Case Studies in Mathematical Modelling for Medical Devices: How Pulse Oximeters, Laser Doppler Flowmeters and Fetal Monitors Work focuses on three medical devices used to monitor some aspect of physiological status: pulse oximetry, laser Doppler flowmetry and Doppler ultrasound fetal heart rate monitoring. The book's three case studies serve as the basis for readers to be able to generalize modeling to other medical devices. It introduces mathematical topics that appear in many areas of science and engineering by demonstrating the value of being able to model how devices work. This requires a brief description of their operating principles before appropriate mathematics.
Containing three parts about each medical device, the book begins with a chapter on probability distributions that will be used in oximetry and laser Doppler flowmetry parts. This book is for MSc and PhD students in biomedical engineering and those interested in the mathematics behind the design of the instrumentation that they use.
- Demonstrates the value of analyzing three medical diagnostic devices and their operation using mathematics
- Looks at numerous, diverse mathematical topics that illustrate how math can be used to completely understand an instrument’s operation
- Models the physical processes that underpin devices operation to the principles of the processing applied to extract clinically relevant data from raw outputs
Preface ix
Acknowledgments xi
Introduction to the book xiii
PART 1 Maths for oximetry
List of symbols and abbreviations 3
1. Introduction 7
2. Discrete probability distributions 9
3. Continuous probability distributions 17
4. Summary statistics, moments, and cumulants 29
5. Commonly encountered distributions 43
6. Shifting and scaling distributions 61
7. Random samples fromdistributions 67
PART 2 Oximeters
8. Introduction: oximetry 79
9. Absorption coefficients 89
10. LambertBeer law 97
11. Oximetry on non-scattering samples 105
12. Scattering and the LambertBeer law 113
13. Attenuation versus absorptiona theoretical derivation 123
14. Pulse oximetry 135
15. Pulse oximetry on a population 147
16. TheMasimo Corporations oximeters 153
17. Modeling light propagation 163
18. The oximeter zoo 177
PART 3 Appendices for oximeters
19. Variance via raw moments 197
20. Taylor series 199
21. Binomial coefficients and series 201
22. Calculus 205
23. Derivatives of attenuation versus absorbance 215
24. Modeling the PPG 217
25. Fluorescence lifetimemeasurements 219
26. Logarithms 223
PART 4 Maths for DUS-FHR
List of symbols and abbreviations 229
27. Introduction 231
28. Waves 237
29. Sinusoids 241
30. Beats 249
31. Fourier analysis 253
32. Frequency domain filtering 263
33. Hilbert transform and the analytic signal 269
34. Convolution 279
35. Modulation 285
36. Sampling 293
37. Autocorrelation 299
PART 5 DUS-FHR
38. Fetal heart rate monitoring 307
39. Ultrasound 313
40. Doppler ultrasound 317
41. Doppler shift extraction 329
42. DUS-FHRmonitoring 355
43. Bandpass sampling 371
44. Pulsed operation 387
PART 6 Appendices for DUS-FHR
45. Compound angle identities 395
46. Complex numbers 397
47. Modeling with Matlab® 399
Bibliography 409
Index 413
John Crowe retired in 2020 after working as a biomedical engineer in academia for 40 years. During this time, he worked on the development of numerous medical devices with a couple leading to the formation of spin out companies. He previously co-authored the undergraduate textbook Introduction to Digital Electronics (1998).
