According to the Centers for Disease Control and Prevention, approximately 610,000 people die of heart disease in the United States each year; this statistic accounts for one in every four deaths, and makes it the leading cause of death in both men and women. Cardiovascular disease, an umbrella term for problems stemming from heart and blood vessel complications, generally signifies conditions in which patients have narrowed or thickened blood vessels, which can lead to heart attack, angina (chest pain), or stroke.
Hemodynamics is the dynamics of blood flow. It explains the physical laws that govern the flow of blood via the vascular system, which is comprised of vessels. These vessels transport the necessary nutrients, hormones, carbon dioxide, and oxygen throughout the body, which work collectively to fight disease; maintain homeostasis; and stabilize temperature, cell-level metabolism, osmotic pressure, and pH levels. Blood enters the heart through two large veins: the inferior and superior vena cava. The right atrium and ventricle pump oxygen-poor blood from the body to the lungs to be reoxygenated, while the left atrium and ventricle pump the freshly-oxygenated blood back to the body; this is called pulmonary circulation. In healthy patients, pulmonary circulation is a low-pressure system. But a massive increase in pressures occurs in patients with pulmonary disease, often affiliated with heart problems. Physicians typically test for pulmonary disease via Swan-Ganz catheterization by inserting a thin tube (catheter) into the ride side of the heart through the anterior jugular vein, a process that is both invasive and unpleasant.
Patient-specific models of hemodynamics and cardiac function are effective tools in analyzing and understanding the progression of cardiovascular disease. Scientists frequently use medical imaging to obtain a patient-specific geometry, which renders a more accurate domain for the models. While image segmentation can be a viable technique to test for the pulmonary issues indicative of cardiovascular disease, researchers do not know much about the process’ specifics as an alternative to catheterization. During a minisymposium at the 2018 SIAM Annual Meeting, currently taking place in Portland, Ore., Mitchel J. Colebank of North Carolina State University incorporated geometric uncertainty into a one-dimensional fluid dynamical model of a pulmonary blood vessel network in search of an alternative, noninvasive test for pulmonary disease. “In an ideal world, if we could do all the diagnostics without catheterization that would be great,” he said.
Colebank works with images of the pulmonary networks in a common strain of laboratory mouse, which resemble human networks in their rapid branching. “We can use this information to understand characteristics of pulmonary networks,” he said. “If we’re going to use computational fluid dynamics, we can study how to incorporate some of these geometries into our model.” The image segmentation technique known as global thresholding adds uncertainty to the geometric parameters that guide model prediction. In global thresholding, a cutoff exists between the foreground of images (things of interest, such as the lungs) and the background. Colebank’s study involves two parameters: the lower threshold (intensities in the foreground) and the smoothness of the transition from background to foreground. He is interested in the types of geometric effects that change these parameters and thus the network’s dimensions. “If you’re going to make physiological conclusions about a patient, these need to be quantified,” Colebank said of the parameters.
Colebank employs open-source image segmentation tools to transform the pulmonary network images into three-dimensional geometries, which yields radius estimates and a collection of centerlines. He then uses MatLab to organize this information into a so-called “tree.” By assuming that blood vessels are more or less cubes that one can transform into cylindrical coordinates, Colebank simplifies the Navier-Stokes equation to yield a one-dimensional fluid model. He then uses a linear elastic wall model to close the system and study the way in which vessel walls react to pressure changes.
Every terminal vessel in the network is connected to a three-element windkessel (directly translated from German as “air chamber,” but more frequently used to mean “elastic reservoir”). To account for images that are no longer within the image resolution, Colebank turns to windkessel boundary conditions and rearranges some of the variables to produce a flow distribution dependent on radius, length, and previous blood vessel flow. He extends the calculation of normal parameter estimates via perturbation analysis, which leads to drastic variation in flow distribution of plus or minus 80 percent.
“We took the same medical image and segmented it 25 times with different segmentation parameters to see the effect of the model,” he said. “We let our segmentation go on for a fixed number of iterations to get as many vessels as it can.” The total cross-sectional area varies among parameters, and a large change in flow profiles occurs downstream. Colebank is particularly interested in how the number of blood vessels change, as large fluctuation occurs further down the network.
Ultimately, Colebank demonstrates that model parameters and predictions are sensitive to the model’s measured geometry. He concludes that vascular geometry is significant in the estimation of nominal model and boundary condition parameters, and shows that the number of generations in the model can dictate prediction shape. Colebank also cautions that researchers should be careful of using the geometry they obtain “as-is,” without checking the model’s sensitivity to geometric parameters.
As he continues his research, Colebank plans to employ more than 25 segmentations because more segmentations might allow for better statistical methods. He also hopes to investigate the direct relationship between segmentation parameters and anatomical/model features, and is curious as to how these dynamics change in human patients and disease. “At the end of the day, we want to know the pressure in the pulmonary arteries so that we don’t need to use invasive techniques,” he said.
||Lina Sorg is the associated editor of SIAM News.