foundational modeling problems, but also generate real world and
relevant optimization and optimal control problems that require
novel analytical tools and careful mathematical approaches to
ensure that their utility is preserved.
5 Notes
1. The majority of mathematical modeling in the cryobiology
literature is concerned with cooling rates, leaving warming
rates relatively unexplored, perhaps because warming rates are
typically an order of magnitude or more greater than the
associated cooling rates.
2. Here we follow the approach and notation outlined in
Benson [45].
3. A table of these parameters is available in [127].
4. The dissociated molality if a salt.
5. Where T indicates the transpose of the row vector.
6. The use of the mathematical term functional is more precise
than function and appropriate here as it is a function that maps
to the real numbers. Often the term functional is used when the
argument is a function on which a global operation such as
integration is performed.
References
1. Mazur P (1963) Kinetics of water loss from
cells at subzero temperatures and the likelihood of intracellular freezing. J Gen Physiol
47:347–369
2. Mazur P, Leibo S, Chu E (1972) A two-factor
hypothesis of freezing injury. Evidence from
Chinese hamster tissue-culture cells. Exp Cell
Res 71:345–355
3. Woelders H, Chaveiro A (2004) Theoretical
prediction of ‘optimal’ freezing programmes.
Cryobiology 49:258–271
4. Liu J, Woods EJ, Agca Y, Critser ES, Critser
JK (2000) Cryobiology of rat embryos II: a
theoretical model for the development of
interrupted slow freezing procedures. Biol
Reprod 63:1303–1312
5. Agca Y, Gilmore J, Byers M, Woods EJ, Liu J,
Critser JK (2002) Osmotic characteristics of
mouse spermatozoa in the presence of extenders and sugars. Biol Reprod 67:1493–1501
6. Morris CE, Homann U (2001) Cell surface
area regulation and membrane tension. J
Membr Biol 179:79–102. https://doi.org/
10.1007/s002320010040
7. Gilmore JA, Liu J, Gao DY, Critser JK (1997)
Determination of optimal cryoprotectants
and procedures for their addition and removal
from human spermatozoa. Hum Reprod
12:112–118
8. Davidson AF, Benson JD, Higgins AZ (2014)
Mathematically optimized cryoprotectant
equilibration procedures for cryopreservation
of human oocytes. Theor Biol Med Model
11:13
9. Benson JD, Kearsley AJ, Higgins AZ (2012)
Mathematical optimization of procedures for
cryoprotectant equilibration using a toxicity
cost function. Cryobiology 64:144–151
10. Benson JD, Chicone CC, Critser JK (2012)
Analytical optimal controls for the state constrained addition and removal of cryoprotective agents. Bull Math Biol 74:1516–1530
11. Levin RL (1982) A generalized method for
the minimization of cellular osmotic stresses
and strains during the introduction and
removal of permeable cryoprotectants. J Biomech Eng 104:81–86
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
167
relevant optimization and optimal control problems that require
novel analytical tools and careful mathematical approaches to
ensure that their utility is preserved.
5 Notes
1. The majority of mathematical modeling in the cryobiology
literature is concerned with cooling rates, leaving warming
rates relatively unexplored, perhaps because warming rates are
typically an order of magnitude or more greater than the
associated cooling rates.
2. Here we follow the approach and notation outlined in
Benson [45].
3. A table of these parameters is available in [127].
4. The dissociated molality if a salt.
5. Where T indicates the transpose of the row vector.
6. The use of the mathematical term functional is more precise
than function and appropriate here as it is a function that maps
to the real numbers. Often the term functional is used when the
argument is a function on which a global operation such as
integration is performed.
References
1. Mazur P (1963) Kinetics of water loss from
cells at subzero temperatures and the likelihood of intracellular freezing. J Gen Physiol
47:347–369
2. Mazur P, Leibo S, Chu E (1972) A two-factor
hypothesis of freezing injury. Evidence from
Chinese hamster tissue-culture cells. Exp Cell
Res 71:345–355
3. Woelders H, Chaveiro A (2004) Theoretical
prediction of ‘optimal’ freezing programmes.
Cryobiology 49:258–271
4. Liu J, Woods EJ, Agca Y, Critser ES, Critser
JK (2000) Cryobiology of rat embryos II: a
theoretical model for the development of
interrupted slow freezing procedures. Biol
Reprod 63:1303–1312
5. Agca Y, Gilmore J, Byers M, Woods EJ, Liu J,
Critser JK (2002) Osmotic characteristics of
mouse spermatozoa in the presence of extenders and sugars. Biol Reprod 67:1493–1501
6. Morris CE, Homann U (2001) Cell surface
area regulation and membrane tension. J
Membr Biol 179:79–102. https://doi.org/
10.1007/s002320010040
7. Gilmore JA, Liu J, Gao DY, Critser JK (1997)
Determination of optimal cryoprotectants
and procedures for their addition and removal
from human spermatozoa. Hum Reprod
12:112–118
8. Davidson AF, Benson JD, Higgins AZ (2014)
Mathematically optimized cryoprotectant
equilibration procedures for cryopreservation
of human oocytes. Theor Biol Med Model
11:13
9. Benson JD, Kearsley AJ, Higgins AZ (2012)
Mathematical optimization of procedures for
cryoprotectant equilibration using a toxicity
cost function. Cryobiology 64:144–151
10. Benson JD, Chicone CC, Critser JK (2012)
Analytical optimal controls for the state constrained addition and removal of cryoprotective agents. Bull Math Biol 74:1516–1530
11. Levin RL (1982) A generalized method for
the minimization of cellular osmotic stresses
and strains during the introduction and
removal of permeable cryoprotectants. J Biomech Eng 104:81–86
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
167