Chapter 1, XRay Imaging and Computed Tomography Video Solutions, Introduction to Biomedical Imaging (IEEE Press Series on Biomedical Engineering)

Problem 1

Figure 1.38 shows the intensity of $X$-rays produced from a source as a function of their energy. With respect to the reference graph shown on the left, one plot corresponds to a decrease in tube current and the other to a decrease in the accelerating voltage $\left(\mathrm{kV}_{\mathrm{p}}\right)$. Explain which plot corresponds to a decrease in which parameter.
(FIGURE CAN'T COPY)

Salamat Ali

Numerade Educator

00:39

Problem 2

The spectrum of $X$-ray energies changes as the $X$-rays pass through tissue due to the energy dependence of the linear attenuation coefficient: this is a phenomenon known as beam hardening. A typical energy distribution of the beam from the X-ray source is shown in Figure 1.39. Sketch the energy spectrum after the beam has passed through the body.

Amrita Bhasin

Numerade Educator

04:06

Problem 2

In Figure 1.40 , calculate the $X$-ray intensity, as a function of the incident intensity $I_0$, that reaches the film for each of the three X-ray beams. The dark-shaded area represents bone and the light-shaded area represents tissue. The linear attenuation coefficients at the effective X-ray energy of 68 keV are 10 and $1 \mathrm{~cm}^{-1}$ for bone and tissue, respectively.
(FIGURE CAN'T COPY)

Prachita Kush

Numerade Educator

01:27

Problem 3

Explain why $\mu_{\text {bone }} \gg \mu_{\text {tissue }}$ at low X-ray energies, but the two values of $\mu$ become closer as the X -ray energy increases.

Ankur S

Numerade Educator

02:36

Problem 4

The linear attenuation coefficient of a gadolinium-based phosphor used for detection of X-rays is $560 \mathrm{~cm}^{-1}$ at an X-ray energy of 150 keV . What percentage of X-rays are detected by phosphor layers of 100,250 and $500 \mu \mathrm{~m}$ thickness? What are the tradeoffs in terms of spatial resolution?

Khoobchandra Agrawal

Numerade Educator

04:06

Problem 5

In Figure 1.41, calculate the relative intensities of the signals $S_1, S_2$, and $S_3$ produced by each crystal. The value of $\mu_{\text {tissue }}$ is $0.5 \mathrm{~cm}^{-1}, \mu_{\text {bone }}$ is $1 \mathrm{~cm}^{-1}$, and $\mu_{\text {crystal }}$ is $2 \mathrm{~cm}^{-1}$.

Prachita Kush

Numerade Educator

Problem 7

Intensifying screens (Section 1.5 .3 ) can be placed on both sides of the X-ray film (double-sided) or on one side only (single-sided). Explain why double-sided screens give a higher image SNR, but single-sided screens have a better spatial resolution.
(FIGURE CAN'T COPY)

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01:48

Problem 8

An $X$-ray with energy 60 keV strikes a gadolinium-based intensifying screen, producing photons at a wavelength of 415 nm . The energy conversion coefficient for this process is $20 \%$. How many photons are produced for each incident X-ray? (Planck's constant $=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s}, 1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J}$.)

Keshav Singh

Numerade Educator

02:51

Problem 9

In mammographic examinations, the breast is compressed between two plates, as shown in Figure 1.42. Answer the following with a brief explanation:
(a) Is the geometric unsharpness increased or decreased by compression?
(b) Why is the image contrast improved by this procedure?
(c) Is the required X-ray dose for a given image SNR higher or lower with compression?

Sri Datta Vikas Buchemmavari

Numerade Educator

03:15

Problem 10

For the two $X$-ray film characteristic curves shown in Figure 1.43:
(a) Which one corresponds to the film with the higher speed?
(FIGURE CAN'T COPY)
(b) Which one corresponds to the film with the broader modulation transfer function?

Keshav Singh

Numerade Educator

00:39

Problem 11

In digital subtraction angiography, two images are acquired, the first before injection of the contrast agent and the other postinjection.
(a) Write an expression for the X-ray intensity $I_1$ in the first scan in terms of $I_0, \mu_{\text {tissue }}, x_{\text {tissue }}, \mu_{\text {blood, }}$, and $x_{\text {vessel }}$, where $x_{\text {tissue }}$ and $x_{\text {vessel }}$ are the dimensions of the respective organs in the direction of X-ray propagation.
(b) Write a corresponding expression for the X -ray intensity $I_2$ for the second scan, replacing $\mu_{\text {blood }}$ with $\mu_{\text {constrast }}$.
(c) Is the image signal intensity from static tissue removed by subtracting the two images?
(d) Show that the signal from static tissue is removed by computing the quantity $\log \left(I_2\right)-\log \left(I_1\right)$.

Amrita Bhasin

Numerade Educator

Problem 12

In digital subtraction angiography, what is the effect of doubling the $X$-ray intensity on the SNR of the image? What would be the effect of doubling the dose of contrast agent on the SNR of the image?

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04:06

Problem 13

For the case of $X$-rays passing through tissue with a constant linear attenuation coefficient ( $\mu_{\text {tissoc }}>\mu_{\text {water }}$ ), does the CT number increase or decrease as a function of distance through the tissue due to beam hardening?

Prachita Kush

Numerade Educator

05:44

Problem 14

Draw the CT projection obtained from the setup shown in Figure 1.44. Assume that the spherical sample has a uniform attenuation coefficient throughout its volume.
(FIGURE CAN'T COPY)

Mayukh Banik

Numerade Educator

Problem 15

Considering the effects of beam hardening, draw the actual CT projection that would be obtained from the sample in Exercise 1.14. Sketch the final image that would be formed from filtered backprojection of all of the projections acquired in a full scan of the sample in Exercise 1.14.

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Problem 16

For the set of projections shown in Figure 1.45, perform one series of a ray-by-ray iteration on the horizontal, the diagonal, and the vertical projections. Calculate the minimum squared error after each iteration.

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Problem 17

For the object shown in Figure B1 (Appendix B), draw the projections that would be acquired at angles $\phi=0,45,90,135$, and $180^{\circ}$.
1.18. For the object shown in Figure 1.46, sketch the sinogram for values of $\phi$ from 0 to $360^{\circ}$.
(FIGURE CAN'T COPY)

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