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MBW:Bacterial Aggregates in the Bloodstream

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Figure 1: Bacterial aggregate

By 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.

Context & Background

Bacterial pathogens like Klebsiella pnuemoniae are causes of community- and hospital-acquired blood stream infections, as well as many other common types of infections. These bacteria are known to form surface-adherent 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 post-fragmentation 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 Modeling

Finding an equivalent ellipsoid

Figure 2: Bacterial aggregate
Figure 3: COMs and ellipse semiaxes

Bacterial 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 side-by-side for comparison.

Figure 4(a): Aggregate
Figure 4(b): Equivalent ellipsoid

Forces on the ellipsoid

Consider the solid ellipsoid determined by the equation Ellipseeqn.png, with the semiaxes aligned as discussed previously. For small Reynolds numbers and the quasi-steady state assumption, the Navier-Stokes equations reduce down to the linear creeping flow equations:

Creep1.png and Creep2.png

where mu is viscosity and p is the pressure of the fluid.

For our ellipsoid we consider the boundary conditions given by:

Bcsorig.png

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 u0 is the constant free-stream velocity of the fluid.

The general solution to the linear creeping flow equations is given by Gensoln.png, where Ut.png is the flow field around a fixed ellipsoid in constant free-stream velocity and Ur.png is the flow field around a rotating ellipsoid in a linear ambient field. This superposition leads us to new boundary conditions:

Bcsnew.png

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 free-stream velocity is given by:

Ft.png

where n is the unit normal to the ellipsoid's surface and s is a vector dependent on u0 and semiaxis length. Similarly, the forces corresponding to a rotating ellipsoid in linear ambient flow are given by:

Fr.png

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]

Figure 5(a)
Figure 5(b)

Application: Flow in a Pipe

Figure 6

One 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 in-class presentation.

Figure 7

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

  • For information on how the body can defend itself from foreign invaders learn more about the Immune Compliment response.


Project Categorization

Mathematics Used

This model uses the Navier-Stokes PDE to describe the effects of fluid flow on bacterial aggregates and the forces acting on an ellipsoid.

Navierstokes.jpg

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 type

This 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 studied

This model studies bacterial aggregates, biofilms, looking at the surface created by individual molecules. Empirical data on the 3-D distribution of actual bacteria within a bacterial aggregate in the blood stream was used to approximate the shape of the clump.

Additional Paper

In "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 steady-state 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.

Figure 8: The light scattering and size distribution of aggregates with different hydrodynamic forces applied on them.

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

  1. D. M. Bortz, T. L. Jackson, K. A. Taylor, A. P. Thompson, and J. G. Younger. Klebsiella pneumoniae flocculation dynamics. Bulletin of Mathematical Biology, 70(3):745-768, April 2008.
  2. Binbing Han, S. Akeprathumchai, S. R. Wickramasinghe, and X. Qian. Flocculation of Biological Cells: Experiment vs. Theory. AIChE Journal, 49(7):1687-1701, July 2003
  3. 3.0 3.1 3.2 3.3 3.4 3.5 Stefan Blaser. Forces on the surface of small ellipsoidal particles immersed in a linear flow field. Chemical Engineering Science, 57 (2002) 515–526.
  4. Jeffery, G. B. (1922). The motion of ellipsoidal particles immersed in viscous fluid. Proceedings of the Royal Society A, 102, 161–179.
  5. Oberbeck, A. (1876). Ueber stationare Flussigkeitsbewegungen mit Berucksichtigung der inneren Reibung. Journal fur die reine und angewandte Mathematik, 81, 62–80 (in German).
  6. Ehrl, Lyonel, Miroslav Soos, and Massimo Morbidelli. "Dependence of aggregate strength, structure, and light scattering properties on primary particle size under turbulent conditions in stirred tank." Langmuir 24.7 (2008): 3070-3081.