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OPTIMAL CONTROL OF THE PRIMARY AND SECONDARY DRYING STAGES OFTHE FREEZE DRYING OF PHARMACEUTICALS IN VIALS

A. I. Liapis1, R. Bruttini2and H. Sadikoglu3

1. Department of Chemical Engineering and Biochemical Processing Institute, University

of Missouri-Rolla, Rolla, Missouri 65401-0249, U.S.A.2. Criofarma-Freeze Drying Equipment, Strada del Francese 97/2L, 10156 Turin, Italy

3. Department of Energy Systems, Gebze Institute of Technology, P.O. Box 141, 41410Gebze, Turkey

Keywords: optimal control of freeze drying in vials, optimal control of lyophilization in vials

ABSTRACT

The process of freeze drying a pharmaceutical product in vials loaded on a tray freeze

dryer to obtain a desired final bound water content in minimum time is formulated as an

optimal control problem and is solved through the use of a rigorous unsteady state modelthat describes the dynamic behavior of the freeze drying process in vials and the

equations of the necessary conditions of optimality for both the primary and secondary

drying stages. The optimal control policy (a) produces drying times for the primary andsecondary drying stages that are significantly smaller than the drying times obtained

from conventional operational policies, (b) makes the geometry of the moving interface

during primary drying to have insignificant curvature and to be, for all practicalpurposes, almost planar, and (c) produces at the end of secondary drying temperature

and concentration of bound water profiles along the length and radius of the vial that are

almost uniform, as would be the desirable result in practice.

INTRODUCTION

The problem of optimal control of the primary and secondary drying stages of bulk solution freeze

drying in trays to obtain a desired final bound water content in minimum time was formulated and solved

by Sadikoglu et al. (1998). The work described here concerns itself with the analysis of the optimal

control problem for the primary and secondary drying stages of the process involving freeze drying ofpharmaceutical products in vials loaded on trays. Sheehan and Liapis (1998) and Sheehan et al. (1998)

have shown that heat input control that runs the process of freeze drying in vials close to the melting and

scorch temperature constraints when the total pressure in the drying chamber is kept constant, provides (i)

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faster movement of the interface during primary drying, (ii) smaller changeover times between primary

drying and the start of the secondary drying stage, (iii) faster drying times, and (iv) more uniformdistributions of temperature and concentration of bound water at the end of the secondary drying stage.

The core of the analysis of the optimal control problem described in this work, employs the rigorous non-

steady state mathematical model of Sheehan and Liapis (1998) that has been found to describe

satisfactorily the dynamic behavior of the freeze drying process in vials, and then, having chosen energyinput and chamber pressure as control variables (Sadikoglu, 1998), with maximum surface (scorch) and

interface (melting) temperatures as constraints (Sadikoglu et al. 1998, 1999; Sadikoglu, 1998; Sheehan

and Liapis, 1998; Rey and May, 1999), variational calculus (Liapis and Litchfield, 1979; Litchfield et al.,1981; Litchfield and Liapis, 1982; Sadikoglu et al., 1998; Sadikoglu, 1998) was used to develop the

necessary conditions (Sadikoglu, 1998) of optimality for both the primary and secondary drying stages.

Then by considering the results of the numerical solution (Sheehan and Liapis, 1998) of the problem ineach of the possible control situations (Sadikoglu, 1998) during the primary and secondary drying stages

of the freeze drying process, the optimal operating conditions of the freeze drying of skim milk in vials are

determined.

MATHEMATICAL FORMULATION

Mathematical model

In this work, freeze drying in vials is considered and the rigorous dynamic model of Sheehan and

Liapis (1998) is used to describe quantitatively the dynamic behavior of the primary and secondary dryingstages of the freeze drying process. The equations of the model are given by equations (1)-(53) in

Sheehan and Liapis (1998) and Figure 1 in Sheehan and Liapis (1998) represents the diagram that

indicates the energy inputs (qI, qII, and qIII) to a material in a vial on a tray during freeze drying; q Idenotesthe heat flux to the top surface of the material being dried, q II is the energy input to the material by the

heating plate at the bottom of the vial and the magnitude of qIIdepends on the value of the temperature,

Tlp, of the lower heating plate, and qIII represents the heat flux to the side surface of the material being

dried. The magnitude of qIalso depends on the value of the temperature, Tlp, of the heating plate, because

current freeze dryer designs use the same temperature for both upper and lower plates (Sadikoglu, 1998;Rey and May, 1999), and thus, Tup= Tlpwhere Tupdenotes the temperature of the upper heating plate.

The equations of the mathematical model were solved numerically (Sheehan and Liapis, 1998; Sadikoglu,1998), after the immobilization of the moving interface during the primary drying stage through the use of

the transformations given by equations (54)-(57) in Sheehan and Liapis (1998).

Control variables, constraints and necessary conditions of optimality

In our freeze drying system which is representative of those used in practice, the control variables arethe heat input qIIand the drying chamber pressure, Po, and both of these variables affect significantly the

mass and heat transfer mechanisms of the freeze drying process. The magnitude of qIIdepends on the

value of Tlpwhich is determined by the heating of the plates; the plates of the freeze dryer can be heatedeither electrically or by using a heating fluid, and the heating of the plates can be controlled by an

appropriate regulator (Liapis and Bruttini, 1994). The value of Po can be changed by changes in the

partial pressure of water vapor in the chamber (the partial pressure of water vapor can be changed bychanges in the temperature of the ice condenser) and by increasing or decreasing the partial pressure of the

inerts in the chamber (the partial pressure of the inerts increases by opening a bleeding valve, and

decreases through the use of a vacuum pump (Mellor, 1978; Bruttini et al., 1991; Liapis and Sadikoglu,

1998)).The controls qII(t) and Po(t) must be selected from a set of admissible controllers

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*

*IIIIII q)t(qq (1)

*

* oooP)t(PP (2)

This set of controllers excludes those which would produce an unacceptable product quality.

During the primary drying stage, important constraints on the product state are that the temperature at anypart of the dried layer must not exceed the scorch temperature, Tscor, and the temperature at any part of the

frozen layer must not exceed the melting temperature, Tm. The temperature constraints during primary

drying are represented by expressions (3) and (4)

Rr0),r,t(HZz0,0tforT)r,z,t(T scorI = (3)

Rr0,Lz)r,t(HZ,0tforT)r,z,t(T mII => (4)

where TIand TIIdenote the temperature in the dried (I) and frozen (II) layers of the material in the vial,

respectively, and H(t, r) represents the geometric shape of the moving interface. The objective during

primary drying is to minimize the drying time, tpd, equivalent to defining a performance index

== pdoIIoII

t

0)t(P),t(qpd

)t(P),t(qpd dtminQminQ (5)

where tpd is defined as the time when the position Z (Z = H(t, r)) of the moving interface, H(t, r), has

reached the bottom surface of the material being dried (Z = H(tpd, r) = L for 0 r R). The problemgiven in equations (1), (2), and (5) along with the dynamic mathematical model, equations (1)-(32) in

Sheehan and Liapis (1998), of the primary drying stage is the standard time optimal control problem. Thenecessary conditions of optimality for the primary drying stage have been presented in Sadikoglu (1998)

and their mathematical expressions are complex and lengthy; these expressions are not reported herebecause of space limitations. The optimal control problem was solved by assuming a control policy based

on the physical mechanisms of the process, for the entire primary drying stage, and integrating the modelequations. Fortunately, the physics of the freeze drying process allows only a limited number of control

possibilities, and therefore, this approach is feasible. When the requirements of the necessary conditions

of optimality were satisfied by the assumed control policy, then it was considered that the assumed controlpolicy is optimum.

During the secondary drying stage, the temperature constraint is represented by equation (6)

Rr0,Lz0,ttforT)r,z,t(T L)r,t(HZscorI == (6)

which requires that the temperature at any part of the dried layer must not exceed the scorch temperature,Tscor, of the dried product. The objective during secondary drying is to minimize the drying time, tsd,equivalent to defining a performance index

== sdoIIoII

t

0)t(P),t(qsd

)t(P),t(qsd dtminQminQ (7)

where tsdis defined as the time when a fixed amount of bound water (solvent) remains in the product. Theproblem given in equations (1), (2) and (7) along with the dynamic mathematical model, equations (33)-

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(53) in Sheehan and Liapis (1998), of the secondary drying stage is the standard time optimal control

problem. The necessary conditions of optimality for the secondary drying stage have been presented inSadikoglu (1998) and their mathematical expressions are complex and lengthy; these expressions are not

reported here because of space limitations. Again, as in the case of primary drying, the optimal control

problem was solved by assuming a control policy based on the physical mechanisms of the process, for

the entire secondary drying stage, and integrating the model equations. When the requirements of thenecessary conditions of optimality were satisfied by the assumed control policy, then it was considered

that the assumed control policy is optimum.

RESULTS AND DISCUSSION

In this work, the optimal control of the freeze drying of skim milk in vials is studied. Skim milk was

selected because it could be considered as a complex pharmaceutical product in the sense that it containsenzymes and proteins; furthermore, the values of the parameters that characterize the heat and mass

transfer mechanisms in the skim milk during freeze drying are well known (Liapis and Bruttini, 1994;

Sheehan and Liapis, 1998). The values of the parameters for skim milk used in the model of Sheehan andLiapis (1998) for the calculations performed in this work, are reported in Table I of the work of Sheehan

and Liapis (1998). The values of the melting and scorch temperatures of skim milk are 263.15 K and313.15 K, respectively, while the temperature at the start of the drying stage is 233.15 K. Sheehan and

Liapis (1998) in their simulations involving the freeze drying of skim milk in vials used a constant dryingchamber pressure, Po, of 5.07 N/m

2 and varied the values of qI and qII by varying the value of the

temperature of the lower heating plate, Tlp (Tup = Tlp in the freeze dyer used by Sheehan and Liapis

(1998)), in order to satisfy the temperature constraints given by equations (3), (4) and (6). Sheehan andLiapis (1998) and Sheehan et al. (1998) have also considered the effect of the location of a vial on a tray

by considering the following two cases: in Case A, the vial is located at the outermost edge of the array of

vials on the tray; in Case B, the vial is placed at the center of the array of vials on the tray. Duringprimary drying, the temperature of the

heating plates is adjusted, for both Cases

A and B, to the highest value which

satisfies the constraints given byequations (3) and (4). When all of the

frozen layer has disappeared, the heating

plates are set, for both Cases A and B, tothe scorch temperature so that the

constraint given by equation (6) is

always satisfied during the secondarydrying stage. The location of the vial on

the tray affects the value of qIII, since,

for example, for vials at the center of thetray radiation from chamber side walls is

negligible. For both Cases A and B,Sheehan and Liapis (1998) kept the

value of the drying chamber pressure,Po, constant and equal to 5.07 N/m

2

during the primary and secondary drying stages of the freeze drying process. The conditions of Cases A

and B correspond to the conditions of Cases I and III, respectively, in Sheehan and Liapis (1998), andprovide for the maximum amount of heating possible at any instant for vials located at the edge and center

of the tray, respectively. Sheehan and Liapis (1998) found that Cases A and B provide heat input control

that runs the process close to the melting and scorch temperature constraints, and yields (i) faster dryingtimes, and (ii) more uniform end-product distributions of temperature and concentration of bound water,

Figure 1. Optimal control policy: time variation of the drying chamberpressure, Po, and of the heat flux, qII, during the primary drying stageof skim milk in a vial.

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especially for Case B (Case B combines high plate heating and low side heating, while in Case A high

side heating is coupled with high plate heating).In the control policies examined in this work, the minimum and maximum values of Powere taken, for

both the primary and secondary drying stages, to be 1 Pa and 1750 Pa, respectively. The maximum value

of the heat input, qII, was taken to be 3.6 kW/m2during primary drying and 1.0 kW/m

2during secondary

drying; the minimum value of qIIwas considered, for both the primary and secondary drying stages, to beequal to zero. Numerous (Sadikoglu, 1998) drying chamber pressure, P

o(t), and heat input, q

II(t), policies

were examined and one control strategy was found that satisfied the necessary conditions for optimality,

and therefore, this control strategy is considered to represent the optimum control policy that minimizesthe total batch time of the freeze drying of skim milk in the freeze dryer considered in this work.

In Figure 1, the dynamic behavior of

the optimal control policies for thedrying chamber pressure, Po, and heat

input, qII, during the primary drying

stage are presented for the case wherethe vial is located at the center of the

array of vials on the tray. The value

of Poremains constant and equal to 1

Pa during primary drying while thevalue of qIIis at its maximum during the initial heating period and, then, it is decreased with time after the

value of TII(t, z = L, r) reaches its melting temperature constraint; furthermore, it was found that the top

surface and interface temperatures never reach their respective scorch and melting temperature constraintsduring primary drying. The value of TII(t, z = L, r) became equal to the value of Tmvery soon after the

start of primary drying and this indicates that the freeze drying system is mass transfer controlled during

primary drying. Therefore, it makes sense for the optimal control policy to require that Po be at itsminimum value during primary drying since a low value of Pokeeps the mass transfer rate in the dried

layer very high, and such a policy outweighs the benefit that a higher value of Pocould provide through

increased heat transfer in the dried layer because a higher value of Powould significantly reduce the mass

transfer rate in the dried layer of this mass transfer controlled freeze drying process during primary drying.It was also found that the optimal control policy during the primary drying stage made the geometry, H(t,

r), of the moving interface to be, for all practical purposes, almost planar, and the radial temperature and

concentration of bound water profiles during the primary drying stage were found to be almost uniform.The drying time obtained for the primary drying stage using the optimal control policy, is equal to 7.21

hours, while the operational policies employed (Sheehan and Liapis, 1998) for Cases A and B required

12.44 hours and 12.73 hours, respectively. In Table 1 the primary drying times and the changeover timesfor Cases A and B as well as for the case where the optimal control policy presented in Figure 1 is

employed, are presented. The changeover time,

tb ta, gives the time period when primary drying

and secondary drying coexist within the vial; tadenotes the time when first H(t, r = R) = L and tbrepresents the time when first H(t, r = 0) = L (asper Figure 1c in Sheehan and Liapis (1998)).The results in Table 1 clearly indicate that the

operational policies employed for Cases A and B

required, respectively, 72.54% and 76.56% moretime for the completion of primary drying than the minimum time of 7.21 hours obtained by using the

optimal control policy. Furthermore, the changeover time for the case employing the optimal control

policy is insignificant and this indicates that the geometric shape of the moving interface was, for all

practical purposes, almost planar. Table 2 gives the value of the angle for Cases A and B as well as for

Case A Case B Optimal Control Policy

Primary Drying time 12.44 12.73 7.21

Changeover time (tb- ta) 0.08 0.002 0.00014

Secondary drying time 12.37 12.43 5.13

Total drying time 24.89 25.16 12.34

Table 1: Drying times (in hours).

Case A Case B Optimal Control Policy

2 hours 1.40 0.03 0.0011

4 hours 1.48 0.04 0.0012

6 hours 1.57 0.03 0.0011

Table 2: Values of the angle (in degrees) during theprimary drying stage.

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the case where the optimal control policy is employed, at times t = 2, 4, and 6 hours. The value of the

tangent of the angle is obtained from equation (8)

Rrr

)r,t(Htan

=

= (8)

where is the angle between the horizontal and a tangent to the interface surface at r = R; thus, the anglegives a quantitative measure of the curvature of the moving interface during primary drying. The valueof the angle is derived from the generated form (Sheehan and Liapis, 1998) of H(t, r) using equation (8).The results in Table 2 clearly show that the geometry of the moving interface for the case where theoptimal control policy is employed, is in effect planar.

In Figure 2, the dynamic behavior of

the optimal control policies for the

drying chamber pressure, Po, and heatinput, qII, during the secondary drying

stage are presented, for the case where

the vial is located at the center of the

array of vials on the tray. The value ofPo is kept constant at its minimum

pressure of 1 Pa while the temperatureat the bottom surface of the material

being dried is heated to its new

permitted maximum of the scorchtemperature, Tscor, using heat input, qII,

at its maximum value, and after about

245 seconds, the chamber pressure, Po,

is raised to its maximum value of 1750Pa and held constant for the remaining

time of secondary drying while the heatinput decreases to maintain thetemperature of the bottom surface of the material being dried at the scorch temperature, Tscor. The drying

time obtained for the secondary drying stage using the optimal control policy, is equal to 5.13 hours, while

the operational policies employed (Sheehan and Liapis, 1998) for Cases A and B required 12.37 and 12.43hours, respectively, as shown in Table 1. This significantly large reduction in the drying time of

secondary drying when the optimal control policy is used, is due to the fact that the bottom temperature,

TII(t, z = L, r), is at its constraint for most of the time of secondary drying and the drying chamber

pressure, Po, is kept at its maximum value, and these operational policies increase significantly theconduction of heat in the dried layer of the material so that the rate of removal of the bound water from the

solid phase of the dried layer becomes very high. It makes sense for the optimal control policy to require

that Poshould be at its maximum value during secondary drying (after the time at which the value of T II(t,z = L, r) reaches the value of Tscor), since a high value of Pokeeps the heat transfer rate in the dried layer

very high, and such a policy outweighs the benefit that a lower value of Po could provide through

increased mass transfer in the dried layer because a lower value of P owould significantly reduce the heattransfer rate in the dried layer of this heat transfer controlled freeze drying process during secondary

drying. It was also found that the optimal control policy during the secondary drying stage produced at

the end of secondary drying temperature and concentration of bound water profiles along the length and

radius of the vial that are, for all practical purposes, almost uniform, as would be the desirable result inpractice.

Figure 2. Optimal control policy: time variation of the drying chamber

pressure, Po, and of the heat flux, qII, during the secondary dryingstage of skim milk in a vial.

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CONCLUSIONS

The process of freeze drying a pharmaceutical product in vials loaded on a tray freeze dyer to obtain a

final bound water content in minimum time was formulated as an optimal control problem and was solvedwith the use of the rigorous unsteady state mathematical model of Sheehan and Liapis (1998) and the

necessary conditions for optimality, for both the primary and secondary drying stages, are presented in

Sadikoglu (1998). The theoretical approach was applied in the freeze drying of skim milk in order toconstruct the optimal control policies of the drying chamber pressure, Po, and heat input, qII, that minimize

the drying times of the primary and secondary drying stages of the freeze drying of skim milk in vials

loaded on trays. The optimal control policy produced drying times for the primary and secondary dryingstages that are significantly smaller than the drying times obtained from conventional operational policies

(Sheehan and Liapis, 1998). Furthermore, the optimal control policy (i) makes the geometry of the

moving interface during primary drying to have insignificant curvature and to be, for all practical

purposes, almost planar, and (ii) produces at the end of the secondary drying stage temperature andconcentration of bound water profiles along the length and radius of the vial that are almost uniform, as

would be the desired in practice result.

ACKNOWLEDGMENTThe authors gratefully acknowledge support of this work by Criofarma and the Biochemical Processing

Institute of the University of Missouri-Rolla.

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NOTATION

H(t, r) geometric shape (as per Figure 1 in Sheehan and Liapis (1998))

of the moving interface, a function of time and radial distance

m

L length of sample mNw water vapor mass flux kg /m

2s

Po drying chamber pressure N/m2

Qpd performance index for primary drying stage s

pdQ defined in equation (5) s

Qsd performance index for secondary drying stage s

sdQ defined in equation (7) s

qI heat flux to the top surface of the material being dried W/m2

qII heat flux to the material by the heating plate at the bottom of

the vial

W/m2

qIII heat flux to the side surface of the material being dried W/m2

R radius of vial m

r space coordinate of radial distance mTI temperature of the dried layer (I) of the material in the vial KTII temperature of the frozen layer (II) of the material in the vial K

Tlp temperature of the lower heating plate K

Tm melting temperature K

Tscor scorch temperature K Tup temperature of the upper heating plate K

t time s

ta t when first H(t, r = R) = L stb t when first H(t, r = 0) = L s

tpd drying time for the primary drying stage s

tsd drying time for the secondary drying stage stZ = H(t, r) = L time at which Z = H(t, r) = L for 0 r R sZ value of z at the moving interface m

z space coordinate of distance along the length of the vial m

Greek Symbols

angle between tangent to moving interface at r = R andhorizontal, as defined in equation (8)

degrees

Subscripts Superscripts* minimum value of qIIand Po * maximum value of qIIand Po

LITERATURE

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lyophilization using a pilot plant, Chemical Engineering Journal, Vol. 45, pp. B67-B77.

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Liapis, A. I., Bruttini, R. (1994), A theory for the primary and secondary drying stages of the freeze-

drying of pharmaceutical crystalline and amorphous solutes: Comparison between experimental dataand theory, Separations Technology, Vol. 4, pp. 144-155.

Liapis, A. I., Litchfield, R. J. (1979), Optimal control of a freeze dryer I: Theoretical development andquasi steady state analysis, Chemical Engineering Science, Vol. 34, pp. 975-981.

Liapis, A. I., Sadikoglu, H. (1998), Dynamic pressure rise in the drying chamber as a remote sensingmethod for monitoring the temperature of the product during the primary drying stage of freeze drying,

Drying Technology, Vol. 16, no. 6, pp. 1153-1171.

Litchfield, R. J., Farhadpour, F. A. and Liapis, A. I. (1981), Cyclical pressure freeze drying, Chemical

Engineering Science, Vol. 36, pp. 1233-1238.

Litchfield, R. J., Liapis, A. I. (1982), Optimal control of a freeze dryer II: Dynamic analysis, ChemicalEngineering Science, Vol. 37, pp. 45-55.

Mellor, J. D., 1978, Fundamentals of freeze drying, Academic Press, London, England.

Rey, L., May, J. C. (Editors), 1999, Freeze-Drying/Lyophilization of Pharmaceutical and BiologicalProducts, Marcel Dekker, Inc., New York, New York, U.S.A.

Sadikoglu, H. (1998), Dynamic modelling and optimal control of the primary and secondary dying stagesof freeze drying of solutions in trays and vials, Ph.D. Dissertation, Department of Chemical

Engineering, University of Missouri-Rolla, Rolla, Missouri, U.S.A.

Sadikoglu, H., Liapis, A. I. and Crosser, O. K. (1998), Optimal control of the primary and secondary

drying stages of bulk solution freeze drying in trays, Drying Technology, Vol. 16, no. 3-5, pp. 399-431.

Sadikoglu, H., Liapis, A. I., Crosser, O. K. and Bruttini, R. (1999), Estimation of the effect of product

shrinkage on the drying times, heat input, and condenser load of the primary and secondary dryingstages of the lyophilization process in vials, , Drying Technology, Vol. 17, no. 10, pp. 2013-2035.

Sheehan, P., Liapis, A. I. (1998), Modeling of the primary and secondary drying stages of the freeze

drying of pharmaceutical products in vials: Numerical results obtained from the solution of a dynamic

and spatially multi-dimensional lyophilization model for different operational policies, Biotechnologyand Bioengineering, Vol. 60, no. 6, pp. 712-728.

Sheehan, P., Sadikoglu, H. and Liapis, A. I. (1998), Dynamic behavior and process control of the primaryand secondary drying stages of freeze drying of pharmaceuticals in vials, Proceedings of the 11th

International Drying Symposium, Vol. C, pp. 1727-1740.