MATHEMATICAL MODELING AND OPTIMIZATION IN CRYOBIOLOGY James Benson, Ph.D. NIST Applied and Computational Math Division Tuesday, October 19, 2010
OUTLINE • Brief introduction to principles of Cryobiology • Model development at three length scales • Optimal control • Current and future directions Tuesday, October 19, 2010
WHY FREEZE BIOSPECIMENS? • Colder temperatures mean longer storage: at least 100 years in LN • Banking, distribution and testing of cells and tissues, maybe organs in the future • Worldwide initiatives to preserve genetic samples • Millennium Seed Bank, Svalbard Seed Bank, UK Biobank (0.5M samples) • JAX Sperm bank ( >10000 strains) • NCRR, MMRRC, MRRRC • NCI-Office of Biorepositories and Biospecimine Research • Kill unwanted cells and tissues in living systems Tuesday, October 19, 2010
Liquid Temperature Liquid+Ice Liquid +Solids Ice+Solids Concentration HOW CELL FREEZING WORKS Tuesday, October 19, 2010
Concentration Temperature HOW CELL FREEZING WORKS Tuesday, October 19, 2010
Salt Salt CPA Alone Concentration 1% 1% 10% 10% 1.5% 15% 30% 3% 30% Temperature - 4% 40% • To reduce the effects of high salt concentrations and to aid in “glass formation” we add cryoprotective agents (CPAs) Tuesday, October 19, 2010
temperature Liquid Liquid+Ice Liquid +Solids Ice+Solids concentration THE TWO FACTOR HYPOTHESIS Tuesday, October 19, 2010
CRITICAL CRYOBIOLOGICAL QUANTITIES Heat Concentration Above 0°C these quantities govern osmotically induced damage Below 0°C these quantities govern the likelihood of intracellular ice Tuesday, October 19, 2010
TRANSPORT PROBLEMS Model Selection Single Cell Larger Tissues Multi Cell Suspensions and Organs Tissues ODE PDE Hybrid ODE/PDE Mass System System System Heat/ Nonlinear heat & Stochastic Large Monte Solidi- stefan problem ODE Carlo System fication Tuesday, October 19, 2010
TRANSPORT PROBLEMS Model Selection All models in cryobiology are coupled systems! Tuesday, October 19, 2010
THE SINGLE CELL PROBLEM Tuesday, October 19, 2010
Volume Time = P 1 ( µ ext − µ int ˙ 1 ) n 1 1 = P 2 ( µ ext − µ int ˙ 2 ) n 2 2 MASS TRANSFER Tuesday, October 19, 2010
MASS TRANSFER THE CHOICE OF µ n N i kT ln N i � Φ ( T, P, N ) = N 1 µ 0 + eN 1 i =2 n n 1 � � + N i ψ i + β ij N i N j 2 N 1 i =2 i,j =2 Differentiating with respect to N 1 or N i and setting β ij /kT = ( B i + B j ) n m i + 1 � � = ( B i + B j ) m i m j µ 1 µ 0 − kT 2 i =2 i,j =2 � . = ln m i + ψ ∗ i + ( B i + B j ) m j µ i kT j =1 JDB. Stability analysis of several non-dilute multiple solute transport equations. J. Math. Chem. , In press. Tuesday, October 19, 2010
Specific Model: set and M i ≈ x 2 /x 1 . B i = 0 Cellular Quantities k n x np x j Water Volume � � x 1 = x 1 ˙ = + M i , − x 1 x 1 Moles of permeating j =2 i =1 x 2 ,...,n = solute � � M 2 − x 2 Moles of x 2 ˙ = b 2 , x np = x 1 nonpermeating solute . . b 2 ,...,n = Relative permeability . � � M n − x n Extracellular Quantities x n ˙ = b n , x 1 Nonpermeating solute M 1 = molality Permeating solute M 2 ,...,n = molality Maximal i th solute ¯ M i = I. Katkov. A two-parameter model of cell membrane permeability molality for multisolute systems. Cryobiology , 40(1):64–83, Feb 2000. Tuesday, October 19, 2010
SINGLE CELLS Specific Model Cellular Quantities Water Volume x 1 = k n 1 � � , Moles of permeating ˙ = x np + x 1 x j − M i x 1 x 2 ,...,n = x 1 solute j =2 i =1 Moles of b 2 ˙ = ( M 2 x 1 − x 2 ) , x np = x 2 nonpermeating solute x 1 . . b 2 ,...,n = Relative permeability . b n Extracellular Quantities ˙ = ( M n x 1 − x n ) , x n x 1 Nonpermeating solute M 1 = molality Permeating solute M 2 ,...,n = molality Maximal i th solute ¯ M i = I. Katkov. A two-parameter model of cell membrane permeability molality for multisolute systems. Cryobiology , 40(1):64–83, Feb 2000. Tuesday, October 19, 2010
We have a system of the form x ( t ) = λ ( x ( t )) f ( x ( t )) , ˙ where is a positive scalar function. In λ ( x ( t )) = 1 /x 1 ( t ) this case, we can define an invertible transformation � τ � τ 1 q ( τ ) = λ ( x ( s )) ds = x 1 ( s ) ds 0 0 and a new system such that w ′ ( τ ) = f ( w ( τ )) w ( τ ) = x ( q ( τ )) meaning that we may, without any penalty, linearize the system by removing the term. 1 /x 1 JDB, C Chicone, J Critser. Exact solutions to a two parameter flux model and cryobiological implications. Cryobiology, 50 , 308-316, 2005 Tuesday, October 19, 2010
n n � � x ′ 1 ( τ ) = x np + M i ( τ ) x 1 , x j − j =2 i =1 x ′ 2 ( τ ) = b 2 ( M 2 ( τ ) x 1 − x 2 ) , . . . x ′ n ( τ ) = b n ( M n ( τ ) x 1 − x n ) . or x ′ = f ( x, M ) := A ( M ) x + x np e 1 , where − � n 1 1 1 i =1 M i . . . b 2 M 2 ( t ) 0 0 − b 2 . . . b 3 M 3 ( t ) 0 0 − b 3 . . . A ( M ) = . . . . . ... . . . . . . . . b n M n ( t ) 0 0 . . . − b n Tuesday, October 19, 2010
Define D := diag(1 , ( b 2 M 2 ) − 1 / 2 , . . . , ( b n M n ) − 1 / 2 ) Then: √ b 2 M 2 √ b 3 M 3 √ b n M n − � i M i . . . √ b 2 M 2 0 0 − b 2 . . . √ b 3 M 3 DA ( M ) D − 1 = 0 0 − b 3 . . . . . . . ... . . . . . . . . √ b n M n 0 . . . 0 − b n is symmetric, negative definite, and our original n-dimensional nonlinear system is globally asymptotically stable. JDB, C Chicone, J Critser. A general model for the dynamics of cell volume, global stability, and optimal control . J. Math Bio., In press Tuesday, October 19, 2010
MASS TRANSPORT IN SMALL TISSUES Layer 1 2 ... n 600 400 4 200 x 10 2 0 200 400 600 800 1000 Cells C e 1 Water Volume (µm3) 4 A 1 A 2 A n − 1 0 x 10 A n 6 0 200 400 600 800 1000 4 B 1 B 2 5 B n 2 x 10 Virtual Cells 2 0 200 400 600 800 1000 Channel/ 1 5 0 x 10 C e 0 200 400 600 800 1000 4 − 2 ( c rr + 2 c r /r ) c t = D 2 5 0 x 10 4 0 200 400 600 800 1000 2 5 0 x 10 r 0 R 6 0 200 400 600 800 1000 Discretization 4 Cells 2 0 200 400 600 800 1000 time (s) Virtual Cells C virtual cell C virtual cell C virtual cell Channel/ k- points k- points k- points JDB, C Benson, J Critser. Submitted to J. Biomech Eng. Tuesday, October 19, 2010
SOLIDIFICATION DURING COOLING, SMALL TISSUES: Monte Carlo Simulation of IIF j + p p p i p j ( δτ ) ≈ j (1 + k j α ) δ t ≈ is the probability of ice p j is the probability of ice p i j forming spontaneously p p is the probability of ice j propagating from neighbor is the number of icy k j neighbors D Iremia, J Karlsson. Biophys. J. 88 647-660, 2005. Tuesday, October 19, 2010
MASS TRANSFER IN LARGE TISSUES Fig. 5. Two sample MR images, with water signal saturated, showing the increasing EG concentration in ovaries during perfusion. Fig. 2. The proton MR spectrum at 7 T from the sample holder loaded with two ovaries and 40% (w/w) EG solution. The excitation frequency was centered at the resonance frequency for the –CH 2 group in EG molecules. ∂ c ∂ t = ∇ · ( D ∇ c ) Fig. 6. The experimental data with their fitted curve for the average EG concen- tration change on the centric cross-section of an ovary with 1.1 mm as its identical radius. X Han, L Ma, A Brown, JDB, J Critser. Cryobiology, 58 (3), 2009 Tuesday, October 19, 2010
MASS TRANSFER IN LARGE TISSUES ∂ c D ( T ) = exp( − E a /RT ) ∂ t = ∇ · ( D ( T ) ∇ c ) X Han, L Ma, A Brown, JDB, J Critser. In review: IJHMT Tuesday, October 19, 2010
temperature Liquid Liquid+Ice Liquid +Solids Ice+Solids concentration WHAT CAN WE CONCLUDE FROM THE ABOVE MODELS? Tuesday, October 19, 2010
HEAT AND MASS TRANSFER LIMIT THE SIZE OF FREEZABLE TISSUE! Tuesday, October 19, 2010
DOES MODELING IN CRYOBIOLOGY WORK? Available online at www.sciencedirect.com Cryobiology 56 (2008) 120–130 www.elsevier.com/locate/ycryo An improved cryopreservation method for a mouse embryonic stem cell line q Corinna M. Kashuba Benson, James D. Benson, John K. Critser * Comparative Medicine Center, Research Animal Diagnostic Laboratory, College of Veterinary Medicine, University of Missouri, 1600 East Rollins Street,Columbia, MO 65211, USA Received 15 May 2007; accepted 3 December 2007 Available online 14 January 2008 • Previous best protocol: 31% recovery • “Optimally” defined new best protocol: 64% recovery Tuesday, October 19, 2010
OPTIMAL CONTROL IN CRYOBIOLOGY: • control quantity to minimize cost J (e.g. time, energy, stress, P IIF or combinations.) • subject to exact and inequality constraints: • exact constraints: governing physical system, (e.g. 2P model, heat equation, diffusion, etc). • inequality constraints: state or control constraints, (e.g. cell volume > 0). Tuesday, October 19, 2010
subject to min M ∈ A s f Cell Volume x np + w 2 w 1 ˙ = − M 1 − M 2 , w 1 w 1 � � M 2 − w 2 w 2 ˙ = b 2 , w 1 Time and w 1 + γ w 2 − k ∗ ≤ 0 , k ∗ − w 1 − γ w 2 ≤ 0 . FIRST CONTROL PROBLEM Tuesday, October 19, 2010
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