Mathematics of microrheology with applications to pulmonary liquids

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  • March 20, 2019
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  • Xu, Ke
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • This thesis results from the Virtual Lung Project at the University of North Carolina at Chapel Hill, whose target is to understand the mechanism of this disease and provide guidance for effective therapeutic strategies. Instead of taking the problem as a whole, the focus here is to develop new methods to characterize rheological properties of low volume biological samples, such as mucus, sputum and their simulants, as well as to work out fluid dynamical behaviors that are associated with experiments using micro-scale beads in biological materials. Classical rheological experiments (creep, relaxation and dynamical) are mostly designed for the averaging steady state properties of milliliter size samples. The difficulties lie in the fact that these biological samples are low-volume (on the order of microliters), highly heterogeneous, sensitive to the surrounding environment and subject to change over time, even during the same course of a constant stress load. The term thixotropy is then used to describe the property of time-dependent change in viscosity. Therefore, in this very first problem, we follow Baravian et al. to exploit inertia in the creep device, which is always present until transients pass, to gain rheological information beyond the typical creep data analysis. A MATLAB graphical user interface (GUI) is developed to allow users to fit different mechanical models to the data by least square fits. In our studies of biological samples, we show that the time average is a poor reflection of data, instead, allowing time-dependence in the material parameters during a constant loading is the correct methodology of studying thixotrophy. We also address the difference of using rheometers of different length scales: cone-and-plate on milliliters and parallel-plate on microliters and associate this difference with the size of macromolecular structures of the materials. As a proof, we show that the CP and PP yield similar rheological properties (same order of magnitudes) for hyaluronic acid but quite different ones (at least one order difference in the magnitudes) for agarose gels. For our second problem, our goal is to develop new methods for characterizing viscoelastic properties of biological liquids by the driven bead experiments done by our collaborators at UNC Physics department. The standard method relies on a force balance argument with an ad hoc geometry factor and fitting with 1D mechanical models. Instead of following this method, we solve the 3D unsteady Stokes equations with specified driving force. These results extend classical solutions of the Stokes equations, called Stokes singularities, from a viscous to a linear viscoelastic medium. With the viscoelastic version of Stokes singularities, we are able to give exact solutions for points, spherical and planar forces, as illustrated in Chapter 3. The difference between a point and a spherical source is also addressed in this chapter.
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  • In Copyright
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  • Forest, M. Gregory
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  • Open access
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  • October 11, 2010

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