Binh Phan's List: Trigonometry in Ultrasound

• all ultrasound imaging is performed by emitting a pulse, which is partly reflected from a boundary between two tissue structures, and partially transmitted (fig. 2). The reflection depends on the difference in impedance of the two tissues.
• Ultrasound is simply sound waves, like audible sound. Although some physical properties is dependent on the frequency, the basic principles are the same. Sound consists of waves of compression and decompression of the transmitting medium (e.g. air or water), travelling at a fixed velocity. Sound is an example of a longitudinal wave oscillating back and forth in the direction the sound wave travels, thus consisting of successive zones of compression and rarefaction. Transverse waves are oscillations in the transverse direction of the propagation. (For instance surface waves on water or electromagnetic radiation.)
   
   

  
   

   Fig. 1. Schematic illustration of  a longitudinal compression wave (top) and transverse wave (bottom). The bottom figure can also represent the pressure amplitude of the sound wave.
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  • Notice how the ultrasound wave moves in a sinusoidal fashion!
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• The reflected signals can be displayed in three different modes. A-mode (Amplitude) shows the depth and the reflected energy from each scatterer.  B-mode (Brightness) shows the energy as the brightness (in this case the higher energy is shown darker, against a light background) of the point. The bottom scatterer is moving. If the depth is shown in a time plot, the motion is seen as a curve, (and horizontal lines for the non moving scatterers) in a M-mode plot (Motion).
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• [0010]The origin of the idea for the new coefficient RSAC to measure the  tooth canal length and to localize the apical foramen is based on the  technique used to measure the ultrasound attenuation within the human  tissue. 
  
  [0011]The technique for the ultrasound attenuation coefficient is called  "Broadband Ultrasound Attenuation" or BUA. As the ultrasound propagates  through the human tissue, its intensity decays exponentially with the  distance. The BUA coefficient is determined by analyzing the logarithm of  the ultrasound signal spectrum. Detailed explanation is beyond the scope  of this patent. The fact is that resistors and capacitors circuits can be  used to model the acoustic and electrical impedance of the tissues. Thus,  we have visualized that a similar procedure, that is, the BUA  measurement, is applicable to determine the tooth canal length and to  localize the apical foramen. 
  
  [0012]Therefore, this patent of invention describes the discovery of a new  coefficient called Radicular Spectral Attenuation Coefficient or RSAC.  The RSAC is directly related with the distance between the tip of the  endodontic file (1.2) and the radicular foramen (1.3). This distance is  called Root Canal Length (RCL). Thus, since the RSAC is directly related  to the RCL, it also can be used as a reference for the localization of  the radicular foramen (LRF). In the following paragraphs it is described  the physical principle involved with the RSAC measurement and how this  coefficient is converted in the RCL and used as reference for the LRF. 
  
  [0013]The process of RSAC calculation is divided into three steps: 1) the  application of a measurement signal; 2) the measurement of an electrical  signal and from this signal the determination of the RSAC and 3) the  conversion of the RSAC into the RCL and the LAF. The first two steps make  use of the already described measurement electrodes (1.1) and (1.4). 
  
  [0014]The measurement signal, applied in the first step of the RSAC  calculation, is composed by a sum of sine waves trigonometric functions,  all them with the same amplitude but different frequencies (or periods)  and initial phases. The measurement signal, represented by f(t), is  determined by equation 1, 
  
  f ( t ) = A i = 1 N sin ( 2 .pi. f i t +  .PHI. i ) ( 1 ) ##EQU00001##

  • The measurement signal function looks very familiar!

• Trigonometry is used in surveying    to determine heights and distances, in navigation to determine location    and distances, and in fields like nondestructive testing for determining    things such as the angle for reflection or refraction of an ultrasound    wave.