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.
1. Introduction to the book - p. xiii 2. Part 1: Maths for oximetry - p. xiii 3. List of symbols and abbreviations - p. 3 4. Chapter 1: Introduction - p. 7 5. Chapter 2: Discrete probability distributions - p. 9 6. Chapter 3: Continuous probability distributions - p. 17 7. Chapter 4: Summary statistics, moments, and cumulants - p. 29 8. Chapter 5: Commonly encountered distributions - p. 43 9. Chapter 6: Shifting and scaling distributions - p. 61 10. Chapter 7: Random samples from distributions - p. 67 11. Part 2: Oximeters - p. 76 12. Chapter 8: Introduction: oximetry - p. 79 13. Chapter 9: Absorption coefficients - p. 89 14. Chapter 10: Lambert–Beer law - p. 97 15. Chapter 11: Oximetry on non-scattering samples - p. 105 16. Chapter 12: Scattering and the Lambert–Beer law - p. 113 17. Chapter 13: Attenuation versus absorption—a theoretical derivation - p. 123 18. Chapter 14: Pulse oximetry - p. 135 19. Chapter 15: Pulse oximetry on a population - p. 147 20. Chapter 16: The Masimo Corporation's oximeters - p. 153 21. Chapter 17: Modeling light propagation - p. 163 22. Chapter 18: The oximeter zoo - p. 177 23. Part 3: Appendices for oximeters - p. 194 24. Chapter 19: Variance via raw moments - p. 197 25. Chapter 20: Taylor series - p. 199 26. Chapter 21: Binomial coefficients and series - p. 201 27. Chapter 22: Calculus - p. 205 28. Chapter 23: Derivatives of attenuation versus absorbance - p. 215 29. Chapter 24: Modeling the PPG - p. 217 30. Chapter 25: Fluorescence lifetime measurements - p. 219 31. Chapter 26: Logarithms - p. 223 32. Part 4: Maths for DUS-FHR - p. 225 33. List of symbols and abbreviations - p. 229 34. Chapter 27: Introduction - p. 231 35. Chapter 28: Waves - p. 237 36. Chapter 29: Sinusoids - p. 241 37. Chapter 30: Beats - p. 249 38. Chapter 31: Fourier analysis - p. 253 39. Chapter 32: Frequency domain filtering - p. 263 40. Chapter 33: Hilbert transform and the analytic signal - p. 269 41. Chapter 34: Convolution - p. 279 42. Chapter 35: Modulation - p. 285 43. Chapter 36: Sampling - p. 293 44. Chapter 37: Autocorrelation - p. 299 45. Part 5: DUS-FHR - p. 304 46. Chapter 38: Fetal heart rate monitoring - p. 307 47. Chapter 39: Ultrasound - p. 313 48. Chapter 40: Doppler ultrasound - p. 317 49. Chapter 41: Doppler shift extraction - p. 329 50. Chapter 42: DUS-FHR monitoring - p. 355 51. Chapter 43: Bandpass sampling - p. 371 52. Chapter 44: Pulsed operation - p. 387 53. Part 6: Appendices for DUS-FHR - p. 392 54. Chapter 45: Compound angle identities - p. 395 55. Chapter 46: Complex numbers - p. 397 56. Chapter 47: Modeling with Matlab® - p. 399 57. Bibliography - p. 409 58. Index - p. 413
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.
1. Introduction to the book - p. xiii 2. Part 1: Maths for oximetry - p. xiii 3. List of symbols and abbreviations - p. 3 4. Chapter 1: Introduction - p. 7 5. Chapter 2: Discrete probability distributions - p. 9 6. Chapter 3: Continuous probability distributions - p. 17 7. Chapter 4: Summary statistics, moments, and cumulants - p. 29 8. Chapter 5: Commonly encountered distributions - p. 43 9. Chapter 6: Shifting and scaling distributions - p. 61 10. Chapter 7: Random samples from distributions - p. 67 11. Part 2: Oximeters - p. 76 12. Chapter 8: Introduction: oximetry - p. 79 13. Chapter 9: Absorption coefficients - p. 89 14. Chapter 10: Lambert–Beer law - p. 97 15. Chapter 11: Oximetry on non-scattering samples - p. 105 16. Chapter 12: Scattering and the Lambert–Beer law - p. 113 17. Chapter 13: Attenuation versus absorption—a theoretical derivation - p. 123 18. Chapter 14: Pulse oximetry - p. 135 19. Chapter 15: Pulse oximetry on a population - p. 147 20. Chapter 16: The Masimo Corporation's oximeters - p. 153 21. Chapter 17: Modeling light propagation - p. 163 22. Chapter 18: The oximeter zoo - p. 177 23. Part 3: Appendices for oximeters - p. 194 24. Chapter 19: Variance via raw moments - p. 197 25. Chapter 20: Taylor series - p. 199 26. Chapter 21: Binomial coefficients and series - p. 201 27. Chapter 22: Calculus - p. 205 28. Chapter 23: Derivatives of attenuation versus absorbance - p. 215 29. Chapter 24: Modeling the PPG - p. 217 30. Chapter 25: Fluorescence lifetime measurements - p. 219 31. Chapter 26: Logarithms - p. 223 32. Part 4: Maths for DUS-FHR - p. 225 33. List of symbols and abbreviations - p. 229 34. Chapter 27: Introduction - p. 231 35. Chapter 28: Waves - p. 237 36. Chapter 29: Sinusoids - p. 241 37. Chapter 30: Beats - p. 249 38. Chapter 31: Fourier analysis - p. 253 39. Chapter 32: Frequency domain filtering - p. 263 40. Chapter 33: Hilbert transform and the analytic signal - p. 269 41. Chapter 34: Convolution - p. 279 42. Chapter 35: Modulation - p. 285 43. Chapter 36: Sampling - p. 293 44. Chapter 37: Autocorrelation - p. 299 45. Part 5: DUS-FHR - p. 304 46. Chapter 38: Fetal heart rate monitoring - p. 307 47. Chapter 39: Ultrasound - p. 313 48. Chapter 40: Doppler ultrasound - p. 317 49. Chapter 41: Doppler shift extraction - p. 329 50. Chapter 42: DUS-FHR monitoring - p. 355 51. Chapter 43: Bandpass sampling - p. 371 52. Chapter 44: Pulsed operation - p. 387 53. Part 6: Appendices for DUS-FHR - p. 392 54. Chapter 45: Compound angle identities - p. 395 55. Chapter 46: Complex numbers - p. 397 56. Chapter 47: Modeling with Matlab® - p. 399 57. Bibliography - p. 409 58. Index - p. 413
