Toward Tomographic Ultrasound Imaging in the ICU

By Jennifer Mueller

A patient lies unconscious in the intensive care unit (ICU) suffering from head and chest trauma and a neck injury sustained in a car accident. The mechanical ventilation keeping him alive also puts him at risk of a tension pneumothorax, a condition in which air enters the space between the lung and chest cavity. If undetected, it is life-threatening. If detected, a practitioner can promptly treat it by inserting a chest tube, and the pneumothorax will heal. When a tension pneumothorax develops, the patient’s oxygen levels drop. Since he is unconscious, he cannot complain of the pain and shortness of breath that accompany a pneumothorax and serve as early indicators of the ailment. Tension pneumothoraces are often undetected by a bedside chest x-ray, but moving the patient to the CT scanner is very risky due to his head and neck injuries. Will the pneumothorax go undetected?

A bedside imaging method for diagnosing lung pathology in the ICU would benefit patients like this. Electrical impedance tomography (EIT) and ultrasound computed tomography (USCT) are two promising techniques. Tomographic imaging is a form of cross-sectional slice imaging, as with computed tomography (CT) and positron emission tomography (PET) scans. In the case of EIT or USCT, doctors place electrodes or transducers respectively around the circumference of the region of interest (such as the patient’s chest), record the transmitted electromagnetic or acoustic signals, and compute a cross-sectional image of the region. In USCT, propagation of the acoustic waves is governed by the wave equation in the time domain, or the Helmholtz equation in the frequency domain (where we assume for simplicity that the medium has constant density):

\[ \triangle p(x)+k^2(x)p(x)= -\phi^{\textrm{inc}}(x), \:\:\:\:\: x \in \Omega. \tag1 \]

Here, \(p(x)\) is the acoustic pressure, \(\phi^{\textrm{inc}}(x)\) is the acoustic source, \(k(x)= \omega/c(x)\) is the wave number (where \(\omega\) is the angular frequency of the incident harmonic wave), and \(c(x)\) is the speed of sound in the medium. USCT for lung imaging differs from the traditional ultrasound, in which images are computed from a probe called a linear array (likely familiar to readers in the context of fetal imaging) with the transducers arranged in a rectangle instead of surrounding the region to be imaged. Also, the acoustic signals are applied at a low frequency (\(<750\) kilohertz, compared to over \(1\) megahertz for the traditional ultrasound) to facilitate the acoustic signal’s penetration of the lung, which is simply reflected if the frequency is too high.

Computing an image from the measured pressures on the transducer is an inverse problem where the objective is to calculate the coefficient of sound speed \(c(x)\) in \((1)\). The method of computing the sound speed distribution inside the body is called the reconstruction algorithm. In our recent work, we applied the distorted Born iterative method (DBIM) to reconstruct simulated data representing pneumothoraces of three sizes and locations in the chest. DBIM is a regularized Newton-based iterative method, and total variation regularization was used to combat the inverse problem’s ill-posedness. The images in Figure 1, computed from simulated data, demonstrate USCT’s potential for detecting a pneumothorax at the bedside.

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The author presented this research during a minisymposium at the 2019 SIAM Conference on Computational Science and Engineering, which took place earlier this year in Spokane, Wash.

Acknowledgments: This project is supported by Award Number R21EB009508 from the National Institute of Biomedical Imaging and Bioengineering (NIBIB). The content is solely the responsibility of the author and does not necessarily represent the official view of the NIBIB or the National Institutes of Health.

Further Reading
[1] Ball, C.G., Hameed, S.M., Evans, D., Kortbeek, J.B., Kirkpatrick, A.W., & Canadian Trauma Trials Collaborative. (2003). Occult pneumothorax in the mechanically ventilated trauma patient. Can. J. Surg., 46(5), 373379.
[2] Mueller, J.L., & Siltanen, S. (2012). Linear and Nonlinear Inverse Problems with Practical Applications. In Computational Science & Engineering. Philadelphia, PA: Society for Industrial and Applied Mathematics.
[3] Mueller, J.L., C´ardenas, D., & Furuie, S. (2019). A preclinical simulation study of ultrasound tomography for pulmonary bedside monitoring. In Proceedings of the Second International Workshop on Medical Ultrasound Tomography. Detroit, MI.

Jennifer Mueller is a professor of mathematics and biomedical engineering at Colorado State University (CSU) with a courtesy appointment in the Department of Electrical and Computer Engineering. She received her Ph.D. from the University of Nebraska-Lincoln and was a National Science Foundation postdoctoral fellow at Rensselaer Polytechnic Institute before joining the faculty at CSU. Her research in inverse problems encompasses development of new hardware, reconstruction algorithms, and clinical applications for electrical impedance tomography and ultrasound computed tomography. Mueller is committed to research with a clinical impact, and her work—supported by the National Institutes of Health—involves close collaborations with mathematicians, engineers, and physicians in the U.S. and abroad. She serves as an associate editor for IEEE Transactions on Medical Imaging; the SIAM Journal on Applied Mathematics; Inverse Problems; and the IEEE Journal of Electromagnetics, RF and Microwave in Medicine and Biology. She is also an editorial board member of the SIAM book series on Advances in Design and Control. She is co-author (with Samuli Siltanen) of Linear and Nonlinear Inverse Problems with Practical Applications.