Comparison of Poroviscoelastic Models for Sound and Vibration in the Lungs

1.1. Background. Like no other anatomical region in the body, the lungs are a unique, multiphase porous structure that has defied conventional noninvasive medical imaging methods and our ability to contrast and quantify changes in its macroscopic properties that can be indicative of disease and that may be fundamentally linked to behavioral and structural changes at the microscopic scale. Patients can suffer from a wide range of pulmonary ailments that result in significant changes, locally or diffusely, to the stiffness or density in the lungs [1]. For example, lung parenchymal stiffness increases with the degree of fibrosis in fibrotic lung [2,3]. While in asthma, increased degree of bronchoconstriction is associated with an increase in parenchymal shear modulus [4]. In contrast, the lung becomes less stiff with the degree of emphysema, primarily due to the remodeling of collagen fibers [5,6]. These changes often are not easily identifiable by most imaging modalities. The utility of conventional ultrasound pulmonary imaging is severely limited, due to the acoustic impedance mismatch between the air in the lungs and soft tissue. X-ray computed tomography (CT) and magnetic resonance imaging (MRI) provide useful anatomic information, but are often limited in their diagnostic accuracy, especially in distinguishing benign, infectious, and malignant pathologies. CT also has the disadvantage of cancer risk associated with ionizing radiation. Spirometry, including the measurement of the volume of inhaled or exhaled air as a function of time, provides a global measure of lung and airway properties but often provides relatively nonspecific findings. Sputum monitoring and respiratory tests before and after the administration of bronchial dilators to assess changes in airway plasticity similarly provide global and, at best, indirect information on spatial extent. MRI using RF tagging techniques has been suggested as a method for assessing the regional mechanical properties of the parenchyma [7,8], but this approach has limitations and does not permit tracking through an entire respiratory cycle. Lung functional and structural imaging based on an array of contact acoustic sensors placed on the back has been researched for the past decade or so [9–11] and has recently gained more prominence through the burgeoning success of such systems as Deep Breeze™, a commercial product utilizing up to 40 vacuum-mounted contact acoustic sensors on the patient's back or integrated into their bed to provide a real-time assessment of lung sound strength, spectral content, and regional variation, all of which may be beneficial to diagnosis [12–14]. Beyond obtaining an image that depicts the distribution of lung sounds on the torso surface, if a better understanding of mechanical wave propagation within the lungs and torso were available, one may be able to reconstruct the wave field within the lungs and torso based on the noninvasive surface measurements. This would take the two-dimensional surface image into three dimensions and could potentially provide not only the location but also more quantitative information about the properties of the lung that can affect how sound and vibration propagate through it [8]. The benefit of coupling an array measurement on the surface with an improved computational model of sound propagation within the torso was demonstrated fundamentally in Ozer et al. [15]. In phantom studies, it was shown that the use of a computational boundary element model of lung acoustics combined with a surface array measurement was significantly superior in identifying the dominant source location of the sound as compared to a simple “ray acoustics” model that neglects the more complex nature of sound transmission in a finite and complex dimensioned structure. Also recently, the phase-contrast-based technique known as magnetic resonance elastography (MRE) has been applied to the lungs in pilot studies with limited success [16–19]. MRE seeks to provide a map of the viscoelastic properties within the region of interest that will affect the shear wave motion that MRE measures. Previously, MRE has been successfully applied to the study of the mechanical properties of a variety of other organs and soft tissue regions in vivo, including the breast, brain, kidney, prostate, liver, and muscle [20–24]. Application to the lungs has proven more challenging, given the poor signal-to-noise available in imaging due to a lower presence of hydrogen in air than in soft tissue (water) and the complex nature of vibratory wave propagation found in the lungs. Again, the authors propose that a better understanding of mechanical wave motion in the lungs would aid in the interpretation of the wave images that are acquired using MRE to reconstruct a quantitative map of variation in mechanical properties that can correlate with injury, the progression of disease, and/or the response to therapy. The lung parenchyma is comprised of soft biological tissue and vasculature, as well as millions of microscopic air sacs (alveoli) that are connected through a complex branching airway structure. Thus, microscopically, the lungs are highly heterogeneous in terms of their physical properties, combining gas (air) that is linked through a complex and tortuous network of channels and microscopic sacs, non-Newtonian liquid (blood) that flows through an equally complex network of vessels of wide-ranging dimensions, and solid tissue structure comprised of a mixture of viscoelastic soft tissues that exhibit nonlinear behavior under large deformation. However, for the purpose of developing a tractable set of equations for predicting small-amplitude mechanical wave motion in the parenchyma for wavelengths larger than the microscopic heterogeneous features of the lung, macroscopic homogenized representations of the lung's physical properties have been proposed. Based on this homogenous or stochastic spatially averaged view, two different models for wave propagation have been put forth. One is sometimes referred to as the “effective medium” or “bubble swarm” theory. It has been prominently used in the literature for modeling lung acoustics since the 1980s [25–27]. More recently, there has been an interest in applying Biot's theory of poroelasticity to the lung [28]. Application of Biot theory leads to a more complex theoretical model that predicts more wave types as compared to the effective medium theory. From a practical and ultimately clinical perspective, questions of interest include: (1) how do these theories compare to each other and to experimental measurements; (2) how complex does the theory need to be to capture the salient phenomena that is measurable and can be linked to disease or injury; and (3) how easily are these theories applied or integrated into computational frameworks that would enable one to better understand and quantify with specificity and significance how mechanical wave phenomena, which may be measured by application of the existing or nascent imaging technologies mentioned above, are affected by disease and pathology.