Gas Motion, Mean Free Path, and Collision Time

Liquid Motion and Viscosity

Diffusion: Fick’s First and Second Laws of Diffusion

Time-Independent Solutions

Steady-State Diffusion to a Spherical Cell and End Effects

Diffusion Through a Collection of Pores

Diffusion from a Sphere

Cell Membranes

Osmotic Pressure in Gas and Liquid

Edema, Nephrotic Syndrome, Liver Disease, and Inflammation

Headaches in Renal Dialysis

Osmotic Diuresis and Fragility

Transport Through a Membrane

Countercurrent Transport

Continuum Model for Volume and Solute Transport in a Pore

Osmotic Pressure in Gas and Liquid

Edema, Nephrotic Syndrome, Liver Disease, and Inflammation

Headaches in Renal Dialysis

Osmotic Diuresis and Fragility

Transport Through a Membrane

Countercurrent Transport

Continuum Model for Volume and Solute Transport in a Pore

Physiology of Nerve and Muscle Cells

Coulomb’s Law, Superposition, and the Electric Field

Gauss’s Law

Potential Difference

Conductors and Capacitance

Current and Ohm’s Law

Charge Distribution in Cells

Hodgkin–Huxley Model for Membrane Current

Voltage Clamp Experiments

Sodium and Potassium Conductance

Voltage Changes in a Space-Clamped Axon

Propagating Nerve Impulse

Myelinated Fibers and Saltatory Conduction

Membrane Capacitance

Rhythmic Electrical Activity

Capacitance, Resistance, and Diffusion

Coulomb’s Law, Superposition, and the Electric Field

Gauss’s Law

Potential Difference

Conductors and Capacitance

Current and Ohm’s Law

Charge Distribution in Cells

Hodgkin–Huxley Model for Membrane Current

Voltage Clamp Experiments

Sodium and Potassium Conductance

Voltage Changes in a Space-Clamped Axon

Propagating Nerve Impulse

Myelinated Fibers and Saltatory Conduction

Membrane Capacitance

Rhythmic Electrical Activity

Capacitance, Resistance, and Diffusion

Electroencephalography & Electrocardiography »

Potentials Heart Electrical Signaling

Electrocardiography Model

Exterior Conductivity

Isotropic and Anisotropic Conductivity

Electrical Stimulation

EEG and EKG

Potentials Heart Electrical Signaling

Electrocardiography Model

Exterior Conductivity

Isotropic and Anisotropic Conductivity

Electrical Stimulation

EEG and EKG

Magnetic Force

Lorentz Force

Cyclotron

Divergence of Magnetic Fields

Ampere’s Circuital Law

Displacement Current

Magnetic Fields Around Axons

Magnetocardiography and Magnetoencephalography

Electromagnetic Induction and Stimulation

Magnetic Materials and Biological Systems

Measuring Magnetic Properties

Magnetic Orientation

Lorentz Force

Cyclotron

Divergence of Magnetic Fields

Ampere’s Circuital Law

Displacement Current

Magnetic Fields Around Axons

Magnetocardiography and Magnetoencephalography

Electromagnetic Induction and Stimulation

Magnetic Materials and Biological Systems

Measuring Magnetic Properties

Magnetic Orientation

Donnan Equilibrium

Potential Changes

Ion Solutions and Saturation

Goldman Equations

Membrane Channels

Noise

Sensory Transducers

Power Frequency Fields

Electromagnetic Interactions and Noise

Microwaves, Mobile Phones, and Wi-Fi

Potential Changes

Ion Solutions and Saturation

Goldman Equations

Membrane Channels

Noise

Sensory Transducers

Power Frequency Fields

Electromagnetic Interactions and Noise

Microwaves, Mobile Phones, and Wi-Fi

Variable Steady-State

Operating Point and Variable Regulation

Equilibria with and without Feedback

Proportional, Derivative, and Integral Control

Nonlinear Differential Equation Models

Nonlinear Systems

Difference Equations and Chaotic Behavior

Examples: Heat Stroke, Pupil Size, White-Blood-Cell Counts

Operating Point and Variable Regulation

Equilibria with and without Feedback

Proportional, Derivative, and Integral Control

Nonlinear Differential Equation Models

Nonlinear Systems

Difference Equations and Chaotic Behavior

Examples: Heat Stroke, Pupil Size, White-Blood-Cell Counts

Least Squares and Signal Modeling »

Least Square Estimation

Polynomial Regression

Nonlinear Least Squares

Least Square Estimation

Polynomial Regression

Nonlinear Least Squares

Medical Imaging »

Fourier Series

Examples: Plant Photosynthesis

Discrete Sampling and Aliasing

Power Spectrum

Correlation Functions (Cross-Correlation, Autocorrelation)

Parseval’s Theorem

Noise and Frequency Spectrum

Chaotic Signals and Stochastic Resonance

Fourier Series

Examples: Plant Photosynthesis

Discrete Sampling and Aliasing

Power Spectrum

Correlation Functions (Cross-Correlation, Autocorrelation)

Parseval’s Theorem

Noise and Frequency Spectrum

Chaotic Signals and Stochastic Resonance

Medical Imaging »
Activity - Clinical-Imaging Kidney Case-Study »

Convolution (1D, 2D, 3D)

Point Spread Function

Spatial Frequencies

Multi-Dimensional Image Reconstructions

Filtered Back Projection Reconstruction

Convolution (1D, 2D, 3D)

Point Spread Function

Spatial Frequencies

Multi-Dimensional Image Reconstructions

Filtered Back Projection Reconstruction

Wave Equation

Plane Waves

Acoustic Impedance

Sound Attenuation

Ultrasound Diagnostics

Plane Waves

Acoustic Impedance

Sound Attenuation

Ultrasound Diagnostics

Light Waves and Particle Duality

Electron Microscope

Atomic Energy and Spectra

Molecular Energy

Radiation Absorption and Scattering

Diffusion Approximation

Infrared Imaging, Near Infrared (NIR), Optical Coherence Tomography (OCT), Raman Spectroscopy

Thermal Radiation

Ultraviolet Radiation, Neonatal Jaundice, Ultraviolet Light, Skin Cancer, Ultraviolet Light Therapy

Radiometry and Photometry

Color Vision

Electron Microscope

Atomic Energy and Spectra

Molecular Energy

Radiation Absorption and Scattering

Diffusion Approximation

Infrared Imaging, Near Infrared (NIR), Optical Coherence Tomography (OCT), Raman Spectroscopy

Thermal Radiation

Ultraviolet Radiation, Neonatal Jaundice, Ultraviolet Light, Skin Cancer, Ultraviolet Light Therapy

Radiometry and Photometry

Color Vision

Atomic Energy and X-ray Absorption

Photon Interactions and Photoelectric Effects

Kinematics, Coherent and Incoherent Scattering

Photon Attenuation Coefficient

Compounds and Mixtures

Atomic Excitation

Photon Interactions and Photoelectric Effects

Kinematics, Coherent and Incoherent Scattering

Photon Attenuation Coefficient

Compounds and Mixtures

Atomic Excitation

Generation and Characterization of X-Rays

Radiation Interactions

X-ray Detectors and Dosimeters

Dosage and kerma calculations

Radiography

X-ray diffraction and phase estimation

Pattern analysis and phase extraction of (X-ray diffraction) XRD datasets

X-ray Imaging

Mammography

Computed Tomography (CT)

Tumor Eradication via Irradiation

Dose Measurement

Radiation Effects (radiation risk - balancing the benefits and detriments)

Radiation Interactions

X-ray Detectors and Dosimeters

Dosage and kerma calculations

Radiography

X-ray diffraction and phase estimation

Pattern analysis and phase extraction of (X-ray diffraction) XRD datasets

X-ray Imaging

Mammography

Computed Tomography (CT)

Tumor Eradication via Irradiation

Dose Measurement

Radiation Effects (radiation risk - balancing the benefits and detriments)

Medical Applications of Nuclear Physics

Radioactive Decay Rate and Half-Life

Gamma and Beta Decays

Absorbed Dose

Beam/dosage modeling calculations

Radiotracers and pharmaceuticals

Gamma Camera Detectors SPECT (Single-Photon Emission Computed Tomography)

Positron Emission Tomography (PET)

Magnetic Resonance Imaging (MRI)

Radioactive Decay Rate and Half-Life

Gamma and Beta Decays

Absorbed Dose

Beam/dosage modeling calculations

Radiotracers and pharmaceuticals

Gamma Camera Detectors SPECT (Single-Photon Emission Computed Tomography)

Positron Emission Tomography (PET)

Magnetic Resonance Imaging (MRI)

The authors are profoundly indebted to all of their mentors,
advisors, and collaborators for inspiring the study, guiding the
courses of their careers, nurturing their curiosity, and providing
constructive and critical feedback. Among these scholars are
Gencho Skordev (Sofia University), colleagues at Burgas Technical University), Kenneth Kuttler (Michigan Tech
University, De Witt L. Sumners and Fred Huffer (Florida State
University), Jan de Leeuw, Nicolas Christou, and Michael Mega
(UCLA), Arthur Toga (USC), Brian Athey, Kathleen Potempa, Janet
Larson, Patricia Hurn, Gilbert Omenn, and Eric Michielssen (University of Michigan).

Many other colleagues, students, researchers, and fellows have shared their expertise, creativity, valuable time, and critical assessment for generating, validating, and enhancing these open-science resources. Among these are Matt Ratanapanichkich and many others. In addition, colleagues from the Statistics Online Computational Resource (SOCR) and the Michigan Institute for Data Science (MIDAS) provided encouragement and valuable suggestions.

The research and development reported in this book was partially supported by the US National Science Foundation (grants 1916425, 1734853, 1636840, 1416953, 0716055 and 1023115), US National Institutes of Health (grants P20 NR015331, U54 EB020406, UL1TR002240, R01CA233487, R01MH121079, T32GM141746), and the University of Michigan.

Many other colleagues, students, researchers, and fellows have shared their expertise, creativity, valuable time, and critical assessment for generating, validating, and enhancing these open-science resources. Among these are Matt Ratanapanichkich and many others. In addition, colleagues from the Statistics Online Computational Resource (SOCR) and the Michigan Institute for Data Science (MIDAS) provided encouragement and valuable suggestions.

The research and development reported in this book was partially supported by the US National Science Foundation (grants 1916425, 1734853, 1636840, 1416953, 0716055 and 1023115), US National Institutes of Health (grants P20 NR015331, U54 EB020406, UL1TR002240, R01CA233487, R01MH121079, T32GM141746), and the University of Michigan.

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