# LyoModelling Calculator

This LyoModelling Calculator has been developed for aqueous product in vials on the shelves of the drying chamber in a lyophilizer. The LyoModelling Calculator estimates the time required for primary drying and the maximum product temperature during primary drying, based on the user’s input. The calculation is intended to be an estimate only, to be used to explore and perhaps compare the effects of combinations of freeze-drying parameters on primary drying time and the highest product temperature that occurs at the very bottom of the frozen product at the end of primary drying.

Freeze-drying consists of three essential steps:

• Freezing – as the freeze-drying shelf is cooled, ice is formed, thereby concentrating the remaining solution. Crystalizable solutes can crystalize out in some cases. Other non-crystalizable solutes become more concentrated as more ice is formed. Any remaining solution eventually solidifies as in the amorphous (noncrystalline) state. However, the major portion of water in the prelyophilized solution is in the crystalline (ice) form that is removed during primary drying. The amorphous solid generally consists of about 20% water which is removed during secondary drying.
• Primary drying – the temperature of the shelf is raised to provide energy required for ice to undergo sublimation. Sublimation requires about 600 calories per gram. The majority of that energy is supplied by the shelf. Care must be taken during primary drying to maintain the temperature of the product in the vial below any critical temperature such as a eutectic melt or collapse temperature. Primary drying is complete when all ice in the product has undergone sublimation.
• Secondary drying – Once all of the ice has undergone sublimation, the shelf temperature is raised to allow desorption of the remaining unfrozen water from the amorphous solid.

The LyoModelling Calculator is based on the steady-state heat and mass transfer model of primary drying developed by Michael Pikal with some modifications by Royce Labriola and Robin Bogner at the University of Connecticut.

User Inputs
[°C]
[mL]
[%]
[cm]
[slices]
[g/mL]
[g/mL]
[g/mL]
[cal/g]
[cal⋅cm-1⋅sec-1⋅K-1]
[cal⋅cm2⋅sec-1⋅K-1]
Report
[Hrs]
[g/Hr]
[°C]
Slice # Drying time for the slice [hr] Cumulative Drying time [hr] Ice Thickness [cm] Resistance to Vapor Flow through Dry Product
Rp [cm2⋅Torr⋅g-1]
Sublimation Rate for the Slice
Δm/Δt [g/hr]
Temperature of Ice at the Sublimation Interface
T0 [°C]
Temp of Frozen Product at the Bottom of the vial
Tb [°C]
Vapor Pressure of Ice at the Sublimation Interface
P0 [Torr]

## Shelf Temperature

Enter either:

1. the shelf temperature set point that will be used for the freeze-drying cycle, or
2. the average of the inlet and outlet fluid temperatures during primary drying (which may be more applicable when the thermal load is high such that the outlet temperature is appreciably lower than the inlet temperature), or
3. the average of the temperature of the shelf surface, measured by themocouples on the shelf surface near the inlet and outlet for the heat transfer fluid flowing within the shelf.

Option 3 is most correct for how the LyoModelling Calculator was developed. However, once one method is selected, that method should be used for all future calculations that will be used to make comparisons.

## Chamber pressure

Chamber pressure $\left({P}_{c}\right)$ influences the driving force for sublimation of ice. Specifically, the sublimation rate $\left(\frac{dm}{dt}\right)$ in grams of ice per hour per vial is directly proportional to the difference between the vapor pressure of water at the sublimation interface $\left({P}_{0}\right)$ and the chamber pressure $\left({P}_{c}\right)$ according to the equation:

$\frac{dm}{dt}=\frac{{A}_{p}\left({P}_{0}-{P}_{c}\right)}{{R}_{p}}$

where Ap is the cross-sectional area of the product parallel to the shelf, in cm2, and Rp is the resistance of the dry layer that remains after sublimation, in units of cm2⋅hr⋅Torr/g, when the pressures are expressed in Torr. Perhaps more importantly, however, the chamber pressure influences the heat transfer to the product in the vial from the shelf. For tubing vials which have a small gap between the bottom of the vial and the shelf, heat transfer from the shelf to the vial by gas conduction increases non-linearly with gas concentration, i.e., chamber pressure (Pc).

 1 atm = 760 Torr (or mm of mercury) 1 Torr = 1.3332 mbar 1 atm = 1.01325 bar 1 Torr = 1000 mTorr 1 atm = 101.325 kPa 1 Torr = 0.1332 kPa

Select your preferred pressure units using the dropdown menu to the right of the box used to enter the chamber pressure.

## Fill Volume

The fill volume is the volume of solution filled into each vial.

The fill volume along with the density of the solution and cross-sectional area of the vial are used to calculate the depth of the frozen solution to be dried.

Higher fill depths can extend the drying process and/or result in higher product temperatures that may exceed product collapse or eutectic melt temperatures, resulting in unacceptable product.

1cc = 1mL

## Solute Concentration

Enter solute concentration of the solution to be filled in the vials in % w/v (i.e., g of solute per 100 mL of solution). The solute concentration refers not only to the drug or main component, but rather to all dissolved solutes.

The LyoModelling Calculator uses the value of solute concentration in the calculation of the amount of ice to be removed by sublimation during primary drying.

If your solute concentration is expressed as % w/w (i.e., g of solute per gram of solution), use that value for the LyoModelling Calculator estimate if the concentration is low (less than about 7-10%). For higher concentrations, multiply the solute concentration in w/w by the solution density, ${\rho }_{solution}$ (in units of g/mL) to obtain the concentration in w/v for better estimates from the LyoModelling Calculator.

## Vial Outer Diameter

The vial outer diameter is used to determine the area of the vial in contact with the shelf. Along with the vial heat transfer coefficient, Kv, the value of the vial outer diameter determines the flow of heat into the product to facilitate sublimation of ice.

Examples of vial dimensions are shown in the table below. The actual dimensions of vials may vary with the manufacturer. The LyoModelling Calculator will provide better estimates of drying time and product temperature when the actual vial dimensions are used.

1cc = 1mL
Vial Size Outer diameter [cm] Inner diameter [cm] Cross-sectional area of the vial [cm2] Cross-sectional area of the product [cm2] Area Ratio [vial]:[product]
5cc 2.2 2.0 3.8 3.3 1.1
10cc 2.4 2.3 4.5 4.0 1.1
20cc 3.0 2.9 7.2 6.6 1.1

## Resistance Parameters

The resistance $\left({R}_{p}\right)$ to vapor flow from the sublimation front through the dried cake has a major influence on the freeze-drying rate and the temperature of the remaining frozen product. It is best to independently measure dried cake resistance as a function of cake thickness. The LyoModelling Calculator uses the function:

where $l$ is the dry layer thickness and ${R}_{p}$ is expressed in units of [cm2⋅hr⋅Torr/g]. The parameters, ${R}_{0}$, ${A}_{1}$, and ${A}_{2}$ can be determined from the data provided by the Auto-MTM function within the SMART™ Freeze-Dryer technology. If the resistance is unknown, several estimates are available to use. Click the button.

## Calculation Tolerance

The "Calculation Tolerance" sets the value for convergence to a solution. The smaller the value of calculation tolerance, the more precise the estimate will be and the longer the calculation will take.

## Solute Material Property

Indicate whether the solutes in the solution being freeze dried are expected to remain AMORPHOUS or will become CRYSTALLINE during the freezing phase prior to primary drying. The LyoModelling Calculator determines the drying time based on the amount of ice in the product, ${m}_{I}$.

If CRYSTALLINE is selected, the amount of ice, ${m}_{I}$, is calculated based on the total mass of the solution, from the fill volume, ${V}_{fill}$; time the solution density, ${\rho }_{soln}$, subtracting out the mass of the solute, ,

where $c$ is the weight fraction of total solute.

If AMORPHOUS is selected, the amount of ice is determined assuming the noncrystallizable solute contains about 20% unfrozen water. That means the amorphous phase consists of 4 parts solute and 1 part water. Thus, the fraction of solute is multiplied by 5/4 to account for the unfrozen water in the amorphous phase.

If the solution has a total solute concentration of about 5% of less, the selection of AMORPHOUS or CRYSTALLINE does not affect the estimate of primary drying time significantly.

## Divisions for Computation

The LyoModelling Calculator determines the heat and mass transfer at steady state in each of a number of slices into which the frozen solution can be divided. For most applications, 10 slices is sufficient to obtain a good estimate of primary drying time and maximum product temperature. A larger number of slices is available to the user in the dropdown menu and may be useful for conditions in which the sublimation rate is very high and the primary drying time is very short.

## Area Ratio

The "Area Ratio", ${A}_{r}$, is the area of the outer horizontal cross-section of the vial, ${A}_{v}$ divided by the area of the inner cross-section of the vial, which is equal to the area of the horizontal cross-section of the product, ${A}_{p}$. For most vials (3cc to 20cc), the area ratio is 1.1 to 1.2. To calculate the area ratio of a specific vial, ${A}_{r}$, use the equation:

where o.d. = outer diameter ofthe vial, and i.d. = inner diameter of the vial, both of which may be listed in the specifications of the vial. Alternatively, the diameters can be directly measured using a micrometer.

The value of Area Ratio is used, along with the value of Fill Volume, to calculate the depth of the frozen product from which sublimation will take place, which affect both the primary drying time and maximum product temperature.

## Solution Density

The Solution Density refers to the total mass of the solution divided by the total volume of the solution to be filled into the vials for lyophilization.

For most solutions with total solute concentrations of about 5% (w/w) or less, the solution density can be approximated as 1 g/mL. Density may be a larger value for solutions of higher total solute concentrations (i.e., >10%).

The value of solution density is used in the calculation of the amount of ice to be removed by sublimation during primary drying.

## Solute Density

Solute density is the mass of all solutes divided by the total volume occupied by those solutes.

• In the case that the solutes crystallize, the weight average crystalline density of the solutes should be inserted in for solute density.
• In the case that the solutes do not crystallize, but instead form an amorphous phase, the density of the dried amorphous phase can be inserted in for solute density. However, it is often difficult to make this measurement without specific equipment. Instead, the weight average crystalline density of the solutes multiplied by 1.05 is an often-used approximation of the amorphous solute density.

The Solute density is used in the calculation of the depth of the frozen product from which ice is removed by sublimation during primary drying.

## Ice Density

Ice Density is lower than the maximum density of water at 4°C (1.00 g/mL) and increases as temperature decreases. However, the change in density of ice with temperature is modest (i.e., 0.6% over the range of 0°C to -50°C). For users who would like to enter a temperature-specific value of ice density in the LyoModelling Calculator, the relevant temperature to reference is the product temperature, not the shelf temperature.

Temperature [°C] Density * [g/cm3]
0 0.9167
-10 0.9182
-20 0.9196
-30 0.9209
-40 0.9222
-50 0.9235

* Harvey, A., Properties of Ice and Supercooled Water, in CRC Handbook of Chemistry and Physic 95th Edition, 2015. p. 6-12.

## Heat of Sublimation

The heat of sublimation refers to the energy required to remove ice by sublimation during primary drying. The value for the heat of sublimation is fairly constant over the relevant temperature range.

Temperature [°C] ΔHsub [J/g]* ΔHsub [cal/g]
0 2834 680.3
-8 2836 680.7
-18 2838 681.1
-28 2839 681.3
-38 2839 681.3
-48 2838 681.1

If vial heat transfer coefficients, Kv , have been carefully measured, use the same value for heat of sublimation in the LyoModelling calculator as was used for Kv determination.

* Feistel, R. and W. Wagner, Sublimation pressure and sublimation enthalpy of H2O ice 1h between 0 and 273.16 K. Geochimica et Cosmochimica Acta, 2007. 71(1): p. 36-45.

## Effective Thermal Conductivity

The effective thermal conductivity value to enter here is for the frozen product. At low concentrations of total solute, the effective thermal conductivity of the frozen product is equal to that of the ice network in the frozen product, which has been measured to equal 5.9 x 10-3 cal⋅cm-1⋅K-1⋅sec-1 *

At higher solute concentrations, the effective thermal conductivity value is expected to be lower. However, the estimates of primary drying time and maximum product temperature are not highly dependent on this value, particularly for the usual fill depths of 0.3 - 0.6cm.

* See Figure 10 and the accompanying text in Pikal, MJ, Roy ML, Shah 5, Mass and Heat Transfer in Vial Freeze-Drying of Pharmaceuticals: Role of the Vial, Journal of Pharmaceutical Sciences, 73(9): 1224-1237 (1984).

## Vial Heat Transfer Coeficient

The vial heat transfer coefficient, ${K}_{v}$, is used to calculate the flow of heat into the vial to supply the energy required for sublimation of ice during primary drying. In general, heat is supplied to the product in the vial by a) contact conduction from the shelf, b) gas conduction from the shelf, and c) radiation from all surfaces with a direct view to the vial. The LyoModelling Calculator’s estimates of primary drying time and maximum product temperature are very sensitive to the value of the vial heat transfer coefficient, ${K}_{v}$. Furthermore, the value of ${K}_{v}$ is highly dependent on chamber pressure, particularly at pressures below 100 mTorr.

### Measuring Kv

A batch average value for ${K}_{v}$ can be determined from the sublimation rate $\left(\frac{dm}{dt}\right)$ measured during a cycle at a relevant shelf temperature, ${T}_{s}$, and chamber pressure, ${P}_{c}$, and during which the temperature of the product (usually only water for these determinations) at the bottom center, ${T}_{b}$, of the vial with a cross-sectional area, ${A}_{v}$ is also monitored. Substituting the measured values in to the equation

${K}_{v}=\frac{\left(\frac{dm}{dt}\right)\left(\Delta {H}_{sub}\right)}{{A}_{v}\left({T}_{s}-{T}_{b}\right)}$

where $\Delta {H}_{sub}$ is the enthalpy of sublimation of ice, values of ${K}_{v}$ for the particular freeze-dryer and conditions can be calculated.

### Roughly Estimating Kv

In the absence of a known value for ${K}_{v}$, a gross estimate for tubing vials may be based on the following broad generalizations, with the understanding that they are only approximate.

For vials in the center of a hexagonal array of vials, the vial heat transfer coefficient, ${K}_{v,c}$, is composed of a pressure-dependent coefficient that is due to gas conduction from the shelf surface to the bottom of the vial, ${K}_{v,c}$ and other non-pressure-dependent heat flow, ${K}_{other}$.

${K}_{v,c}={K}_{g,c}\left({P}_{ch}\right)+{K}_{other}$

**

Vials at the edge of the array are known to have higher heat transfer coefficients. The difference between the edge and center vial heat transfer coefficent, $\Delta {K}_{edge}$, can be added to the value for the center coefficient, ${K}_{v,c}$

$\Delta {K}_{v,edge}={K}_{v,edge}-{K}_{v,c}\approx 0-5$***

* where chamber pressure, ${P}_{ch}$, is expressed in units of Torr and ${K}_{v}$ as cal⋅cm-2⋅K-⋅sec-1
** where 1.2 is more applicable to low shelf temperatures (${T}_{s}$) and 3 is for high ${T}_{s}$
*** where 0 applies to low ${P}_{ch}$ and high ${T}_{s}$(shelf temp); 5 applies to high ${P}_{ch}$ and low ${T}_{s}$

## Method for Estimating Kv

The three options for providing a value for ${K}_{v}$ are:

1. DEFAULT VALUE - use the default value of 4x10-4 cal⋅cm-2⋅K-1⋅sec-1
2. ESTIMATED based on ${P}_{c}$ - the value is based on the chamber pressure entered in the set of USER INPUTS using the formula:

${K}_{v}=\left[\frac{\left(6.4×{10}^{-4}\right)\left({P}_{c}\right)}{1+\left(9×{P}_{c}\right)}\right]+\left(1×{10}^{-4}\right)$ cal⋅cm-2⋅K-1⋅sec-1
where ${P}_{c}$ is in units of Torr. This expression allows the value of ${K}_{v}$ to be adjusted realistically when the chamber pressure input is changed.
3. EXTRAPOLATED based on ${P}_{c}$ - having entered one value of chamber pressure, ${P}_{c,1}$, and a value of ${K}_{v,1}$ by any of the four methods, when EXTRAPOLATION is selected, the new value of vial heat transfer coefficient, ${K}_{v,2}$ is based on the first chamber pressure, ${P}_{c,1}$, and vial heat transfer coefficient, ${K}_{v,1}$ , and the new chamber pressure, ${P}_{c,2}$ based on the formula

This equation finds a pressure-dependent portion of ${K}_{v}$ and adjusts it for the new pressure.
4. USER DEFINED - allows the advanced user to enter a particular value of ${K}_{v}$ in units of cal⋅cm-2⋅K-1⋅sec-1

If "ESTIMATED based on Pc" or "EXTRAPOLATED based on Pc" are selected, the estimated or extrapolated value of Kv will appear when the CALCULATE button is clicked.

## Primary Drying Time

This is an estimate based on assumptions. Good freeze-drying practice is to use an additional 20% "soak" time to the batch average estimate from the LyoModelling Calculator.

## Average Sublimation Rate

To obtain the sublimation rate for the entire batch of vials, multiply the "Average Sublimation Rate per Vial" by the total number of vials. Check the total sublimation rate with the maximum that can be handled by the freeze-dryer to be used. There is a maximum sublimation rate beyond which water vapor begins to accumulate in the drying chamber. The maximum decreases with lower chamber pressure and is dependent on freeze-dryer design. If the maximum sublimation rate is reached, the chamber pressure will rise above the set point and the process will be considered "out of control". Two reasons for loss of control are (1) condenser overload and (2) choked flow – vapor velocity cannot exceed the Mach 1 limit. If the calculated average sublimation rate x batch size exceeds the capacity of the freeze-dryer, adjust the shelf temperature, chamber pressure and/or batch size to lower the average sublimation rate to maintain control during the process. Note that when using a constant shelf temperature, the sublimation rate decreases as primary drying proceeds, due to the increase in dry layer resistance. Thus, the calculated average sublimation rate is lower than the sublimation rate expected early in primary drying.

## Maximum Product Temperature

The temperature of the product is highest at the bottom of the vial and increases as primary drying proceeds, due to the increase in dry layer resistance. The Maximum Product Temperature calculated by the LyoModelling Calculator is the temperature of the product adjacent to the bottom of the vial at the end of primary drying. For most products, there is a critical product temperature beyond which (1) the amorphous product collapses, (2) crystalline or partially crystalline product undergoes eutectic melting, and/or (3) product chemical or physical stability declines to an unacceptable level. Adjust the shelf temperature and/or chamber pressure to maintain the product temperature below the critical temperature.