Case Studies in Mathematical Modeling for Medical Devices

How Pulse Oximeters and Doppler Ultrasound Fetal Heart Rate Monitors Work

1st Edition - November 12, 2024
Imprint: Academic Press
Author: John Crowe
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 9 5 4 7 2 - 3
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 9 5 4 7 3 - 0

Case Studies in Mathematical Modelling for Medical Devices: How Pulse Oximeters and Doppler Ultrasound Fetal Heart Rate Monitors Work focuses on two medical devices: pulse oximet… Read more

Case Studies in Mathematical Modeling for Medical Devices

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Case Studies in Mathematical Modelling for Medical Devices: How Pulse Oximeters and Doppler Ultrasound Fetal Heart Rate Monitors Work focuses on two medical devices: pulse oximeters and Doppler ultrasound fetal heart rate monitors. The mathematical topics needed to explain their operation from first principles are introduced. These broadly cover the statistics of random processes and Fourier based signal processing. They are used to explain the devices’ operation from first principles to how clinically relevant information is extracted from the devices’ raw outputs. .

The book is for MSc and PhD students working in the area who want a quick, clear introduction to the topics, upper-division undergrads as part of biomedical engineering or applied math degree courses, biomedical engineers looking for a quick "refresher course" and clinicians interested in the operation of the instruments they use.

Title of Book
Cover image
Title page
Table of Contents
Copyright
Preface
Acknowledgments
Introduction to the book
Part 1: Maths for oximetry
List of symbols and abbreviations
Acronyms
Probabilities
Symbols
Absorption cross-section and coefficients
Hemoglobin
Attenuation
Pathlengths
Light energy
Chapter 1: Introduction
1.1. Oximetry
1.2. Probability
Chapter 2: Discrete probability distributions
2.1. Dice throwing and the probability mass function
2.1.1. Notation
2.1.2. Some “rules” of probability
2.2. Example: coin tossing
2.3. Cumulative distribution function
2.4. Summary
2.5. Problems
Chapter 3: Continuous probability distributions
3.1. Pathlengths through a scattering sample
3.1.1. Bar chart of counts
3.1.2. Probability mass
3.1.3. Probability density
3.2. Probability density function
3.2.1. Pathlength simulation
3.3. Cumulative distribution function
3.4. Example: uniform distribution
3.5. Summary
3.6. Problems
Chapter 4: Summary statistics, moments, and cumulants
4.1. The mean and its synonyms
4.2. Calculating the mean: discrete data
4.2.1. Two dice
4.2.2. Three coins
4.3. Higher order and central moments
4.3.1. Raw moments
4.3.2. Central moments
4.4. Variance: discrete data
4.4.1. Variance – the second central moment
4.4.2. Variance examples: dice and coins
4.5. Mean and variance: continuous data
4.5.1. Example: uniform distribution
4.6. Moment generating function
4.6.1. Introduction
4.6.2. MGF in series form
4.6.3. Discrete example: coin tossing
4.6.4. Continuous example: uniform distribution
4.7. Cumulant generating function
4.8. Moments and cumulants
4.9. Summary
4.10. Problems
Chapter 5: Commonly encountered distributions
5.1. Introduction
5.2. Uniform distribution
5.3. Binomial distribution
5.3.1. Moment generating function
5.3.2. Mean and variance
5.4. Poisson distribution
5.4.1. Derivation
5.4.2. Mean and variance
5.4.3. Moment generating function
5.5. Exponential distribution
5.5.1. Mean and variance
5.5.2. Cumulants
5.5.3. Exponential and Poisson relationship
5.6. Gaussian distribution
5.6.1. Moment and cumulant generating function
5.7. Wald distribution
5.7.1. Gaussian and Wald relationship
5.8. Summary
5.9. Problems
Chapter 6: Shifting and scaling distributions
6.1. Summary statistics—PDF and CDF
6.1.1. Expectation value: E(Y)
6.1.2. Variance
6.1.3. CDF
6.1.4. PDF
6.2. Example 1: uniform distribution
6.3. Example 2: Gaussian distribution
6.4. Summary
Chapter 7: Random samples from distributions
7.1. Sampling from a discrete distribution
7.2. Sampling from a continuous distribution
7.2.1. How it works
7.2.2. Observations to use
7.2.3. Derivation of the inverse CDF
7.3. Examples
7.3.1. Example: uniform random variates
7.3.2. Example: simple continuous distribution
7.3.3. Example: exponential random variates
7.4. Summary
7.5. Problems
Part 2: Oximeters
Chapter 8: Introduction: oximetry
8.1. What oximetry measures
8.2. Oximetry—an overview
8.2.1. Absorption coefficients
8.2.2. Lambert–Beer law
8.2.3. Oximetry on a non-scattering sample
8.2.4. Light scattering
8.2.5. Attenuation of an absorbing and scattering sample
8.2.6. Pulse oximetry
8.2.7. Oximetry on a population
8.2.8. Masimo® and the Discrete Saturation Transform (DST)®
8.2.9. Modeling light propagation
8.2.10. A zoo of oximeters
8.3. Summary
Chapter 9: Absorption coefficients
9.1. Chromophores
9.2. Chromophores in tissue
9.3. Absorption cross-sections and coefficients
9.4. Molar absorption coefficients
9.4.1. Things to note
9.5. Summary
9.6. Problems
Chapter 10: Lambert–Beer law
10.1. Lambert–Beer law derivation
10.1.1. Via integral calculus
10.1.2. ⋯ and a probability based approach
10.2. Graphical display of the Lambert–Beer law
10.3. Microscopic Lambert–Beer law
10.4. Practicalities
10.4.1. Pathlength
10.4.2. Optical density
10.5. Summary
Chapter 11: Oximetry on non-scattering samples
11.1. Definition of oxygen saturation
11.2. Standard oximetry equation via [HbO2] and [Hb]
11.2.1. Cramer's rule
11.3. Standard oximetry equation via equating (d[Htot])
11.4. The theoretical SO2 calibration equation
11.4.1. When A1A2=0
11.5. Incremental measurements
11.6. Summary
Chapter 12: Scattering and the Lambert–Beer law
12.1. Scattering model
12.2. Lambert–Beer law revisited
12.3. Multiple pathlength model
12.4. A versus μa gradient ≡ mean pathlength
12.5. Summary
12.6. Problems
Chapter 13: Attenuation versus absorption—a theoretical derivation
13.1. Introduction
13.2. Transmittance of a purely scattering sample
13.3. Transmittance upon adding absorber
13.3.1. Transmittance and the MGF
13.3.2. Transmittance and the CGF
13.4. Attenuation of a scattering and absorbing sample
13.5. Effect of a base and incremental absorber
13.6. The advantages of incremental measurements
13.7. Examples
13.7.1. Light impulse
13.7.2. Gaussian
13.7.3. Poisson distribution
13.7.4. Fluorescence lifetime measurements
13.8. Heterogeneous media
13.9. Summary
Chapter 14: Pulse oximetry
14.1. Incremental Lambert–Beer law
14.2. Measuring incremental attenuation
14.2.1. Ear oximeter
14.2.2. Pulse oximeter
14.3. The (AC/DC) and R ratios
14.4. Wavelength choice
14.4.1. The size of the AC signal
14.5. Calibration and B
14.6. Summary
Chapter 15: Pulse oximetry on a population
15.1. Invariance of S between individuals
15.2. The significance of the invariance of SR=0
15.3. Summary
Chapter 16: The Masimo Corporation's oximeters
16.1. Conventional pulse oximetry
16.2. The Masimo Corporation's approach
16.3. Processing
16.3.1. Error signal: e˜
16.3.2. Adaptive filter
16.3.3. Discrete Saturation Transform (DST)®
16.3.4. From r(s) to SO2
16.4. A simple math description
16.5. Arterial and venous component model
16.5.1. r(s)=ra
16.5.2. r(s)≈ra
16.5.3. r(s)≠ra≠rv
16.5.4. r(s)=rv
16.6. Summary
Chapter 17: Modeling light propagation
17.1. Modeling the PPSF
17.1.1. Photon diffusion
17.1.2. Monte Carlo modeling
17.2. Applying MC methods to tissue modeling
17.2.1. Method of photon propagation
17.2.2. Model parameters
17.3. Absorption and scattering mean-free path
17.3.1. Absorption MFP
17.3.2. Scattering MFP
17.4. Scattering direction
17.4.1. Deflection angle ϕ
17.4.2. Azimuthal angle θ
17.5. Example
17.6. Summary
Chapter 18: The oximeter zoo
18.1. Introduction
18.1.1. Oximetry: a recap
18.1.2. Why oximetry is difficult
18.1.3. Pulse oximetry—a recap
18.2. Sites, spectra, and substances
18.2.1. Sites
18.2.2. Spectra
18.2.3. Substances
18.3. Constant intensity (modified Lambert–Beer) oximeters
18.3.1. Removal of offset G
18.3.2. Hewlett Packard (HP) ear oximeter
18.3.3. Near infrared spectroscopy
18.4. Non-constant intensity oximeters
18.4.1. Use of the PPSF
18.4.2. Frequency domain analysis
18.4.3. Time integrated spectroscopy
18.5. Hyper-spectral processing
18.5.1. Methods
18.5.2. Forms of A versus μa
18.6. Summary
Part 3: Appendices for oximeters
Chapter 19: Variance via raw moments
Chapter 20: Taylor series
20.0.0.1. Exponential
20.0.0.2. Binomial series
20.0.0.3. Logarithmic series
20.0.0.4. Arctan series
Chapter 21: Binomial coefficients and series
21.1. Binomial theorem
21.2. Binomial coefficient
21.3. n choose b
Chapter 22: Calculus
22.1. Introduction
22.2. Calculus of natural logarithms
22.3. Derivative of a product
22.4. Derivative of a quotient
22.5. Integration by parts
22.6. Chain rule—differentiation using a substitution
22.7. L'Hopital's rule
22.8. First-order differential equation: for light attenuation
22.8.1. The integral of ∫dII
22.8.2. Boundary condition
22.9. Second-order differential equation
Chapter 23: Derivatives of attenuation versus absorbance
Chapter 24: Modeling the PPG
Chapter 25: Fluorescence lifetime measurements
25.1. Introduction
25.2. Theory
25.2.1. Modulation depth
25.2.2. Phase delay
Chapter 26: Logarithms
26.1. Introduction
26.2. Common bases
26.3. Logarithmic manipulations
26.4. Examples
Part 4: Maths for DUS-FHR
List of symbols and abbreviations
Acronyms
Greek symbols – [units: example]
Symbols – [units: example]
Chapter 27: Introduction
27.1. Doppler ultrasound fetal heart rate monitoring
27.2. Contents
27.2.1. Waves
27.2.2. Sinusoids
27.2.3. Beats
27.2.4. Fourier analysis
27.2.5. Frequency domain filtering
27.2.6. Hilbert transform
27.2.7. Convolution
27.2.8. Modulation
27.2.9. Sampling
27.2.10. Autocorrelation
27.3. Summary
Chapter 28: Waves
28.1. A boat at sea
28.1.1. Wavelength λ
28.1.2. Period T and frequency f
28.1.3. Wave speed v
28.2. Summary
28.3. Exercises
Chapter 29: Sinusoids
29.1. Ball on a string model
29.1.1. Period T and frequency f
29.1.2. Angular frequency ω
29.1.3. Examples
29.2. Trigonometric form
29.2.1. Example: boat at sea
29.3. Cartesian form
29.4. Comparison of forms
29.4.1. Example
29.5. Frequencies and harmonics
29.6. Summary
29.7. Problems
Chapter 30: Beats
30.1. Pictorial description
30.2. Math
30.3. Summary
30.4. Exercises
Chapter 31: Fourier analysis
31.1. Cartesian “real” Fourier analysis
31.1.1. Example: time domain
31.1.2. Example: frequency domain
31.2. Exponential “complex” form
31.2.1. Derivation
31.2.2. Example
31.3. Trigonometric frequency spectrum
31.4. Summary
31.5. Problems
Chapter 32: Frequency domain filtering
32.1. Procedure
32.2. Example
32.3. Summary
32.4. Problems
Chapter 33: Hilbert transform and the analytic signal
33.1. Introduction
33.2. Defining the Hilbert transform
33.2.1. Illustration of phase shift
33.3. Deriving the HT
33.3.1. HT transformation
33.4. Analytic signal
33.4.1. Definition
33.4.2. Obtaining the analytic signal
33.5. Instantaneous amplitude and frequency
33.5.1. Examples
33.6. Summary
33.7. Problems
Chapter 34: Convolution
34.1. The process
34.2. Convolving in frequency
34.3. Examples
34.3.1. Example 1: cos⁡⊗cos
34.3.2. Example 2: cos⁡⊗sin
34.3.3. Example 3: sin⁡⊗sin
34.3.4. Verification
34.4. Trigonometric form
34.5. Summary
34.6. Problems
Chapter 35: Modulation
35.1. Modulation process
35.2. Amplitude modulation
35.2.1. Modulation
35.2.2. Demodulation
35.3. Homodyne or heterodyne
35.4. DUS-FHR
35.5. Summary
35.6. Problems
Chapter 36: Sampling
36.1. Sampling with a comb
36.1.1. Sampling comb
36.2. Sampling via convolution
36.3. Sampling rate limits
36.3.1. Anti-aliasing filters
36.4. Summary
Chapter 37: Autocorrelation
37.1. Simple example
37.2. Periodicity
37.3. Examples
37.3.1. Pulse train
37.3.2. Periodicity of a noisy signal
37.4. Summary
37.5. Problems
Part 5: DUS-FHR
Chapter 38: Fetal heart rate monitoring
38.0.1. Fetal heart rate format
38.1. Ways of monitoring FHR [26]
38.1.1. Doppler ultrasound-based monitoring
38.2. Improving DUS-FHR monitoring
38.3. Chapter structure
38.3.1. Ultrasound
38.3.2. Doppler shifts
38.3.3. Extraction of Doppler shifts
38.3.4. DUS-FHR monitoring
38.3.5. Band-pass filtering
38.3.6. Pulsed systems
38.4. Summary
Chapter 39: Ultrasound
39.1. Sound
39.1.1. Waves
39.1.2. Quantification
39.2. Ultrasound
39.2.1. Impedance matching
39.3. Summary
Chapter 40: Doppler ultrasound
40.1. Insonation
40.2. Received ultrasound
40.2.1. Static reflector
40.2.2. Steady motion
40.2.3. Sinusoidal motion
40.2.4. Motion of cardiac structure
40.3. Summary
40.4. Problems
Chapter 41: Doppler shift extraction
41.1. Introduction
41.2. Down-conversion via modulation
41.2.1. Static and Doppler shifted components
41.2.2. Clutter
41.3. Homodyne modulation: ω′=ωc
41.3.1. Filtering
41.3.2. Example
41.4. Lack of directional information
41.4.1. Heterodyne demodulation solution: ω′<ωc
41.5. In-phase and quadrature representation
41.5.1. I and Q derived using straight multiplication
41.5.2. Modeling I and Q
41.6. Directional down-shifting
41.6.1. Example: two targets with constant velocities
41.7. Further examples
41.7.1. Sinusoidal motion
41.7.2. Cardiac simulation
41.8. Summary
41.9. Problems
Chapter 42: DUS-FHR monitoring
42.1. Introduction
42.2. The heart
42.2.1. The heart as a pump
42.2.2. Systolic timing intervals
42.2.3. Heart timings and motions
42.3. Modeling a single cardiac structure
42.3.1. I and Q generation and directional processing
42.3.2. Filtering and instantaneous amplitude
42.3.3. Instantaneous frequency
42.4. Multiple cardiac structures
42.5. FHR via autocorrelation
42.5.1. Double beating
42.6. Summary
Chapter 43: Bandpass sampling
43.1. Baseband and bandpass signals
43.1.1. Baseband signals revisited
43.1.2. Bandpass signals
43.1.3. Integer band sampling
43.2. Bandpass sampling requirements
43.2.1. Constraint on fH
43.2.2. Constraint on fL
43.2.3. Overall constraints
43.3. Illustration of acceptable sampling rates
43.3.1. Optimal sampling rate
43.4. Examples
43.4.1. Example 1
43.4.2. Example 2: Doppler ultrasound
43.5. Direct digital demodulation
43.5.1. Samples from a sinusoid
43.5.2. Optimal sampling revisited
43.5.3. Decimation
43.6. Summary
43.7. Problems
Chapter 44: Pulsed operation
44.1. Introduction
44.2. Pulsed timings
44.3. Sampling rate: fp
44.4. VmaxZmax constraint
44.5. Summary
44.6. Problems
Part 6: Appendices for DUS-FHR
Chapter 45: Compound angle identities
45.1. Sums to products
45.2. Mixing
Chapter 46: Complex numbers
46.1. Arithmetic
Chapter 47: Modeling with Matlab®
47.1. Processing code
47.1.1. FFT and related functions
47.1.2. Convolution
47.2. Discrete modeling of analogue signals
47.3. Using harmonics
47.3.1. From harmonics to frequencies
47.4. Spectral leakage
47.4.1. Non-integer harmonics
47.4.2. Windowed
47.5. Sampling
47.6. Hilbert transform: instantaneous amplitude and frequency
47.6.1. Instantaneous amplitude
47.6.2. Instantaneous frequency from instantaneous phase
Bibliography
Index

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Health

Case Studies in Mathematical Modeling for Medical Devices

How Pulse Oximeters and Doppler Ultrasound Fetal Heart Rate Monitors Work

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