Systems Model of Mucociliary Clearance: Fluid-Structure Interaction Sorin Mitran Applied Mathematics UNC

Overview Applied mathematician - don’t know anything about: cilia, PCL, mucus However, please tell me what cilia, PCL, mucus are and: –A computational model will verify that they can work together as thought –Restrict parameter ranges –Suggest experimental measurements

Example Model Mucus – Upper Convected Maxwell viscoelastic fluid PCL – Newtonian fluid (saline) Cilia – Flexible, large deformation active beams If the above are not true don’t blame the computational results! Change the model.

Mucus – Viscoelastic fluid There are no sinks/sources in the mucus Mucus momentum is changed by: pressure, friction, elastic forces Elastic forces are transported by fluid velocity, decay at rate and arise from fluid strain

PCL – Newtonian fluid There are no sinks/sources in the PCL PCL momentum is changed by: pressure, friction, elastic forces

Cilia – Active, flexible axoneme structure Clear internal structure Undergoes large deformation Is fixed onto a flexible substrate Introduce finite element model: –Large deformation beams for microtubules –Elastic springs for nexin links, spokes –Elastic cilium membrane

Overall model Once components are defined: –Devise numerical algorithms for the equations (e.g. viscoelastic fluid simulation still a research topic) –Assess difficulties and chances of successful computational modeling: Very many cilia – but they might be simulated in parallel Cilia undergo large deformations – use Lagrangean finite element model

Overall Model – Computational Setup Overlaid grids Deforming grid around each cilium Cartesian grid within overall domain

Overall Model – The Result

Overall Model – Or Some More

Overall Model – The Result

What’s it good for? Pretty pictures? Sure, if it’s prerequisite to funding More seriously, from the model one can: –Impose a beat pattern in a known PCL and extract axoneme forces –Impose axoneme forces in a known PCL and obtain a beat pattern –Compute: The total force transmitted to fluid by cilia The unsteady PCL+mucus velocity field

Forces near cilium – as it beats

Forces from cilium surface outwards

Unsteady velocity field – near cilium

Unsteady velocity field – cilium surface

Unsteady velocity field – go around cilium

Impose beat pattern – obtain axoneme forces

Show effect of axoneme structural defects

What else can be done? As applied mathematicians we don’t know what questions to ask – please forward some. We have some ideas on system characterization – Sensitivity analysis –A rigorous technique for assessing influence of any one parameter or group of parameters on a system outcome

It’s just a tool … As computational scientists we’re in the tool-building business – let us know what you need. –PCL is not Newtonian! – OK, what is it? As soon as that’s established and put into equations a new systems model component will replace the current Newtonian fluid –The model is too expensive to run! – OK, what kind of data do you need? The unsteady velocity field? Tough luck, the expense is imposed by the system you’re studying. An average transport rate? Great, that can be extracted with little computational effort.

What’s on offer Gory description of details: S. Mitran, Computers & Structures, in press, July 2007 Single cilium computation – Stand-alone program almost complete. Will easily run on a laptop. Allows fooling around with axoneme structure. Send requesting code on Mar. 15 Pretty pictures: –A Gallery of Ciliary Motion (