Medical imaging and mathematics
Medical imaging combines advanced technology and mathematical principles to visualize the internal structures of the human body for diagnostic purposes. Since the introduction of X-rays by Wilhelm Conrad Röntgen in 1885, medical imaging has evolved significantly, leading to the development of techniques such as ultrasound, computed tomography (CT), and magnetic resonance imaging (MRI). These imaging modalities rely on mathematical models to interpret data and reconstruct images, utilizing algorithms that involve signal processing, noise analysis, and geometric techniques.
Medical images play a crucial role not only in diagnosis but also in guiding surgical procedures and planning interventions, particularly in minimally invasive surgeries. Additionally, understanding and improving image quality necessitates careful manipulation of various parameters to minimize risks associated with ionizing radiation, such as cancer. Techniques like bone densitometry and scintigraphy further illustrate the diverse applications of medical imaging, serving specific functions like assessing bone health and evaluating cardiovascular conditions, respectively.
Mathematical definitions and image processing enhance the clarity and clinical utility of medical images, providing critical information about anatomical structures and conditions. As technology advances, the interplay between mathematics and medical imaging continues to evolve, offering new solutions to meet the growing demands of healthcare.
Medical imaging and mathematics
Summary: Mathematical models interpret measurements, and algorithms construct images used in the health industry.
Until the late nineteenth century, the structures of the human body were represented only by illustrations found in medical books. However, in 1885, Wilhelm Conrad Röntgen introduced the humanity into a new path in the world of images: the access to visual information from inside the human body. He used X-rays, which pass through objects with different densities producing images on photographic plates. Since the insertion of radiographic diagnosis, new technologies have brought great progress for medical diagnosis, such as ultrasound, computed tomography, and magnetic resonance imaging (MRI). Furthermore, medical images are currently used for navigation systems that guide surgeons during surgical interventions or aid in surgical planning, for example, in minimally invasive operations. Mathematical models interpret measurements and algorithms reconstruct images. Signal processing and noise analyses, as well as geometric, statistical, and algebraic techniques, are fundamental in this area. Mathematicians have also been the subjects of medical imaging studies. For example, one study found that mathematicians had an increased gray matter density in the cortical regions.
X-Rays
When a physician performs a radiograph on an arm, the image is obtained in different shades of gray, aiding the identification of different anatomical structures. This identification is possible only because the arm is a structure formed by tissues of different densities, such as muscles, bones, and cartilage. The possibility of differentiation of these tissues occurs because of attenuations caused by a partial or total absorption of the rays before the formation of the image. Since X-rays are a type of ionizing radiation, they can cause damage to the human body, such as cancer, if used in excess. Modern equipment has been developed to minimize this risk. On the other hand, it is necessary to manipulate parameters that affect image quality and at the same time to control the amount and the dose distribution of this material on the patient.
Other Medical Imaging Devices
While X-rays detail the morphology of bone structures, bone densitometry provides the mineral content of the bone. This technique is used to control and to prevent osteoporotic fractures. With the advent of computerized tomography and magnetic resonance imaging, the human body is being studied in a segmented way. These advanced imaging techniques are especially useful in the study of central nervous system disturbances. Ultrasound is a diagnostic tool that, like magnetic resonance, does not use ionizing radiation. It is used to investigate soft tissues and is based on reflection of high-frequency sound waves to form two- and three-dimensional images, for example, in monitoring fetal development. Some diagnostic imaging techniques require the use of tracer substances. Scintigraphy, for instance, is a technique used for the evaluation of the cardiovascular system. This procedure uses the injection of radioactive substances to provide a two-dimensional image through the use of radioisotopes.
Resolution and Software
The spatial resolution of a digital image refers to the amount of points per unit of measure that allows the perception of details of a structure. Each point or constituent element of the array image is called a “pixel” (an abbreviation of “picture element”). The pixel is the smallest unit that can conduct operations. Colored or gray levels inform the size and location of the structure analyzed. Image processing is used to reduce interference and to increase the contrast to aid the analysis of the structures. It is possible to use mathematical techniques to manipulate the pattern of gray pixels. The interaction with neighboring pixels highlights structures of interest.
The mathematical definition of the images provides important clinical information, such as the size of lesions or fetal structure length, as well as morphology of structures, gland volume, blood supply area, and the monitoring of prostheses. Without these appropriate tools to analyze medical data, the images could be devoid of concrete meaning and require the use of complex computing resources to process the data. To achieve a medical image in real time, complex mathematical algorithms are needed. Diverse software has been appearing to meet the growing demands in the medical field, as well as needs concerning the storage and handling of patient data. Innovations continue to meet the growing challenges in this dynamic field.
Bibliography
Epstein, Charles. Introduction to the Mathematics of Medical Imaging. 2nd ed. Philadelphia: Society for Industrial Mathematics (SIAM), 2007.
Gonzalez, R. C., and R. E. Woods. Digital Image Processing. 3rd ed. Upper Saddle River, NJ: Pearson Education, 2008.
Natterer, Frank. The Mathematics of Computerized Tomography. Philadelphia: SIAM, 2001.