Biomedical Instrumentation

Transducers and sensors

A transducer converts a physical quantity such as displacement, temperature, pressure or flow into an electrical signal, and the sensing element that performs the first step of this conversion is called the sensor. For displacement, the linear variable differential transformer (LVDT) is an inductive sensor whose secondary output varies linearly with core position. It combines high sensitivity with a comparatively large linear measurement range, which is why it is preferred over strain gauges and capacitive sensors when displacements of millimetres to centimetres must be measured, since those two are sensitive but restricted to very small displacements.

For body temperature, contact-type digital thermometers normally use a negative temperature coefficient (NTC) thermistor. Its resistance falls sharply and nonlinearly as temperature rises, which gives high sensitivity over the narrow clinical range and a fast response in a small probe. Thermocouples give small voltages that need reference junctions and amplification, while resistance temperature detectors are stable but less sensitive and slower, and infrared detectors are non-contact devices.

In a Hall effect flow meter, a voltage V = BvD is induced across the vessel perpendicular to both the magnetic field B and the flow, where v is the mean velocity and D the diameter. The volumetric flow Q = \frac{\pi D^2}{4}v = \frac{\pi D V}{4B} is therefore proportional to the product of the diameter and the measured voltage, so an error in either quantity propagates directly into Q.

Piezoelectric transducers convert mechanical pressure or vibration into an electrical charge and the reverse, so they are the usual elements in ultrasonography, echocardiography and phonocardiography. Electroencephalography instead records ionic potentials through metal scalp electrodes and involves no piezoelectric transduction.

Measurement accuracy and precision

Accuracy is the closeness of the mean of repeated readings to the true value and is governed by systematic error or bias, whereas precision is the repeatability of the readings and is measured by their spread, such as the standard deviation. An instrument can be precise but inaccurate if all its readings cluster tightly away from the true value, and an accurate instrument can be imprecise if its mean is correct but its readings scatter widely. Comparing the mean error with the spread of readings from two instruments therefore identifies which one is more accurate and which one is more precise.

Percentage error is defined as \frac{|\text{measured} - \text{reference}|}{\text{reference}} \times 100, so the reference (true) value must always be the denominator, not the measured value.

For a quantity computed as a quotient, such as impedance Z = V/I, the maximum relative error is the sum of the relative errors of the numerator and denominator in the worst case. Thus two instruments each with 1 % uncertainty give a maximum 2 % error in Z, and the absolute error follows by multiplying this fraction by the computed value.

Biopotential amplifiers

Surface EMG amplifiers and ECG amplifiers handle signals of similar amplitude, so their required gains are of the same order, typically around 10^3. The difference lies in bandwidth: the useful surface EMG spectrum extends to several hundred hertz, well beyond the ECG band of roughly 0.05 to 150 Hz, so an EMG amplifier needs a larger bandwidth at a comparable gain.

The common mode rejection ratio is CMRR = A_d / A_{cm}, usually expressed in decibels as 20\log_{10}(A_d/A_{cm}). The common-mode gain is the output common-mode noise divided by the input common-mode noise, so a differential gain of 2000 with a common-mode gain of 0.2 gives CMRR = 10^4, that is 80 dB. The signal amplitude itself does not enter this calculation.

In an instrumentation amplifier, the input stage uses matched gain-setting resistors. If one resistor is slightly mismatched, the differential gain changes by a small fraction, which is a gain error, and a nonzero common-mode gain appears, which lowers the CMRR. With ideal op-amps and zero input the output is still zero, so a mismatch of this kind does not create an offset error.

Strain gauge Wheatstone bridge

A full Wheatstone bridge with four active gauges, arranged so that opposite arms are in tension and the other opposite pair is in compression, gives the largest output. Each gauge changes by \Delta R/R = G\varepsilon, and the bridge output is V_o = V_s G \varepsilon, where V_s is the excitation voltage, G the gauge factor and \varepsilon the strain. With Hooke's law \varepsilon = \sigma/E, the output per unit stress is V_s G/E, and the nominal gauge resistance cancels out.

Each half of the bridge is a voltage divider across the excitation, and the output is the difference between the two divider voltages. When two arms increase and the other two decrease by the same fraction, the output equals the excitation times the fractional resistance change, V_{out} = V_{in}\,\Delta R/R. For example, a 10 % change in each arm with 10 V excitation gives 1 V.

In a quarter bridge with only one active arm, the output is approximately V_{OUT} = V\,\frac{\Delta R}{4R} for small changes, and the exact expression is V_{OUT} = V\frac{\Delta R}{2(2R+\Delta R)}. The bridge is balanced at zero strain, and the strain is recovered by dividing the fractional resistance change by the gauge factor.

Pulse oximetry principle

A pulse oximeter determines arterial oxygen saturation non-invasively by measuring the light transmitted through a fingertip or earlobe. The pulsatile (AC) component of the transmitted light, which is strongest during systole when arterial blood volume is largest, is separated from the steady (DC) absorbance of tissue, skin and venous blood. Only the arterial pulsatile signal is used to compute saturation.

Oxyhemoglobin (HbO_2) and deoxyhemoglobin (Hb) have different absorption spectra. At red wavelengths near 600 to 660 nm deoxyhemoglobin absorbs more strongly, whereas in the near-infrared near 900 to 940 nm oxyhemoglobin absorbs more strongly. The ratio of the pulsatile absorbances at a red and an infrared wavelength is calibrated against saturation, and this contrast is the basis of the measurement.

The isosbestic wavelength, near 805 nm, is the point where Hb and HbO_2 have equal absorbance, so it is independent of oxygenation and is used as a reference. The conventional oximetry pair is red light near 660 nm and infrared light near 905 or 940 nm, and a pair such as 850 nm and 950 nm is not the standard choice.

Einthoven's triangle lead vectors

In Einthoven's triangle, lead I, lead II and lead III are bipolar leads whose axes form a triangle, and the cardiac dipole is projected onto each axis. The signal recorded in a lead equals the projection of the dipole vector on that lead axis, so its magnitude is |\vec{p}|\cos\theta, where \theta is the angle between the dipole and the lead axis. A dipole perpendicular to a lead axis produces zero voltage in that lead.

For a dipole that is oriented at a small angle to the lead II axis, the lead II voltage is close to the full dipole magnitude, because \cos\theta is near one for small \theta. Adding the angle between the lead axis and the dipole is the key step, since the axis direction of each lead is fixed by the triangle and the sign convention for angles must be applied consistently.

Blood pressure measurement techniques

Sphygmomanometry measures arterial blood pressure with an inflatable cuff. During gradual deflation, Korotkoff sounds are heard through a stethoscope over the brachial artery; their onset marks systolic pressure and their disappearance marks diastolic pressure. Other techniques, such as audiometry, spirometry and tonometry, measure hearing, lung volumes or intraocular pressure respectively and do not use these arterial sounds.

In a catheter-transducer system, the liquid resistance of the catheter arises from viscous friction in laminar flow, and it follows the Poiseuille relation R_L = \frac{8\eta L}{\pi r^4}, where \eta is the viscosity, L the length and r the radius. For a fixed pressure difference, R_L depends on the radius to the fourth power, so doubling the radius reduces the resistance by 2^4 = 16 times.

Thermodilution cardiac output

In thermodilution, a known volume of cold injectate is introduced into the right atrium and the resulting temperature change in the pulmonary artery is recorded. The heat carried by the injectate is Q = V_i \rho_i c_i \Delta T_i, and cardiac output is F = Q/(\rho_b c_b \int \Delta T_b\,dt). When the density and specific heat of blood are taken equal to those of the injectate, the properties cancel and F = V_i\Delta T_i / \int \Delta T_b\,dt.

Because F is proportional to the injectate volume and inversely proportional to the integral of the blood temperature change, an underestimated injectate volume or an overestimated magnitude of the integral both lead to a cardiac output that is too low. Errors of this kind propagate linearly through the formula, so each quantity must be measured accurately.

Pacemaker battery life estimation

The lifetime of a pacemaker battery is estimated from its charge capacity in ampere-hours and the average current it delivers. Because the stimulus is a short pulse repeated at a fixed interval, the average current is the pulse current multiplied by the duty cycle, that is the pulse width divided by the period. The pulse current itself follows from Ohm's law applied to the stimulus voltage and the electrode-tissue resistance.

The lifetime is then the capacity divided by the average current, and the result is converted from hours to years using 8760 hours per year. Because the duty cycle is very small, the average current is far lower than the pulse current, which is why a small battery can last for many years.

Medical instrument electrical safety

Let-go current is the maximum current at which a person grasping an energized conductor can still release it voluntarily, and it is a standard threshold in electrical hazard assessment. Microshock is a hazard in which a very small current, often in the microampere range, passes directly through the heart and induces ventricular fibrillation. It occurs when an electrode or catheter is placed in contact with the myocardium or inside the vascular system near the heart, because such internal conductors bypass the high skin resistance. Patients in intensive care units with internal conductive electrodes are therefore exposed to this risk.

For a direct cardiac path, the safe 50 Hz current limit is on the order of 10 \muA, which is far below the milliampere-level limits that apply to surface contact. Confusing these two limits is a common error in safety questions.

Voltmeter range extension

A moving-coil meter gives full-scale deflection at a fixed current I_{fs}, which is found from the original range and the internal resistance as I_{fs} = V_{range}/R_m. To extend the range to a higher full-scale voltage V_{new} with the same deflection, the total circuit resistance must be R_{total} = V_{new}/I_{fs}.

The series multiplier resistor is then the difference between this total resistance and the internal resistance, R_s = V_{new}/I_{fs} - R_m. The meter's internal resistance must be subtracted, not ignored, because it forms part of the total circuit resistance.

Measurement noise averaging

For band-limited white noise with one-sided bandwidth B, samples taken at intervals of 1/(2B) are uncorrelated, so an averaging time T contains N = 2BT independent samples. Averaging reduces the standard deviation of the residual noise by \sqrt{N}, so the residual becomes \sigma/\sqrt{2BT}, where \sigma is the RMS noise.