
MBW:Bacterial Aggregates in the BloodstreamFrom MathBioBy Erin Byrne Understanding the hydrodynamic forces acting on an aggregate suspended in a fluid is vital to characterizing their possible fragmentation. We consider bacterial aggregates suspended in the bloodstream, discuss a way to determine an appropriate equivalent ellipsoid, and look at the hydrodynamic forces exerted on them. ContentsContext & BackgroundBacterial pathogens like Klebsiella pnuemoniae are causes of community and hospitalacquired blood stream infections, as well as many other common types of infections. These bacteria are known to form surfaceadherent biofilm colonies, and are common contaminants of indwelling catheters. Shedding from these colonies leads to bacterial aggregates suspended in the bloodstream, and could lead to systemic infection. Understanding the eventual fate of a initial population of aggregates in a host bloodstream is integral to developing effective treatment methods. Bortz, et al (2008)^{[1]} developed a PDE model to describe the flocculation dynamics of Klebsiella pneumoniae in suspension. Three important phenomena influence the behavior of the population of aggregates: proliferation, aggregation, and fragmentation. Fragmentation is the least understood of the three for any type of system, and recent literature has used at least 12 different functional forms to describe the postfragmentation distribution^{[2]}. For bacterial aggregates, there is 3D positional data for the individual bacteria in actual aggregates, and this information can be used to investigate fragmentation at a deeper level than has previously been explored. Understanding the hydrodynamic forces exerted on aggregates in the bloodstream by fluid motion is integral to studying their fragmentation. Blaser (2002) ^{[3]} outlines the analysis for determining the forces on the surface of an ellipsoid in a linear flow field based on previous work done by Jeffrey ^{[4]} and Oberbeck ^{[5]}. In the following sections I will demonstrate a method for finding an equivalent ellipsoid for an example aggregate, reproduce the results from Blaser ^{[3]} for the hydrodynamic forces using that ellipsoid, and discuss the motion of and forces on an aggregate in the flow in a pipe.
Mathematical ModelingFinding an equivalent ellipsoidBacterial aggregates have a variety of shapes and sizes, as can be seen in Figures 1 and 2. In general, they are roughly ellipsoidal, or even spherical. For this analysis, the aggregate in Figure 2 are used. Since the centers of mass (COMs) of the bacteria that compose this aggregate are known, Principle Components Analysis (PCA) can be applied to find the three primary directions in which the COMs vary spatially. These three directions will be the three semiaxes of the equivalent ellipsoid (Figure 3): The standard deviation of the COMs in each principle direction define the ratios of the three semiaxes to each other, and without loss of generality the axes are aligned such that the largest semiaxis coincides with the x axis, the second largest with the y axis, and the smallest with the z axis. Figure 4 displays the original aggregate with its equivalent ellipsoid sidebyside for comparison. Forces on the ellipsoidConsider the solid ellipsoid determined by the equation , with the semiaxes aligned as discussed previously. For small Reynolds numbers and the quasisteady state assumption, the NavierStokes equations reduce down to the linear creeping flow equations: where mu is viscosity and p is the pressure of the fluid. For our ellipsoid we consider the boundary conditions given by: where x is the center of mass of the ellipsoid, Gamma is the spatially independent velocity gradient tensor, omega is the angular velocity vector, and u_{0} is the constant freestream velocity of the fluid. The general solution to the linear creeping flow equations is given by , where is the flow field around a fixed ellipsoid in constant freestream velocity and is the flow field around a rotating ellipsoid in a linear ambient field. This superposition leads us to new boundary conditions: which allow us to consider the two flow fields separately. The forces on the surface on the ellipsoid corresponding to a fixed ellipsoid in constant freestream velocity is given by: where n is the unit normal to the ellipsoid's surface and s is a vector dependent on u_{0} and semiaxis length. Similarly, the forces corresponding to a rotating ellipsoid in linear ambient flow are given by: where chi is determined by semiaxis length and A is a tensor dependent on Gamma. For the full equations, I refer the reader to Blaser 2002 ^{[3]}. For our particular problem, we may assume the aggregate is traveling with the fluid and therefore only consider the forces on a rotating ellipsoid in linear ambient flow. Implementing the formulation in matlab and calculating the hydrodynamic forces on the example ellipsoid yields the image in Figure 5, recreating the shear stress shown in Blaser.^{[3]} Application: Flow in a PipeOne vital part of the analysis done by Blaser is that the velocity gradient tensor must be spatially independent. For flow in a pipe or tube this will not be the case. The general velocity profile for laminar flow in a cylindrical tube is shown in Figure 6. There are two important features of this type of velocity profile: fluid velocity is zero at the walls and the maximum velocity is reached at the center of the pipe, but the fluid shear is greatest at the walls and goes to zero at the center. For aggregates whose size is significantly smaller than the radius of the pipe, we can approximate the velocity gradient tensor with a spatially independent one locally and use it to determine the forces on the ellipsoid's surface as well as its motion. Due to the features mentioned previously, it can be easily shown that an ellipsoid close to the center of the pipe will have a high translational velocity and a low angular velocity, while an ellipsoid close to the pipe's wall will have a high angular velocity and a low translational velocity. Simulations of two ellipsoids were shown in the inclass presentation. In order to consider the forces on the surface of the ellipsoid in these two different motion scenarios, we need to understand the relationship between the shear rate and the magnitude of the forces. From the above equations and those in Blaser^{[3]}, we can calculate how the magnitude of the largest force vector on the ellipsoid's surface changes with the fluid's shear rate. The result in Figure 7 shows the magnitude varies linearly with the shear rate. Thus the aggregate located closer to the wall will have stronger forces acting on it from the fluid and will be more likely to fragment than one traveling closer to the center of the pipe. See Also
Project CategorizationMathematics UsedThis model uses the NavierStokes PDE to describe the effects of fluid flow on bacterial aggregates and the forces acting on an ellipsoid. where v is the flow velocity, ρ is the fluid density, p is the pressure, T is the (deviatoric) component of the total stress tensor, and f is forces acting on the fluid. Model typeThis model focuses on individual molecules as well as their aggregates. It looks at the different components of forces acting on an elliptical object that is flowing within a cylindrical pipe with laminar fluid flow. Biological system studiedThis model studies bacterial aggregates, biofilms, looking at the surface created by individual molecules. Empirical data on the 3D distribution of actual bacteria within a bacterial aggregate in the blood stream was used to approximate the shape of the clump. Additional PaperIn "Dependence of Aggregate Strength, Structure, and Light Scattering Properties on Primary Particle Size under Turbulent Conditions in Stirred Tank,"^{[6]} the authors validate the conclusions that Blaser ^{[3]} comes to in his study via empirical evidence. The authors ran experiments in which they observed the steadystate structure of particle aggregates under various conditions of shear stress and solid volume fraction. Using light scattering, they were able to describe the mass distribution of each of the aggregates and obtain a critical aggregate size for breakage for lower solid volume fractions. For large and compact aggregates, it was found that there is a clear relationship between the maximal size of the stable mass after fragmentation and the hydrodynamic stresses applied to the system. These results are depicted in figure 8. These results match the expectations that Blaser had using his model on the effects of shear stress on the fragmentation of elliptical aggregates under different flow conditions. References
