CONSTRUCTS

AFP Modeling

When cryopreserving, cells are exposed to temperatures of -80°C that favor the growth of ice crystals. Ice crystals growth is a phenomenon that occurs at different rates depending on how pure is the water or which substances are in solution, and it’s a major factor that determines the loss of cell viability. The kinetics of loss of cell viability in cryopreservation have not being studied much, to the extent where only some graphs exists but no mathematical model that describes them.

Some problems when modeling the cell viability are that ice growth is different for each cell culture and some minimum changes on this rate of mortality due to the variable temperature inside the freezer.

Taking in account these problems we concluded that the best modeling would be done if we have some previous results to do a semi-rational modeling (different than our other models, which were done only rationally). Semi-rational modeling, which is a term that we just made up for this case, is based on observing the behavior of previous data and propose the body of the model which would adjust the best to experimental data.

First of all, we know that cell viability would drop, so we expected one of the following scenarios.

• a) The scenario where cells start by dying slow and then as ice grows faster viability suddenly drops to 0.
• b) Cell death occurs at a constant rate
• c) Cell death occurs at an initial rate that lowers after time
• d) Cell death starts slow, and then increases over time to finally reach an apparent stability point.

Doing some research, we found out that the ice growth normally starts slow by nucleation and then accelerates, but all of this information was for ice on lakes and oceans. In our case, we are treating with small containers so we concluded their information adding that it reaches a limit to this acceleration. Based on this, we think that our best model will be in c) or d).

To prove which model was the best, we minimized the squares of the differences between experimental data and two proposed models; a logarithmic model and an adaptation of a Michaelis-Menten equation for c), and an adaptation of the form of a Hill equation for model d). As the logarithmic model can be determined by just asking a program to do it, we concentrate our efforts on the Michaelis-like model for c) and the Hill-like for d) that ended up as follows.

We compared the r2 values from each model and also to some basic models as logarithmic and polynomial, this is what we got.

Logically, if we continued with the polynomial model, it would have been the best, but that is just a mathematical adjustment, not a model. The model c) and d) gave the best results. We had a fair idea of what each parameter would have meant, but it was only after we saw the parameter optimization that we realized what they actually meant.

We have three parameters for model c): Cm, K and Cell0. Each of them has a physical meaning. First, Cm is the average amount of lost viability, this is, how many cells would die until the reach of equilibrium. K is the time in cryopreservation that it takes to kill the half of the Cm. Finally Cell0 is the initial amount of cells before cryopreservation.

Model d) has these same parameters but includes an “n” parameter. This n has been included so that it can adjust better to the model; it has no rational meaning as for now. Model d) has the greatest r2 of all models but its difference with c) is less than 5%, so any of both should adjust just fine.

What would happen if a greater cryopreservant effect occurs? Loss of viability is expected to lessen if a better cryopreservant is added to the culture. With the anti-freeze protein from Raghium inquisitor, this is expected to happen, so some differences in the model would have to be made. Based on the Michaelis-Menten model in which we based our model, we infer that the principal differences will occur in the parameters Cm and k.

Depending on how the cryopreservant acts, two different phenomena can occur. One of the possibilities is the absolute value of Cm (the amount of cells that die until equilibrium is reached) lowers. Another possibility is that the value of k (the time it is needed for the half of the cells to die) gets greater. So when our model applies to a case when no cryopreservant is added 1) and when it does, either the amount of cells that die are less than normal 2), or it takes longer for them to die 3), or both 4) (Shacham, 2004).

• d(Cell)/d(t) = (k * Cm) / ((k + t) * (k + t))
• V = (Cell) / 37725000
• t(0) = 0.00001
• Cell(0) = 37725000
• t(f) = 5
• Cm = -31755909.1 for 1) and 3) and -11755909.1 for 2) and 4)
• k = 0.0161627719088921 for 1) and 2) and 1 for 2) abd 4)