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We study the mathematical modeling and numerical simulation of the motion and deformation of red blood cells (RBC) and vesicles subject to an external incompressible flow in a microchannel. RBC and vesicles are viscoelastic bodies consisting of a deformable elastic membrane enclosing an incompressible fluid. We provide an extension of the Finite Element Immersed Boundary Method based on a model for the membrane that additionally accounts for bending energy and also consider inflow/outflow conditions for the external fluid flow. The stability analysis requires both the approximation of the membrane by cubic splines (instead of linear splines without bending energy) and an upper bound on the inflow velocity. In the fully discrete case, the resulting CFL-type condition on the time step size is also more restrictive. We perform numerical simulations for various scenarios including the tank treading motion of vesicles in microchannels, the behavior of 'healthy' and 'sick' RBC which differ by their stiffness, and the motion of RBC through thin capillaries. The simulation results are in very good agreement with experimentally available data.
The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multi-scale, multi-physics problem. Therefore, reduced order modeling techniques such as balanced truncation model reduction, proper orthogonal decomposition, or reduced basis methods are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. In particular, we are interested in such shape optimization problems where the design issue is restricted to a relatively small portion of the computational domain. In this case, it appears to be natural to rely on a full order model only in that specific part of the domain and to use a reduced order model elsewhere. A convenient methodology to realize this idea consists in a suitable combination of domain decomposition techniques and balanced truncation model reduction. We will consider such an approach for shape optimization problems associated with the time-dependent Stokes system and derive explicit error bounds for the modeling error. As an application in life sciences, we will be concerned with the optimal design of capillary barriers as part of a network of microchannels and reservoirs on microfluidic biochips that are used in clinical diagnostics, pharmacology, and forensics for high-throughput screening and hybridization in genomics and protein profiling in proteomics.
We consider the mathematical modeling and numerical simulation of high throughput sorting of two different types of biological cells (type I and type II) by a biomedical micro-electro-mechanical system (BioMEMS) whose operating behavior relies on surface acoustic wave (SAW) manipulated fluid flow in a microchannel. The BioMEMS consists of a separation channel with three inflow channels for injection of the carrier fluid and the cells, two outflow channels for separation, and an interdigital transducer (IDT) close to the lateral wall of the separation channel for generation of the SAWs. The cells can be distinguished by fluorescence. The inflow velocities are tuned such that without SAW actuation a cell of type I leaves the device through a designated outflow channel. However, if a cell of type II is detected, the IDT is switched on and the SAWs modify the fluid flow such that the cell leaves the separation channel through the other outflow boundary. The motion of a cell in the carrier fluid is modeled by the Finite Element Immersed Boundary Method (FE-IB) featuring a coupled system consisting of the incompressible Navier-Stokes equations with respect to a Cartesian coordinate system and the equation of motion of the cell described in a Lagrangian framework. The generation of the SAWs is taken care of by the linearized equations of piezoelectricity, and the impact of the SAWs on the fluid flow is realized by means of a boundary condition for the Navier-Stokes equations. The discretization in space is done by P2/P1 Taylor-Hood elements for the fluid flow and periodic cubic splines for the immersed cell, whereas for discretization in time we use the backward Euler scheme for the Navier-Stokes equations and the forward Euler scheme for the equation of motion of the immersed cell. This backward Euler/forward Euler Finite Element Immersed Boundary Method (BE/FE FE-IB) requires a CFL-type condition for stability. Numerical results are presented that illustrate the feasibility of the surface acoustic wave actuated cell sorting approach.
Comparative Study of the Dynamics of Lipid Membrane Phase Decomposition in Experiment and Simulation
(2012)
Phase decomposition in lipid membranes has been the subject of numerous investigations both experimentally and by theoretical simulation. Yet quantitative comparisons of the simulated data to the experimental results are rare. In this work we present a novel way of comparing the temporal development of liquid ordered domains obtained from numerically solving the Cahn–Hilliard equation and by inducing phase transition in giant unilamellar vesicles (GUVs).
Quantitative comparison is done by calculating the structure factor of the domain pattern. It turns out that the decomposition takes place in three distinct regimes both in experiment and simulation. These regimes are characterized by different rates of growth of the mean domain diameter and there is a quantitative agreement between experiment and simulation as to the duration of each regime and the absolute rate of growth in each regime.
The sorting of biological cells using biological micro-electro-mechanical systems (BioMEMS) is of utmost importance in various biomedical applications. Here, we consider a new type of devices featuring surface acoustic wave (SAW) actuated cell sorting in microfluidic separation channels. The SAWs are generated by an interdigital transducer (IDT) and manipulate the fluid flow such that cells of different type leave the channel through designated outflow boundaries. The operation of the device can be formulated as an optimal control problem where the objective functional is of tracking type, the state equations describe the fluid-structure interaction between the carrier fluid and the cells, and the control is the electric power applied to the IDT.
Comparative study of the dynamics of lipid membrane phase decomposition in experiment and simulation
(2013)
High-Voltage Dielectrophoretic and Magnetophoretic Hybrid Integrated Circuit/Microfluidic Chip
(2009)
