DFG Priority Program SPP 1253 Optimization with partial differential equations Mathematical modeling and control in cryopreservaton of living cells and tissues Prof. Dr. K-H. Hoffmann. Technical University of Munich, Center of Mathematical Sciences, Chair of Applied Mathematics Dr. N. D. Botkin. Technical University of Munich, Center of Mathematical Sciences, Chair of Applied Mathematics Annual Meeting Bad Honnef, October 4 and 5, 2007
Louis Leopold Boilly 1825
Postponed repair of teeth using stored stem cells Tooth tooth follicle Tooth Repair (Connective Tissue) Ectomesenchymal Stem Cell Tissue Engineering Cytodifferentiation cryopreservaton Cell Culture C. Morsczeck, T, W. Götz, J. Schierholz, F. Zeilhofer, U. Kühn, C. Möl, C. Sippel, K.-H. Hoffmann. Isolation of precursor cells (PCs) from human dental follicle of wisdom teeth. Matrix Biology 24 (2005) 155-165.
Towards a full coupled controlled model Implemented Optimal cooling protocol Control of the temperature response using averaged models (reported techniques). 1 Used in experimental environment IceCube 14/15M! ice solute tissue ampoule Ice formation in the milieu. Controlled Stefan problems and phase field models (Hoffmann, Niezgodka, Pawlow, Sprekels) 2 In work! Ice formation in the extracellular matrix of tissue. Phase-field models. (van Duijn, Frémond, Garcke) 3 Formation of intracellular ice. Thermodynamic models of ice nucleation and cellular dehydration (Karlsson, Mazur, Toner ) 4
Averaged model of the cooling process 1 λ, λ( - the averaged enthalpy density - the H -T material law (from experiment) - the temperature in the chamber - the overall heat conductivity coefficient - the volume and surface area of the ampoule Identification of : Per unit volume : For common materials without phase changes:, - the density - the specific heat sample sensor chamber sensor known measured measured λ!! Ampoule
Freezer IceCube 14/15M Cooperation partner: SY-LAB Geräte GmbH, Austria. Freezing chamber Sample sensor Chamber sensor Rack Ampoules Computer Main features: - cooling system is based on gas nitrogen - chamber and sample sensors measure the temperature responses - cooling protocols are being entered either manually or as a file or can be computed and optimized using our averaged model! - chamber temperature tracks the cooling protocol
Control panel of IceCube 14/15M with the new option Recent news from SY-LAB Geräte GmbH, Austria.
Ice formation in the milieu 2 Caginalp, Hoffmann, Niezgodka, Pawlow, Sprekels in plastic λ, on plastic or metal Ampoule with a tissue and a milieu λ is the scaled temperature the phase function; the scaled latent heat the scaled heat conductivity with control liquid, without control solid the scaled overall heat conductivity the boundary control (add-on to the nominal profile optimal ) temperature the nominal cooling profile profile
From 2D to 3D plastic or metal plastic Ampoule with a tissue and a milieu
Some results of 3D simulations temperature not optimized time temperature not optimized optimized time optimized temperature time
Ice formation in the extracellular matrix 3 Michel Frémond. Non-Smooth Thermomechanics. Springer. 2001. Water-ice phase change in a porous medium. - the liquid water volume fraction - the ice volume fraction - the volume fraction of the solid matrix - the density - the latent heat - the heat capacity -the freezingpoint scaling Free energies:
Derivation of model equations The 2 nd law of thermodynamics: Unfrozen water content in a porous medium: It is assumed that the phase transition itself is not dissipative, and the dissipation results from thermal conditions only. This results in the equilibrium of the driving forces of the phase change. This yields that the unfrozen water content is a function of the temperature. The (maximal) unfrozen water content as a function of the temperature is a material property and depends on the pore size distribution and the material of the solid matrix. In freezing soil science, it is measured directly by NMR (nuclear magnetic resonance). There are a lot of theoretical works on derivation of this function.
Final model equations Let be the Celsius temperature. Heat balance: 1 Boundary conditions: on Constitutive law: 0 where
Objective functional and the adjoint equation Active averaged tracking the nominal slope :, Averaging the model equations yields: Adjoint equation: Boundary and initial conditions: on,
θ Freezing of a porous medium: uncontrolled o chamber temperature porous medium temperature response Freezing occurs without any significant supercooling because the solid matrix effects as a seed providing homogeneous nucleation or very rapid crystallization. t temperature unfrozen water content
Freezing of a porous medium: controlled cooling impulse temperature unfrozen water content
Stresses in a frozen porous medium Stress fee strain (inflation) p Homogenization of the stated elasticity equation yields a material with growing internal pressure when the ice content grows! p
Stresses in a frozen porous medium Simulation is based on homogenization techniques which allow us to consider a porous medium as a homogeneous material with growing internal pressure when the ice content grows! pressure [Pa] (1bar=100.000 Pa) shear stress (disappears)
Stress exerted on a single living cell pore with a cell inside hydro-carbon chains lipid double layer very slow heat conductivity protein molecules
Temperature and phase jumps on the cell boundary temperature unfrozen water content
Pressure exerted on the cell boundary unfrozen water pressure [Pa] (1bar=100.000 Pa) 1.4 bar ice
Balancing the pressures on the cell boundary Liquids inside and outside the cell must freeze simultaneously. This can be achieved through reducing the freezing point of the outside liquid. That is: The control. parameter: unfrozen water content pressure [Pa]
Conclusion Done since November 21, 2006 I. Joint work with cooperation partners (SY-LAB Geräte GmbH, Austria) on the practical implementation of the averaged model has been continued. II. Simulations of ice formation in the milieu are extended to three dimensions. III. Simulations of ice formation in the extracellular matrix are performed. 1. Models of water-ice phase transition in porous media are adapted to the description of ice formation in the extracellular matrix. 2. Control techniques for reducing negative effects of the latent heat are developed. 3. Stresses caused by the volume change during the water-ice phase transition in a porous medium are accounted for. Balancing the pressures exerted on living cells is simulated. What is to do to enhance III? 1. Accounting for salt release due to ice formation, coupling the present model with mass balance equations for fluid and salt. 2. More precise description of porous media using stochastic homogenization. 3. Including the balancing of pressures exerted on living cells into the objective functional.