Mathematical principles of ultrasound
Ultrasound is a medical imaging technique that employs sound waves at frequencies higher than human hearing (above 20,000 Hz) to visualize internal structures of the body. It operates by sending sound waves into the body, which then reflect off tissues and return to the device, enabling real-time imaging. This method has been effectively used in various fields since its inception in the 1920s, with significant advancements in the medical domain since the 1960s. One of its key advantages is that it is non-invasive, poses no harm to patients, and is relatively cost-effective compared to other imaging technologies.
Mathematical principles play a vital role in ultrasound, particularly in analyzing how sound waves propagate through different materials, such as human tissues. The speed of sound in a medium can be calculated using equations that incorporate factors like elasticity and density, allowing healthcare providers to assess conditions such as bone quality and fracture risk. Through sophisticated algorithms, ultrasound can generate detailed two- and three-dimensional images, aiding in the detection of various medical conditions, including fetal development issues and musculoskeletal problems. This makes ultrasound an invaluable tool in modern diagnostics, continually evolving with advances in computer science and engineering.
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Mathematical principles of ultrasound
Summary:Ultrasound uses mathematical principles to create images of the human body.
Although ultrasound cannot be heard by humans, it has been produced and used for a vast number of applications in many different fields. In industry, ultrasound has been used as a technique to assess the structural integrity of materials. The interaction between ultrasound and live systems has been studied since the 1920s. During the 1960s, it was used in medicine, initially as a therapeutic option and then later as a diagnostic resource. In the twenty-first century, ultrasound is a major medical imaging technology widely used in clinical facilities around the world because it causes no harm to the human body and results can be achieved in real time, besides the fact it is considerably cheap and easy to use. The available technologies using ultrasound are in constant development. Every new application depends on the advance of computer sciences that work with many concepts of physics and the solution of mathematical problems in this field seems inexhaustible.
Sound is a form of energy consisting of the vibration of molecules of an environment that can be air, water, solid, or biological tissues (such as bones and muscles). This kind of energy propagates across the medium in the form of waves. Sound is a mechanical wave whose fundamental characteristics are amplitude, which is the distance between the highest and lowest point of the wave and frequency, which is the number of cycles that occur in a second, measured in hertz (Hz). Humans are able to detect sounds with a frequency of 20–20,000 Hz–the normal limits of the human hearing. The term “infrasound” refers to sound waves that have a frequency lower as 20 Hz, and sounds with a frequency higher than 20,000 Hz are called “ultrasound.” Unlike humans, some animals, such as bats, dolphins, whales, dogs, cats, and mice can hear ultrasound.
Imaging the Human Body
While traversing a material, the properties of ultrasound change in intensity and speed of propagation, which means that ultrasound waves travel at different speeds depending on the material. Consider two samples of human bone, one from a 30-year-old person and the other from an 80-year-old person. If ultrasound waves cross these two bony samples, the speed at which the sound propagates in the bones can be represented algebraically by the following equation:
where ν is the speed of ultrasound in the bone sample, E is the modulus of elasticity of the bone sample, and ρ is the density of the bone sample.
The speed of sound (ν) can be calculated by measuring the time required for the wave to propagate through the bone and then dividing by the width of the bone. Knowing the density of the bones (ρ), this equation could be used to determine the values of the modulus of elasticity (E) that indicates the elastic properties of the bone. In a 30-year-old person, the speed of the sound through the bone is approximately 4000 m/s. In an 80-year-old person, this rate drops to 3800 m/s. This fact means that the higher the speed of the sound through the bone, the better is the quality of bone. A low speed could reveal a bony fragility and a fracture probability. This principle is used in ultrasonometry, a technique used to estimate the bony fracture or osteoporosis risk in patients. Ultrasound medical imaging is one of the most powerful diagnostic tools in modern medicine. Along with other imaging methods, it is based on advanced mathematical techniques and numerical algorithms that are necessary to analyze the data and produce readable pictures or three-dimensional images of inner body structures without surgery or use of radiation. It has been widely used to identify the sex or to detect malformations in fetuses during gestation.
Bibliography
Ammari, Habib. An Introduction to Mathematics of Emerging Biomedical Imaging. Berlin: Springer, 2009.
Gibbs, Vivien, et al. Ultrasound: Physics and Technology. Philadelphia: Churchill Livingstone, 2009.