S15Ultrasound
-- Main.paya0009 - 09 May 2015
Measuring Resolution in Ultrasound Imaging
Andrew Payant & Alexandra Swancutt
University of Minnesota
Minneapolis, MN 55455
Abstract
We determined the resolution of an ultrasound image from a single piezoelectric transducer. The transducer created an ultrasound signal that propagates through the object and reflects off internal and external boundaries. In order to transform the raw data into an image of a cross section, we needed to find the attenuation of the different materials the signal was passed through. We wrote a program in Labview to produce two dimensional images based on the timing and amplitude of the received reflection signals. Objects of known internal structure, called phantoms, were used to confirm the accuracy of our software. We found that the axial resolution is 0.8+-0.1 mm and the lateral resolution is 2.0+-0.2 mm.
Introduction
Today, ultrasound imaging is an essential diagnostic tool in the medical world. However, understanding the physical processes that underlay ultrasound techniques is essential to the success of ultrasound imaging as a tool. The principles that govern ultrasound signal propagation are similar to signal analysis used in other fields, such as light passing through mediums of different indices. In this experiment, we will explore how to take the raw signal, transform it into an image, and find the limiting resolution.
Theory
Ultrasound waves, as the name suggests, <!--StartFragment-->propagate in such a way that can be described by a wave equation. However, when the pressure wave passes through real material, the amplitude of the wave decreases exponentially. As a result of this decay, the ultrasound wave's propagation can be described in the following:
This describes the pressure wave traveling in the z-direction with a frequency of omega and a wave number of k. the rho naught is the density of the material before the pressure wave interacts and creates periodic changes in the density. When the wave passes through real material, it experiences a a decay that is related to the amount of material the wave propagates through, and each material has a unique alpha coefficient.
Depth resolution, also called axial resolution, is directly proportional to the length of the pulse. The shorter the time length the transducer is sending a pulse, the longer the transducer can receive reflected pulse data. However, the wavelength is the dominant factor when it comes to depth resolution. If the wavelength of the pulse is larger than two boundaries that are some distance apart that is smaller than the wavelength, the two boundaries will appear in the image as one. Taking these factors into account, we find that our depth resolution should be limited at:
Where is the axial resolution, lambda is the wavelength between pulses, and is the length of the pulse compared to the measuring length[6].
In order to measure resolution we will be utilizing Rayleigh's criterion. In essence, Rayleigh's criterion states that two point sources can be resolved at the limit that the central peak of the diffraction pattern of one is located on the first diffraction minimum of the other[2]. If the two point sources are closer than the Rayleigh's criterion, the superposition of the diffraction patterns looks the same as a single strong point source would. By making point sources in only one of the dimensions we are interested in, we are able to measure the resolution of either axial or lateral directions.
Experimental Setup
The ultrasound machine we were using is comprised of a pulser receiver and a single piezoelectric transducer. The pulser receiver sent a voltage pulse to the transducer which through the piezoelectric effect caused the material in the transducer to compress and decompress fast enough to create an ultrasound pulse. However, the piezoelectric not only created signals when a voltage was applied to it, it also created voltage differences when it was compressed or stressed. So, our transducer was the source of ultrasound signals and was a sensor to them. For this experiment the pulser receiver was always set at 10 MHz and the pulse width was found to be 5. A stepper motor was used to move the transducer to effectively simulate having multiple transducers producing a cross-section once the stepper motor had finished moving. [1] The stepper motor was comprised of a motor shaft that rotates a precision screw which drives the carriage that is attached to the transducer.
This is an image of our experimental setup. Each component of the setup is labeled.:
A phantom is an object where the entire structure is known. Phantoms are what we used throughout the whole project. By using phantoms, it was easier to troubleshoot the Labview code to create an image because we had a good idea of how the image was supposed to appear. Phantoms can be made with agar, gelatin or some sort of medium that provides substantial attenuation. We sometimes wanted the impedance mismatching to be smaller than it would be for water or air, for example. The process for making a phantom is explained below.
This is an image of a part of the process when making a phantom. Water was heated to near boiling, then agar was added and stirred until the mixture was clear. After it was clear, an item was added. (For example a screw, apple, or something where the structure was known.) The entire phantom was then set and cooled with ice.:
Results
We found that the axial resolution to be 0.8+-0.2 mm and the lateral resolution to be 2.0+-0.2 mm, both at a frequency of 10 MHz. Since wavelength is inversely proportional to frequency, an increase in frequency corresponds to a general increase in resolution. Attached bellow are some of the cross-sections used to determine resolution.
Above is a figure showing a cross-section or two wires positioned vertically with a 1.1 mm gap between the two of them. The following cross-section was picked because it was a typical cross-section that is well resolved. There are two clear peaks were the wire is located and there is not much overlap.:
Above is a figure showing a cross-section or two wires positioned horizontally with a 1 mm gap between the two of them. The following cross-section was picked because it was a typical cross-section that is not resolved. There is only one clear peak and the peak's width is larger to the size of the gap between the two wires:
Above is a figure showing a cross-section or two wires positioned horizontally with a 2 mm gap between the two of them. The following cross-section was picked because it was a typical cross-section that is very near the Rayleigh's criterion. While there is one overall peak there are definite 'sub-peaks' with the distance between the two peaks being similar to the gap distance:
From theory, we expected that the lateral resolution would be on the order of the wavelength of the ultrasound pulse, since our wavelength was about 100 times the size of our steps. For the axial resolution, theory suggests it to be 0.37mm. For lateral resolution, our experimental value was at least 2 times larger than the theoretical(5%$\sigma$%).Similarly, for axial resolution, our experimental value was 2 times larger than the theoretical(2%$\sigma$%).
Conclusions
There are a few factors that could worsen our resolution. One issue that is hopefully not a problem is that our oscilloscope does not have enough data points to describe the point sources accurately. It is hopefully not a problem because before measurements began, we increased the number of data points the oscilloscope would break the signal into and the computer we were using was having memory overflow. One of the main factors that affect the actual resolution is that our transducers had a focal point. The focal point was 76.2 mm from the transducer's surface. If the boundary was not on the focal point, the ultrasound pulse would have a certain beam width. This beam width would cause the resolution to lower dramatically.
Another minor factor is that our transducer is likely not perfectly aligned with the dimension of the gap we are interested in. This would cause the gap to become smaller by a factor of %$cos(\theta)$% where %$\theta$% is the angle the transducer is makes with the plane the two wires create. But since this angle at most was 10 degrees, the gap distance would have changed at most to 98.5% its real value. So, the contribution to the uncertainty of misalignment is negligible.
More Images
This is an image of a wire phantom that was made and the corresponding ultrasound image.:
This is an ultrasound image of two wires on top of each other.:
This is an ultrasound image of two wires right next to each other.:
This is an ultrasound image of a hollow prism structure. The echo is also present.:
This is an ultrasound image of diagonal prism supported by wires.:
References
This experiment was inspired by:
1. Stiles, Timothy A. "Ultrasound Imaging as an Undergraduate Physics Laboratory Exercise." American Journal of Physics (2014) http://scitation.aip.org/content/aapt/journal/ajp/82/5/10.1119/1.4868000
2. Juvells, I., A. Carnicer, J. Ferré-Borrull, E. Martín-Badosa, and M. Montes-Usategui. "Understanding the Concept of Resolving Power in the Fabry–Perot Interferometer Using a Digital Simulation." European Journal of Physics 27.5 (2006): 1111-119. Web.