Workbook of Medical Devices, Engineering and Technology

Basic Concepts and Applications in Medical Physics, Engineering and Science

The intention of this workbook is to provide exercises and corresponding solutions to several subjects in medical technology and engineering. Thereby the reader can learn how to solve problems in this field on the basis of mathematical formulas and calculations. In order to provide a better understanding, the physical background for the solutions is shortly explained. If the reader likes to customise the solutions using other input parameters, he can download the Mathcad program code on the web site __https://drive.google.com/file/d/1VgAdHPLLUjEJY6SD2BqXQ7BD9OyJBASƯview?usp=sharing __and the needed free Mathcad-Prime software on__ https://www.ptcde.com/software-iuer-konstruktionsberechnungen/mathcad/free-download.__

For the advancement of medical devices a thorough understanding of the physical principles of operation is needed together with a mathematical formulation thereof. On the basis of this formulation it is possible to predict and optimize the performance of these devices in the medical application.

In this workbook it is shown by exercises how the outcome is related to the physical processes and the corresponding parameters. This is done for a variety of physical methods that are applied in different medical devices. Of course no completeness can be achieved regarding all medical devices that are in use at present.

Prof. Dr. Dr. habil Michael Thoms was born in 1963 in Eckernförde, Germany, studied physics and received his Ph. D. at the University of Heidelberg in 1991. In 1998 he habilitated in material science at the University of Erlangen and is at present Professor at both the University of Ansbach and the University of ErlangenNürnberg, giving lectures in physics and medical technology.

Besides his academic career, he worked in industry on the research and development of medical devices. For a long time he was in charge as a director of research and development. He did several inventions mainly in the field of medical diagnostics and medical devices and holds several patents.

1. X-RAYS

1.1 Attenuation of X-rays

1.1.1 Exercise: X-ray transmission of lead

1.1.2 Solution

1.1.3 Exercise: X-ray attenuation of silver bromide

1.1.4 Solution

1.1.5 Exercise: X-ray absorption of film and intensifying screen

1.1.6 Solution

1.1.7 Exercise: X-ray transmission of bone

1.1.8 Solution

1.1.9 Exercise: X-ray contrast of muscle and adipose

1.1.10 Solution

1.1.11 Exercise: K-absorption edge of Calcium

1.1.12 Solution

1.2 X-ray tubes

1.2.1 Exercise: X-ray spectrum of an X-ray tube

1.2.2 Solution

1.2.3 Exercise: Relation of high voltage setting and dose

1.2.4 Solution

1.2.5 Exercise: Relation of distance and exposure time

1.2.6 Solution

1.2.7 Exercise: Characteristic K*a*-radiation of molybdenum

1.2.8 Solution

1.2.9 Exercise: Characteristic K*a*-radiation radiation of tungsten

1.2.10 Solution

1.3 X-ray scattering

1.3.1 Exercise: Energy of Compton-scattered X-ray radiation

1.3.2 Solution

1.3.3 Exercise: Cross sections of photoelectric absorption and Compton scattering of water

1.3.4 Solution

1.3.5 Exercise: Cross sections of photoelectric absorption and Compton scattering of Calcium

1.3.6 Solution

1.4 X-ray dosimetry

1.4.1 Exercise: Number of X-rays per area in a chest radiograph

1.4.3 Exercise: Relation of the thickness of X-ray shielding and X-ray energy

1.4.4 Solution

1.5 X-ray statistics

1.5.1 Exercise: Statistical X-ray noise in a chest radiograph

1.5.2 Solution

1.5.3 Exercise: Image noise of an integrating Germanium detector

1.5.4 Solution

1.5.5 Exercise: Relation of the error of the estimated path integral of the attenuation coefficient on the number of irradiated and transmitted X-rays

1.5.6 Solution

1.5.7 Exercise: The path integral of X-ray absorption coefficient µ and the error thereof due to the statistics of irradiated and transmitted X-rays

1.5.8 Solution

1.5.9 Exercise: Signal, signal noise, and signal to noise ratio in a computed tomography system

1.5.10 Solution

1.5.11 Exercise: Signal, noise, signal to noise ratio and DQE of a CCD-based X-ray sensor

1.5.12 Solution

1.5.13 Exercise: Signal to noise ratio of an integrating X-ray detector in the case of a continuous energy spectrum

1.5.14 Solution

1.5.15 Exercise: Signal to noise ratio of the integrated X-ray signal of an X-ray source with continuous X-ray spectrum

1.5.16 Solution

1.5.17 Exercise: Probability to absorb a specific number of X-ray quanta

1.5.18 Solution

1.5.19 Exercise: Probability to absorb a specific number of X-ray quanta for given number of irradiated quanta

1.5.20 Solution

1.5.21 Exercise: Standard deviation of the number of absorbed X-ray quanta

1.5.22 Solution

2. ULTRASOUND

2.1 Ultrasound Waves

2.1.1 Exercise: Wavelength of sinusoidal ultrasound waves

2.1.3 Exercise: Reflected intensity at an interface

2.1.4 Solution

2.1.5 Exercise: Reflected intensity at an interface of muscle and bone

2.1.6 Solution

2.1.7 Exercise: Change of direction of a sound wave traversing an interface

2.1.8 Solution

2.1.9 Exercise: Transversal deflection of an ultrasound beam

2.1.10 Solution

2.1.11 Exercise: Displayed size of tissues in ultrasound images

2.1.12 Solution

2.1.13 Exercise: Frequency shift in Doppler mode

2.1.14 Solution

2.2 Ultrasound scanners

2.2.1 Exercise: Beam focusing by delaying elements in a linear array

2.2.2 Solution

2.2.3 Exercise: Best size of focus

2.2.4 Solution

2.2.5 Exercise: Depth of focus

2.2.6 Solution

2.2.7 Exercise: Longitudinal resolution of an ultrasound pulse

2.2.8 Solution

3. ELECTROCARDIOGRAPHY (ECG)

3.1 Dipole fields

3.1.1 Exercise: Potential of an electric dipole along the dipole axis

3.1.2 Solution

3.1.3 Exercise: Potential of an electric dipole in the symmetry plane

3.1.4 Solution

3.1.5 Exercise: Component of the electric dipole vector

3.1.6 Solution

3.2 ECG instrumentation

3.2.1 Exercise: Heart rate in an ECG paper print

3.2.2 Solution

3.2.3 Exercise: Angle of the heart electrical axis

3.2.4 Solution

3.2.5 Exercise: Equation to calculate *U*iii from *U*i and *U*ii

3.2.6 Solution

3.2.7 Exercise: Angle of the heart electrical axis for given *U*i and *U*ii

3.2.9 Exercise: The signal lead augmented vector foot *aV*f

3.2.10 Solution

3.2.11 Exercise: Voltage ratios of Einthoven and Goldberger signal leads

3.2.12 Solution

3.2.13 Exercise: Precordial leads according to Wilson

3.2.14 Solution

4. LASER

4.1 Interaction of laser light with matter

4.1.1 Exercise: Energy density and time range of laser radiation to coagulate soft tissue

4.1.2 Solution

4.1.3 Exercise: Energy density and time range of laser radiation to vaporize soft tissue

4.1.4 Solution

4.1.5 Exercise: Energy density and time range of laser radiation to photoablate soft tissue

4.1.6 Solution

4.1.7 Exercise: Energy density and time range of laser radiation to photodisrupt soft tissue

4.1.8 Solution

4.1.9 Exercise: Power density of a laser diode

4.1.10 Solution

4.1.11 Exercise: Energy density and beam diameter of a pulsed laser

4.1.12 Solution

4.1.13 Exercise: Ablation depth of a laser pulse

4.1.14 Solution

4.1.15 Exercise: Ablation depth versus energy density of laser pulses

4.1.16 Solution

4.1.17 Exercise: Beam diameter and depth of focus of a focused laser

4.1.18 Solution

4.1.19 Exercise: Irradiation time in photodynamic therapy

4.1.20 Solution

4.1.21 Exercise: Therapeutic window

4.1.22 Solution

5. PULSE OXYMETRY

5.1 Interaction of light with blood

5.1.1 Exercise: Optical density of blood

5.1.3 Exercise: Isobestic point of light absorption in blood

5.1.4 Solution

5.1.5 Exercise: Maximum difference of light absorption in hemoglobin

5.1.6 Solution

5.1.7 Exercise: Optical densities of oxy- and deoxygenated hemoglobin

5.1.8 Solution

5.2 Analysis of oxygen saturation

5.2.1 Exercise: Variation of the optical path length

5.2.2 Solution

5.2.3 Exercise: Ratio of optical density differences during a heartbeat

5.2.4 Solution

5.2.5 Exercise: Ratio of optical density differences at specific oxygen saturation

5.2.6 Solution

6. HIGH-FREQUENCY SURGERY

6.1.1 Exercise: Current densities around a spherical electrode

6.1.2 Solution

6.1.3 Exercise: Electrical potentials around a spherical electrode

6.1.4 Solution

6.1.5 Exercise: Current between a spherical and a large neutral electrode

6.1.6 Solution

6.1.7 Exercise: Current between a spherical and a large neutral electrode for a given set of parameter values

6.1.8 Solution

6.1.9 Exercise: Power density caused by current flow

6.1.10 Solution

6.1.11 Exercise: Supplied heat energy and rise of temperature

6.1.12 Solution

6.1.13 Exercise: Ratio of peak and average power

6.1.14 Solution

6.1.15 Exercise: Dissipated power versus specific resistance

6.1.16 Solution

6.1.17 Exercise: Dissipated power at different orientations of tissues

6.1.18 Solution

6.1.19 Exercise: Dissipated power at different orientations of tissues with specific resistances

6.1.20 Solution

7. COMPUTED RADIOGRAPHY (CR)

7.1.1 Exercise: Number of generated photostimulable storage centers per X-ray quantum

7.1.2 Solution

7.1.3 Exercise: Wavelength of maximum photostimulability

7.1.4 Solution

7.1.5 Exercise: Crosstalk of subsequently scanned pixel

7.1.6 Solution

7.1.7 Exercise: Probability of F-center electrons to escape to the conduction band

7.1.8 Solution

7.1.9 Exercise: Schottky defect pair concentration in NaCl

7.1.10 Solution

7.2 CR scanner

7.2.1 Exercise: Diffraction limited spot size of a CR scanner

7.2.2 Solution

7.2.3 Exercise: Maximum scan speed at specific pixel size and crosstalk

7.2.4 Solution

7.2.5 Exercise: Rotational speed of a mirror and scan speed of laser beam

7.2.6 Solution

7.2.7 Exercise: Bearing play and projected beam positioning

7.2.8 Solution

7.2.9 Exercise: Readout time and efficiency of information readout

7.2.10 Solution

7.2.11 Exercise: DQE of a CR-system

7.2.12 Solution

8. COMPUTED TOMOGRAPHY (CT)

8.1 Tomographic Reconstruction

8.1.1 Exercise: Number of X-ray projections and number of voxels

8.1.2 Solution

8.1.3 Exercise: Point spread function using unfiltered backprojection

8.1.4 Solution

8.1.5 Exercise: Ideal filter function in filtered backprojection

8.1.6 Solution

8.1.7 Exercise: Transmitted dose signals in real and in Fourier space

8.1.8 Solution

8.1.9 Exercise: Grid pattern of Fourier transformed absorption data

8.1.10 Solution

8.2 Instrumentation

8.2.1 Exercise: Acceleration of a rotated X-ray tube

8.2.2 Solution

8.2.3 Exercise: Data rate of a CT scanner

8.2.4 Solution

8.2.5 Exercise: Decay time of luminescence and crosstalk

8.2.6 Solution

8.2.7 Exercise: Number of angular positions of X-ray exposures and number of pixel elements in a sectional image

8.2.8 Solution

8.2.9 Exercise: Acquisition time of a tomogram and pixel rate of a CT scanner

8.2.10 Solution

8.2.11 Exercise: CT number of adipose tissue

8.2.12 Solution

8.2.13 Exercise: CT numbers of cortical bone

8.2.14 Solution

8.2.15 Exercise: CT numbers in dual Energy CT

8.2.16 Solution

8.2.17 Exercise: CT artefacts of a metal sphere

8.2.18 Solution

8.2.19 Exercise: Number of photons and electrons per absorbed X-ray

8.2.20 Solution

8.2.21 Exercise: Photodiode current in a detector element of a CT scanner

8.2.22 Solution

8.3 X-ray Dose

8.3.1 Exercise: Error of measured absorption coefficients and X-ray dose

8.3.2 Solution

9. NUCLEAR MAGNETIC RESONANCE IMAGING

9.1 Nuclear magnetic resonance

9.1.1 Exercise: Energy levels of hydrogen nuclei in a magnetic field

9.1.2 Solution

9.1.3 Exercise: Frequency of a nuclear spin flip in a magnetic field

9.1.4 Solution

9.1.5 Exercise: Relative occupation difference of energy levels in a magnetic field

9.1.6 Solution

9.1.7 Exercise: Required field direction to induce spin flips

9.1.8 Solution

9.1.9 Exercise: Nuclear spin quantum numbers in the ground state

9.1.10 Solution

9.1.11 Exercise: Number of energy levels of nuclei in a magnetic field

9.1.12 Solution

9.1.13 Exercise: Influence of the electron shell on nuclear energy levels

9.1.14 Solution

9.1.15 Exercise: Types of nuclear spin relaxations and relaxation times

9.1.16 Solution

9.1.17 Exercise: Mechanism of contrast agents in NMR

9.1.18 Solution

9.1.19 Exercise: Decay of the transversal magnetizations after a pulse sequence

9.1.20 Solution

9.1.21 Exercise: Transversal magnetizations after different pulse sequences

9.1.22 Solution

9.1.23 Exercise: Time interval between 180° and 90° pulses to get transversal magnetization down to zero

9.1.24 Solution

9.1.25 Exercise: Spin echo signals of different tissues at a specific pulse sequence

9.1.26 Solution

9.1.27 Exercise: TR and TE values in proton density weighted MRI

9.1.28 Solution

9.2 Magnetic resonance imaging instrumentation

9.2.1 Exercise: Number of gradient coils in an MRI scanner

9.2.2 Solution

9.2.3 Exercise: Magnetic flux of MRI scanners using normally conducting electro magnets

9.2.4 Solution

9.2.5 Exercise: Waveform of the high frequency pulse to excite spins in a plane

9.2.6 Solution

9.3 Image reconstruction

9.3.1 Exercise: Relation between spin signals in real and Fourier space

9.3.2 Solution

9.3.3 Exercise: Location of the Fourier transforms of nuclear resonance signals in Fourier space

9.3.4 Solution

10. NUCLEAR MEDICAL IMAGING

10.1 Radionuclides

10.1.1 Exercise: Half-life and decrease of activity

10.1.2 Solution

10.1.3 Exercise: Amount of decays within a time period after incorporation of the radionuclide

10.1.4 Solution

10.2 Instrumentation

10.2.1 Exercise: Radius of field of a circular collimator

10.2.2 Solution

10.2.3 Exercise: Efficiencies of circular collimators

10.2.4 Solution

10.2.5 Exercise: Amount of γ-absorption within a body using ^{99m}Tc as radioactive emitter

10.2.6 Solution

10.2.7 Exercise: Amount of γ-absorption within a body in PET

10.2.8 Solution

10.2.9 Exercise: Probability of coincident photoabsorption of two γ quanta in PET

10.2.10 Solution

10.2.11 Exercise: Energy resolutions of scintillation detectors

10.2.12 Solution

10.2.13 Exercise: Compton scattering angles of counted γ quanta for a given energy window in PET

10.2.14 Solution

11. RECEIVER OPERATOR CHARACTERISTIC (ROC)

11.1 Binary classification

11.1.1 Exercise: Tables of confusion for different threshold values

11.1.2 Solution

11.1.3 Exercise: Sensitivity and specificity for different threshold values

11.1.4 Solution

11.2 ROC curves

11.2.1 Exercise: ROC curve for different threshold values

11.2.2 Solution

12. MODULATION TRANSFER FUNCTION (MTF)

12.1.1 Exercise: Evaluation of the MTF using a sinusoidal test pattern

12.1.2 Solution

12.1.3 Exercise: Relation between two MTFs corresponding to PSFs of different width

12.1.5 Exercise: Standard deviation of the PSF of a CR scanner comprising two processes of spatial broadening of information

12.1.6 Solution

12.1.7 Exercise: Spatial frequency at a specific value of the MTF of an X-ray detector having a PSF with Gaussian profile

12.1.8 Solution

12.1.9 Exercise: Calculation of the MTF at a specific spatial frequency for a PSF with Gaussian profile

12.1.10 Solution

12.1.11 Exercise: MTF of a second process that broadens the PSF

12.1.12 Solution

12.1.13 Exercise: Fourier expansion of a rectangular grid pattern

12.1.14 Solution

13. LIST OF ABBREVIATIONS

14. LIST OF IMPORTANT SYMBOLS

15. SUBJECT INDEX

**1.1 Attenuation of X-rays**

1.1.1 **Exercise: X-ray transmission of lead**

What is the percentage of X-ray radiation that is transmitted through a lead sheet having a thickness of *d* = 1 mm at an X-ray energy of 50 keV?

1.1.2 **Solution**

X-rays of a specific X-ray energy are exponentially attenuated by uniform matter according to the law

with

*x* = penetration depth [cm],

*µ* = X-ray attenuation coefficient [1/cm] of the matter at the specific energy of the X-rays,

*D*o = incident dose,

*D* = dose at penetration depth x.

The X-ray attenuation coefficient is dependent on the energy *E* of the X-rays, the elemental composition of the material and the density *ρ* of the material. In this exercise the material is composed only of one element: Lead (Pb). In order to calculate the transmitted percentage of the dose D(*x* = *d*)/*D*_{o}, the attenuation coefficient *µ* has to be known. Usually instead of the attenuation coefficient *µ,* the mass attenuation coefficient *µ/ρ* is tabulated, because the attenuation coefficient *µ* is proportional to the density *ρ* and thus the mass attenuation coefficient *µ/ρ* depends no longer on the density *ρ.* The mass attenuation coefficient *µ/ρ* can be found on the NIST web page

__https://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z82.html__. At an Xray energy of *E* = 50 keV, it has a value of 8.041 cm^{2}/g. Using this value the attenuation coefficient can be calculated by the equation

The density of the lead sheet can also be found on the NIST web page __https://physics.nist.gov/PhysRefData/XrayMassCoef/tab1.html__to be *ρ*=

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