Stochastic modeling and control of particle size in crystallization of a pharmaceutical

Transcription

1 Stochastic modeling and control of particle size in crystallization of a pharmaceutical Daniel B. Patience, Eric L. Haseltine, Philip Dell Orco, and James B. Rawlings Department of Chemical Engineering University of Wisconsin, Madison, WI GlaxoSmithKline Pharmaceuticals King of Prussia, PA World Congress On Particle Technology 4, Sydney, Australia July 24, 2002 World Congress On Particle Technology 4, Sydney, Australia July 24,

2 Outline 1. Experimental apparatus experimental apparatus and measurements video microscopy pharmaceutical in alcohol solvent system 2. Stochastic simulation of particulate systems random diffusion growth-dependent dispersion 3. Application to pharmaceutical system parameter estimation cooling profiles 4. Conclusions World Congress On Particle Technology 4, Sydney, Australia July 24,

3 Experimental apparatus Densitometer Colorimeter 670nm Out In Photomicroscope TT Image Analysis Hot Stream TT Cold Stream Controller For the seeded pharmaceutical crystallization system, no secondary nucleation occurs. World Congress On Particle Technology 4, Sydney, Australia July 24,

4 Video microscopy flow cell World Congress On Particle Technology 4, Sydney, Australia July 24,

5 Pharmaceutical crystal measurements 3 min 11 min Images show no apparent secondary nucleation, only growth of seeds 350 Experiment min 140 min Crystal length [µm] min 1 mm Time [min] Images show the pharmaceutical system has growth-dependent dispersion World Congress On Particle Technology 4, Sydney, Australia July 24,

6 Classes of dispersion Class Mechanism Author Random Growth Same mean growth rate Kashchiev (2000) Diffusivity but each crystal s growth Randolph & White (1977) f t = G f L + D 2 f L 2 randomly fluctuates White & Wright (1971) Intrinsic Growth Each crystal exhibits Berglund & Larson (1984) an individual growth rate Jones & Larson (1999) f (L, G, t)dg f (L, t) = 0 Size-dependent Growth rate increases Kashchiev (2000) Growth with more units Abegg et al. (1968) G(L, t) = k g S g (1 + γ 1 L) γ 2 Growth-dependent Crystals grow in McCoy (2001) Dispersion discrete sizes Randolph & White (1977) f t f = G L + G 2 f 2 L 2 find D = 50G World Congress On Particle Technology 4, Sydney, Australia July 24,

7 Random diffusion mechanism f (L, t) Gf (L, t) L Gf (L, t) L+ L D f L L D f L L+ L L L Population Balance Equation: f (L, t) t f (L, t) = G L + D 2 f (L, t) L 2 in which f (L, t) is the number of crystals of size L at time t, G is the crystal growth rate, and D the diffusivity. World Congress On Particle Technology 4, Sydney, Australia July 24,

8 Growth-dependent dispersion mechanism + N n is the number of crystals of size n per volume. Population Balance Equation: + f (L, t) t f (L, t) = G L + G 2 2 f (L, t) L 2 Reaction Mechanism for Nucleation and Growth in which f (L, t) is the number of crystals of size L at time t, and G is the crystal growth rate. 2M kn N 1 N n + M k g Nn+1 World Congress On Particle Technology 4, Sydney, Australia July 24,

9 Isothermal, size-independent nucleation and growth Stochastic Solution Average of 100 Simulations Deterministic Solution Via Orthogonal Collocation World Congress On Particle Technology 4, Sydney, Australia July 24,

10 Nonisothermal, size-independent nucleation and growth Stochastic Solution Average of 500 Simulations Deterministic Solution Via Orthogonal Collocation World Congress On Particle Technology 4, Sydney, Australia July 24,

11 Modeling crystal habit Crystal habit affects the predictions through shape factors Transmittance = exp k v = l w d w ( k ) al L 2 f (L, t)dl 2 0 k a = 2(l w d w + l w + d w ) in which k v and k a are volume and area shape factors and l w and d w are the length-to-width and depth-to-width ratios. Assuming parallelepiped crystals d w l World Congress On Particle Technology 4, Sydney, Australia July 24,

12 Measurements C : solution concentration I/I 0 : slurry transmittance L : CSD mean length σ : CSD standard deviation Parameters k g : growth rate constant g : growth rate order d w : depth-to-width ratio L max : maximum seed size : size of crystal growth unit World Congress On Particle Technology 4, Sydney, Australia July 24,

13 Model predictions - GSK pharmaceutical Concentration Transmittance Concentration [g pharmaceutical/g solvent] Experiment 16 Experiment 16 fit Transmittance [%] Experiment 16 Experiment 16 fit Time [min] Time [min] World Congress On Particle Technology 4, Sydney, Australia July 24,

14 Model predictions - GSK pharmaceutical CSD mean length CSD standard deviation 200 Experiment 16 Experiment 16 fit Experiment 16 Experiment 16 fit Mean length [µm] Standard deviation [µm] Time [min] Time [min] World Congress On Particle Technology 4, Sydney, Australia July 24,

15 Parameter estimates ln(k g ) g ln(d w ) L max ln( /2) Estimate Interval ± 0.2 ± 0.08 ± 0.1 ± 0.5 ±0.2 World Congress On Particle Technology 4, Sydney, Australia July 24,

16 Finding the optimal cooling profile For this system, the crystals with the greatest bioavailability are the ones with the greatest solubility in administered form For this system, the crystals with the greatest stability are the ones with the lowest solubility The desired CSD for this pharmaceutical is the one with a µm range f (L, t) f (L, t) t=0 desirable CSD t=1 undesirable CSD requires milling and sieving t=2 L L = 110 L World Congress On Particle Technology 4, Sydney, Australia July 24,

17 Finding the optimal cooling profile For this system, the crystals with the greatest bioavailability are the ones with the greatest solubility in administered form For this system, the crystals with the greatest stability are the ones with the lowest solubility The desired CSD for this pharmaceutical is the one with a µm range f (L, t) f (L, t) t=0 desirable CSD undesirable CSD requires milling and sieving t=1 t=1 t=2 t=2 t=3 L L = 110 L World Congress On Particle Technology 4, Sydney, Australia July 24,

18 Cooling profiles to produce L=110µm Temperature Supersaturation Temperature [ C] no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S Concentration [g pharmaceutical/g solvent] no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S Time [min] Time [min] World Congress On Particle Technology 4, Sydney, Australia July 24,

19 Resulting L and CSD coefficient of variation from cooling profiles to produce L=110µm CSD mean length CSD coefficient of variation no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S 0.1 Crystal length [µm] Coefficient of variation Time [min] Time [min] World Congress On Particle Technology 4, Sydney, Australia July 24,

20 Implemented cooling profiles Concentration CSD mean length Concentration [g pharmaceutical/g solvent] Experiment 18 model prediction Experiment 18 Experiment 20 model prediction Experiment Mean length [µm] Experiment 18 model prediction Experiment 18 Experiment 20 model prediction Experiment Time [min] Time [min] World Congress On Particle Technology 4, Sydney, Australia July 24,

21 Conclusions Established a connection between stochastic chemical kinetics and the population balance for growth-dependent dispersion. New use of video images in model identification and parameter estimation for crystallization kinetics. Successfully identified kinetics for a pharmaceutical crystallization, with narrow parameter confidence intervals. For this seeded pharmaceutical crystallization without secondary nucleation, the optimization of CSD properties is infeasible. Instead we minimize the batch operating time. Calculated optimal cooling profiles to manufacture crystals within a desired size range by minimizing t f. World Congress On Particle Technology 4, Sydney, Australia July 24,

22 Acknowledgments The authors gratefully acknowledge the financial support of GlaxoSmithKline and NSF through grant #CTS The authors also gratefully acknowledge the financial support of the industrial members of the Texas-Wisconsin Modeling and Control Consortium. All simulations were performed using Octave ( Octave is freely distributed under the terms of the GNU General Public License. World Congress On Particle Technology 4, Sydney, Australia July 24,

23 References [1] C. F. Abegg, J. D. Stevens, and M. A. Larson. Crystal size distributions in continuous crystallizers when growth rate is size dependent. AIChE Journal, 14(1): , January [2] K. A. Berglund and M. A. Larson. Modeling of growth rate dispersion of citric acid monohydrate in continuous crystallizers. AIChE Journal, 30: , [3] C. M. Jones and M. A. Larson. Characterizing growth-rate dispersion of NaNO 3 secondary nuclei. AIChE Journal, 45(10): , [4] D. Kashchiev. Nucleation. Butterworth-Heinemann, Oxford, England, 1st edition, [5] B. J. McCoy. A new population balance model for crystal size distributions: reversible, sizedependent growth and dissolution. Journal of Colloid and Interface Science, 240(1): , [6] A. D. Randolph and E. T. White. Modeling size dispersion in the prediction of crystal-size distribution. Chemical Engineering Science, 32: , [7] E. T. White and P. G. Wright. Magnitude of size dispersion effects in crystallization. Chemical Engineering Progress Symposium Series, 67(110):81 87, World Congress On Particle Technology 4, Sydney, Australia July 24,