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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)

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

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

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

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

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

Electric Impulses »
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

Electroencephalography & Electrocardiography »
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

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

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

Least Squares and Signal Modeling »
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

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

Wave Equation
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

Atomic Energy and X-ray Absorption
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)

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)

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.