S E C O N D
E D I T I O N
Biomedical
Sensors
and
Instruments
Tatsuo Togawa
Toshiyo Tamura
¨
P. Åke Oberg
Boca Raton London New York
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Contents
Preface...............................................................................................................................................xi
Authors ........................................................................................................................................... xiii
Chapter 1
Fundamental Concepts ................................................................................................. 1
Signals and Noise in the Measurement ............................................................. 1
1.1.1 Measurement ........................................................................................1
1.1.2 Signals and Noise ................................................................................. 1
1.1.3 Amplitude and Power ........................................................................... 2
1.1.4 Power Spectrum ................................................................................... 2
1.1.5 Signal-to-Noise Ratio ........................................................................... 3
1.1.6 Different Types of Noise ...................................................................... 4
1.1.6.1 Thermal Noise ......................................................................4
1.1.6.2 1/f Noise ................................................................................ 4
1.1.6.3 Interference ...........................................................................4
1.1.6.4 Artifact.................................................................................. 5
1.2 Characteristics of the Measurement System .....................................................5
1.2.1 Sensor and Measurement System ......................................................... 5
1.2.2 Static Characteristics ............................................................................6
1.2.2.1 Sensitivity, Resolution, and Reproducibility......................... 6
1.2.2.2 Measurement Range .............................................................7
1.2.2.3 Linearity or Nonlinearity .....................................................7
1.2.2.4 Hysteresis ..............................................................................7
1.2.3 Dynamic Characteristics ......................................................................8
1.2.3.1 Linear and Nonlinear Systems ............................................. 8
1.2.3.2 Frequency Response ............................................................. 8
1.2.3.3 Time Constant, Response Time, Rise Time,
and Settling Time ............................................................... 10
1.3 Determination of Absolute Quantity ............................................................... 10
1.3.1 Standard and Calibration.................................................................... 11
1.3.2 Accuracy and Error ............................................................................ 11
1.3.3 Types of Error..................................................................................... 11
1.3.3.1 Random Error ..................................................................... 12
1.3.3.2 Systematic Error ................................................................. 12
1.3.3.3 Quantization Error .............................................................. 12
1.3.3.4 Dynamic Error .................................................................... 13
1.4 Units of Measurement Quantities .................................................................... 13
1.4.1 The International System of Units ..................................................... 13
1.4.1.1 Base Units and Derived Units............................................. 13
1.4.1.2 Dimension of a Quantity..................................................... 14
1.4.1.3 Recommendations for the Use of SI Units and Symbols.... 15
1.4.2 Non-SI Units ....................................................................................... 17
References .................................................................................................................. 17
1.1
iii
iv
Chapter 2
Contents
Pressure Measurements .............................................................................................. 19
Object Quantities ............................................................................................. 19
2.1.1 Units of Pressure ................................................................................ 19
2.1.2 Requirements for Pressure Measurement........................................... 19
2.1.2.1 Physiological Pressure Ranges and Measurement Sites ..... 19
2.1.2.2 Reference Point for Pressure Measurement ........................ 22
2.2 Direct Pressure Measurement .........................................................................24
2.2.1 Catheters and the Diaphragm-Type Pressure Sensor .........................24
2.2.1.1 Catheters for Pressure Measurements ................................24
2.2.1.2 Diaphragm Displacement Sensor .......................................24
2.2.2 Dynamic Response of Catheter–Sensor Systems............................... 29
2.2.2.1 Evaluation of Dynamic Response of the Catheter–
Sensor System ..................................................................... 30
2.2.2.2 Improvement of Dynamic Response .................................. 33
2.2.3 Catheter-Tip Pressure Sensor .............................................................34
2.2.4 Implantable Pressure Sensors............................................................. 39
2.2.4.1 Absolute Pressure Sensors for Implantable Devices .......... 39
2.2.4.2 Pressure Monitoring by Implantable Devices .................... 41
2.2.5 Pressure Measurements in Small Vessels .......................................... 45
2.2.5.1 Highly Rigid Pressure Sensor System ................................ 45
2.2.5.2 Servo-Controlled Pressure-Measuring System ..................46
2.2.6 Pressure Measurements in Collapsible Vessels and Interstitial
Spaces ................................................................................................. 47
2.2.6.1 Pressure Measurements in Collapsible Vessels .................. 47
2.2.6.2 Interstitial Pressure Measurements..................................... 49
2.2.7 Differential Pressure Measurements .................................................. 53
2.3 Indirect Pressure Measurement ....................................................................... 54
2.3.1 Indirect Measurement of Systolic, Diastolic, and Mean Blood
Pressure .............................................................................................. 54
2.3.1.1 Cuff Design for Indirect Blood Pressure
Measurements .............................................................. 55
2.3.1.2 Detection of Korotkoff Sounds ........................................... 57
2.3.1.3 Mean Blood Pressure Measurements
by the Oscillometric Method .............................................. 59
2.3.1.4 Blood Pressure Measurements by Doppler Ultrasound ...... 61
2.3.2 Indirect Measurements of Instantaneous Arterial Pressure............... 63
2.3.3 Indirect Pressure Measurements by Reaction Forces ........................66
2.3.3.1 Applanation Method ...........................................................66
2.3.3.2 Intraocular Pressure Measurements ................................... 67
2.3.3.3 Intra-Amniotic and Intra-Abdominal Pressure
Measurements ..................................................................... 71
2.3.3.4 Intracranial Pressure Measurement in Newborn Infants.... 72
2.3.3.5 Arterial Tonometry ............................................................. 73
References .................................................................................................................. 74
2.1
Chapter 3
Flow Measurement ..................................................................................................... 81
3.1
Object Quantities ............................................................................................. 81
3.1.1 Units in Flow Measurements.............................................................. 81
3.1.2 Requirements for Measurement Ranges ............................................ 82
v
Contents
3.2
3.3
3.1.2.1 Blood Flow in a Single Vessel ............................................ 82
3.1.2.2 Tissue Blood Flow .............................................................. 83
3.1.2.3 Respiratory Gas Flow .........................................................84
Blood Flow Measurements in Single Vessels .................................................. 85
3.2.1 Electromagnetic Flowmeter ............................................................... 85
3.2.1.1 Principle .............................................................................. 85
3.2.1.2 Factors Affecting the Measurements .................................. 86
3.2.2 Ultrasonic Blood Flowmeters............................................................. 86
3.2.2.1 Propagation of Ultrasound in the Tissue ............................ 86
3.2.2.2 Transit Time and Phase Shift Ultrasound Flowmeters ....... 88
3.2.2.3 Ultrasonic Doppler Flowmeters.......................................... 93
3.2.2.4 Methods of Range Discrimination .....................................97
3.2.2.5 Perivascular and Intravascular Doppler Probes................ 101
3.2.3 Indicator Dilution Method................................................................ 102
3.2.3.1 Principle ............................................................................ 102
3.2.3.2 Dye Dilution Method ........................................................ 104
3.2.3.3 Thermodilution Method.................................................... 106
3.2.3.4 The Fick Method............................................................... 109
3.2.3.5 Other Dilution Methods .................................................... 110
3.2.4 Flow Velocity Measurements by Heat Dissipation .......................... 111
3.2.4.1 Thermistor Velocity Probes .............................................. 111
3.2.4.2 Hot Film Velocity Probes ................................................. 112
3.2.5 Impedance Cardiography ................................................................. 113
3.2.6 Blood Flow Recording in Single Vessels by Laser Doppler
Flowmetry ........................................................................................ 116
3.2.6.1 Introduction ...................................................................... 116
3.2.6.2 Airborne Beams................................................................ 116
3.2.6.3 Microscope-Based Instrument.......................................... 116
3.2.6.4 Catheter-Based Instruments.............................................. 117
3.2.7 Correlation Methods for Microvascular Red Blood Cell Velocity
Measurement .................................................................................... 119
3.2.8 Miscellaneous Mechanical Flowmeters ........................................... 122
3.2.9 Implantable Flow Sensors ................................................................ 124
Tissue Blood Flow Measurement .................................................................. 125
3.3.1 Venous Occlusion Plethysmography ................................................ 125
3.3.1.1 Venous Occlusion Method ................................................ 125
3.3.1.2 Displacement Plethysmographies ..................................... 126
3.3.1.3 Impedance Plethysmographies ......................................... 129
3.3.2 Clearance Technique ........................................................................ 131
3.3.2.1 Principle ............................................................................ 131
3.3.2.2 Use of Radioactive Indicators ........................................... 133
3.3.2.3 Hydrogen Gas Clearance Method..................................... 134
3.3.3 Tissue Blood Flow Measurements by Indicator Dilution
Method ......................................................................................137
3.3.4 Tissue Blood Flow Measurements by Heat Transport ..................... 138
3.3.5 Tissue Blood Flow Measurement by Laser Doppler Flowmetry ..... 138
3.3.5.1 Introduction ...................................................................... 138
3.3.5.2 Interaction between Light and Tissue ............................... 139
3.3.5.3 Instrument Design Principles ........................................... 143
3.3.5.4 Fiberoptic Arrangements .................................................. 144
3.3.5.5 Self-Mixing Laser Diode Technology .............................. 146
vi
Contents
3.3.5.6
3.3.5.7
Signal Processing Principles............................................. 146
Calibration and Standardization of Laser Doppler
Flowmeters ....................................................................... 147
3.3.5.8 Laser Doppler Perfusion Imaging..................................... 148
3.3.6 Tissue Blood Flow Measurements by Nuclear Magnetic
Resonance......................................................................................... 151
3.3.6.1 Principle of Nuclear Magnetic Resonance ....................... 151
3.3.6.2 Magnetic Resonance Tissue Blood Flow Measurement
Methods ............................................................................ 155
3.4 Respiratory Gas Flow Measurements............................................................ 157
3.4.1 Gas Flow Sensors ............................................................................. 157
3.4.1.1 Rotameter.......................................................................... 157
3.4.1.2 Pneumotachograph ........................................................... 159
3.4.1.3 Hot-Wire Anemometer ..................................................... 161
3.4.1.4 Time-of-Flight Flowmeter ................................................ 165
3.4.1.5 Ultrasonic Flowmeter ....................................................... 167
3.4.1.6 Vortex Flowmeter ............................................................. 169
3.4.1.7 Acoustical Respiratory Flow Measurements .................... 170
3.4.2 Volume Measuring Spirometers ....................................................... 171
3.4.3 Lung Plethysmography ..................................................................... 172
3.4.3.1 Body Plethysmography ..................................................... 172
3.4.3.2 Inductance Plethysmography ............................................ 174
3.4.3.3 Impedance Pneumography ............................................... 175
References ................................................................................................................ 177
Chapter 4
Motion and Force Measurement............................................................................... 191
4.1
4.2
4.3
Objects of Measurement ................................................................................ 191
4.1.1 Units of Quantities ........................................................................... 191
4.1.2 Objects of Measurements ................................................................. 191
4.1.3 Coordinate System ........................................................................... 192
Motion Measurements ................................................................................... 193
4.2.1 Displacement and Rotation Measurements by Contact Sensors ...... 194
4.2.1.1 Displacement and Rotation Measurements
of the Body and an Extracted Tissue ................................ 194
4.2.1.2 Displacement Measurements in Vivo................................200
4.2.2 Noncontact Measurement of Displacement and Rotation ................ 203
4.2.2.1 Optical Methods ...............................................................203
4.2.2.2 Magnetic Methods ............................................................206
4.2.3 Linear and Angular Velocity Measurements ...................................207
4.2.3.1 Electromagnetic Velocity Sensors ....................................208
4.2.3.2 Doppler Methods ..............................................................209
4.2.3.3 Angular Velocity Sensors ................................................. 210
4.2.4 Translational and Angular Acceleration Measurements .................. 211
4.2.4.1 Translational Accelerometers ........................................... 212
4.2.4.2 Angular Accelerometers ................................................... 215
Force Measurements ...................................................................................... 216
4.3.1 Muscle Contraction Measurements .................................................. 216
4.3.1.1 Design of the Elastic Beam for Muscle Contraction
Measurements ................................................................... 216
vii
Contents
4.3.1.2 Force Measurements in Isolated Muscles ......................... 217
4.3.1.3 In Vivo Measurements of Muscular Contraction .............. 218
4.3.2 Measurements of Stresses in the Bone ............................................. 219
4.3.3 Ground Force Measurements ........................................................... 221
4.3.3.1 The Force Plates ............................................................... 221
4.3.3.2 Stabilometers .................................................................... 222
4.3.3.3 Instrumented Shoe ............................................................224
4.3.3.4 Foot Force Distribution Measurements ............................ 227
References ................................................................................................................ 228
Chapter 5
Temperature, Heat Flow, and Evaporation Measurements....................................... 235
5.1
5.2
5.3
5.4
5.5
5.6
Object Quantities ........................................................................................... 235
5.1.1 Units of Thermal Quantities............................................................. 235
5.1.2 Requirements for Measurement Ranges .......................................... 236
Temperature Sensors ..................................................................................... 238
5.2.1 Thermistors ...................................................................................... 238
5.2.2 Thermocouples .................................................................................240
5.2.3 Wire and Thin Film Thermoresistive Elements ............................... 242
5.2.4 p–n Junction Diodes and Transistors ............................................... 243
5.2.5 Crystal Resonators ........................................................................... 245
5.2.6 Temperature Sensors for Use in Strong Electromagnetic Fields ......246
Noncontact Temperature Measurement Techniques .....................................248
5.3.1 Infrared Measurements .................................................................... 249
5.3.1.1 Infrared Radiation Thermometers .................................... 249
5.3.1.2 Infrared Detectors............................................................. 250
5.3.1.3 Infrared Thermography .................................................... 253
5.3.2 Noninvasive Thermal Imaging in Deep Tissue................................254
5.3.2.1 Magnetic Resonance Thermal Imaging ........................... 255
5.3.2.2 Active and Passive Microwave Thermal Imaging ............ 255
5.3.2.3 Ultrasonic Thermal Imaging ............................................ 256
5.3.2.4 Thermal Imaging by Tissue Electrical Impedance .......... 257
Clinical Thermometers .................................................................................. 257
5.4.1 Indwelling Thermometer Probes...................................................... 257
5.4.1.1 Rectal Temperature Measurement .................................... 257
5.4.1.2 Esophageal Temperature Measurement ............................ 258
5.4.1.3 Bladder Temperature Measurement.................................. 258
5.4.2 Tympanic Thermometers ................................................................. 258
5.4.2.1 Contact Probes .................................................................. 258
5.4.2.2 Noncontact Tympanic Thermometers .............................. 259
5.4.3 Zero-Heat-Flow Thermometer .........................................................260
5.4.4 Telemetering Capsules ..................................................................... 261
5.4.4.1 Radio Pill .......................................................................... 261
5.4.4.2 Ultrasonic-Coupling Crystal Resonator ........................... 263
Heat Flow Measurements .............................................................................. 263
5.5.1 Heat Flow Sensors ............................................................................ 263
5.5.2 Heat Flow Measurement at the Body Surface.................................. 265
5.5.3 Direct Calorimetry ........................................................................... 267
Evaporation Measurement ............................................................................. 268
5.6.1 Introduction ...................................................................................... 268
viii
Contents
5.6.2
Humidity Sensors ............................................................................. 269
5.6.2.1 The Electrolytic Water Vapor Analyzer ........................... 269
5.6.2.2 Thermal Conductivity Cell ............................................... 269
5.6.2.3 Infrared Water Vapor Analyzer ........................................ 270
5.6.2.4 Dew-Point Hygrometer ..................................................... 270
5.6.2.5 Electrohygrometer ............................................................ 270
5.6.2.6 Capacitive Humidity Sensor ............................................. 271
5.6.2.7 Impedance Humidity Sensor ............................................ 271
5.6.2.8 Thermoelectric Psychrometer ........................................... 271
5.6.3 Measurement of Evaporative Water Loss from Skin and Mucosa ... 272
5.6.3.1 Unventilated Chamber ...................................................... 272
5.6.3.2 Ventilated Chamber .......................................................... 272
5.6.3.3 Gradient Estimation Method ............................................ 273
References ................................................................................................................ 275
Chapter 6
Bioelectric and Biomagnetic Measurements ............................................................ 281
6.1
6.2
6.3
6.4
6.5
Objects of Measurements .............................................................................. 281
6.1.1 Units of Electromagnetic Measurements ......................................... 281
6.1.1.1 Electrical Units ................................................................. 281
6.1.1.2 Magnetic Units.................................................................. 282
6.1.2 Requirements for Measurement Ranges .......................................... 282
6.1.2.1 Bioelectric Events ............................................................. 282
6.1.2.2 Biomagnetic Events .......................................................... 283
Electrode Theory ........................................................................................... 283
6.2.1 Electrode–Electrolyte Interface ....................................................... 283
6.2.2 Liquid Junction Potentials ................................................................284
6.2.3 Double Layer .................................................................................... 285
6.2.4 Electrode Potential ........................................................................... 287
Surface Potential Electrodes.......................................................................... 288
6.3.1 ECG Electrodes ................................................................................ 288
6.3.1.1 Silver–Silver Chloride Electrode ...................................... 289
6.3.1.2 Stainless Steel Electrodes ................................................. 290
6.3.1.3 Electrode Impedance ........................................................ 290
6.3.1.4 Motion Artifacts ............................................................... 294
6.3.1.5 Dry Contactless or Capacitive Electrodes ........................ 294
6.3.2 Electromyogram Electrodes ............................................................. 295
6.3.2.1 Surface Electrodes ............................................................ 295
6.3.2.2 Needle and Wire Electrodes ............................................. 296
6.3.3 Electroencephalogram Electrodes.................................................... 296
Micro and Suction Electrodes ....................................................................... 298
6.4.1 Glass Microelectrodes ...................................................................... 298
6.4.2 Metal Microelectrodes ..................................................................... 299
6.4.3 Suction Electrodes ............................................................................300
6.4.4 Patch Clamping ................................................................................ 301
Biomagnetism ................................................................................................ 301
6.5.1 Biomagnetic Fields ........................................................................... 301
6.5.1.1 Magnetopneumography ....................................................302
6.5.1.2 Magnetoencephalography ................................................. 303
ix
Contents
6.5.2 Induction Coil Measurements .......................................................... 303
6.5.3 Fluxgate Magnetometer....................................................................304
6.5.4 Squid Systems ..................................................................................306
6.5.5 Magnetic Noise and Shielding .........................................................308
References ................................................................................................................ 311
Chapter 7
Chemical Measurement ............................................................................................ 315
7.1
7.2
7.3
Objects of Measurements .............................................................................. 315
7.1.1 Units of Chemical Quantities ........................................................... 315
7.1.2 Objects of Chemical Measurement .................................................. 316
7.1.3 Requirements and Limitations in Chemical Measurements ............ 317
Chemical Sensors .......................................................................................... 317
7.2.1 Electrochemical Sensors .................................................................. 317
7.2.1.1 Electrode Potential and Reference Electrode ................... 318
7.2.1.2 Potentiometric Sensors ..................................................... 321
7.2.1.3 Amperometric Sensors ..................................................... 325
7.2.1.4 Electrochemical Gas Sensors ........................................... 328
7.2.1.5 Impedimetric Sensors ....................................................... 333
7.2.2 Optically Based Chemical Sensors .................................................. 334
7.2.2.1 Spectrophotometric Chemical Analysis ........................... 334
7.2.2.2 Chemical Sensors Based on Surface Plasmon Resonance ... 337
7.2.2.3 Fiber Optic Chemical Sensors .......................................... 339
7.2.3 Chemical Sensors of Acoustic and Thermal Principles ................... 341
7.2.4 Biosensors......................................................................................... 345
7.2.4.1 Enzyme-Based Biosensors................................................ 345
7.2.4.2 Immunosensors .................................................................348
7.2.4.3 DNA Sensors .................................................................... 350
7.2.4.4 Microbial Sensors ............................................................. 352
7.2.5 Other Analytical Methods and Instrumentations ............................. 353
7.2.5.1 Mass Spectrometry ........................................................... 353
7.2.5.2 Chromatography ............................................................... 356
7.2.5.3 Electrophoresis ................................................................. 357
7.2.5.4 Magnetic Resonance ......................................................... 359
7.2.5.5 Other Analytical Methods Based on Physical Material
Properties .......................................................................... 361
Continuous Measurements of Chemical Quantities ...................................... 363
7.3.1 Measurements by Indwelling Sensors ..............................................364
7.3.1.1 Intravascular Measurements .............................................364
7.3.1.2 Tissue Measurements........................................................ 367
7.3.2 Ex Vivo Measurements ..................................................................... 368
7.3.2.1 Measurements by Blood Drainage.................................... 368
7.3.2.2 Microdialysis .................................................................... 371
7.3.3 Transcutaneous Measurements ........................................................ 373
7.3.3.1 The Skin as a Membrane in Transcutaneous
Measurements ................................................................... 373
7.3.3.2 Transcutaneous Blood Gas Measurements ....................... 375
7.3.3.3 Transcutaneous Arterial Oxygen Saturation Monitoring ...378
7.3.3.4 Transcutaneous Glucose Monitoring ................................ 383
x
Contents
7.3.4
Respiratory Gas Analysis ................................................................. 384
7.3.4.1 Monitoring Ventilation ..................................................... 385
7.3.4.2 Estimation of Metabolic Rate ........................................... 386
7.3.5 Electronic Noses ............................................................................... 388
References ................................................................................................................ 389
Index .............................................................................................................................................. 399
Preface
In medicine and healthcare, acquisition and assessment of reliable physiological data are essential.
Many kinds of sensors and instruments are routinely used in the clinical field, and many instruments are becoming automatic, fail safe, and easy to use. However, it is still important to know how
the sensor or instrument works when one is responsible for presenting the obtained data in scientific
meetings or explaining the evidence correctly to patients.
The first edition of this book was written to meet these demands and it was published 14 years
ago. Since that time, many new sensors and instruments have been developed. In the field of sensor
application where advanced technology was introduced, old techniques were replaced by new ones.
Such changes in measurement technologies and their applications are covered in the second edition. Although the chapter titles are unchanged, their contents have been updated, new topics have
been added, and details about techniques that are becoming obsolete have been reduced to keep the
book compact. On the other hand, there are potentially promising techniques that have already been
developed but their applications are limited or have not been attempted at all due to some practical
difficulties. However, if such problems can be solved, new possibilities will appear. This book also
covers these kinds of techniques.
Micro- and nanotechnologies have been introduced in almost all kinds of sensor devices. While
this is topical, the fundamental sensing principles of most sensors remain unchanged. This book
does not go into much detail about downscaling efforts except when essentially different phenomena appear in micro- and nanostructures.
In the second edition, the book title was changed from Biomedical Transducers and Instruments
to Biomedical Sensors and Instruments because the term sensor is becoming used more in the field
of biomedical engineering, even though the term transducer continues to be used.
We hope that this new edition will be of interest to university students, industrial companies
in the sensor field, as well as serve as a platform for courses in this particular field of biomedical
engineering.
Tatsuo Togawa
Toshiyo Tamura
P. Åke Öberg
xi
Authors
Tatsuo Togawa received his PhD in applied physics in 1965 from the University of Tokyo. He
spent three years at the Institute of Medical Electronics, University of Tokyo, as a research assistant. He then moved to the Institute for Medical and Dental Engineering, Tokyo Medical and
Dental University, and served as professor of the Division of Instrumentation Engineering, Tokyo
Medical and Dental University, from 1972 to 2003. Subsequently, he moved to the School of Human
Sciences, Waseda University, and served as a professor until 2008. He is now a guest researcher
at the Advance Research Center for Human Sciences, Waseda University. Throughout his career,
Togawa has been involved mostly in research in biomedical measurements and instrumentation.
His scientific works have been published in various books and as more than 110 articles in international journals. Since 1973, he has been cooperating with Professor P. Åke Öberg, Department
of Biomedical Engineering, Linköping University, Sweden. In 1994, he received the degree of
Doctor honoris causa from Linköping University. He is a foreign member of the Polish Academy of
Sciences and a fellow of the Institute of Physics, United Kingdom.
Toshiyo Tamura received his PhD in 1980 from the Tokyo Medical and Dental University.
Subsequently, he served as a research associate at the Institute for Medical and Dental
Engineering. From 1993 to 1998, he was an associate professor at the Institute for Medical and
Dental Engineering, Tokyo Medical and Dental University. From 1998 to 2003, he served as a
d­ irector, Department of Gerontechnology, National Institute for Longevity Sciences. From 2003,
he served as a professor in the Department of Biomedical Engineering, Chiba University Graduate
School of Engineering. His research and teaching activities have focused on biomedical ­transducers
involving noninvasive apparatus and biosignal analysis. His scientific work is represented in more
than 100 peer-reviewed articles in national and international journals. He has presented numerous
lectures at international meetings and is also an active member of several national and international
societies. He has served as a chair of the IEEE/EMBS Tokyo Chapter from 1996 to 2000 and as the
Asian Pacific representative for the EMBS from 2000 to 2004. He is currently serving as president
of the Japanese Society of Medical Electronics and Biological Engineering (2009–2010) and has
previously served as president of the Japanese Society of Life Support Engineering (2008–2009).
P. Åke Öberg received his PhD in biomedical engineering from the Department of Physiology and
Medical Biophysics, Uppsala University, Sweden, in 1971. He became the founding professor of the
Department of Biomedical Engineering at Linköping University, Sweden, in 1972. He also served
as the director of clinical engineering at the Linköping University Hospital. His research interests include sensors and instruments in medicine, biomagnetics, and clinical engineering. Professor
Öberg has published 560 papers and conference presentations and a number of books, many of them
with Japanese colleagues as coauthors. He is a fellow of the Royal Swedish Academy of Engineering
Sciences, the Royal Swedish Academy of Sciences, an honorary fellow of the Hungarian Academy
of Engineering Sciences, American Institute of Medical and Biological Engineering, and other
international learned societies. He has chaired the Swedish Society for Medical Engineering
and Medical Physics for two periods and the International Academy for Medical and Biological
Engineering for three years. Professor Öberg also has served as a professor emeritus since 2002.
xiii
1 Fundamental Concepts
This chapter provides brief definitions and explanations of some fundamental concepts in biomedical
measurement techniques. We recommend the use of this chapter as a checklist when doing research
or in medical practice involving measurements employing sensors and measurement systems.
Different definitions for the same technical terms often appear in different fields. In such
cases, the content of this chapter is limited to the situation where physiological measurements are
concerned.
1.1
1.1.1
SIGNALs AND NOIsE IN THE MEAsUREmENT
MEASUREmENT
A measurement is a procedure by which an observer determines the quantity that characterizes
the property or state of an object. The quantity to be determined is the object quantity of the measurement. In this book, physical or chemical quantities that contain physiological information are
considered as the object quantities. Sometimes such quantities can be estimated by human sense, for
instance through visible observations. But, to obtain objective, reproducible, and quantitative results,
instruments should be used where the results are given as the output of the measurement system.
The physical characteristics of the output depend on the type of instrument used. When electronic instruments are used, the output will be in the form of an electric potential. This output can
also be converted into digital values if desired. In any case, the original physical or chemical quantity is converted into convenient forms such as an electric potential or a numerical value. To describe
the object quantity correctly from the output of an instrument, the relation between the output of the
instrument and the object quantity must be defined. To obtain this, adequate instrument calibration
procedure is required as discussed in Section 1.3.1. The term measurement always implies the whole
procedure by which the object quantity is correctly determined.
1.1.2
SIGNALS AND NOISE
In a measurement, the signal is defined as the component of a variable that contains information
about the object quantity, whereas noise is defined as a component unrelated to the object quantity. Thus, the signal is the component that an observer wants to obtain, and noise is the unwanted
component.
Therefore, signals and noise are not uniquely defined by their physical nature, but depend on the
intention of the observer. For example, the electromyogram (EMG), which is the potential generated
by muscles, can be regarded as a signal by an observer who is interested in muscle activity. But the
EMG is an unwanted component for another observer who is interested in obtaining nerve action
potentials. In this situation, the EMG component is considered to be a kind of noise. In this sense,
the definition of signals and noise is arbitrary in general measurement situations. The situation is
different in communication technology, where a sender sends signals to a receiver, so that the signal
is fully determined by the intention of the sender.
In actual measurement situations, there is no general rule by which signals and noise can be
distinguished. Only detailed knowledge about the nature of the measurement object and possible
disturbances in the system can help to distinguish signals from noise. Typical sources of noise and
their characteristics are discussed in Section 1.1.6.
1
2
1.1.3
Biomedical Sensors and Instruments
AmPLITUDE AND POWER
In biomedical measurements, object quantities are usually time-varying. Usually, unwanted timevarying components are superimposed on the signal. If an electronic instrument is used to measure
the object quantity, it provides an electric potential output containing the signal and noise components both of which have to be regarded as time-varying.
To describe the range of variation of a time-varying signal, the concepts amplitude and power
are commonly used. Sinusoidal variation is considered a fundamental pattern of variation, because
any time-varying variable in a definite time interval can be expressed as the sum of sinusoidal variations as described later.
Where sinusoidal variations are concerned, the difference between the maximal positive and
the maximal negative peaks is a measure of the amplitude, which is called the peak-to-peak value.
The peak-to-peak value can also be defined for any time-varying signal which is not sinusoidal or
periodic.
The root-mean-square amplitude is a convenient measure of the variability of the signal in a
definite time period. If the signal is denoted x(t), then the root-mean-square amplitude is defined as
x(t )2 . Both peak-to-peak values and root-mean-square amplitudes have the same dimensions as
that of the variable itself.
Power is a quantity defined as the time average of the square of the variable. For a variable x(t),
the power is given as x(t )2 . If x(t) corresponds to the electric potential developed over a resistor R,
the power dissipation in the resistor is given as x(t )2 /R. In this case, x(t )2 has a physical meaning
of power which is defined as the amount proportional to energy dissipation per unit time. But the
concept of power defined as x(t )2 is used more widely even though it does not directly correspond
to the physical concept of power. As described in Section 1.1.4, the quantity x(t )2 is useful when
frequency components of the variable are considered.
1.1.4
POWER SPECTRUm
The power spectrum is the distribution of power of the variable corresponding to frequency components. As rigorously treated by the theory of Fourier series, any periodic function of time, x(t),
having a period, T, and time average equal to zero, can be expressed as the sum of sinusoidal components of different frequencies as
∞
x (t ) =
∑ ( A cos(nω t ) + B sin(nω t ))
0
n
n
0
(1.1)
n =1
where
ω0 is equal to 2π/T
An and Bn are coefficients called the Fourier coefficients
defined as
An =
2
T
2
Bn =
T
+T / 2
∫
x(t ) cos(nω 0t )dt
(1.2)
x(t )sin(nω 0t )dt
(1.3)
−T / 2
+T / 2
∫
−T / 2
3
Fundamental Concepts
Then the total power is given as
x (t )2 =
1
2
∞
∑ (A + B )
2
n
2
n
(1.4)
n =1
This expression implies that the power can be expressed as the sum of all the power of the sinusoidal
components of different frequencies.
A similar expression can be given for nonperiodic functions using the Fourier transform. Similar
to Equation 1.1 for a periodic function, the nonperiodic function x(t) can be written as
1
x (t ) =
2π
+∞
∫ X (ω)e
− i ωt
dω
(1.5)
0
where X(ω) is the Fourier transform of x(t) defined as
+∞
X (ω ) =
∫ x(t )e
i ωt
dt
(1.6)
−∞
It can be shown that the power is given by
1
x (t ) =
2π
2
+∞
∫ X (ω ) d ω
2
(1.7)
0
In this equation, the sum in Equation 1.1 is replaced by an integral, and Fourier coefficients are
replaced by a continuous function of the angular frequency. But the function ∙X(ω)∙2 can be understood as a component of power corresponding to the angular frequency ω. This function is called
the power density function, and the power is expressed by the integral of power density over whole
angular frequency range.
1.1.5
SIGNAL-TO-NOISE RATIO
The signal-to-noise ratio is generally defined as the ratio of the value of the signal to that of the
noise, and often denoted simply by S/N (or SNR). Commonly, the ratio of the power of the signal to
that of the noise is used, but the ratio of peak-to-peak values or also root-mean-square amplitudes
of the signal to that of the noise may be used as the signal-to-noise ratio, if it is previously defined.
In practical situations, the signal-to-noise ratio is considered in limited frequency ranges, and its
value is always different for different frequency ranges because the signal and the noise always have
different frequency spectra.
The ratio is often expressed in decibels (dB), which is defined as
10 log10
S
N
(1.8)
when S and N denote the powers of the signal and the noise, respectively, while it is
20 log10
S
N
(1.9)
4
Biomedical Sensors and Instruments
when S and N denote the root-mean-square amplitudes of the signal and the noise, respectively. Due
to these different definitions of signal-to-noise ratio for power and amplitude, the value expressed
in decibels can be the same regardless of whether S and N are expressed in power or in amplitude.
1.1.6
DIFFERENT TYPES OF NOISE
Different types of noise arising from different sources may appear in actual measurement situations
and can be characterized by their power spectra, and can be treated theoretically. Some types of noise
are discussed briefly in the following sections, but more detailed treatment can be found in other textbooks (e.g., van der Ziel 1976). In addition to these types of noise, all unwanted signals superimposed
on the signal are classified as noise, irrespective of whether the sources are identified or not.
1.1.6.1 Thermal Noise
Thermal noise is a kind of noise that is caused by random thermal agitation. The power density
of the thermal noise is uniformly distributed through the whole frequency range, and its power is
proportional to the temperature. For example, across the terminals of a resistor of resistance R, the
noise potential v(t) appears in a frequency range Δf, so that it obeys the relation
v(t )2 = 4kTR ∆f
(1.10)
where
k is the Boltzmann constant (1.38 × 10 −23 J/K)
T is absolute temperature
This relation is known as Nyquist’s theorem.
1.1.6.2 1/f Noise
1/f noise is a kind of noise characterized by its power spectrum so that the power density is inversely
proportional to the frequency in the lower frequency range. The 1/f noise may have different origins.
When a current is passing through a semiconductor device, the 1/f noise is caused by the fluctuation
of carriers in the semiconductor. This type of noise is also called flicker noise. Flicker noise is also
generated in a resistor when a current is flowing through it.
In many unstable quantities, which may change over a long time interval up to days, months, or
even years, it is often observed that the power density of the fluctuation in the quantity is almost
inversely proportional to the frequency, and thus it is considered to be the 1/f noise. The drift is also
considered as a very low frequency component of the 1/f noise.
Fluctuations in which the power density is inversely proportional to the frequency are found in
physiological quantities such as fluctuation in the heart rate, which is considered to reflect physiological activities (Musha et al. 1983). If this measurement is used to study physiological activities,
it cannot be considered as a noise but rather as a signal.
1.1.6.3 Interference
Interference is a kind of noise caused by physical or chemical events outside the object and the measurement system. The interference is sometimes caused by natural phenomenon such as lighting, but
more commonly by artificial sources. The power line often causes interference by electromagnetic
coupling to the object of measurement and for the measurement system. Sources other than electromagnetic equipments can cause interference. For example, fluorescent lamps may cause noise in
optical measurement systems.
The power spectrum of noise due to interference depends on the source. The power line frequency (50/60 Hz) and its harmonics often appear in the power spectrum of the noise, when electric
5
Fundamental Concepts
equipments supplied from the power line are used near the object and for the measurement system.
Electric equipments which utilize pulse or switching operations generate noise in a wider frequency
range. Machines having mechanical moving parts may generate vibration which interferes with
mechanical measurement systems. The power spectrum of the mechanical interference may have
peaks corresponding to mechanical resonance frequencies of the machine itself or materials excited
mechanically.
Details of actual situations of interference and practical techniques for reducing noise can be
found in appropriate textbooks (e.g., Morrison 1991).
1.1.6.4 Artifact
The term artifact usually refers to a component of noise superimposed on the object quantity and
caused by external influences such as movement. The motion artifact often appears in biopotential measurement when using skin surface electrodes. It is considered that this is due in part to
the potential generated by the epidermal layers of the skin and in part to the change in electrode
potentials generated at the interfaces of the electrolyte and the metal. The waveform of the artifact
depends on the nature of the external influence. Sometimes, the motion artifact resembles biopotential signals such as EEG and ECG, and thus it is difficult to remove the noise from the signal by
simple methods such as the use of a band-pass filter.
Artifacts can be reduced by suppressing the process which couples the interference to the object
or the measurement system. For example, motion artifacts at the recording electrodes can be reduced
by the use of nonpolarizing electrodes. While the electrode potential varies due to the variation of
the ionic concentration near the electrode surface, that variation is suppressed in nonpolarizing electrodes by employing adequate dissociation equilibrium. The abrasion or puncture of the epidermal
part of the skin also reduces motion artifacts; as a result the potential developed in the skin is suppressed by short-circuiting the stratum corneum (Tam and Webster 1977, Ödman 1981).
1.2
CHARAcTERIsTIcs OF THE MEAsUREmENT SYsTEm
1.2.1 SENSOR AND MEASUREmENT SYSTEm
In a measurement procedure, the observer obtains information about the object by using an appropriate measurement system. Usually, a measurement system consists of the sensor and the electronic
instrument, as shown in Figure 1.1. The physical or chemical quantity which characterizes the
object is detected by a sensor, and is converted into an electric quantity, which is displayed by using
an appropriate electronic instrument which will transfer obtained results to the observer.
Sometimes, measurements require active procedures to be applied to the object such as excitation, illumination, irradiation, stimulation, application, or injection. Such procedures are considered
a part of the measurement process and are provided by the sensor or another part of the measurement system.
Active procedures such as excitation, transmission,
illumination, irradiation, stimulation, application, or injection
Object
of
measurement
FIGURE 1.1
Input
signal
as
physical
or
chemical
quantity
Sensor
Signal
as
electric
quantity
Electronic
instrument
Measurement system
The general structure of a measurement system.
Output
signal
Observer
6
Biomedical Sensors and Instruments
Although an active procedure is unavoidable in some measurements, its influence on the object
should be minimized for two reasons, i.e., to minimize hazard and to minimize the change in the
object quantity due to the active procedure. On the other hand, a situation often occurs in which
measurement becomes much easier and more accurate if the applied energy or strength of the active
procedure is increased. The level of the active procedure should then be determined as a compromise
between minimized influence on the object and maximized performance of the measurement system.
The sensor is an essential part of the measurement system, because the quality of the measurement system is determined mostly by the sensor used. Signal-to-noise ratio, for example, is always
determined mainly by the sensor, as long as adequately interfaced electronic circuits are employed.
Different types of object quantities require different kinds of sensors. Also different kinds of
sensors are required according to the requirements of different measurement situations, such as
different signal amplitudes and frequency ranges, accuracy requirements, limitations in size, shape
or materials, or invasiveness of the measurement procedure. In biomedical measurements, sensors
designed for other purposes are often unsuitable even if fundamental characteristics such as type
of object quantities, measurement ranges, or frequency responses are acceptable. Actually, most
sensors used in biomedical measurements are designed so that they can be applied to the body with
minimum subsidiary effects, obtaining desired biological information correctly. Most parts of this
book concern such sensors that are applicable to biomedical measurements.
1.2.2 STATIC CHARACTERISTICS
In most measurement systems, the output of the measurement system at every moment can be fully
determined by its input as the object quantity at that moment, if the change of the object quantity
is slow enough. In such a situation, the input–output relationship of the measurement system can
be uniquely determined without depending on the time course. The object quantity and the characteristics that represent the relation between the object quantity and the output of the measurement
system are called the static characteristics. Fundamental features of a measurement system can be
characterized by the static characteristics.
1.2.2.1 Sensitivity, Resolution, and Reproducibility
The term sensitivity is always used in such a way that the sensitivity of a sensor or a measurement
system is high when a small change in object quantity causes a large change in its output. But quantitative definition of sensitivity is not unique. In some fields, the sensitivity is defined as the ratio of
the output to the input. In this definition, the numerical value that represents the sensitivity is large
when the sensitivity is high. In other fields, the sensitivity is defined as the ratio of the input to the
output. This factor corresponds to the amount of change in the object quantity that produces a unit
change in the output. By this definition, the numerical value is small when the sensitivity is high.
The sensitivity has a dimension when the dimension of the object quantity and that of the output
are different. Sensitivities for different object’s quantities are represented in different units such as
mV/kPa, μA/K, mV/pH, etc.
Sensitivity can be a constant value when the change in the output is linearly related to the change
in the object quantity, whereas it cannot be constant when the response is nonlinear. In such a case,
sensitivity depends on the absolute value of the object quantity.
The resolution is the least value of the object quantity that can be distinguished at the output of
the measurement system. A change in the object quantity, which is smaller than the resolution of the
measurement system, will not produce a detectable change in its output which can be distinguished
from noise. The numerical value of the resolution is small when the resolution is high. The resolution has the same dimension as that of the object quantity.
The reproducibility describes how close to one another repeated outputs are when the same
quantity is measured repeatedly. Quantitatively, the reproducibility of a measurement system is
defined as a range in the object quantity so that the results of successive measurements for the same
Fundamental Concepts
7
quantity fall into that range with a given probability. If the probability level is not specified, it is usually understood to be 95%. When the range is narrow, the reproducibility is high. The term repeatability is also used to express the similar concept of reproducibility, but repeatability is understood
as the reproducibility in a short time interval, when these two terms are distinguished.
1.2.2.2 Measurement Range
The measurement range is the total range of the object quantity within which the measurement
system works so as to meet the nominal performance of the measurement system. Thus the measurement range depends on performance requirements such as sensitivity, resolution, or reproducibility. If the requirements are severe, the measurement range will be narrow. Sometimes different
measurement ranges are specified for different requirements. For example, in a thermometer, measurement range is from 30°C to 40°C for reproducibility of 0.1°C, and that from 0°C to 50°C for
reproducibility of 0.5°C.
The measurement range states the maximum allowable change of the object quantity as long as
the nominal performance of the measurement system is expected. On the other hand, the minimum
detectable change of the object quantity is stated by the resolution. The ratio of measurement range
to resolution is called the dynamic range. The dynamic range is a nondimensional value and is
sometimes expressed in decibels (dB).
The dynamic range has to be considered when the signal is converted into a digital quantity. For
example, the number of bits of an analog-to-digital converter and the data format or number of digits of the digital display are determined so that the maximum digital number usable by these devices
is large enough compared with the dynamic range.
1.2.2.3 Linearity or Nonlinearity
The linearity describes how close the input–output relationship of the measurement system is to
an appropriate straight line. Different definitions of linearity are used depending on which straight
line is considered. The straight line defined by the least square fit to the input–output relation can
be used, whereas other straight lines defined by the least square fit positioned to pass either the
origin, or the terminal point, or both, can also be used. When the straight line passing through the
origin is used, the linearity of this specific definition is sometimes called zero-based linearity or
proportionality.
As a quantitative measure of the linearity, the ratio of the maximum deviation from the input–
output relation curve from a straight line is used. However, the term nonlinearity is conventionally
used to indicate this value, because the numerical value when using this definition is large when the
deviation of the input–output relation curve from the straight line is significant.
When the linearity is high (or the nonlinearity is small), the input–output relationship can be
regarded as a straight line, and thus the sensitivity can be regarded as constant. On the other hand,
when the linearity is low (or nonlinearity is large), the sensitivity depends on the input level.
Although higher linearity is desired in most measurement systems, accurate measurement is
possible even if the response is nonlinear as long as the input–output relation is fully determined.
By using a computer, the input can be estimated at every sampling interval from the output and
knowledge of the input–output relation.
1.2.2.4 Hysteresis
Hysteresis is a kind of phenomenon in which different output values appear corresponding to the
same input no matter how slow the speed of change of the input is. If a measurement system has
hysteresis, the input–output relation curve is not unique but depends on the direction and the range
of successive input values.
There are different causes of hysteresis, such as backlash in mechanical coupling parts, viscoelasticity or creep of mechanical elements, magnetization of ferromagnetic materials, or adsorption and desorption of materials on electrochemical devices. While hysteresis due to backlash is
8
Biomedical Sensors and Instruments
independent of the range of variations of the input, hysteresis because of other causes depends on
the variations of the input, so that large variations cause large hysteresis. To minimize hysteresis,
larger inputs appearing at a transient, or by the artifacts beyond the normal variation range of the
signal, should be avoided. The sensor design in which the input is limited within the normal measurement range is advantageous not only to protect sensing elements in the sensor from destruction
but also to reduce hysteresis. For example, the stopper for the diaphragm or beam in mechanical
sensors is employed for this reason.
1.2.3 DYNAmIC CHARACTERISTICS
The dynamic characteristics of a measurement system describe the input–output relation in a transient, whereas the static characteristics describe the relation when the input remains constant or
changes slowly. The dynamic characteristics are required where the response to time-varying inputs
is of concern. The time-varying pattern of the object quantity is observed as a waveform, but true
waveforms will not be reproduced unless its dynamic characteristics are excellent.
The dynamic characteristics are particularly important when the sensor consists of a part of a
control system. Instability or oscillation may occur due to a poor dynamic response of the sensor.
The most common cause that affects the dynamic characteristics of the measurement system is
the presence of elements which store and release energy when the object quantity varies. For example, inertial elements such as masses, capacitances and inductances, and compliant elements such as
springs, electric and heat capacitances are such system components. If the displacements of mechanical parts and fluids cause significant time delays, they will also affect the dynamic characteristics of
the system. Besides the measurement system, the object matter of the measurement and the interfacing media may also affect the dynamic characteristics of the whole measurement process. In such
a situation, the dynamic characteristics should be discussed, including the object or the interfacing
media, as in the case of the catheter-sensor systems in pressure measurements (see Section 2.2.2).
Although some important concepts and terms regarding the dynamic characteristics of the measurement systems are explained briefly in the following sections, we also recommend the study of
other textbooks (e.g., Doebelin 1990, Pallas-Areny and Webster 1991).
1.2.3.1 Linear and Nonlinear Systems
The term linear system, or the expression that a system is linear, always refers to a system in which
the response to simultaneous inputs is the sum of their independent inputs. A system which does not
meet this condition is called a nonlinear system.
In linear systems, the dynamic characteristics are the same regardless of the input’s amplitude. The
amplitude of the response is simply proportional to the input amplitude, because a large input can be
regarded as a sum of smaller inputs, and hence the response corresponding to the large input is the
sum of the responses corresponding to the small inputs. This property is important because many convenient parameters, which characterize the system, can be defined regardless of the signal amplitude.
Real systems cannot always be linear when the input increases far beyond the measurement
range. On the other hand, most measurement systems can be regarded as linear if the variations of
the input are small. Even in a nonlinear system, an appropriate linear system can be assumed as a
result of the approximation in a small measurement range.
In a linear system, the response to a sinusoidal input is also sinusoidal with the same frequency
as that of the input, while other frequency components such as higher harmonics may appear in a
nonlinear system.
1.2.3.2 Frequency Response
The frequency response refers to the distribution of the amplitude and the phase shift of the output
to sinusoidal inputs of unit amplitude over the whole frequency range in which the dynamic characteristics are considered. Usually, the frequency response is defined only for linear systems.
9
Fundamental Concepts
The output of a linear system can be described as the sum of the responses corresponding to
sinusoidal inputs having different frequencies, because the input is expressed as the sum of sinusoidal functions such as Equation 1.1 or 1.5. Therefore, the frequency response provides complete
information about the output of the system for any input.
When the input–output relation of a system is described by a constant-coefficient first-order
differential equation, the system is called a first-order system. The differential equation which
describes a first-order system is written as
a1
dy(t )
+ a0 y(t ) = x(t )
dt
(1.11)
where
x(t) and y(t) are the input and the output of the system
a 0 and a1 are constants
Then the frequency response of this system can be represented as shown in Figure 1.2a and b, where
fc is given as a 0/2πa1 and called the cut-off frequency.
The second-order system is a system which can be described by a second-order constant-coefficient
differential equation written as
a2
d 2 y(t )
dy(t )
+ a1
+ a0 y(t ) = x(t )
2
dt
dt
(1.12)
where a 0, a1, and a2 are constants. The frequency response of this system can be represented as
shown in Figure 1.2c and d, where
1
2π
0
–5
–10
–15
–20
–25
(a)
0
1
f/fc
10
Relative amplitude (dB)
Relative amplitude (dB)
f0 =
20
h = 0.5
10
–1
0
1
f/fc
10
–10
h = 1.0
–20
–30
–40
0
1
10
f/fc
h = 0.2
h = 0.5
–1
h = 1.0
–2
–3
(d)
h = 0.2
0
0
Phase shift (rad)
Phase shift (rad)
(b)
(1.13)
(c)
0
–2
a0
a2
0
1
f/fc
10
FIGURE 1.2 Frequency responses of first- and second-order linear systems. Amplitude and phase responses
of the first-order system (a) and (b), and those of the second-order system (c) and (d).
10
Biomedical Sensors and Instruments
2
0.8
0.6
0.4
0.2
0
(a)
Normalized response
Normalized response
1.0
0
FIGURE 1.3
1
2
t/τ
3
4
h = 0.2
h = 0.5
1
h = 1.0
0
(b)
0
4
8
2πf0t
12
16
Responses to a step input in the first-order system (a) and that of the second-order system (b).
which is called the natural frequency, and
h=
a1
2 a0 a2
(1.14)
which is called the damping coefficient.
1.2.3.3 Time Constant, Response Time, Rise Time, and Settling Time
When the input of a system changes abruptly from one level to another, the behavior of the output
can be characterized by some specific parameters according to the type of the system. Such parameters can be determined by the response to a unit step input in which the input is zero before a
specific time, and unity after that time.
The time constant is defined in the first-order system. As shown in Figure 1.3a, the response of a
first-order system to a unit step input is a process approaching exponentially to the final value, and
the time constant τ is defined as the time required for the output to reach to 1 – 1/e ≈ 0.673 of the
final value, and is given as a1/a 0 for the system represented by the Equation 1.11.
In the second-order system, the response to a unit step input varies with the damping coefficient
as shown in Figure 1.3b. Some parameters are used to express how quickly the system can follow
the input. The response time is usually defined as the time needed to reach 90% of the final value,
and the rise time is usually defined as the time that the output changes from 10% to 90% of the final
value to a unit step input. The settling time is defined as the time required for the output to settle
within a definite range near the final value, for example, the range is defined as ±5% of the final
value to a unit step input.
1.3 DETERmINATION OF ABsOLUTE QUANTITY
Usually, measurements are performed to determine the absolute values of the object quantities in
physical or chemical units, while only relative values are required in some situations. The measurement of absolute quantities requires calibration of the measurement system unless the measurement
system has the standard in itself. The accuracy of the measurement is assessed by the amount of
error which is the deviation of the measured value from the true value. There are many sources of
error, such as the use of unreliable standards, inadequate calibration procedure, noise contamination of the signal, poor static and dynamic characteristics of the measurement system, and unsuitable data processing. Errors are sometimes classified according to their nature such as random,
systematic, quantization, and dynamic errors.
Fundamental Concepts
1.3.1
11
STANDARD AND CALIBRATION
Any measurement system can be calibrated by comparing with either an intrinsic standard or a reliable standard instrument. There are some convenient standards for different quantities. For example,
a mercury column can be a standard of pressure. Because the density of pure mercury is known, and
the gravity of the earth is also known, the pressure developed at the bottom of a mercury column
of a definite height can be a standard of pressure. Ice point of pure water, which is 0°C, or melting
point of gallium, which is 29.771°C, are used as intrinsic temperature standards.
An instrument which is stable enough and correctly calibrated can also be a standard, so that
other measurement systems can be calibrated comparing with the standard instrument. For example, commercial standard thermometers having crystal-resonator temperature sensors can be used
to determine temperatures absolutely within a deviation of 0.01°C, which is precise enough for most
biomedical measurements.
When the measurement system to be calibrated is nonlinear, calibration has to be performed at
many points in the measurement range. On the contrary, when the measurement system is linear,
calibration at two points is enough. Even if slight nonlinearity exists in the whole measurement
range of the system, as long as the variation of the object quantity is limited to a narrow range and
the system can be regarded as linear in that limited range, two-point calibration in that range is
enough for all practical purposes. In a case where the measurement system is linear and its sensitivity is stable but drift remains to some extent, occasional one-point calibration after initial two-point
calibration is recommended.
Even if a measurement system is nonlinear but is stable enough and its input–output relation can be
approximated by a simple formula having few parameters, what is required in calibration is to determine all parameters in the formula. For example, if the input–output relation is well approximated by
a quadratic formula, the number of parameters to be determined is three, and thus the fitting curve
can be determined by three-point calibration. Such a curve-fitting computation can be performed easily even in real time when the output of the measurement system is connected to a computer.
1.3.2
ACCURACY AND ERROR
The term accuracy describes how close the measured value is to the true value. The difference
between them is termed the error.
The error may depend on the level of the object quantity, especially when the measurement range
includes small to large values of the object quantity. The error may be small when the object quantity is small, while it may be large when the object quantity is large. In such a situation, the relative
error, which is defined as the ratio of the error to the true value, can be a convenient figure of the
performance of the measurement system.
When the measurement system is calibrated adequately, the error will be reduced to a limit
which is determined by the reproducibility. Sometimes, the accuracy of the measurement system
becomes poor due to the drift in a long period of time even though the reproducibility in a short
period of time has not changed very much. In such a situation, the accuracy can be recovered to the
initial level by a recalibration. By repeated calibration procedures, the accuracy can be maintained
within a definite range for a long time period.
1.3.3
TYPES OF ERROR
In actual measurement situations, there are always many sources of errors. The errors arising from
different sources are classified into different types. The nature of typical types of error is discussed
briefly in the following sections, but more detailed explanations can be found in standard textbooks
(e.g., Sydenham 1999).
12
Biomedical Sensors and Instruments
1.3.3.1 Random Error
The random error is a kind of error which appears unpredictably in repeated measurements.
Random noise superimposing on the signal and short-term fluctuations of the measurement system
may cause random errors. Deviations of the measured values are distributed on the positive and
negative sides of the true value with equal probabilities, and the mean of the deviations is zero.
Thus, if the measurements are repeated while the object quantity is unchanged, the average of measured values approaches the true value.
The statistical property of random errors can be determined from the distribution of measurement values when the same object quantity is measured repeatedly. In practice, it is often assumed
that random errors are distributed according to a normal distribution. That assumption is secured
theoretically when the errors are considered to be the result of a large number of small uncorrelated
component errors. According to the central-limit theorem, it is expected that the sum of many
uncorrelated random values forms a normal distribution.
For the normal distribution, the standard deviation of measurement values of n measurements,
which is the standard error, is reduced inversely proportional to the square root of n. Consequently,
repeated measurement combined with signal averaging is an effective way to reduce random
errors.
1.3.3.2 Systematic Error
The systematic error is the bias from the true value appearing equally in repeated measurements of
the same object quantity. Systematic errors have different origins, such as very low frequency components of noise contaminating the signal, drift of the measurement system, inadequate calibration,
uncorrected nonlinearity, or round down in digital data processing.
When the same object quantity is measured by two different measurement systems, the difference between averages of measurement values of these two systems indicates the presence of a
systematic error.
Systematic errors cannot be eliminated by repeated measurements and averaging. It is also difficult to identify all causes of systematic errors. One practical way of reducing systematic errors
is recalibration of the measurement system under the measurement conditions over the required
measurement range. As long as recalibrations are performed repeatedly at adequate intervals, the
systematic errors can be significantly reduced for a long period of time even though the drift of the
measurement system exists to some extent.
Sometimes, when a systematic error is suspected due to the influence of the environment such
as climatic change, two identical measurement systems are used, so that the object and a standard
are measured simultaneously, and the measurement value is obtained from the difference of their
outputs. If the systematic errors in both outputs are the same, the error will be cancelled out in the
difference between these two outputs, and then the object quantity can be measured correctly as the
deviation from the standard value.
1.3.3.3 Quantization Error
The quantization error is an error caused by conversion of an analog value to a corresponding
digital value, as in an analog-to-digital converter. The error is the difference between the original
analog value and the converted digital value.
When an analog value is converted into a digital value, an ambiguity corresponding to 1 in the
least-significant bit remains. The fraction of error due to this ambiguity depends on the level of the
measurement value. If measurement values always remain at a higher level so that 1 in the leastsignificant bit is negligible, quantization error is insignificant. On the other hand, if measurement
values stay near the least-significant bit, quantization error will be serious even though a precision
analog-to-digital converter is used. To reduce quantization error, it is important to adjust the input
level so as to fit the input range of the analog-to-digital converter.
Fundamental Concepts
13
1.3.3.4 Dynamic Error
The dynamic error is a kind of error occurring from imperfect dynamic characteristics of the measurement system. It occurs when the object quantity varies so quickly that the output of the measurement
system does not follow the change of the input. The difference between the output value and the object
quantity is the dynamic error. When the dynamic error exists, the instantaneous output of the measurement system cannot reflect the input at that moment, and depends on the past history of the input.
The possible dynamic error of a system is estimated from the response corresponding to a stepwise input. As discussed in Section 1.2.3.2 and shown in Figure 1.3, the response curve for a stepwise
input can be determined by a few parameters in simple systems, such as the first- or second-order
linear systems, and the dynamic error for a stepwise input is the deviation of the response curve from
the final steady-state value. In higher order systems, some parameters, such as the settling time, can
still be defined in the response curve corresponding to a stepwise input. It can be expected that the
dynamic error will remain in a definite range if a time longer than the settling time passes after an
abrupt change in the object quantity.
1.4 UNITs OF MEAsUREmENT QUANTITIEs
Quantitative measurement results are always represented in numerical form using some unit so that
anyone can compare them to other measurement results. While any unit can be used in each measurement as long as it is well defined, inconveniences will arise when different units for the same
quantity are used in different measurement results even though unit conversion is possible. Even
if the same unit is used for each quantity, if the units of interrelating quantities are defined independently, numerical factors may appear in each equation representing the relation between these
quantities. For example, consider the heat produced by mechanical work. If the units of mechanical work and heat are defined independently, the developed heat cannot be obtained immediately
from the mechanical work, but a numerical coefficient has to be used for unit conversion. In order
to eliminate such inconvenience, a coherent system of units is used in which each unit is defined
systematically so that the numerical coefficient becomes unnecessary. While the coherent system is
not unique as different coherent systems of units such as CGS and MKS systems existed, it is now
recommended to use the International System of Units (SI).
1.4.1
THE INTERNATIONAL SYSTEm OF UNITS
The International System of Units (SI) was established by the International Organization of
Standard (ISO). SI is a coherent system of units of physical quantities. The International Standard
of quantities and units includes not only the definitions of quantities and units but also the rules
for writing them such as the symbols to be used to express quantities and units. Information about
the International Standard of quantities and units is available in the documents published by ISO
referred to as ISO 1000 (International Standard ISO 1000 1992) and ISO 31—Part 0 to Part 13
(International Standard ISO 31 1992).
1.4.1.1 Base Units and Derived Units
The SI is based on the seven base quantities and their units, called base units, which are length in
meter, mass in kilogram, time in second, electric current in ampere, thermodynamic temperature
in kelvin, amount of substance in mol, and luminous intensity in candela, as listed in Table 1.1. The
base units are defined rigorously as follows (Units of Measurements 1982):
• The meter is the length equal to 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton-86 atom.
• The kilogram is equal to the mass of the international prototype of the kilogram.
14
Biomedical Sensors and Instruments
TABLE 1.1
SI Base Units
SI Base Unit
Base Quantity
length
mass
time
electric current
thermodynamic temperature
amount of substance
luminous intensity
Name
Symbol
meter
kilogram
second
ampere
kelvin
mole
candela
m
kg
s
A
K
mol
cd
• The second is the duration of 9,192,631,770 periods of the radiation corresponding to the
transition between two hyperfine levels of the ground state of the caesium-133 atom.
• The ampere is that constant current which, if two straight parallel conductors of infinite
length, of negligible cross section, and placed 1 m apart in vacuum, would produce between
these conductors a force equal to 2 × 10 −7 newton per meter of length.
• The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point
of water.
• The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12.
• The candela is the luminous intensity, in a given direction of a source which emits monochromatic radiation of frequency 540 × 1012 Hz and of which radiant intensity in that
direction is 1/683 W/sr.
Derived quantities are quantities expressed algebraically in terms of base quantities, and their units,
called derived units, are expressed in terms of base units. For example, velocity is expressed as
length divided by time, and the unit of velocity is meter per second.
For some derived quantities, special names and symbols are given in SI, as listed in Table 1.2.
Radian and steradian are called supplementary units which are dimensionless. Sometimes derived
units are expressed by other derived units having special names. For example, moment of force is
usually expressed as N · m instead of kg m2/s2.
1.4.1.2 Dimension of a Quantity
If a quantity Q is expressed as a product of powers of base quantities a, b, c,…, then Q can be
­ ritten as
w
Q = xa α bβc γ ...
(1.15)
where x is a numerical factor. The dimension of the quantity Q is then expressed as
dim Q = Aα BβC γ ...
where
A, B, C, … denote the dimensions of base quantities a, b, c,…
α, β, γ are called the dimensional exponents
(1.16)
15
Fundamental Concepts
TABLE 1.2
SI-Derived Units with Special Names
SI-Derived Unit
Special Name
plane angle
solid angle
frequency
force
pressure, stress
energy, work quantity of heat
power, radiant flux
electric charge, quantity of electricity
electric potential, potential difference,
tension, electromotive force
capacitance
electric resistance
electric conductance
magnetic flux
magnetic flux density
inductance
Celsius temperature
radian
steradian
hertz
newton
pascal
joule
watt
coulomb
volt
rad
sr
Hz
N
Pa
J
W
C
V
1 rad = 1 m/m = 1
1 sr = 1 m2/m2 = 1
1 Hz = 1/s
1 N = 1 kg m/s2
1 Pa = 1 N/m2
1 J = 1N·m
1 W = 1 J/s
1C=1As
1 V = 1 W/A
farad
ohm
siemens
weber
tesla
henry
degree Celsius
F
Ω
S
Wb
T
H
°C
luminous flux
illuminance
activity (of a radio nuclide)
absorbed dose, specific energy
imparted, kerma, absorbed dose index
dose equivalent, dose equivalent index
lumen
lux
becquerela
graya
lm
lx
Bq
Gy
1 F = 1 C/V
1 Ω = 1 V/A
1 S = 1/Ω
1 Wb = 1 V s
1 T = 1 Wb/m2
1 H = 1 Wb/A
1°C = 1 K (in temperature
difference)
1 lm = 1 cd sr
1 lx = a l m/m2
1 Bq = 1/s
1 Gy = 1 J/kg
sieverta
Sv
1 Sv = 1 J/kg
a
Symbol
Expressed in Terms of SI Base
Units and SI-Derived Units
Derived Quantity
SI-derived units with special names admitted for reasons of safeguarding human health.
The dimensions of the seven base units, which are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, are conventionally denoted by
L, M, T, I, Θ, N, and J, respectively, and the dimension of a quantity Q becomes in general
dim Q = Lα Mβ T γ I δΘ ε N ς J η.
(1.17)
Some examples of quantities, their units, and dimensions are shown in Table 1.3.
While the unit of a quantity, such as N, J, and V, is not unique in some derived units in Table 1.3
the dimension is uniquely determined. Therefore, it is helpful to compare the dimensions of each
expression for finding errors in the derivation of a mathematical expression of a quantity.
1.4.1.3 Recommendations for the Use of SI Units and Symbols
In order to express quantities and units correctly and avoid confusion, the ISO provides recommendations for the use of SI units and symbols.
The symbol of the unit shall be placed after the numerical value in the expression for a quantity,
leaving a space between the numerical value and the unit symbol. The exceptions to this rule are
for the units of degree, minute, and second for plain angles, in which case there shall be no space
between the numerical value and the unit symbol.
16
Biomedical Sensors and Instruments
TABLE 1.3
Some Examples of Quantities, Units,
and Their Dimension
Quantity
velocity
force
energy
electric potential
heat capacity
molar concentration
illuminance
Unit
Dimension
m/s
kg m/s2 or N
kg m2/s2 or J
kg m2/(s2 A) or V
kg m2/(s2 K) or J/K
mol/m3
cd sr/m2 or lx
LT−1
LMT−2
L2MT−2
L2MT−3I−1
L2MT−2Θ−1
L−3N
L−2J
The symbols for quantities are generally single letters of the Latin or Greek alphabet, sometimes
with subscripts or modifying signs. The symbols for quantities are printed in italic (sloping) type,
while the symbols for units shall be printed in roman (upright) type. For example,
v = 1.2 m/s
A subscript that represents a symbol for a physical quantity is printed in italic type, while other
subscripts are printed in roman type. For example,
C p ( p: pressure),
T1/ 2 (1/2: half)
Numbers should generally be printed in roman type. To print a number
with many digits, put a small space at every three digits, and not a comma
or point.
As the decimal sign, a comma on the line is used in ISO documents.
A dot on the line is often used instead of a comma.
Product of quantities may be indicated in one of the following ways:
ab, a ⋅ b, a × b
The sign for multiplication of numbers, a cross (×) or a half-high dot (·)
can be used. But if a half-high dot is used as the multiplication sign for
numbers, a comma should be used as the decimal sign.
Division of one quantity by another may be indicated in one of the
following ways:
a
, a /b, ab −1, a ⋅ b −1, a × b −1
b
Any quantity, whatever large or small, can be expressed by a numerical value of significant digits with powers of 10. The prefixes given in
Table 1.4 can be used instead of powers of 10, such as nm instead of 10 −9
m, or GHz instead of 109 Hz.
The symbol of a prefix is considered to be combined with the single
unit symbol to which it is directly attached, forming with it a new symbol
which can be raised to a positive or negative power. For example,
1 µs −1 = (10 −6 s)−1 = 106 s −1
TABLE 1.4
SI Prefixes
Prefix
Factor
Name
Symbol
24
10
1021
1018
1015
yotta
zetta
exa
peta
Y
Z
E
P
1012
109
106
103
tera
giga
mega
kilo
T
G
M
k
102
10
10−1
10−2
hecto
deca
deci
centi
h
da
d
c
10−3
10−6
10−9
10−12
milli
micro
nano
pico
m
μ
n
p
10−15
10−18
10−21
10−24
femto
atto
zepto
yocto
f
a
z
y
Fundamental Concepts
17
Symbols for chemical elements shall be written in roman (upright) type. The nucleon number (mass
number) of a nuclide is shown in the left superscript position, for example, 14N. The number of atoms
of a nuclide in a molecule is shown in the right subscript position, for example, 14N2. The proton
number (atomic number) may be indicated in the left subscript position, for example, 64Gd. A state
of ionization may be indicated in the right superscript position, for example, Na+, PO43−.
1.4.2
NON-SI UNITS
There are certain units which are outside the SI but the ISO International Committee for Weights
and Measurements has recognized them to be retained because of their practical importance. These
are as follows: For time, minute (min, 1 min = 60 s), hour (h, 1 h = 60 min), and day (d, 1 d = 24 h),
for plane angle, degree (°, 1° = (π/180) rad), minute (′, 1′ = (1/60)°), and second (″, 1″ = (1/60)′), for
volume, liter (l or L, 1 l = 1 dm3), for mass, tonne (t, 1 t = 103 kg) and unified atomic mass unit (u, 1 u ≈
1.660 540 × 10 −27 kg), for energy, electron volt (eV, 1 eV ≈ 1.602 177 × 10 −19 J).
Many units outside the SI have been used traditionally in many documents and papers in different fields. In medicine, for example, mmHg and cmH2O are still commonly used for pressure
instead of Pa, and cal for metabolic energy instead of J. Because immediate conversion into SI units
will cause great inconvenience, it may be recommended to show values in SI and traditional unit,
such as 16 kPa (≈120 mmHg), or simply 16 kPa (120 mmHg) when it is obvious that the converted
value is an approximation. In this book, this simple style is employed.
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