ENGG 4620/6620: HOMEWORK 1

A. Quantitative X-ray Imaging

(Application of Lambert-Beer's Law)

We have discussed X-ray imaging mostly from a qualitative perspective. However, X-ray imaging can be used for quantitative measurements, and this homework provides an example. Bone-tissue contrast is high, and x-ray-based methods to determine bone density are widely used. One such method, called dual-energy x-ray absorptiometry (DEXA), uses two absorption measurements at different energies and makes use of the different energy dependency of the absorption of soft tissue and bone to increase measurement accuracy. More specifically, if we use a two-compartment model (i.e., soft tissue and bone), we obtain two equations for two unknowns. The unknowns in this case are the combined thickness of soft tissue and that of bone along an x-ray path. Please review pages 26--27 in our book for the principle of DEXA.

For quality control, phantoms are available that mimic bone with soft tissue. Three such phantoms with different bone mineral content have been imaged and their absorption measured. One image, taken at 35kVp, is displayed below. The vertical intensity profiles of three images (35kVp, 25kVp, 20kVp) are plotted as well. It is evident that the densest phantom saturates the detector at 20kVp, and we will not use those data.

At the thickest point (maximum height of the phantoms: 0.46cm), absorption is highest. Average absorption values are given below for 25kVp and 35kVp. "kVp" stands for kilovolt-peak and indicates the highest energy present. Since the x-ray energies are disrtibuted over a broad spectrum, we will use 17keV as the representative energy for 25kVp and 22keV for 35kVp.

Energy Phantom 1 Phantom 2 Phantom 3
25kVp 0.94 1.73 2.75
35kVp 0.50 0.88 1.38

Task 1:

  1. Obtain the linear attenuation coefficients for compact bone and for muscle tissue at 17keV and 22keV, respectively.
  2. Determine the weight factor w that is needed to eliminate the influence of soft tissue in a two-compartment model (Equations 2.19 and 2.20)
  3. Determine for each of the phantoms the equivalent thickness of compact bone as part of the 0.46cm total thickness.
Hints: If you run into the NIST tables for X-ray absorption (part a), please note that you need to find the linear attenuation coefficient in inverse cm. Also, in part (b), you only need to determine w, nothing else. In addition, part (c) does not necessarily make use of w, although this is the quickest way.

B. Radiation Dose and Exposure

Task 2: Units related to radiation. There are a number of units that describe radiation strength and exposure. Create a bulleted list in which you explain the significance of the following units: Roentgen, Sievert, rad, rem, Curie, Becquerel. Provide your sources.

Task 3: Typical radiation exposure What is the typical background radiation we are exposed to? What additional exposure would we get from a chest x-ray? A CT scan of the torso? A coast-to-coast flight? A year of living in Denver, CO? Please provide representative values, and stay within the same unit system (i.e., choose either Sievert or rem) for easy comparison.

Task 4: CRT-based TV sets. Before the flat-screen TV became ubiquitous, TV screens were based on cathode-ray tubes (CRT). In a CRT, electrons are accelerated in a high-voltage field onto a phosphor target (namely, the visible part of the TV screen itself). What are typical acceleration voltages of large color TV sets? How are they related to the diagnostic x-ray energy range? Assess the risk of unintended x-ray production and describe protective measures taken: Why was watching TV on a CRT set never a radiation risk?

Hint: Federal regulation exists for limiting x-ray emission from CRT sets. However, this is not a valid reason for the low radiation risk of CRT TV sets, because it is merely regulation, not an engineering solution.

Task 5: Diagnostic X-ray Imaging (Qualitative). Neodymium magnets, if swallowed, pose a significant hazard due to mechanical forces in the intestinal tract, see, for example, Kim Y, Hong J, Moon S-B. Ingestion of multiple magnets: The count does matter. Journal of Pediatric Surgery Case Reports 2014; 2: 130--132 . With sequential abdominal x-ray images, the progress of the magnet in the intestines can be observed. This allows to make a decision whether the magnet exits the body by natural means or if surgery is needed.

Your x-ray machine allows several kVp setting between 50kV and 120kV. (a) Is x-ray imaging a suitable modality in the sense that good contrast can be achieved between the magnet and surrounding tissue/bone? (b) To minimize exposure, what kVp setting would you choose? Provide brief reasons!


This homework will give you a maximum of 25 points toward your total score (10 points for part A and 15 points (3+3+4+5) for tasks 2,3,4,5 in part B).

Turnin and Due date

Homework is due on Sept. 7, 2017.

The turnin may either be a typewritten printed paper or a file sent by e-mail. If you decide to send an e-mail, you MUST use the word "HOMEWORK" in capital letters in the subject line. Otherwise I won't guarantee that I received your homework. Also note that I will grade electronic submissions on-screen. Only typewritten and printed turnins can be returned to you with comments.

Late turnin penalty: For each late day, you lose 1 point from your total score.

Early turnin bonus: If your homework is done before the due date, you may present your work for a brief evaluation. If I find significant errors, I will point those out and return the homework to you for revision. You may turn in a revised version by the due date.