G.J. Schwartz
Research Assistant.
K. R. Diller
Associate Professor of Mechanical
Engineering and
Biomedical Engineering
Mem. ASME
Department of Mechanical
Engineering,
Biomedical Engineering Center,
The University of Texas
at Austin,
Austin, Tex. 78712
Analysis of the Water Permeability
of Human Granulocytes at Subzero
Temperatures in the Presence of
Extracellular Ice1
The plasma membrane water permeability of human granulocytes in the presence of
extracellular ice was determined experimentally on a cryomicroscope. Transient
volumes of individual cells were measured at constant subzero temperatures subsequent to ice nucleation. Permeability values were deduced by adjustment of
multiple parameters in a model to obtain an optimal fit to the data. The permeability was determined to be a function of both temperature and intracellular
solute osmolality, with a reference value at 0°C of 0.407 pmlatm'min and temperature and solute coefficients of 2l8kJ I mol and 1.09 Osm/kg.
Introduction
Living cells may be exposed to a large osmotic stress during
freezing due to a concentrating of extracellular electrolytes
resulting from the selective initial formation of ice exterior to
the cells. The subsequent response of the cells to the rigors of
freezing is determined in large part by the magnitude of water
flux which may occur across the plasma membrane as the
temperature is lowered. Thus, the membrane water permeability and its thermal coefficient at subzero temperatures
are among the most important physiological parameters
which govern cell freezing. Measurement of those properties
at freezing temperatures present a very challenging task, and
is the problem addressed in the present study.
The physio-chemical basis of cell freezing can be modeled
in simple terms according to the following scenario. Upon
nucleation at a subfreezing temperature ice is formed initially
in the extracellular solution giving rise to a water activity
difference across the plasma membrane. The ice depresses the
extracellular water activity in comparison to the liquid intracellular water. In response to this imbalance of activities,
water is expressed from the intracellular volume so as to
osmotically relax toward a state of thermodynamic
equilibrium. As the temperature decreases progressively, the
extracellular ice mass continues to grow, resulting in a further
reduction of the water activity and a concomitant cell volume
diminution. At relatively fast cooling rates the extracellular
activity drops more rapidly than can be compensated by
cellular dehydration. Consequently the intracellular water
activity remains at an elevated level, thereby increasing the
likelihood of intracellular ice formation [1, 2] with its
associated injury [3]. Conversely, at relatively slow cooling
This investigation was sponsored by the National Science Foundation,
GrantNo.ECS 8021511.
Contributed by the Bioengineering Division for publication in the JOURNAL
OF BIOMECHANICAI ENGINEERING. Manuscript received by the Bioengineering
Division, February 4, 1983; revised manuscript received July 11, 1983.
rates the intracellular and extracellular water activities can be
maintained nearly equal via continual adjustment of the
intracellular solution concentration by dehydration. Under
these conditions intracellular freezing is avoided, although
injury may occur by concentrated solution effects [4] or
excessive cell shrinkage [5]. In either case, the membrane
permeability is a governing factor dictating the cellular
response to freezing.
Several quantitative models have been published describing
the volumetric behavior of cells during freezing [6-10].
Models are useful for explaining observed freezing
phenomena, and potentially, for designing optimal
cryopreservation protocols. However, quantitative predictions are of questionable validity at this point of time due to a
lack of information detailing the magnitude of the membrane
permeability to water in frozen cells.
Presently, there exist no measured values for the water
permeability of human granulocytes at subzero temperatures.
For the entire spectrum of research involving freezing cells,
there are only a few reports of subzero temperature permeability measurements, with two studies for yeast [11, 12], a
report detailing erythrocyte permeability [13], and a study
involving a determination of lymphocyte permeability [14].
The current study involves an analysis of the membrane
water permeability of human granulocytes at subfreezing
temperatures. Permeability values are deduced by matching a
cell dehydration model to measured transient cell volumes
under osmotic stress induced by extracellular ice formation.
The permeability represents an unknown quantity which is
adjusted until the model most accurately matches the experimental data. Results indicate that the permeability is a
function of both temperature and intracellular solution
concentration.
Experimental Procedure
Human granulocytes were
3 6 0 / V o l . 105, NOVEMBER 1983
harvested
by
dextran
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Copyright © 1983 by ASME
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GRANULOCYTES IN PLASMA
NUCLEATION TEMP(C)--2.00
GRANULOCYTES IN PLASMA
NUCLEATION TEMP (C) = -3.00
UJ .
LU-.
2:
<;'
> I
Oil
UJ
<_>
to
o<>
Oc
UJl
rvi.
Mr
d
CC
2:
OZtn
CD (M
^
O CM
NO. OF TRIALS
NO. OF CELLS -
00
0.50
1.00
TIME
1.50
2.00
2.50
NO.
OF TRIALS
NO.
OF CELLS =
"bToo"
0.50
1.00
TIME
(MIN)
1. 50
2.00
2. 50
(MIN)
Fig. 1 Transient cell volumes at a nucleation temperature of - 2 C
representing an average of 2 trials with 7 cells
Fig. 2 Transient cell volumes at a nucleation temperature of
representing an average of 2 trials with 7 cells
sedimentation. Venous blood was drawn and mixed with 6
percent T-500 dextran (suspended in citrated, modified
Hank's medium). The suspension was maintained at 4 C for
60 min, during which time the various cellular elements
differentially settled in the dextran solution. The supernatant
containing the granulocytes was pipetted and centrifuged at
lOOg for 10 min. The cells were washed once with citrated
modified Hank's medium (M.H.M.), followed by centrifugation at lOOg for 10 min. The cell pellet was resuspended
with 2 parts autologous plasma and 1 part M.H.M. The cell
separation technique reduced the population ratio of
erythrocytes to leukocytes from 700:1 to 3:1, with 92 percent
of the leukocytes identified as granulocytes. The granulocyte
suspension was viable according to diacetyl fluorescien assay
for up to 4 hr of liquid storage at near ice-water temperature.
The experimental trials were conducted on a cryomicroscope stage [15] equipped with a programmable,
microprocessor based temperature regulation system [16]. A
l-/xL sample of the granulocyte suspension was positioned on
the top surface of the cryostage. Approximately 2 to 6 cells
were viewed during each freezing experiment using transmitted bright field illumination. The specimen temperature
was monitored via a 25-/xm-dia copper-constantan thermocouple embedded in the cryostage, and was regulated by
the microprocessor controller. Specimen cooling was
provided by the flow of low temperature nitrogen gas through
the cryostage. Complete descriptions of the cryomicroscope
and the control system are given by Schwartz and Diller [15]
and by Evans and Diller [16], respectively.
A series of experiments was conducted in which the temperature of the granulocyte sample was lowered and maintained at a specified constant subzero temperature. The
suspension was independently nucleated by briefly spraying a
stream of liquid nitrogen onto the surface of the cryostage
after which the temperature was continuously maintained at
the same constant value. It was possible to freeze the cell
•3 C
Nomenclature
2
A = area(/xm )
a = water activity
B = computational function used
in curve-fitting algorithm
£,; = intracellular concentration
coefficient (Osm-kg - ' )
activation energy (kJ-mol ~')
ELP
membrane permeability (/xmLn
atm "' - min ~')
reference permeability (/xmatm "' - min ~')
number of moles
n
ideal gas const (cm3 - atm R
mol-'-K-1)
T = temperature (K)
t = time (s)
V = volume (ftm3)
v„ = partial molar water volume
(/xm3 -mol~')
X = osmole fraction
Journal of Biomechanical Engineering
a = generalized
permeability
parameters
5a = increment in permeability
parameters
A = computational function used
in curve-fitting algorithm
X = interpolating factor for
biasing curve-fitting
algorithm between gradient
search and linearizing
techniques
7r = intracellular solution concentration (Osm-kg ~')
a = uncertainties in experimentally measured cell
volumes
2
X = chi-squared statistic
Ax2 = variation in x2 due to increment in permeability
parameter values
Subscripts
b = bound
e = electrolyte
/ = free
g = reference value
h = hydrated
j = iteration parameter for the
curve-fitting algorithm
k = iteration parameter for the
curve-fitting algorithm
/ = iteration parameter for serial
experimentally measured cell
volumes
s = solute
w = water
Superscripts
;' = intracellular
o = extracellular
NOVEMBER 1983, Vol. 105/361
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GRANULOCYTES IN PLASMA
NUCLEATION TEMP (C) =-6. 00
GRANULOCYTES IN PLASMA
NUCLEATION TEMP (C) = -4. 00
i°4
>
>• m
IT)
UJ
(_)
O o
LU m
O
CD CM
CM
NO.
NO.
^3. 00
OF TRIALS = 2
OF CELLS = 5
* -
1. 00
1.50
2.00
2. 50
TIME ( M I N )
Fig. 3 Transient cell volumes at a ncleation temperature of -4 C
representing an average of 2 trials with 8 cells
0.50
NO.
NO.
i—i
^Too
1.50
2.50
1.00
2.00
TIME (MIN)
Fig. 5 Transient cell volumes at a nucleation temperature of -6 C
representing an average of 3 trials with 14 cells
GRANULOCYTES IN PLASMA
NUCLEATION TEMP (C) = -5. 00
0.50
GRANULOCYTES IN PLASMA
NUCLEATION TEMP IC)=-7. 00
„,
%
o
o
OF TRIALS = 3
OF CELLS = 10
o
(I
LU
()<>
(_)
<>
Oo
LU I/)
O o
Mr,'
cr
2:
D CM
O CM
NO. OF T R I A L S
NO. OF C E L L S =
"bTw
NO.
NO.
^.00
0.50
1.00
1.50
2.00
2.50
TIME (MIN)
Fig. 4 Transient cell volumes at a nucleation temperature of -5 C
representing an average of 2 trials with 5 cells
suspension under these conditions without spontaneously
nucleating intracellular ice for temperatures between - 2 and
- 10°C. It was possible to effect a continuously isothermal
process due in particular to the ability of the cryostagethermal controller apparatus to compensate for the release of
latent heat during the phase change process.
The sequential images of the freezing granulocytes were
recorded on ASA 400 black and white negative film using a
motor-driven 35-mm camera. The resultant photomicrographs were analyzed to determine individual cell
volumes. The planar area was measured by overlaying a
transparent rectangular grid over a projection of the cell
362/Vol. 105, NOVEMBER 1983
OF TRIALS = 3
OF CELLS = 10
0. 50
1.50
2.00
2. 50
(MIN)
Fig. 6 Transient cell volumes at a nucleation temperature of - 7 C
representing an average of 3 trials with 10 cells
1.00
TIME
image. The measured projected area consisted of the number
of rectangles enclosed by the cell boundary. The area was
converted to volume by assuming a spherical geometry, so
that
V/V,= (A/Ai)i/2
(1)
where the volume and area are normalized relative to the prefreeze values.
Figures 1 through 9 present experimental data illustrating
the transient nature of the granulocyte volume as a function
of extracellular nucleation temperature. Each experimental
volume curve represents the average behavior of a minimum
of 2 trials, each trial consisting of 2 or more cells. The
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GRANULOCYTES IN PLASMA
NUCLEflTION TEMP ld = -8. 00
GRANULOCYTES IN.PLASMA
NUCLEATION TEMP (C) =-9. 00
<!>
Hlli
-Jo
> in
\\
u
I
(
m
M^1
UJ
NO. OF TRIALS = 6
NO. OF CELLS = 18
U
Q
00
0. 50
l. 00
TIME
O CM
l. 50
2. 00
2. 50
"b 00
(MIN)
Fig. 7 Transient cell volumes at a nucleation temperature of - 8 C
representing an average of 6 trials with 18 cells
NO. OF TRIALS = 3
NO. OF CELLS = 7
0.50
1.00
TTIME
IME
1.50
2.00
2.50
(MIN)
(MIN)
Fi
9- 8 Transient cell volumes at a nucleation temperature Of - 9 C
representing an average of 3 trials with 7 cells
GRANULOCYTES IN PLASMA
NUCLEATION TEMP (C) = -10. 00
standard deviations of the measured volumes are indicated by
the bars associated with each data point.
Cell Dehydration Model
The dehydration model is designed to consider an idealized
biologic cell as an open thermodynamic system having a semipermeable membrane boundary. The system is constrained
such that: (a) no pressure gradients exist, (b) temperature
differentials are negligible, and (c) no concentration
gradients exist except across the plasma membrane. The cell is
suspended in a binary solution consisting of water and sodium
chloride, whereas the intracellular solution is ternary due to
the presence of proteins. The membrane is permeable only to
water.
Water flux across the semi-permeable membrane can be
expressed by
dnu
~LpARTln aH
(2)
dt
wherein molar flux is defined in terms of the membrane
permeability (Lp), the membrane surface area (A), the ideal
gas constant (R), the absolute temperature (7), the partial
molar water volume (D"„), and the logarithm of the transmembrane water activity ratio.
Chemically, the sodium chloride in solution is assumed to
undergo complete ionic dissociation, thus providing two
osmoles for every mole of salt. The extracellular solution is
assumed ideal, from which the water activity is defined in
terms of the osmole water fraction by
« l v°=*,/=«,v 0 /(n l v 0 +2V)
(3)
The extracellular osmole water fraction is determined as a
function of temperature from phase data for the watersodium chloride system [17].
The intracellular solution is composed of water, salt, and
proteins. The proteins are assumed to exist in a hydrated
form, which results in the exclusion of some of the cell water
from being available for transport. Levin [18] illustrated that
a hydrated water-salt-protein solution, when treated by excluding the bound water from the water activity, could acJournal of Biomechanical Engineering
If II, t T
o
O CN
NO. OF TRIALS = 4
NO. OF CELLS = 16
"TJ 00
0.50
1.00
1.50
2.00
2.50
TIME (MIN)
Fig. 9 Transient cell volumes at a nucleation temperature of -10 C
representing an average of 4 trials with 16 cells
curately describe changes in erythrocyte volume. For an ideal,
hydrated solution, the water activity is defined as [19]
aj =Xv)h> = (nw''--«„&'V(«w' -nwb''+ns')
= nv///(nw/ + nj)
(4)
The free water content is determined from an inactive cell
volume of 35 percent of the initial value [20, 21] and the
criterion for pre-freezing equality of intracellular and extracellular water activities.
The water flux is transformed from a mole to a volume
basis by definition
nw = V„/vw, or - — =————
A dt
Av„ dt
(5)
NOVEMBER 1983, Vol. 105/363
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Table 1 Measured permeability coefficients for frozen granulocytes.
The estimated error is computed from the error matrix of the least
squares curve fit of the experimental data. The volume variance is
calculated between the experimental and model values.
Coefficient
L
Value
pg
E
LP
E
ci
FROZEN GRANULOCYTES
IN PLASMA
Estimated Error
.407 jjm-atnf -min"
.033
218 k j - m o l " 1
14,5
1.09 Osm-kg"1
.174
Overall normalized volume variance = 1.65*10 -3
and the membrane permeability is defined as a function of
temperature and intracellular solution concentration by
Lp=Lpgexp[(ELp/R)(l/Tg-l/D
+ Ed(l/T-l/irg)]
(6)
Substituting equations (3)-(6) into (2) yields the equation
used to fit the experimental data.
dVw
dt
_ J
ART
txV[(ELp/R)(\/Tg-\/T)
+ £' c / (l/-jr-l/7r„)] In
Xw
(7)
Xw
An iterative process was employed to obtain a best fit
between experimental transient cell volume data and the cell
dehydration model described by equation (7). The applied
fitting technique minimized the difference between the
simulated and experimental volume curves as judged by the
magnitude of the chi-squared statistic, x 2 • An initial estimate
for the values of the unknown transport parameters (Lpg,
ELP, Eci) was used to solve the model equation, and to
calculate the chi-squared statistic, x 2 • Based on these results, a
new estimate was then made for the unknown coefficients,
generating an updated, lower value of x 2 - Values of the
transport coefficients were adjusted by iteration until x2 could
no longer be reduced [12]. Practically, the chi-squared
statistic, x 2 , can only be made to approach a minimum value
as opposed to zero [11, 23, 24]. The iterative solution for the
transport coefficients is
3
da.
dE,Lp
-0,
jt
dEr,
-0
4.00
(KG/0SM)
Fig. 10 Membrane permeability of granulocytes to water, calculated
as a function of intracellular solution concentration and temperature
Permeability Fitting Technique
-0,
3.00
1/ 3 0 0 - 1 / T T
This nonlinear first-order differential equation is solved by a
fourth-order Runge-Kutta algorithm [22].
-o,
2.00
1. 00
0. 00
FROZEN GRANULOCYTES IN PLASMA
COOLING RATE (C/MIN) = 5. 00
U J LT>
(8)
where
X 2 («i
• < X m ) = X 2 ( £^Pg>
p ^Lpi
f^ci
0. 00
-10.00
=
-20.00
-30.00
TEMPERATURE
n
Ti\.v„i-vw(tl)}/al2
Saj=A'- B
where
364/Vol. 105, NOVEMBER 1983
- 5 0 . 00
Fig. 11 Comparison of model to experimental transient cell volumes
for granulocytes cooled at a rate of 5 C/min and with an extracellulai
nucleation temperature of - 2 C
Determination of the transport coefficients involved two
methods for minimizing x 2 • The first technique, referred to as
the gradient method, defined convergence in the direction of
steepest descent by -dx 2 /do,-. The second approach was to
linearize x2 as dx2/d8aj. For the present application the
gradient method was applied for conditions far from convergence, and the linearization method for conditions near
convergence [23]. This procedure defined a system of
equations for the increments, 5a,,
l
- 4 0 . 00
(C)
A',* = Ajk for
A'it Jk = A,jk for
y*
/=1 ^
OUj
j*k
j*k
dak
(9)
and
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(C)
TEMPERATURE
FROZEN
GRANULOCYTES
IN
PLRSHfl
1
•w
-r
30.
~:
-
zi.—-^
«
IT IN
1 o_
z-E
i—
r
10.
\
:
•N,
T" .
^.°-.
1_1 - :
\
\
" a '
— o_
0Q-:
Q;
;
v
IE
UNFROZEN
VALUES
0-0_
U"U
oc
- ::
H
.
UJ
h-
\
\
\\
\S
v
:
oc
-10.
FROZEN VALUES
5 :
>t—
0 '0
\
*
\
\\
\
<n
o_
n
r
20.
\x
*v
:
'
\
\
\\
\
s
\
\ \ ^ _ FROZEN
VALUES
\\
s
\
UNFROZEN VALUES
(2 9 1
- \
A
"L
-:
1. 00
7 "
O
1
0.30
0.32
1
0.34
—,—
r
0.36
1/TEMPERATURE (1/rO
.
0.38
0. 40
• 10"
Fig. 12 Comparison of the temperature coefficients of membrane
permeability at suprazero and subzero temperatures. Solid lines indicate the temperature range for which permeability values are confirmed by experimental data, whereas broken lines represent extrapolations.
2.00
• 3.00
4.00
5.00
1/. 300-1/tt (KG/0SM)
Fig. 13 Comparison of membrane permeability as a function of intracellular solution concentration for coefficients measured at
suprazero temperatures with values extrapolated from subzero temperature measurements.
Figure 10 illustrates a family of isothermal permeability
curves generated from the coefficients given in Table 1 as a
-Vw(a,
am, ti)}
function of intracellular solution concentration. It is apparent
that the permeability decreases for higher solute concentrations. In addition, the permeability exhibits the exdv»
pected reduction concomitant with decreasing temperature.
(ai, . . . ,am, ti).
dot:
The validity of the deduced permeability coefficients was
The computational procedure followed that used by Levin tested by comparing the simulated volumetric behavior of
[11, 24]. It began with a determination of x2 using an initial granulocytes with the measured transient volume of cells
estimate of the transport coefficients and of X = 0.001. These frozen for a defined thermal protocol. Figure 11 shows the
values were used to compare increments defining a new set of results for a cooling rate of 5 C/min in which the simulated
coefficients. If the x2 resulting from the new coefficients behavior agreed with the measured values to within the range
decreased, then X was decremented; otherwise X remained of experimental uncertainty.
constant. This procedure was repeated until the change in x2
between successive iterations converged to a preset minimum Discussion of Permeability Coefficients
(in this study Ax2/x2 = 0.05).
Prior investigators have characterized the osmotic coefficient of membrane permeability as a function of exPermeability Estimation
tracellular solution concentration [14, 25, 26]. In the current
The membrane permeability coefficients, Lpg, ELp, and Eci, study the permeability was initially also correlated
were determined from the cell volume curves shown in Figs. 1 simultaneously with the temperature and extracellular
through 9. The composite set of permeability coefficients, as solution concentration. It was found that, although the
described by (6), were deduced by determining the composite derived permeability coefficients simulated granulocyte
best fit for the complete set of all experimental data and volume behavior quite well for the constant temperature
applying the matching technique described by equations (8) trials, the cell volume response was underpredicted for
and (9). Consequently, the coefficients were established over a transient thermal histories of actual preservation protocols
spectrum of temperatures from - 2 to -10 C, instead of [27]. Neither was an alternate analysis of the experimental
deriving the permeability values at discrete, subzero tem- data based on a permeability dependency on both temperature
peratures. In this manner, the permeability was weighted and cooling rate able to provide a satisfactory correlation.
towards those experimental trials which included large data
The most accurate description of cell water transport for
sets and small values of experimental variance.
both constant and variable temperature freezing processes
Values of the permeability coefficients, Lpg, ELp, and Ech was obtained for permeability defined in terms of the absolute
as calculated using the granulocyte solids content and isotonic temperature and the intracellular solution concentration.
volume described by Hempling [20], are listed in Table 1. The Intuitively, it may be possible that the intracellular comcoefficients were estimated for temperatures between - 2 and position would directly affect the granulocyte water transport
-10 C, with a reference state of Tg = 273.15 K and IT/ = system, due to the presence of extensive intracellular morphology. The granulocyte is a complex cell, containing in.300Osm-kg'.
1
Journal of Biomechanicat Engineering
NOVEMBER 1983, Vol. 105/365
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tracellular membranes whhich may contribute to the transport
of water [28], and which may be affected by the intracellular
solution composition.
Hempling [20] determined values for Lpg (at a reference
temperature of 298.15 K) and ELp for granulocytes at
suprazero temperatures as 0.65 ^m-atm ~' - min ~' and 77 kJmol - ', respectively. Bradley and Diller [29] measured Eci for
granulocytes as 1.30 osm-kg"1 at 25°C. Compared to nonfrozen granulocytes, the frozen cells exhibit a significantly
higher activation energy, 218 kJ/mol, and a similar
magnitude for the intracellular concentration coefficient, 1.09
Osm-kg ~'.
Figure 12 illustrates a comparison of permeabilities
determined at suprazero and subzero temperatures as a
function of temperature. It is clear that the permeability
determined from freezing data and extrapolated to suprazero
temperatures is significantly greater than that actually
measured by Hempling [20] at suprazero temperatures.
Furthermore, extrapolation of the nonfrozen permeability
values to subzero temperatures indicates a lesser permeability
than deduced from the data in this study. It is obvious that
application of activation energies obtained at temperatures
above freezing is not acceptable for modeling the freezing
process. Alternatively, granulocyte membrane permeability
can be viewed as a function of intracellular concentration as
in Fig. 13, comparing permeabilities deduced from frozen cell
data and extrapolated to a temperature of 25 C with values
determined at 25 C [29]. As expected, the two curves are
parallel, with the values measured at room temperature being
smaller than the permeability values deduced from frozen cell
data.
Conclusions
The analysis presented illustrates that constant temperature
freezing of cells provides an effective method of measuring
the plasma membrane water permeability at subzero states.
The technique is readily adapted to any cell type which can be
prepared as a thin suspension for the cryomicroscopic stage.
Prerequisite to the application of this procedure is the
availability of a cryomicroscope for which the degree of
extracellular supercooling can be regulated as an independent
parameter and which has a temperature control system
capable of effecting isothermal processes. The method
enables a more direct determination of permeabilities than
can be obtained from measurement and correlation of cell
volumes from actual transient temperature preservation
protocols [11,12].
For this and previous studies, the plasma membrane
permeability to solutes was assumed negligible. In situations
for which the solute flux is nonzero, the solute permeability
and the relation of solute transport to water transport can be
ascertained by using an expanded version of the presented
fitting technique. The transient cell volume curve would
represent a superposition of water and solute volume
alterations, which can be modeled by irreversible thermodynamics in terms of conjugate fluxes and forces [30]. The
transmembrane transport would be governed not only by the
permeability to water, but also the solute permeability and the
reflection coefficient representing the membrane selectivity.
Quantification of membrane properties at subzero temperatures provides necessary data for modeling the
dehydration of freezing cells. The model could be used to
design thermal protocols that would avoid freezing states that
may cause injury by either excessive cell shrinkage or concentration of electrolytes.
366/Vol. 105, NOVEMBER 1983
References
1 Toscano, W. M., Cravalho, E. G., Silvares, O. M., and Huggins, C. E.,
"The Thermodynamics of Intracellular Ice Nucleation in the Freezing of
Erythrocytes," ASME Journal of Heat Transfer, Vol. 97, 1975, pp. 326-332.
2 Diller, K. R., "Intracellular Freezing: Effect of Extracellular Supercooling," Cryobiology, Vol. 12, 1975, pp. 480-485.
3 Mazur, P., "The Role of Intracellular Freezing in the Death of Cells
Cooled at Supraoptimal Rates," Cryobioology, Vol. 14, 1977, pp. 251-272.
4 Mazur, P., "Cryobiology: The Freezing of Biological Systems,"
Science, Vol. 168, 1970, pp. 939-949.
5 Meryman, H. T., Williams, R. J., and Douglas, M. St. J., "Freezing
Injury from Solution Effects and Its Prevention by Natural and Artificial
Cryoprotection," Cryobiology, Vol. 14, 1977, pp. 287-302.
6 Mazur, P., "Kinetics of Water Loss from Cells at Subzero Temperatures
and the Likelihood of Intracellular Freezing," Journal of General Physiology,
Vol. 47, 1963, pp. 347-369.
7 Silvares, O. M., Cravalho, E. G., Toscano, W. M., and Huggins, C. E.,
"The Thermodynamics of Water Transport from Biological Cells during
Freezing," ASME Journal of Heat Transfer, Vol. 97, 1975, pp. 582-588.
8 Mansoori, G. A., "Kinetics of Water Loss from Cells at Subzero Centigrade Temperatures," Cryobiology, Vol. 12,1975, pp. 34-45.
9 Levin, R. L., Cravalho, E. G., and Huggins, C. E., "A Membrane Model
Describing the Effect of Temperature on the Water Conductivity of
Erythrocyte Membranes at Subzero Temperatures," Cryobiology, Vol. 13,
1976, pp. 415-429.
10 Knox, J. M., Schwartz, G. J., and Diller, K. R., "Volumetric Changes in
Cells During Freezing and Thawing," ASME JOURNAL OF BIOMECHANICAL
ENGINEERING, Vol. 102, 1980, pp. 91-97.
11 Levin, R. L., "Water Permeability of Yeast Cells at Subzero Temperatures," Journal of Membrane Biology, Vol. 46,1979, pp. 191-214.
12 Schwartz, G. J ., and Diller, K. R., "Osmotic Response of Individual
Cells During Freezing. II. Membrane Permeability Analysis," Cryobiology,
1983, in press.
13 Papanek, T. H., The Water Permeability of the Human Erythrocyte in
the Temperature Range +25°C to -10°C, Ph.D. thesis, Massachusetts Institute of Technology, 1978.
14 Scheiwe, M. W., Untersuchungen zum Verfahren der Langzeitknonservierung Lebender Blutzellen durch Gefrieren, doctoral dissertation, der
Rheinisch-Westflischen Technischen Hoschule Aachen, 1981.
15 Schwartz, G. J., and Diller, K. R., "Design and Fabrication of a Simple
Versatile Cryomicroscopy Stage," Cryobiology, Vol. 19, 1982, pp. 529-538.
16 Evans, C. D., and Diller, K. R., " A Microprocessor Based, Programmable, Controlled, Temperature Microscope Stage for Microvascular Studies,"
Microvascular Research, Vol. 24,1982, pp. 314-325.
17 Handbook of Chemistry and Physics, 54th Edition, The Chemical
Rubber Co., Cleveland, Ohio, 1973.
18 Levin, R. L., Cravalho, E. G., and Huggins, C. E., "Effect of Hydration
on the Water Content of Human Erythrocytes," Biophysical Journal, Vol. 16,
1976, pp.1411-1426.
19 Prausnitz, J. M., Molecular Thermodynamics of Fluid-Phase Equilibria,
Prentice-Hall Inc., Englewood Cliffs, 1969.
20 Hempling, H. G., "Heats of Activation for the Exosmotic Flow of Water
Across the Membrane of Leukocytes and Leukemic Cells," Journal of Cellular
and Comparative Physiology, Vol. 81, 1973, pp. 1-9.
21 Armitage, W. J., and Mazur, P., "The Response of Granulocytes to
Osmotic Shrinkage," Cryobiology, Vol. 19, p. 678.
22 Carnahan, B., Luther, H. A., and Wilkes, J. O., Applied Numerical
Methods, Wiley, New York, 1969.
23 Stusnick, E., and Hurst, R. P., "Numerical Determination of Membrane
Permeability Parameters," Journal of Theoretical Biology, Vol. 37, 1972, pp.
261-271.
24 Bevington, P. R., Data Reduction and Error Analysis for the Physical
Sciences, McGraw-Hill, New York, 1969.
25 Rich, G. T,, Sha'afi, R. I., Romualdez, A., and Solomon, A. K., "Effect
of Osmolality on the Hydraulic Permeability Coefficient of Red Cells,"
Journal of General Physiology, Vol. 52,1968, pp. 941-954.
26 Vieira, F. L., Sha'afi, R. I., and Solomon, A. K., "The State of Water in
Human and Dog Red Cell Membranes," Journal of General Physiology, Vol.
54, 1970, pp.451-466.
27 Schwartz, G. J., and Diller, K. R., "Cryomicroscopic Measurement and
Interpretation of Transient Granulocyte Volumes During Freezing,"
Cryobiology, 1983, submitted.
28 Schmid-Schonbein, G. W., Shih, Y. Y., and Chien, S., "Morphometry of
Human Leukocytes," Blood, Vol. 56, 1980, pp. 866-875.
29 Bradley, D. A., and Diller, K. R., "Membrane Permeability
Measurement in Isolated Cells," 1982 Advances in Bioengineering, ed., L. E.
Thibault, ASME, New York, 1982, pp. 115-118.
30 Lynch, M, E., and Diller, K. R., "Analysis of the Kinetics of Cell
Freezing with Cryophylactic Additives," 1981 Advances in Bioengineering, ed.,
D. C. Viano, ASME, New York, 1981, pp. 229-232.
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