cubic in molality, therefore the linear model is likely only valid for
small molalities (i.e., in the dilute region where m
2
s n
( 1 ). This
being said, however, if m
e
s n
% m
i
s n
(e.g., the cell is near equilibrium),
then the expansion of ln x in terms of its power series in 1 À x gives
ln ðm
e
s n
=m
i
s n
Þ ¼
P 1
n¼1
ðÀ1Þ
nþ1
n
ðm
e
s n
=m
i
s n
À 1Þ
n :
ð16Þ
This power series has unit radius of convergence corresponding to
the interval 0 < m
e
s n
=m
i
s n
2. Now, taking only the first term yields
ln m
e
s n
À ln m
i
s n
% ðm
e
s n
=m
i
s n
À 1Þ,
¼
m
e
s n
À m
i
s n
m i
s n
:
ð17Þ
The error using this approximation after one term is bounded by
1
2 ðm
e
s n
=m
i
s n
À 1Þ
2 , yielding an estimate of when the approximation
will be valid.
This approximation, however, is less than ideal as it contains m
i
s n
in the denominator and is functionally undefined when m
i
s n
¼ 0, a
very common cryobiological initial condition. For example, during
CPA equilibration protocols, m
i
s n
ðtÞ ¼ 0 at time t ¼ 0. Therefore, a
different approximation must be used. In this case, let m
avg
s n
¼
ðm
i
s n
þ m
e
s n
Þ=2 and apply the approximation from Eq. 17 to arrive at
ln m
e
s n
À ln m
i
s n
% ðm
e
s n
=m
avg
s n
À 1ÞÀðm
i
s n
=m
avg
s n
À 1Þ ¼
1
m
avg
s n
ðm
e
s n
À m
i
s n
Þ:
ð18Þ
As above, the error from truncating the power series after the first
term is bounded by
1
2 ðm
e
s n
=m
avg
s n
À 1Þ
2 þ ðm
i
s n
=m
avg
s n
À 1Þ
2
. To
illustrate the error from this approximation, Fig. 5 shows
ln m
e
s n
À ln m
i
s n
and its approximation by Eq. 18 in two forms.
First is the pointwise error at any given r, shown by the solid line.
The dashed line shows the error assuming that m
avg
s n
¼ 5 which
would be relevant in the case where, for example, the cell initially
has no intracellular permeating solutes (m
i
s n
ð0Þ ¼ 0) and the extracellular molality of permeating solutes is 10 mol/kg (m
e
s n
10).
Finally, note, however, that μ s n in fact contains additional terms
(using the virial expansion in Eq. 13) therefore this approach
approximates the approximation.
Additionally, note that in this formulation, there is an implicit
concentration dependence in the solute permeability term where
~
P s ðm
e
s , m
i
s Þ ¼ P s RT =m
avg unless m
avg
s
is fixed at 1 (see Fig. 5 and
caption for discussion). In this case, system (14) becomes
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
143
small molalities (i.e., in the dilute region where m
2
s n
( 1 ). This
being said, however, if m
e
s n
% m
i
s n
(e.g., the cell is near equilibrium),
then the expansion of ln x in terms of its power series in 1 À x gives
ln ðm
e
s n
=m
i
s n
Þ ¼
P 1
n¼1
ðÀ1Þ
nþ1
n
ðm
e
s n
=m
i
s n
À 1Þ
n :
ð16Þ
This power series has unit radius of convergence corresponding to
the interval 0 < m
e
s n
=m
i
s n
2. Now, taking only the first term yields
ln m
e
s n
À ln m
i
s n
% ðm
e
s n
=m
i
s n
À 1Þ,
¼
m
e
s n
À m
i
s n
m i
s n
:
ð17Þ
The error using this approximation after one term is bounded by
1
2 ðm
e
s n
=m
i
s n
À 1Þ
2 , yielding an estimate of when the approximation
will be valid.
This approximation, however, is less than ideal as it contains m
i
s n
in the denominator and is functionally undefined when m
i
s n
¼ 0, a
very common cryobiological initial condition. For example, during
CPA equilibration protocols, m
i
s n
ðtÞ ¼ 0 at time t ¼ 0. Therefore, a
different approximation must be used. In this case, let m
avg
s n
¼
ðm
i
s n
þ m
e
s n
Þ=2 and apply the approximation from Eq. 17 to arrive at
ln m
e
s n
À ln m
i
s n
% ðm
e
s n
=m
avg
s n
À 1ÞÀðm
i
s n
=m
avg
s n
À 1Þ ¼
1
m
avg
s n
ðm
e
s n
À m
i
s n
Þ:
ð18Þ
As above, the error from truncating the power series after the first
term is bounded by
1
2 ðm
e
s n
=m
avg
s n
À 1Þ
2 þ ðm
i
s n
=m
avg
s n
À 1Þ
2
. To
illustrate the error from this approximation, Fig. 5 shows
ln m
e
s n
À ln m
i
s n
and its approximation by Eq. 18 in two forms.
First is the pointwise error at any given r, shown by the solid line.
The dashed line shows the error assuming that m
avg
s n
¼ 5 which
would be relevant in the case where, for example, the cell initially
has no intracellular permeating solutes (m
i
s n
ð0Þ ¼ 0) and the extracellular molality of permeating solutes is 10 mol/kg (m
e
s n
10).
Finally, note, however, that μ s n in fact contains additional terms
(using the virial expansion in Eq. 13) therefore this approach
approximates the approximation.
Additionally, note that in this formulation, there is an implicit
concentration dependence in the solute permeability term where
~
P s ðm
e
s , m
i
s Þ ¼ P s RT =m
avg unless m
avg
s
is fixed at 1 (see Fig. 5 and
caption for discussion). In this case, system (14) becomes
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
143