Dead-Time and Beam-Hardening correction methods

1. BMIL Timing Pulse Measurement and Detector Calibration for the OsteoQuant® Advisor: Dr. Thomas N. Hangartner By: Binu Enchakalody 1

2. The OsteoQuant® pQCTscanner which can provide precise density assessment of the trabecular and cortical regions of bone. Currently being upgraded to a x-ray tube source and a CZT semiconductor detector. 2 Ref: http://www.wright.edu/academics/bmil/bmil1.htm

3. Objective Implement a system capable of registering the motor- and detector timing-pulses of the Osteoquant® using a common time-base in microsecond resolution. 2a. Correct the photon-count loss due to dead time to an error level of less than 0.5% of the maximum expected photon counts. 2b. Correct the non-linearity of the projection values due to beam hardening to an error level of less than 1% of the expected maximum projection value. 3

4. Timing-Pulse Measurement for the OsteoQuant® 4

5. Need for Timing-Pulse Measurement Data collected is only useful if there is synchronization between the motor- and detector-timing pulses. The motor- and detector timing-pulses have to be well synchronized to assure correlation between the data collected and the measurement interval. Aim Each detector frame should be accurately related to a motor position. Solution Measure time stamps using a common time base to relate a motor position to a detector frame. 5

6. Timing-Pulse Measurement: Analyzing the problem Sample Detector Timing-Pulse 0 2 4 6 8 10 12 ms Sample Motor Timing-Pulse 0 3 6 9 12 ms Common Time-Base (1/1000 resolution) 1 µs µs 3000 6000 8000 12000 0 2000 4000 10000 9000 6

7. Timing-Pulse Measurement: Requirement Requires counters working at clock frequencies of 1 MHz or above. High data transfer speed Fast event notification Solution The USB-4301 is a low-power USB-2.0-compliant, 16-bit, 5 channel, up-down binary counter, capable of operating at frequencies as high as 5 MHz can be used in event counting and pulse generating applications. 7

9. Modules are daisy chained to form 32-bit counters

10. Terminal count is achieved in ~72 minutes8

11. External Circuit Components Synchronization Circuit 9

12. External Circuit Components 10

13. Timing Diagram 11

15. Start Pulse

16. Synchronization Circuit

17. Interrupt Mask

19. Configuration

20. Counter Save

21. Time Stamp12

22. Error Range of Measured Time Periods Interrupt signal of 2.22 ± 0.003 ms Relative error : 0.13% 13

23. Sample Spreadsheet 14

24. Detector Calibration for the OsteoQuant® 15

25. Detector Calibration: Dead Time Dead Time An event occurs at the detector is converted into an electrical signal depending on the intensity and duration of the event. The time required to collect this charge depends on certain characteristics (mobility, distance to collection electrodes, etc.) of the detector itself and on the subsequent electronics. Due to the random nature governed by Poisson statistics, there is always a probability that the detector misses a true event that follows a recorded event. The missed true events are called dead-time losses 16

26. Need for Dead-Time Correction Operating Parameters: Tube Voltage : 45 kVp Tube Current: 0 – 1 mA Photon accumulation time : 50 ms Aim: Correct the photon-count loss due an error level of less than 0.5% of the maximum expected photon counts (0.5% of 72,000) ± 350 counts. The photon counts-vs.-tube current response plot is mathematically modeled and then linearized. The measured non-linear and the expected linear photon counts for varying tube currents of one detector element from a sample data set 17

27. Detector Calibration: Beam Hardening Beam Hardening X-ray beams used in CT are usually polychromatic When the photon beam passes through material, it tends to preferentially loose its lower energy photons, hardening the beam in the process. Lower-energy x-rays are more prone to attenuation, and the average energy of a polychromatic beam varies with increasing thickness of the material, thereby making the beam harder. 18

28. Beam Hardening Operating Parameters: Tube Voltage : 45 kVp Tube Current: 1 mA Photon accumulation time : 50 ms Aim: Correct the non-linearity of the projection values due to beam hardening to an error level of less than 1% of the expected maximum projection value (1% of 5 projection value units) ± 0.05. The projections-vs.-thickness plot is mathematically modeled and then linearized. The measured non-linear and expected linear projection values for varying absorber thicknesses. 19

29. Experimental Setup Source : The tube can operate at a maximum anode voltage of 50 kVp and a maximum anode current of 1 mA, at a maximum operating temperature of 55oC. Detector : The detector is a CZT semi-conductor cuboid of 64 pixilated elements composed of 50% tellurium, 5% zinc and 45% cadmium. Slabs : Slabs are used in the beam hardening experiment as a substitute for the bone and soft-tissue in human-body (aluminum and Plexiglas). A total on 19 pairs of these slabs were used. Data was collected on 22 dates over a span of nine months. 20

30. Dead-Time Correction: Steps Modeling using the fourth-degree polynomial function Linearization of the model Drift and stability analysis of the correction Same-date Correction Different-date Correction 21

31. Dead-Time Correction: Modeling and linearization For dead-time correction 22

33. Corrections applied to the data collected at the same date (same-date corrections) and a different-date (different-date corrections).

34. The stability was analyzed by comparing the residuals

35. The corrections were declared stable if their residuals < 350.

36. The least expected residuals are those from the same-date corrections23

37. Beam-Hardening Correction: Steps Fifth-Degree Polynomial Model Modeling using the polynomial function for 10- and 19-plate data sets Linearizing the mathematical model Corrections applied on the 10- and 19-plate data sets Bimodal-Energy Model Modeling using the bimodal-energy model for 10- and 19-plate data sets Linearizing the mathematical model Corrections applied on the10- and 19-plate data sets Compare the stability analysis between both models 24

38. Beam-Hardening Correction According to Beer’s Law, the projection values of the x-ray beam passing through an object are linearly proportional to μ(E). The projection values are generally linearized to a linear line. i: 1, 2, 3, ... 19 for the number of slab pairs used I0 : counts collected with no object in the beam path Ii: counts collected with i number of slab pairs µeff: linear attenuation coefficient for Al and Pl di : thickness of i number of slab pairs 25

39. Linearization Using the Polynomial Model Based on the projection value-vs.-slab thickness plots, it was decided that an 5th-degree polynomial fit can model these data. Each detector element’s data was fitted using a second to a sixth-degree polynomial function. 26

41. μ2is the slope by E2 for large thicknesses

42. μ1 is the slope by E1 for smaller thicknesses.To solve for the unknowns, the non-linear least squares method is used. The equation system was iteratively solved using a Matlab script by assuming initial values for the unknown fitting parameters. Ref: de Casteele, E. V., Dyck, D. V., Sijbers, J., and Raman, E. 2002. An energy-based beam hardening model in tomography. Phys Med Biol 47, 23–30. 27

44. Energies E2 and E1 are not predetermined.

45. Estimate of the expected lower and upper bound for μ1 and μ2.

46. E2 proved to be greater than E1Linear attenuation coefficient (μ(E)) vs. energy for the Al/Pl slabs. 28

48. Linearization using both models satisfy the required error criterion

49. The stability of the beam-hardening corrections were analyzed by evaluating the error statistics of the same-date and different-date corrected values.29

50. Secondary Correction Primary correction every day is a tedious process. Apply primary corrections from one particular date to the data sets collected from other dates and following this with a secondary correction (3rd degree polynomial) based only on a few plates (0, 6, 14, 19 or 0, 3, 7, 9) measured on the specific date. Without Secondary Correction With Secondary Correction 30

51. Stability analysis for beam-hardening corrections Same-date and different-date corrections. Same-date primary Different-date primary Different-date-primary followed by secondary Data collected on 22 dates, during nine months. 134 coefficient matrices for the 10-plate data sets. 5 coefficient matrices for the 19-plate data sets. Stability for the correction method is assumed if the residuals of the same-date and different-date corrections are less than 0.05. 31

52. Results: Dead-Time Correction Linearization using the fourth-degree model for a data set. Histogram of the individual residuals using a fourth-degree polynomial correction for a data set 32

53. Same-Date Corrections: 10 plate data-set 33 Corrected projection values and their histograms for the polynomial and the bimodal-energy model

54. Different-date corrections: 10 plate data-set Corrected projection values and their histograms for the polynomial and the bimodal-energy model 34

55. Different-date primary followed by secondary correction: 10 plate data-set 35 Corrected projection values and their histograms for the polynomial and the bimodal-energy model

56. Summary: Correction Methods Summary of the same-date and different-date primary (Prim) corrections using the fifth degree polynomial and bimodal-energy models, and the same-date and different-date primary corrections using both models followed by the secondary (Sec) corrections. The checked cells represent the methods that produced residuals lower than 0.05. 36

57. Detector Calibration: Summary The same-date dead-time corrections were all within the expected residual value. The data collection required for the dead-time corrections can be automated. Same-date primary corrections consistently produced corrected projection values that were well within the expected residual of 0.05. Most of the residuals for the different-date bimodal corrections were below 0.05, whereas the residual values for the different-date polynomial corrections were above 0.05. Future Work Using the non-paralyzable dead-time model. Beam hardening corrections should be studied with different tube voltages. Stability of the corrections over a shorter time period can be studied 37