Analytical optimal controls for the state constrained addition and removal of cryoprotective agents.

Cryobiology is a field with enormous scientific, financial, and even cultural impact. Successful cryopreservation of cells and tissues depends on the equilibration of these materials with high concentrations of permeating chemicals (CPAs) such as glycerol or 1,2 propylene glycol. Because cells and tissues are exposed to highly anisosmotic conditions, the resulting gradients cause large volume fluctuations that have been shown to damage cells and tissues.

A general model for the dynamics of cell volume, global stability, and optimal control.

Cell volume and concentration regulation in the presence of changing extracellular environments has been studied for centuries, and recently a general nondimensional model was introduced that encompassed solute and solvent transmembrane flux for a wide variety of solutes and flux mechanisms. Moreover, in many biological applications it is of considerable interest to understand optimal controls for both volume and solute concentrations. Here we examine a natural extension of this general model to an arbitrary number of solutes or solute pathways, show that this system is globally asymptotically stable and controllable, define necessary conditions for time-optimal controls in the arbitrary-solute case, and using a theorem of Boltyanski prove sufficient conditions for these controls in the commonly encountered two-solute case.

Exact solutions of a two parameter flux model and cryobiological applications.

Solute-solvent transmembrane flux models are used throughout biological sciences with applications in plant biology, cryobiology (transplantation and transfusion medicine), as well as circulatory and kidney physiology. Using a standard two parameter differential equation model of solute and solvent transmembrane flux described by Jacobs [The simultaneous measurement of cell permeability to water and to dissolved substances, J. Cell.