On the Use of Mathematical Models to Build the Design Space for the Primary Drying Phase of a Pharmaceutical Lyophilization Process

On the Use of Mathematical Models to Build the Design Space for the Primary Drying Phase of a Pharmaceutical Lyophilization Process ANNA GIORDANO, ANTONELLO A. BARRESI, DAVIDE FISSORE Dipartimento di Scienza dei Materiali e Ingegneria Chimica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy Received 3 February 2010; revised 7 April 2010; accepted 11 May 2010 Published online 23 June 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jps.22264 ABSTRACT: The aim of this article is to show a procedure to build the design space for the primary drying of a pharmaceuticals lyophilization process. Mathematical simulation of the process is used to identify the operating conditions that allow preserving product quality and meeting operating constraints posed by the equipment. In fact, product temperature has to be maintained below a limit value throughout the operation, and the sublimation flux has to be lower than the maximum value allowed by the capacity of the condenser, besides avoiding choking flow in the duct connecting the drying chamber to the condenser. Few experimental runs are required to get the values of the parameters of the model: the dynamic parameters estimation algorithm, an advanced tool based on the pressure rise test, is used to this purpose. A simple procedure is proposed to take into account parameters uncertainty and, thus, it is possible to find the recipes that allow fulfilling the process constraints within the required uncertainty range. The same approach can be effective to take into account the heterogeneity of the batch when designing the freeze-drying recipe. ß 2010 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 100:311–324, 2011

Keywords:

freeze-drying; design space; process modeling; dynamic parameters estimation

INTRODUCTION Freeze-drying is a process widely used to remove water from a solution containing an active pharmaceutical ingredient (API), thus providing long term stability. In fact, the process can be carried out in sterile conditions and at low temperatures, thus preserving product quality. Besides, a freeze-dried product has a high surface area that makes the rehydration very easy. The batch of vials, filled with the solution containing the API, is generally placed over the shelves in a drying chamber. At the beginning of the operation product temperature is lowered below the freezing point of the solvent (generally water): during this step not all the solvent freezes, forming ice crystals, but a certain amount remains bound to the product. Also the API and the excipients often do not crystallize, and form an amorphous glass that can retain a high amount of solvent. After freezing, the pressure in the Abbreviations: API, active pharmaceutical ingredient; DPE, dynamic parameters estimation; MTM, manometric temperature measurement; PRT, pressure rise test. Correspondence to: Davide Fissore (Telephone: 39-11-0904693; Fax: 39-11-0904699; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 100, 311–324 (2011) ß 2010 Wiley-Liss, Inc. and the American Pharmacists Association

drying chamber is reduced, thus causing ice sublimation (primary drying): during this step the product is maintained at a low temperature (typically in the range from
40 to
108C) and is continuously heated, as the sublimation is endothermic. Finally, when all the ice has sublimated, product temperature is increased (e.g., up to 20–408C) so that the bound water is desorbed (secondary drying) and the target value of residual water is reached.1,2 The operating conditions, that is, the temperature of the shelf and the pressure in the chamber, determine the evolution of the temperature and of the residual water content of the product during the process and, thus, they have to be carefully selected in order to preserve the API.3,4 Product temperature has to be maintained below a limit value corresponding to the eutectic temperature in case of a product that crystallizes during freezing: this is required to avoid the formation of a liquid phase, and the successive boiling due to the low pressure. In case of a product that remains amorphous, the limit value corresponds to the glass transition temperature: in this case the goal is to avoid shrinkage and collapse of the dried cake, as they can be responsible for a higher amount of residual water in the final product, and for a higher reconstitution time. Moreover, a collapsed product is often rejected because of the unattractive physical

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appearance. Beside shelf temperature and chamber pressure, the other parameter that has to be carefully selected is the duration of the two drying steps. In fact, if secondary drying is started before the end of primary drying, product temperature may become higher than the maximum allowed value, while if secondary drying is delayed, the cycle is not optimized. Finally, the duration of the secondary drying has to be selected in order to reach the desired value of residual moisture in the product.5,6 The equipment used to carry out the operation poses other constraints that have to be taken into account when looking for the best recipe. In particular, there is an upper limit to the vapor flow rate that can be condensed. A further limitation is due to the fluid-dynamics of the vapor in the duct connecting the chamber to the condenser. In fact, the velocity of the vapor cannot be higher than the speed of sound in water vapor: as the velocity of vapor approaches this limit the flow of water is choked, and any further reduction in the downstream pressure has no influence on the mass flow rate through the duct.7 Primary drying is generally recognized to be the longest and the most risky phase of the whole process. This is due to the higher amount of water in the partially dried product that requires maintaining the product below a very low temperature. Moreover, the sublimation flow rate is higher during primary drying. The operating conditions (shelf temperature, chamber pressure, and drying time) for this phase are generally identified through an extended experimental campaign with the goal to obtain a product with acceptable quality. This approach can be really expensive and time consuming, even if the number of tests required to design the recipe can be reduced by using the Experimental Design Technique8 and the Multi-Criteria Decision Making method,9,10 as it has been shown by Baldi et al.11 This motivates the research for most efficient methods to design the recipe. In this framework, the ‘‘Guidance for Industry PAT—A Framework for Innovative Pharmaceutical Development, Manufacturing, and Quality Assurance’’ issued by US-FDA in September 2004 encourages to implement a true quality-by-design manufacturing principle, rather than the classical quality-by-testing approach, to have safe, effective and affordable medicines: the goal is that quality is no longer tested into products, but it is built-in or is by design. Various techniques have been proposed to get quality-by-design in a pharmaceutical freeze-drying process. They can be roughly divided into two groups, depending on whether the recipe is obtained in-line or off-line. The methods of the first group are based on a monitoring system that provides information about the state of the system (i.e., product temperature and JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

amount of water), and of a control system that manipulates the shelf temperature and, eventually, the chamber pressure. Fissore et al.12 proposed a simple system based on a soft-sensor, that is, a device that uses the temperature measurement provided by a thermocouple and a mathematical model of the process, to provide a real-time estimation of the state of the product.13–18 As an alternative, the state of the product can be estimated from the measurement of the pressure rise occurring in the drying chamber when the valve placed in the duct connecting the chamber to the condenser is periodically closed for a short time interval, for example, 5–30 s (pressure rise test, PRT). A mathematical model is used to interpret the pressure rise, and the state of the product and the parameters of the model are estimated looking for the best fit between measured and calculated values of chamber pressure. Various algorithms have been proposed to this purpose, and thus used in a control loop, for example, the Manometric Temperature Measurement (MTM) algorithm proposed by Milton et al.19 and the dynamic parameters estimation (DPE) algorithm proposed by Velardi et al.20 Results obtained using the MTM algorithm, with some empirical and good practice rules, have been used to manipulate the shelf temperature and the chamber pressure by an expert system, the SMARTTM Freeze-Dryer, recently proposed and patented by Tang et al.21 and by Pikal et al.22 and Barresi and coworkers14,17,18,23,24 developed a control system, called LyoDriver, that uses DPE algorithm as measuring device, and a mathematical model to calculate the shelf temperature required to keep product temperature at the desired value. Both the soft-sensor and the PRT-based methods pose some problems that have to be solved. In fact, the soft-sensor requires using a thermocouple to measure the temperature of the product, and this can affect the dynamics of the product, thus making the monitored vial not representative of the whole batch. The use of the measurement of the external temperature at the bottom of the vial can solve this problem: this is the ‘‘smart vial’’ recently patented by Barresi and coworkers.13,15,23 Another problem is posed by the wires of the thermocouples, but this can be solved by means of a wireless system, as it has been recently proposed.25,26 With respect to PRT-based methods, product temperature increase during the test has to be taken into account when setting the duration of the test in order to preserve product quality, in particular when the process is run close to the limit temperature. Moreover, the PRT requires a fast-closing valve, and in the last part of the drying the estimations of the state of the product are not reliable, even if Fissore et al.27 have recently proposed a new method that can solve, at least partially, this problem. A detailed discussion about the application of these DOI 10.1002/jps

ON THE USE OF MATHEMATICAL MODELS TO BUILD THE DESIGN SPACE

methods to recipe design can be found in Barresi et al.28 A different approach can be used to design the recipe for a given formulation and is based on the use of the design space concept. According to ‘‘ICH Q8 Pharmaceutical Development Guideline’’29 a design space is the multidimensional combination of input variables and process parameters that have been demonstrated to provide assurance of quality. The design space can be useful as it does not provide only the best recipe, but it gives also information about the effect of the operating conditions (shelf temperature, chamber pressure, and process time) on the process (i.e., on product temperature, sublimation flux, and residual water content). Although the design space has recently attracted significant interest, there are only a handful of publications focused on the design space as it is defined in Ref.29, for example, the article of Chang and Fischer,30 where the design space is obtained by means of an extended experimental campaign, and the articles of Nail and Searles31 and Hardwick et al.32 focused on the constraints imposed on the design space by the product and by the equipment, and on the in-line measurement of the sublimation flow in order to reduce the work required by the experimental campaign. Anyhow, even if a design space was not shown, the influence of the operating conditions on the drying process was studied extensively in the past, using both mathematical simulation and experimental investigation (see, among the others, Ref.33) and recently, Bogner et al.34 proposed a sophisticated treatment, based on the analysis of propagation of errors, to assess the impact of variation in input parameters on primary drying time and maximum product temperature. In this framework, our article is focused on the use of a mathematical model to build the design space for the primary drying of a pharmaceutical formulation. Process simulation allows calculating quickly the design space: the drawback of this approach is that the result is a function of the values of model parameters, and the accuracy of the result is affected by model accuracy and by parameters uncertainty. As a consequence, as it will be discussed in next section, the model has to be accurate and it should involve few parameters that could be easily measured (or estimated) with few experimental runs. The PRT, interpreted with DPE algorithm can be used to this purpose. A simple method is proposed and discussed to take into account parameters uncertainties in the design space, thus avoiding the use of safety factors that would not be very efficient and would not guarantee preserving product quality. Finally, the possibility of using the same approach to cope with the heterogeneity of the batch of vials will be briefly discussed at the end of the article. DOI 10.1002/jps

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MATERIALS AND METHODS Mathematical Model of the Process Various mathematical models have been proposed in the literature to calculate the evolution of the state of the product (the temperature and the amount of water) as a function of the operating conditions: they are based on mass and heat balance equations for the frozen and the dried product in each vial. Multi-dimensional models (e.g., that of Sheehan and Liapis35) are complex and their numerical solution is highly time-consuming. Moreover, they involve a high number of parameters: most of them are very often unknown, and could be estimated only with high uncertainty. Besides, it appears from published articles that in typical freeze-drying conditions a mono-dimensional model can be adequate to represent the system as radial thermal gradients are usually small, even when radiation from the environment is taken into account.33,35 Various mono-dimensional models were proposed in the past:33,36 most of them did not include the effect of heat transfer in the sidewall of the vial, although it has been argued that this could play an important role.37,38 In fact, the energy coming from the shelf is provided to the product mainly at the bottom of the sample, but, to some extent, it is transferred to the product from the vial wall too, as a consequence of conduction through the glass. Recently, a mono-dimensional model including the transient energy balance to describe the heat transfer in the vial glass has been proposed by Velardi and Barresi.39 In order to be useful for building the design space a mathematical model has to be simple, so that the time required by the calculations is short: this is due to the procedure that will be described in the following and that requires carrying out a high number of simulations. To this purpose we propose to use the simplified model of Velardi and Barresi:39 it is a monodimensional model constituted by the energy balance for the frozen product and the mass balance for the water vapor inside the dried product, both taken in pseudo-stationary conditions because of the slow dynamics of the process. According to these assumptions, the evolution of the thickness of the frozen layer is given by the following equation: dLfrozen 1 1 ¼ ½pw;i
pw;c  rfrozen
rdried Rp dt

(1)

while product temperature at the bottom of the vial can be calculated by means of the following equation:
 1 1 Lfrozen
1 TB ¼ TS
þ ðTS
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Finally, it is assumed that the heat flux at the moving interface is only due to ice sublimation:
 1 Lfrozen
1 1 þ ðTS
Ti Þ ¼ DHs ½pw;i
pw;c  (3) Kv kfrozen Rp Eqs. (1–3) are similar to those presented by Pikal,33 but in this case vapor flux from the sublimating interface to the drying chamber is modeled by means of an overall resistance to vapor flow (Rp), instead of taking into account the various contribution of the dried layer, of the stopper, and of the shelf. By this way the number of parameters of the model is reduced and, as it will be shown in the following, in Eqs. (1–3) there are the same parameters that can be estimated in-line. The key parameters of the model are Kv, the overall heat transfer coefficient between the shelf and the product at the bottom of the vial, and Rp, the resistance of the dried layer to vapor flow. As the role of the vial wall is not explicitly accounted for in this simplified model, as well as that of radiation from chamber wall (that can affect the dynamics of the product in the vials located near the edge of the shelves), the coefficient Kv should be regarded as an effective heat transfer coefficient, weighing up the additional heat input due to heat transfer from the vial wall and from radiation. Taking into account that also the parameter Kv estimated from experiments is an effective coefficient, it comes that this model is very well suited for our application. Calculation of the Parameters Kv and Rp In case the vial is placed directly over the shelf, that is, if no tray is used, various equations have been proposed in the literature to calculate the overall heat transfer coefficient between the shelf and the bottom product in the vial (Kv). Firstly, the heat transfer coefficient between the shelf and the bottom of the vial ðKv0 Þ is calculated as the sum of three terms: Kv0 ¼ Kc þ Kr þ Kg

(4)

corresponding to the various mechanisms of heat transfer, namely the direct conduction from the shelf to the glass at the points of contact, the radiation, and the conduction through the gas between the shelf and the vial bottom. According to Smoluchowski theory, as outlined by Dushman and Lafferty40 and reported by Pikal et al.,41 Kg is a function of the distance between the bottom of the vial and the shelf (as the bottom of the vial is not flat, and the curvature depends on the type of vial), and of chamber pressure: Kg ¼

aL0 p 1 þ lðaL0 =l0 Þp

(5)

The parameter a is a function of the energy accommodation coefficient and of the absolute JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

temperature of the gas: ac a¼ 2
ac

rffiffiffiffiffiffiffiffiffiffiffiffi 273:2 T

(6)

In case other gases, beside water vapor, are present in the drying chamber (e.g., nitrogen entering the chamber when controlled leakage is used to regulate the pressure) then Eqs. (5) and (6) have to be modified to take into account gas composition, as total pressure is due to the sum of the contribution of water and nitrogen partial pressures (see Ref. [38] for the equations that have to be used in this case). With respect to radiation flux, we have to take into account two fluxes, one from the shelf upon which the vials rest, and the other from the top. Each flux is proportional to the difference in the forth powers of the absolute temperatures of the two surfaces, and to the effective emissivity for the heat exchange, which depends on the relative areas of the two surfaces, their emissivities, and a geometrical view factor. According to Pikal et al.41 the radiative heat transfer coefficient can be written as: Kr ¼ 4ðeS þ ev ÞkT

3

(7)

While the values of L0, l0, and eS, are available from the literature, the parameters Kc, ac, l, and ev in case of radiation from upper shelf, should be determined by regression analysis of experimental data, even if some values can be found in the literature for some types of vials.41 Finally, the coefficient Kv can be calculated by taking into account the conduction in the glass at the bottom of the vial: Kv ¼

1 ð1=Kv0 Þ þ ðsg =lg Þ

(8)

The other parameter of the model, that is, Rp, has been correlated to the specific surface area of the product:42 assuming that the porous cake is a collection of capillary tubes of radius r and tortuosity t (given by the ratio of the total channel length to the thickness of the porous system), the dried layer resistance may be written as: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s
ffi 3 t2 Ldried pRT Rp ¼ (9) 4 " 2Mw r In real systems, there can be a nonlinear dependence of the dry layer resistance on the thickness of the cake, as implied by Eq. (9), and the following empirical equation is often proposed in order to express the average product resistance as a function of cake thickness: Rp ¼ Rp;0 þ

P1 Ldried 1 þ P2 Ldried

(10) DOI 10.1002/jps

ON THE USE OF MATHEMATICAL MODELS TO BUILD THE DESIGN SPACE

where Rp,0, P1, and P2 are obtained by regression analysis of experimental data.42 It has to be remarked that the parameter Rp used in Eqs. (1)–(3) has to be regarded as the sum of the various resistances to vapor flow, as the driving force used to calculate the sublimation flux is the difference between solvent partial pressure at the interface of sublimation and in the drying chamber. Experimental Determination of the Parameters Kv and Rp Various methods can be used to estimate Kv and Rp. Recently, the use of the tunable diode laser absorption spectroscopy (TDLAS) has been proposed to this purpose. This device uses Doppler-shifted near infrared absorption spectroscopy to measure water vapor concentration and gas flow velocity in the duct connecting the freeze-drying chamber and condenser.43,44 The vapor flux can be calculated given an estimation of the velocity profile in the duct. This technique is very interesting for measuring in-line the sublimation flow,31 and it can be used also for measuring the parameters Kv and Rp:45,46 Kv ¼

Jw DHs ; ðTS
TB ÞAv

Rp ¼

pw;i
pw;c Jw

(11)

The calculation of the parameters requires knowing the value of shelf temperature and of product temperature at the bottom and at the interface of sublimation, from which the vapor pressure at the interface is calculated. Thus, the use of TDLAS sensor to estimate model parameters requires additional experimental work in order to be applicable. Monitoring systems bases on the PRT, for example, the MTM, the DPE, or the PRA,47 can be used to estimate Kv and Rp during a freeze-drying cycle. These measurements are not made continuously, but typically every 30–60 min to minimize the disturb to the process dynamics: in fact, during the PRT product temperature increases due to the increase of chamber pressure and this can be responsible for shrinkage/ collapse in case product temperature is close to the limit value. In this work we used the dynamic parameters estimation (DPE) algorithm; the details of the algorithm can be found in Velardi et al.20 The duration of the test is 30 s, with a sampling frequency of 10 Hz, and the test is repeated every 30 min in order to provide values of Rp as a function of Ldried, thus estimating the parameters Rp,0, P1, and P2 in Eq. (10). The freeze-drying cycle is then repeated at different values of pressure, and the values of Kv are used to estimate Kc, ac, l, and ev. Experimental Validation of the Mathematical Model The mathematical model has been validated by means of an extended experimental campaign carried DOI 10.1002/jps

315

out in our Lab at Politecnico di Torino. Both placebo formulations (mainly aqueous solutions of sucrose, mannitol, dextran, and their mixtures), as well as pharmaceutical products have been processed, in a wide range of operating conditions, that is, chamber pressure, type of vials, equipment loading, and total solutes concentration. Results discussed in the following section have been obtained in a prototype freeze-dryer (LyoBeta25TM by Telstar) with a chamber volume of 0.2 m3 and equipped with thermocouples, capacitance (MKS Type 626A Baratron) and thermal conductivity (Pirani PSG-101-S) gauges, and with LyoMonitor,14 a system that allows for process monitoring using various devices (including DPE algorithm), and LyoDriver, for process control. The pressure in the chamber has been generally regulated by controlled leakage, but runs using only the valve on the vacuum pump have been also carried out. The case study that is discussed in the following is the freeze-drying of a 10% w/w sucrose aqueous solution in tubing vials ISO 8362-1 2R: each vial has an internal diameter of 14  0.25 mm, with a wall thickness of 1  0.08 mm, a bottom thickness of 0.7 mm, and the maximum gap between the bottom and the shelf of 0.4 mm.48 In order to assess the adequacy of the mathematical model to describe the dynamics of the product, the following measurements have been considered: – The ratio of the measurements given by Pirani and Baratron gauges: this value approaches unity at the end of the primary drying as at that point the concentration of water into the drying chamber becomes very low, and the pressure measured by Pirani (which is generally calibrated for air) approaches that of the capacitive gauge.49 – Thermocouples: they are used to measure the temperature of the product at the bottom of the vial. Unfortunately, the drying kinetics is generally faster in the monitored vials and the results are not representative of the whole system. – PRT with DPE algorithm: it is used to estimate product temperature at the interface of sublimation and at the bottom of the vial, and the thickness of the frozen layer. Unfortunately, these estimations are not reliable in the last part of the main drying, when the estimated product temperature generally decreases: this drop may be an artifact because a fraction of vials, the edge-vials, has already finished sublimating while DPE (as well as all the other algorithms proposed in the literature) continues interpreting pressure rise curves assuming a constant number of sublimating vials. Thus, a decrease JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

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in pressure rise, corresponding to a lower sublimation rate, is interpreted by DPE as a reduction of the front temperature. As far as DPE and other PRT-based algorithms are compared, it could be argued that the estimations of DPE can be consistent for a larger fraction of the primary drying as a more accurate model is used to describe the pressure rise during the PRT.18 Recent studies show that batch heterogeneity is surely responsible, at least partly, for the observed behavior, but this can be also due to the optimization algorithm and caused by problems of ill-conditioning. In fact, all the PRTbased methods solve a nonlinear optimization problem that can become ill-conditioned when the measured signals do not allow for an accurate estimation of all the model parameters: in this case parameter estimates become very sensitive to measured data.27,50,51

– Given the values of maximum allowed product  Þ and maximum allowed subtemperature ðTmax  limation flux ðJmax Þ, that is, the constraints posed by the product and by the equipment, it is possible to identify the design space for the process considered. Generally the constraint  on Tmax is much more demanding than that  on Jmax .

It has to be remarked that such design space is obtained using few experimental runs (three tests at different pressure is the minimum number). Moreover, parameters uncertainty has to be taken into account. To this purpose it is possible to assume that the values of Kv and Rp are distributed around their mean values according to a density function fKv ;Rp , where fKv ;Rp dKv dRp ¼ NKv ;Rp

Calculation of the Design Space The procedure required to build the design space using the mathematical model of the process can be summarized in the following steps: – Choice of the minimum and maximum values of shelf temperature and chamber pressure to be considered in the design space. – Calculation of two arrays of values of TS and pc: pc ðjÞ ¼ pc;min þ ðj
1ÞDpc TS ðkÞ ¼ TS;min þ ðk
1ÞDTS

with

j ¼ ½1; np 

with

(12)

k ¼ ½1; nT  (13)

where DTS and Dpc are, respectively, the sampling intervals for TS and pc in the design space and nT and np are the dimensions of the two arrays: TS;max
TS;min þ1 DTS

(14)

pc;max
pc;min þ1 np ¼ Dpc

(15)

nT ¼

The two arrays define a grid of nT  np points, having coordinates (TS,k, pc,j). – For each couple of values (TS,k, pc,j) the dynamics of the product can be simulated using the selected model, with the values of the parameters previously calculated. Maximum reached values of product temperature and sublimation flux, and the time required to complete the primary drying can thus be calculated. These results can be shown using diagrams where the values of the output variables are plotted as a function of the operating conditions. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

(16)

identifies the probability that the value of Kv is in the interval [Kv, Kv þ dKv] and that of Rp is in the interval [Rp, Rp þ dRp]. Due to the high number of vials of the batch, NKv ;Rp corresponds to the percentage of the vials of the batch having a value of Kv and Rp, respectively, in the intervals [Kv, Kv þ dKv] and [Rp, Rp þ dRp]. In this work we assume a Gaussian density function, characterized by mean values of the two parameters (K v and Rp ) and by their standard deviations (s Kv and s Rp ): fKv ;Rp ¼

1 2ps Kv s Rp

1

2

e
2½ððKv
K v Þ=sKv Þ

þððRp
Rp Þ=s Rp Þ2 

(17)

The procedure required to build the design space is now the following: – Choice of the values of TS,min, TS,max, pc,min, pc,max, and of the sampling intervals DTS and Dpc, thus obtaining the two arrays of values of TS and pc. – Given the values of K v and Rp , and their standard deviations s Kv and s Rp , it is possible to calculate the density function fKv ;Rp . Given the minimum and maximum values of Kv and Rp, and the sampling intervals DKv and DRp it is possible to calculate two arrays of values of Kv and Rp:

Kv ðf Þ ¼ Kv;min þ ðf
1ÞDKv

with

f ¼ ½1; nKv 

Rp ðgÞ ¼ Rp;min þ ðg
1ÞDRp

with

g ¼ ½1; nRp  (19)

(18)

DOI 10.1002/jps

ON THE USE OF MATHEMATICAL MODELS TO BUILD THE DESIGN SPACE

where nKv and nRp are the dimensions of the two arrays: n Kv ¼

Kv;max
Kv;min þ1 DKv

(20)

nRp ¼

Rp;max
Rp;min þ1 DRp

(21)

The two arrays define a grid of nKv  nRp points, having coordinates (Kv,f, Rp,g). – Given a couple of values of operating conditions (TS,k, pc,j) and of parameters (Kv,f, Rp,g) it is possible to simulate the dynamics of the product using the previously described model, thus obtaining Tmax,calc, Jmax,calc and td,calc. It is possible to assume that the probability of obtaining these values of Tmax,calc, Jmax,calc and td,calc in the batch is equal to NKv ;Rp . – The mathematical simulation has to be repeated for all the nKv  nRp values of (Kv, Rp). It is thus

317

possible to calculate the probability to get these values of Tmax,calc, Jmax,calc and td,calc for the selected operating conditions (TS,k, pc,j). – Previous calculations are then repeated for all the values of the operating conditions (a total number of nT  np  nKv  nRp simulations are carried out to build the design space, thus motivating the need for a simple model). – Finally, it is possible to identify the design space that allows to fulfill the operating constraints   Tmax and Jmax with a given probability level.

RESULTS Figure 1 shows a comparison between process measurements and model predictions for certain key variables, namely sublimation flux, thickness of the frozen layer, and product temperature (at the bottom of the vial and at the interface of sublimation) during a freeze-drying cycle. The values of the

Figure 1. Example of model validation. Comparison between experimental measurements and process simulation using the simplified model ( pc ¼ 10 Pa, number of vials in the batch ¼ 98). (A) Ratio between the signals of Pirani and Baratron gauges. (B) Sublimation flux (referred to the sublimating surface): values estimated by means of DPE (symbols) and calculated (line). (C) Evolution of the thickness of the frozen layer: DPE estimations (symbols) and calculated values (line). (D) Shelf temperature versus time. (E) Evolution of the product temperature at the bottom of the vial: thermocouple measurements (dashed lines) and calculated values (solid line). (F) Evolution of the product temperature at the interface of sublimation: DPE estimations (symbols) and calculated values (solid line). DOI 10.1002/jps

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parameters of the model (i.e., Kv and Rp) have been calculated using DPE algorithm. In this test shelf temperature is not constant (graph D), but it has been manipulated by LyoDriver in order to maintain product temperature close to the limit value of 241 K. By this way it is possible to test the ability of the model to track the dynamics of the product in presence of time-varying boundary conditions. As it has been previously discussed the duration of the primary drying can be assessed using the ratio of the signals of the Pirani and Baratron gauges, that approaches unity at the end of the primary drying (see graph A), and is in a fairly good agreement with the value calculated using the mathematical model, corresponding to the time when the thickness of the frozen layer becomes zero (see graph C). Similarly, a good agreement is obtained between the measurements of product temperature using thermocouples in close contact with the bottom of the vial, and the values calculated using the mathematical model (graph E), at least until ice is no longer present in the monitored vial. The values of interface product temperature, sublimation flux and frozen layer thickness are compared also with those calculated using DPE algorithm at each sampling time: these values are reliable in the first part of the drying (the arrows in the graphs indicate this point), as it has been previously discussed, and, up to that moment a good agreement with calculated values is obtained both for sublimation flux (graph B), frozen layer thickness (graph C), and interface temperature (graph F). With respect to the sublimation flux calculated by means of mathematical simulation of the process we have to highlight that in the last part of primary drying (e.g., when the thickness of the frozen layer is less than 10% of the initial value) it does not decreases towards zero as it is expected. In fact, as the following equation is used to calculate the flux: Jw ¼

1 ½pw;i
pw;c  Rp

analysis have been carried out in a wide range of operating conditions and product characteristics, thus confirming the adequacy of the model to describe the dynamics of the process. Figure 2 (upper graph) shows values of Kv obtained by means of DPE in three cycles carried out at three different values of chamber pressure. These data have been used to calculate the parameters of the theoretical model previously presented looking for the values of the parameters of Eqs. (4)–(8) that give best fit with experimental data. The following values are obtained: Kc ¼ 4.9 W m
2 K
1, ac ¼ 0.85, l ¼ 3  10
4 m (vials are considered to be shielded by

(22)

and, in the last part of the drying, product temperature and Rp, that has reached the asymptotic value, do not change, it follows that Jw remain almost constant until the end of sublimation. Actually, the dependence of Rp on the thickness of the dried layer is much more complex than that of Eq. (10), as Rp increases in the last part of the drying (and, thus, Jw decreases). Unfortunately, the values of Rp calculated using various algorithms to interpret the PRT are not reliable after 1/2–2/3 of primary drying; moreover, also the mono-dimensional approach used to describe the moving interface could be not adequate at the end of primary drying. In any case, model assumptions do not seriously impact on the estimated value of the time required to complete the primary drying. Similar JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

Figure 2. Upper graph: comparison between values of Kv versus chamber pressure calculated using Eqs. (4)–(8) after evaluation of parameters by best fitting (dashed line) and values obtained by means of DPE (symbols; each point and error bar indicate the average and standard deviation based on three experiments). Lower graph: product resistance to vapor flow versus thickness of the dried layer. DOI 10.1002/jps

ON THE USE OF MATHEMATICAL MODELS TO BUILD THE DESIGN SPACE

the stopper from upper shelf radiation). It is thus possible to calculate the coefficient Kv for all the values of pressure that will be used to build the design space. Figure 2 (lower graph) shows an example of values of Rp as a function of dried layer thickness estimated by means of DPE during one experimental run. This curve fits with Eq. (10) with the following parameters: Rp,0 ¼ 1.15  104 m s
1, P1 ¼ 2.65  108 s
1, P2 ¼ 2.5  103 m
1. The curve of Figure 2 (lower graph) shows a plateau value of about 1.1  105 m s
1, corresponding to about 2.3 cm2 Torr h g
1. Recently, data about dry product resistance to vapor flow have been published by Schneid52 for a 5% sucrose solution (the plateau value is 2.0 cm2 Torr h g
1) and for a 20% sucrose solution (the plateau value is 4.5 cm2 Torr h g
1) in case filling height is 5 mm. In our experiments we used an intermediate sucrose concentration (10%), with a slightly higher filling height, and, thus, our data can be considered in good agreement with the values obtained by Schneid. Using the mathematical model previously validated (see Fig. 1) and the parameters Kv and Rp previously determined (see Fig. 2), it is possible to calculate the maximum product temperature (Fig. 3, upper graph) and the maximum sublimation flux (Fig. 3, lower graph) as a function of the operating conditions (shelf temperature and chamber pressure). As far as Tmax,calc is concerned, if TS is increased also product temperature rises, as it can be expected. Chamber pressure has the same effect on product temperature. With respect to the sublimation flow the influence of TS and pc on Jw,max,calc is much more complex as the sublimation flux is a function of a driving force that is increased when product temperature increases, and that is decreased when chamber pressure increases. Moreover, chamber pressure affects also the heat transfer coefficient between the shelf and the product. Thus, when pc is increased the flux increases when shelf temperature is high, while it decreases when the temperature of the shelf is low. Results shown in Figure 3 can be used to calculate the design space when the maximum allowed values of product temperature and sublimation flux are given. An example is given in Figure 4 (upper graph) where the boundary of the design space, identified by the thick line in the graph, corresponding to a maximum product temperature of 241 K, is shown: only couples of values (TS, pc) corresponding to points below this line are able to preserve product quality. Sublimation flux ranges from 0.3 to 0.8 kg h
1 m
2 and these values are adequate with respect to equipment characteristics. In the same diagram it is possible to add lines corresponding to operating conditions giving the same drying time: this allows selecting operating conditions that not only avoid DOI 10.1002/jps

319

Figure 3. Maximum product temperature (upper graph) and maximum sublimation flux (lower graph) versus shelf temperature and chamber pressure.

product overheating, but also reduce the duration of the primary drying. Figure 4 (lower graph) shows similar results: in this case various curves corresponding to various maximum product temperatures are shown (curves corresponding to different values of drying time are not shown in this graph). A curve  versus chamber pressure could corresponding to Jmax be added, if available. Finally, parameters uncertainty has to be taken into account when building the design space. Due to this uncertainty, it is possible to calculate the probability that product temperature, and/or sublimation flux, remains below a maximum allowed value. In Figure 5 (upper graph) various boundaries of the design space are shown, corresponding to different probability that the constraint on product temperature is satisfied. It is evident that the higher is the target value for this probability, the smaller is the range of operating conditions that can be used, that is, lower values of shelf temperature and chamber pressure are required to preserve product JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

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Figure 4. Examples of design space. Upper graph: design space obtained in case the maximum allowed product temperature is 241 K (thick line). Thin solid lines indicate operating conditions (TS and pc) resulting in the same drying time, while dotted lines refer to operating conditions giving the same maximum sublimation flux. Lower graph: maximum sublimation flux versus chamber pressure for various values of shelf temperature (solid lines). Dashed lines identify the maximum product temperature.

quality within the specified probability value. As the value of the parameters Kv and Rp is not known, there is a not null probability to have values of these parameters higher than the estimated values, and this may be responsible for higher product temperature. A diagram like that of Figure 5 (upper graph) can be calculated for a given value of uncertainty of the JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

Figure 5. Examples of design space in case the uncertainty of the model parameters is taken into account (dashed lines identify the boundary of the design space when parameter uncertainty is not taken into account;  Tmax ¼ 241 K). Upper graph: Solid lines: design space boundary for various values of probability of success ðs Kv ¼ s Rp ¼ 10%Þ. Lower graph: Solid lines: design space boundary in case of various values of parameters uncertainty (s Kv ¼ s Rp , probability of success ¼ 99.95%).

parameters: if the exact value is not known, the probability, that is, the variance of the distribution of Kv and Rp around the estimated values, can be used as a parameter of the design space. Figure 5 (lower graph) shows the boundaries of the design space calculated for various values of the uncertainty parameter, given the value of the desired probability of success. As it should be evident, the higher is the DOI 10.1002/jps

ON THE USE OF MATHEMATICAL MODELS TO BUILD THE DESIGN SPACE

uncertainty on the values of the two parameters, the smaller is the region of the design space where process constraints can be fulfilled. This analysis can be used as a guideline for the experimental investigation required to determine the two parameters Kv and Rp as it evidences how the uncertainty of the two parameters can affect the recipe for the freeze-drying. Both graphs of Figure 5 show also the curve corresponding to the boundary of the design space in case parameters uncertainty is not considered: this points out that if the operator selects the operating conditions without taking into account parameters uncertainty, then product can be damaged.

DISCUSSION In order to use the mathematical model to build the design space we need to estimate model parameters: to this purpose the use of DPE algorithm, as well as of other PRT-based algorithms, can be really effective, as it requires carrying out few cycles at different values of chamber pressure. If the drying time is relatively long, it could be even possible to use only one experimental run, where chamber pressure is changed with two or more steps. The estimations of Kv can be used to calculate the parameters appearing in Eqs. (4)–(8). With this respect, it must be pointed out that DPE provides a sort of ‘‘effective’’ heat transfer coefficient, taking into account also the role of the glass wall and, eventually, of radiation, while in the theoretical model for the calculation of Kv (Eqs. 4–8) it appears the ‘‘true’’ heat transfer coefficient between the shelf and the product at the bottom of the vial. In any case, as the parameters required to calculate Kv versus chamber pressure (i.e., Kc, ac, l and, eventually, ev) are retrieved by means of best-fit with the values obtained using DPE, their values take into account the ‘‘additive’’ contributions. Once the mathematical model is available, the influence of shelf temperature and chamber pressure on the maximum values of product temperature and sublimation flux can be quickly calculated, thus providing a huge amount of information about the process. Finally, the design space is obtained when process constraints are considered: these constraints can have two sources, namely the product (i.e., the maximum allowed product temperature) and the equipment (i.e., the maximum sublimation flux). In the previous section we focused on the constraint imposed by product temperature, but also the characteristics of the equipment have to be taken into account.31,32 In any case, once diagrams like those of Figure 3 have been obtained, it is straightforward to add the curves representing the constraints. The design space can thus be used to find out the operating conditions that allow preserving product DOI 10.1002/jps

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quality, and investigating what are the effects of variations of shelf temperature and chamber pressure on the dynamics of the process. This type of diagram is not really new in the freezedrying domain, as a very similar one was proposed by Chang and Fischer30, and, recently, by Nail and Searles31 and Hardwick et al.32 The time consuming experimental investigation can be replaced by the mathematical simulation of the process, with few experimental runs (with PRT and DPE), to build quickly and reliably the design space for the specific product and the specific equipment that will be used in production. In fact the design space is a function of the system that is investigated, as the curves are affected not only by the formulation (product concentration, type of solvent), but also by the geometrical characteristics of the vial and of the equipment, even if this is often neglected when using this diagram to find a recipe. Moreover, while in the diagram of Chang and Fischer we can find information about product temperature versus operating conditions in a well defined time instant of the process (or average values over a certain time interval, depending on the way experimental data have been obtained), in order to obtain a design space we need the values of maximum product temperature and sublimation flux, as both values change during the cycle, even when constant operating conditions are used, due to changes in product resistance. When using a mathematical model, parameters uncertainty has to be taken into account: our approach allows modifying the design space according to this issue without using large safety factors (i.e., a lower value of maximum allowed product temperature). To this purpose the diagram shown in Figure 5 (bottom graph) provides useful information. In fact, the probability of preserving product quality, that is, of maintaining product temperature below the maximum allowed value, is generally specified and, according to this value, we are able to calculate the proper operating conditions, without using excessively large safety factors that can increase the duration of the primary drying. Beside parameters uncertainty, there are other two issues that have to be taken into account when building the design space, namely the stochastic variability among the vials of the batch, and the heterogeneity of the batch due to nonuniform operating conditions. In fact, the coefficient Kv can be different among the various vials of the batch because of nonuniform geometrical characteristics of the vials (especially small differences in the gap can be influent). Similarly, the coefficient Rp can be different from vial to vial due to the fact that this parameter is a function mainly of the shape and the dimension of the ice crystals that form the structure of the frozen product, and ice nucleation is a stochastic JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 1, JANUARY 2011

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process. In addition, if the stopper resistance is not negligible, differences in stopper positioning can be responsible for nonuniform values of Rp. Both issues can be accounted for by considering a proper distribution of the two parameters (in case the distribution of Kv and Rp is measured experimentally, for example, by using a soft-sensor, the true distribution could be available). In this case, the same procedure previously described can be used: in fact, it is possible to know the percentage of the vials of the batch with a given value of Kv and of Rp and, thus, for a given couple of values of TS and pc it is possible to calculate the percentage of the vials of the batch where maximum temperature remains below the upper limit. With respect to batch heterogeneity, this can be due to various causes, namely radiation from chamber walls and door, vapor (and inert gas, in case it is used for pressure control) fluid dynamics in the drying chamber, and nonuniform shelf temperature.53 Also in this case it is possible to build the design space considering the proper boundary conditions for all the vials of the batch. The proposed approach is therefore able to manage also the heterogeneity of the batch, thus providing an effective tool to reach the ‘‘six-sigma’’ goal (i.e., 3–4 defects in a batch of one million vials) recently proposed by the International Society of Lyophilization—Freeze Drying.54

eS ev fKv ;Rp DHs J  Jmax Kc

Kg

Kr Kv Kv0 Kv

DKv kfrozen

CONCLUSION AND FINAL REMARKS l The design space is an effective tool for finding a freeze-drying recipe that allows preserving product quality. Moreover, it shows how the operating conditions affect process dynamics. The method proposed to build the design space requires a mathematical model to simulate the dynamics of the product as a function of the operating conditions, and few experimental runs to get model parameters. Mathematical simulation provides quickly and reliably the design space: this is of outmost importance as the design space has to be built for the specific characteristics of the product, of the vial and of the equipment used in the freezedrying operation. Moreover, the uncertainty of the model parameters has to be taken into account, without requiring too large safety margin when choosing the operating conditions to preserve product quality. Finally, also the variability among the vials in the drying chamber of the parameters, and the heterogeneity of the batch can be accounted for if the proposed methodology is used.

NOMENCLATURE ac Av

energy accommodation coefficient cross sectional area of the vial (m2)

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Ldried Lfrozen M nKv np nRp nT NKv ;Rp

p Dpc P1 P2 R Rp DRp

emissivity for radiation heat exchange from the shelf to the bottom of the vial emissivity for radiation heat exchange from the shelf to the top of the vial density function describing the distribution of the values of Kv and Rp heat of sublimation (J kg
1) sublimation flux (kg s
1 m
2) maximum allowed sublimation flux (kg s
1 m
2) heat transfer coefficient due to direct conduction from the shelf to the glass at the points of contact (J m
2 s
1 K
1) heat transfer coefficient due to conduction in the gas between the shelf and the vial bottom (J m
2 s
1 K
1) heat transfer coefficient between the shelf and the vial due to radiation (J m
2 s
1 K
1) overall heat transfer coefficient between the shelf and the product (J m
2 s
1 K
1) overall heat transfer coefficient between the shelf and the bottom of the vial (J m
2 s
1 K
1) batch mean value of the overall heat transfer coefficient between the shelf and the vial (J m
2 s
1 K
1) sampling interval for Kv in the design space (J m
2 s
1 K
1) thermal conductivity of the frozen layer (J K
1 s
1 m
1) constant effective distance between the bottom of the vial and the shelf (m) thickness of the dried layer (m) thickness of the frozen layer (m) molecular weight (kg mol
1) dimension of the array of values of overall heat transfer coefficient between the shelf and the vial dimension of the array of values of chamber pressure dimension of the array of values of dried product resistance to vapor flow dimension of the array of values of shelf temperature probability that the vial has a value of Kv in the interval [Kv, Kv þ dKv], and the dried product in the vial has a value of Rp in the interval [Rp, Rp þ dRp] pressure (Pa) sampling interval for pc in the design space (Pa) parameter used in Eq. (10) (s
1) parameter used in Eq. (10) (m
1) ideal gas constant (J kg
1 K
1) product resistance to vapor flow (m s
1) sampling interval for Rp in the design space (m s
1) DOI 10.1002/jps

ON THE USE OF MATHEMATICAL MODELS TO BUILD THE DESIGN SPACE

Rp Rp,0 r sg T T  Tmax TB

DTS t td

batch mean value of product resistance to vapor flow (m s
1) parameter used in Eq. (10) (m s
1) radius of the capillary tubes in the dried layer (m) thickness of the glass at the bottom of the vial (m) temperature (K) mean temperature between the shelf and the vial bottom (K) maximum allowed product temperature (K) temperature of the frozen product at the bottom (K) sampling interval for TS in the design space (K) time (s) time required to complete the primary drying (h)

Greeks a e k L0 l0 lg rfrozen rdried s t

parameter used to calculate Kg porosity of the dried layer Stefan–Boltzman constant (J s
1 m
2 K
4) free molecular heat conductivity at 08C (J s
1 m
1 K
1) heat conductivity at ambient pressure (J s
1 m
1 K
1) heat conductivity of the glass (J s
1 m
1 K
1) density of the frozen product (kg m
3) apparent density of the dried product (kg m
3) standard deviation tortuosity of the channel in the dried layer

Subscripts and Superscripts c calc i max min S w

drying chamber calculated value interface, moving front maximum value minimum value heating shelf water vapor

ACKNOWLEDGMENTS The authors would like to acknowledge Dr. Roberto Pisano (Politecnico di Torino) for his valuable support in the experimental investigation.

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DOI 10.1002/jps