Cryopreservation of engineered tissue (ET) has achieved limited success due to limited understanding of freezing-induced biophysical phenomena in ETs, especially fluid-matrix interaction within ETs. To further our understanding of the freezing-induced fluid-matrix interaction, we have developed a biphasic model formulation that simulates the transient heat transfer and volumetric expansion during freezing, its resulting fluid movement in the ET, elastic deformation of the solid matrix, and the corresponding pressure redistribution within. Treated as a biphasic material, the ET consists of a porous solid matrix fully saturated with interstitial fluid. Temperature-dependent material properties were employed, and phase change was included by incorporating the latent heat of phase change into an effective specific heat term. Model-predicted temperature distribution, the location of the moving freezing front, and the ET deformation rates through the time course compare reasonably well with experiments reported previously. Results from our theoretical model show that behind the marching freezing front, the ET undergoes expansion due to phase change of its fluid contents. It compresses the region preceding the freezing front leading to its fluid expulsion and reduced regional fluid volume fractions. The expelled fluid is forced forward and upward into the region further ahead of the compression zone causing a secondary expansion zone, which then compresses the region further downstream with much reduced intensity. Overall, it forms an alternating expansion-compression pattern, which moves with the marching freezing front. The present biphasic model helps us to gain insights into some facets of the freezing process and cryopreservation treatment that could not be gleaned experimentally. Its resulting understanding will ultimately be useful to design and improve cryopreservation protocols for ETs.

References

References
1.
Karlsson
,
J. O. M.
, and
Toner
,
M.
, 1996, “
Long-Term Storage of Tissues by Cryopreservation: Critical Issues
,”
Biomaterials
,
17
(
3
), pp.
243
256
.
2.
Pancrazio
,
J. J.
,
Wang
,
F.
, and
Kelley
,
C. A.
, 2007, “
Enabling Tools for Tissue Engineering
,”
Biosens. Bioelectron.
,
22
(
12
), pp.
2803
2811
.
3.
Mazur
,
P.
, 1970, “
Cryobiology: The Freezing of Biological Systems
,”
Science
,
168
(
934
), pp.
939
949
.
4.
Mazur
,
P.
, 1984, “
Freezing of Living Cells: Mechanisms and Implications
,”
Am. J. Physiol.
,
247
(
3 Pt 1
), pp.
C125
142
.
5.
Mazur
,
P.
, 1988, “
Stopping Biological Time. The Freezing of Living Cells
,”
Ann. N.Y. Acad. Sci.
,
541
, pp.
514
31
.
6.
Zhmakin
,
A. I.
, 2009,
Fundamentals of Cryobiology: Physical Phenomena and Mathematical Models
,
Springer
,
New York
.
7.
Discher
,
D. E.
,
Janmey
,
P.
, and
Wang
,
Y. L.
, 2005, “
Tissue Cells Feel and Respond to the Stiffness of Their Substrate
,”
Science
,
310
(
5751
), pp.
1139
1143
.
8.
Pedersen
,
J. A.
, and
Swartz
,
M. A.
, 2005, “
Mechanobiology in the Third Dimension
,”
Ann. Biomed. Eng.
,
33
(
11
), pp.
1469
1490
.
9.
Kumar
,
S.
,
Ulrich
,
T. A.
, and
Pardo
,
E. M. D.
, 2009, “
The Mechanical Rigidity of the Extracellular Matrix Regulates the Structure, Motility, and Proliferation of Glioma Cells
,”
Cancer Res.
,
69
(
10
), pp.
4167
4174
.
10.
He
,
X.
, and
Bischof
,
J. C.
, 2003, “
Quantification of Temperature and Injury Response in Thermal Therapy and Cryosurgery
,”
Crit. Rev. Biomed. Eng.
,
31
(
5–6
), pp.
355
422
.
11.
He
,
X.
, and
Bischof
,
J. C.
, 2005, “
Analysis of Thermal Stress in Cryosurgery of Kidneys
,”
J. Biomech. Eng.
,
127
(
4
), pp.
656
661
.
12.
Rabin
,
Y.
, and
Steif
,
P. S.
, 1996, “
Analysis of Thermal Stresses Around a Cryosurgical Probe
,”
Cryobiology
,
33
(
2
), pp.
276
290
.
13.
Rabin
,
Y.
, and
Steif
,
P. S.
, 2000, “
Thermal Stress Modeling in Cryosurgery
,”
Int. J. Solids Struct.
,
37
(
17
), pp.
2363
2375
.
14.
Zhang
,
J.
,
Sandison
,
G. A.
,
Murthy
,
J. Y.
, and
Xu
,
L. X.
, 2005, “
Numerical Simulation for Heat Transfer in Prostate Cancer Cryosurgery
,”
J. Biomech. Eng.
,
127
(
2
), pp.
279
294
.
15.
Hoffmann
,
N. E.
, and
Bischof
,
J. C.
, 2001, “
Cryosurgery of Normal and Tumor Tissue in the Dorsal Skin Flap Chamber: Part II–Injury Response
,”
J. Biomech. Eng.
,
123
(
4
), pp.
310
316
.
16.
Shi
,
X.
,
Datta
,
A. K.
, and
Mukherjee
,
Y.
, 1998, “
Thermal Stresses From Large Volumetric Expansion During Freezing of Biomaterials
,”
J. Biomech. Eng.
,
120
(
6
), pp.
720
726
.
17.
Rabin
,
Y.
, and
Steif
,
P. S.
, 1998, “
Thermal Stresses in a Freezing Sphere and its Application to Cryobiology
,”
ASME Trans. J. Appl. Mech.
,
65
(
2
), pp.
328
333
.
18.
Mow
,
V. C.
,
Kuei
,
S. C.
,
Lai
,
W. M.
, and
Armstrong
,
C. G.
, 1980, “
Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments
,”
J. Biomech. Eng.
,
102
(
1
), pp.
73
84
.
19.
Simon
,
B. R.
, 1992, “
Multiphase Poroelastic Finite Element Models for Soft Tissue Structures
,”
Appl. Mech. Rev.
,
45
(
6
), pp.
191
218
.
20.
Cowin
,
S.
, and
Doty
,
S. B.
, 2007,
Tissue Mechanics
,
Springer
,
New York
.
21.
Biot
,
M. A.
, 1941, “
General Theory of Three-Dimensional Consolidation
,”
J. Appl. Phys.
,
12
, pp.
155
164
.
22.
Netti
,
P. A.
,
Baxter
,
L. T.
,
Boucher
,
Y.
,
Skalak
,
R.
, and
Jain
,
R. K.
, 1997, “
Macro- and Microscopic Fluid Transport in Living Tissues: Application to Solid Tumors
,”
AIChE J.
,
43
(
3
), pp.
818
834
.
23.
Lanir
,
Y.
, 1987, “
Biorheology and Fluid Flux in Swelling Tissues. 1. Bicomponent Theory for Small Deformations, Including Concentration Effects
,”
Biorheology
,
24
(
2
), pp.
173
187
.
24.
Lanir
,
Y.
, 1987, “
Biorheology and Fluid Flux in Swelling Tissues. 2. Analysis of Unconfined Compressive Response of Transversely Isotropic Cartilage Disk
,”
Biorheology
,
24
(
2
), pp.
189
205
.
25.
Gu
,
W. Y.
,
Lai
,
W. M.
, and
Mow
,
V. C.
, 1998, “
A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-Electrolytes: Passive Transport and Swelling Behaviors
,”
ASME J. Biomech. Eng.
,
120
(
2
), pp.
169
180
.
26.
Oomens
,
C. W. J.
,
Vancampen
,
D. H.
, and
Grootenboer
,
H. J.
, 1987, “
A Mixture Approach to the Mechanics of Skin
,”
J. Biomech.
,
20
(
9
), pp.
877
885
.
27.
Swartz
,
M. A.
,
Kaipainen
,
A.
,
Netti
,
P. A.
,
Brekken
,
C.
,
Boucher
,
Y.
,
Grodzinsky
,
A. J.
, and
Jain
,
R. K.
, 1999, “
Mechanics of Interstitial-Lymphatic Fluid Transport: Theoretical Foundation and Experimental Validation
,”
J. Biomech.
,
32
(
12
), pp.
1297
1307
.
28.
Chen
,
X.
, and
Sarntinoranont
,
M.
, 2007, “
Biphasic Finite Element Model of Solute Transport for Direct Infusion Into Nervous Tissue
,”
Ann. Biomed. Eng.
,
35
(
12
), pp.
2145
2158
.
29.
Lu
,
Y. L.
, and
Wang
,
W.
, 2008, “
Interaction Between the Interstitial Fluid and the Extracellular Matrix in Confined Indentation
,”
ASME J. Biomech. Eng.
,
130
(
4
), p.
041011
.
30.
Lai
,
W. M.
,
Hou
,
J. S.
, and
Mow
,
V. C.
, 1991, “
A Triphasic Theory for the Swelling and Deformation Behaviors of Articular-Cartilage
,”
ASME J. Biomech. Eng.
,
113
(
3
), pp.
245
258
.
31.
Myers
,
E. R.
,
Lai
,
W. M.
, and
Mow
,
V. C.
, 1984, “
A Continuum Theory and an Experiment for the Ion-Induced Swelling Behavior of Articular Cartilage
,”
J. Biomech. Eng.
,
106
(
2
), pp.
151
158
.
32.
Spilker
,
R. L.
,
Nickel
,
J. C.
, and
Iwasaki
,
L. R.
, 2009, “
A Biphasic Finite Element Model of In Vitro Plowing Tests of the Temporo Mandibular Joint Disc
,”
Ann. Biomed. Eng.
,
37
(
6
), pp.
1152
1164
.
33.
Teo
,
K. Y.
,
Dutton
,
J. C.
, and
Han
,
B.
, 2010, “
Spatiotemporal Measurement of Freezing-Induced Deformation of Engineered Tissues
,”
ASME J. Biomech. Eng.
,
132
(
3
), p.
031003
.
34.
Fung
,
Y. C.
, 1965,
Foundations of Solid Mechancis
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
35.
Espinoza Vallejos
,
P. A.
, and
Dustin
,
C.
, 2005, “
Carbon Foam Filled With Phase Change Materials for Passive Temperature Management
,”
Proceedings of the COMSOL Multi-Physics Users Conference
, Boston, MA.
36.
Raffel
,
M.
, 2007,
Particle Image Velocimetry: A Practical Guide
,
Springer
,
New York
.
37.
Teo
,
K. Y.
,
DeHoyos
,
T. O.
,
Dutton
,
J. C.
,
Grinnell
,
F.
, and
Han
,
B.
, 2011, “
Effects of Freezing-Induced Cell-Fluid-Matrix Interactions on the Cells and Extracellular Matrix of Engineered Tissues
,”
Biomaterials
,
32
(
23
), pp.
5380
5390
.
38.
Han
,
B.
,
Miller
,
J. D.
, and
Jung
,
J. K.
, 2009, “
Freeezing Induced Fluid-Matrix Interaction in Poroelastic Material
,”
AMSE J. Biomech. Eng.
,
131
, p.
021002
.
39.
Taylor
,
M. J.
,
Song
,
Y. C.
, and
Brockbank
,
K. G. M.
, 2004, “
Vitrification in Tissue Preservation: New Developments
,”
Life in the Frozen State
,
E.
Benson
,
B.
Fuller
, and
N.
Lane
, eds.,
CRC Press
,
Boca Raton, FL
, pp.
603
641
.
40.
Rabin
,
Y.
, and
Steif
,
P. S.
, 2005, “
Letter to the Editor: Analysis of Thermo-Mechanical Stress in Cryopreservation
,”
CryoLetters
,
26
(
6
), pp.
409
412
.
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