Mass conservation equations are well established and have been applied to many biological systems such as bioreactors and isolated tissues. These conservation equations are typically written in terms of partial differential equations with boundary conditions that depend on the system dimensions, internal interfaces, and interactions with the external environment. For example, the concentration boundary conditions for an extracellular matrix at the surfaces of cells are described by cellular uptake and efflux rate expressions. Since the determination of velocities, diffusion coefficients, and cellular uptake rates is fairly standard (e.g., Albright et al, 1999; Brune & Kim, 1993; Truskey et al, 2004), this effort focuses on how to use this information to control growth factor release rates.
The problem is formulated as a distributed parameter optimal control problem in which the system is expressed by a reaction-diffusion-convection equation (RDCE), and the surfaces of the source particles are treated as unknown time-varying boundary conditions that are determined by the optimization. This formulation is flexible enough to consider length scales from smaller than the diameter of a cell to the distance across a tissue scaffold. Four algorithms were developed to solve the optimal control problem: (1) a Fourier Series Approach (FSA), 2) a Method of Moments (MoM) approach, 3) Internal Model Control (IMC), and 4) Model Predictive Control (MPC). At the writing of this abstract, we have already solved problems that are dominated by transport phenomena in one or two spatial dimensions. The presentation will cover these cases as well as summarizing our progress on solving the fully three-dimensional problem with manipulatable boundary conditions. Below is more information on the four optimal control approaches.
Methods (1), (2), and (4) are all time-domain methods. The FSA method expands the concentrations on the to-be-determined boundaries with the same basis function and then calculates the coefficients in front of the sinusoidal basis functions so that the objective function is optimized. This straightforward method results in a numerical algorithm that is computationally efficient but suffers from the Gibbs effect along the time axis (this Gibbs effect is a well-known phenomena that arises when applying Fourier series expansions to functions with sharp gradients). Alternatively, by assuming that the desired temporal and spatial cellular uptake rate can be well parameterized in terms of the moments (e.g., mean position, variance about a position, etc.) of simple functions such as Gaussian distributions, the MoM method gives very fast analytical estimates of the moments of the optimal control trajectory. Provided that the optimal control trajectories are also well parameterized by such functions, the parameters defining the spatial and temporal characteristics of the Gaussians can be computed analytically. While the MoM method is not a general approach, for some problems it can be used to compute quick estimates of optimal control trajectories that can be used as initial guesses in a more thorough optimization such as Model Predictive Control.
The Internal Model Control method is a Laplace transform approach that applies for linear or linearized cellular uptake kinetics. The transfer function obtained by taking the Laplace transform of the CDRE is irrational but useful for performing frequency analysis to obtain insight into the dynamics of the optimal control problem. To apply IMC, a rational transfer function is obtained by discretizing the spatial variables. Analysis of the Bode plots of the exact transfer function and approximate transfer function verifies the validity of the approximate transfer function near the time-scales of the interest. Because the system is minimum phase, the controller used to compute the optimal control trajectory is given by the inverse of the plant transfer function, with a filter included so that the overall controlled system is proper. An advantage of the IMC method is that it is very computationally efficient and fairly general. A weakness of the IMC method that is shared by the FSA and MoM methods is that it does not take constraints on the manipulated variables into account.
The standard MPC formulations assume a zero-order hold on the manipulated variable so that the control trajectories have a staircase temporal profile. In this problem, it is desired for the manipulated variable to be smooth, so the state-space equations for the system were augmented with an integrator before computing the optimal control trajectory via MPC. Then the manipulated variable trajectory to the original system is given by the integral of the control trajectory obtained for the augmented system. In simulations using experimentally determined values of kinetic parameters reported in the literature (Truskey et al, 2004), the MPC algorithm gives nearly optimal control trajectories even for cases in the control and prediction horizons are relatively small, provided that the sampling time is sufficiently small. This is true even when manipulated variable constraints are taken into account. The computational cost is high for the most naïve general MPC formulation, so we discuss methods to reduce the computational cost using good initial guesses obtained by Methods (1), (2), and (3) as well as a restructuring of the MPC formulation for important classes of three-dimensional tissue scaffolds so as to bootstrap the solution to the MPC problem when diffusion and convection occur primarily in only one spatial dimension.
References:
Albright, J. G., O. Annunziata, D. G. Miller, L. Paduano, & A. J. Pearlstein (1999). JACS, 121, 3256-3266.
Brune, D., & S. Kim (1993). PNAS, 90, 3835-3839.
Langer, R. (1998). Nature, 392(suppl.), 5-10.
Truskey, G. A., F. Yuan, & D. F. Katz (2004). Transport Phenomena in Biological Systems, Prentice Hall, Upper Saddle River, NJ.