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Simulation-Based Design of an Efficient Control System for the Continuous Purification and Processing of Active Pharmaceutical Ingredients

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Abstract

In this study, an efficient system-wide controlsystem has been designed for the integrated continuous purification and processing of the active pharmaceutical ingredient (API). The control strategy is based on the regulatory PID controller which is most widely used in the manufacturing industry because of its simplicity and robustness. The designed control system consists of single and cascade (nested) control loops. The control system has been simulated in gPROMSTM (Process System Enterprise). The ability of the control system to track the specified set point changes as well as to reject disturbances has been evaluated. Results demonstrate that the model shows an enhanced performance in the presence of random disturbances under closed-loop control compared to an open-loop operation. The control system is also able to track the set point changes effectively. This proves that closed-loop feedback control can be used in improving pharmaceutical manufacturing operations based on the Quality by Design (QbD) paradigm.

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Acknowledgments

This work is supported by the National Science Foundation Engineering Research Center on Structured Organic Particulate Systems, through Grant NSF-ECC 0540855.

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Corresponding author

Correspondence to Rohit Ramachandran.

Appendix and Mathematical Model

Appendix and Mathematical Model

The equations used for developing the mathematical model for each unit operation of the flowsheet have been listed.

Crystallizer

In this study, a 2-D population balance model considering growth in two directions has been implemented for the crystallization process, as shown below.

$$\begin{array}{@{}rcl@{}} &&\frac{\partial{F(L_{1},L_{2},t)}}{\partial{t}}+\frac{\partial({G_{1}(L_{1},t)F(L_{1},L_{2},t)})}{\partial{L_{1}}} \notag \\ &&+\frac{\partial({G_{2}(L_{2},t)F(L_{1},L_{2},t)})}{\partial{L_{2}}} =B_{0}(C,t)\delta(L_1)\delta(L_2) \\ &&+ Inflow - Outflow \end{array} $$
(3)

The growth and the nucleation terms can be written as follows:

$$ G_1=k_{g1} \left(\frac{C-C_{sat}}{C_{sat}}\right)^{g_{1}} $$
(4)
$$ G_2=k_{g2} \left(\frac{C-C_{sat}}{C_{sat}}\right)^{g_{2}} $$
(5)
$$ B_0=k_{b} \left(\frac{C-C_{sat}}{C_{sat}}\right)^b $$
(6)

Another internal coordinate (Liq(L eq )) is added to the PBE which tracks the amount of liquid present in the crystals based on the works of Miki et al. [42]. Considering the width of the crystal to be L 1, the equivalent length of the crystal can be expressed as follows:

$$ L_{eq}(L_{1},L_2)=\left(\frac{6}{\pi}\times L_{1}^{2} L_{2}\right)^{\frac{1}{3}} $$
(7)
$$ Liq(L_{eq})=2 \times 10^{-5} L_{eq}^{2} -3 \times 10^{-9} L_{eq}+ 10^{-13} $$
(8)

The mass balance equation can be written as shown in the equation below [43]:

$$\begin{array}{@{}rcl@{}} &&\frac{dC}{dt}=-\rho_{c} \int_{0}^{\infty} \int _{0}^{\infty} F(L_{1},L_{2},t)(2G_{1}(L_1L_2-L1^2) \notag\\ &&+G_2L1^2)dL_1dL_2 \end{array} $$
(9)

Since this is a case of cooling crystallization, the temperature cooling schedule can be expressed as a function of time. C sat has been expressed as a function of temperature. The expression has been obtained by fitting experimental data as obtained from Bristol-Myers Squibb Co., NJ.

$$ C_{sat}=2.7357T-40.925 $$
(10)

The heat transfer between the coolant and crystallizer can be represented as follows:

$$ U A_{c} (T-T_c)=M_{w} C_{pw} (T_c-T_{in}) $$
(11)

The controller equations for the crystallization process are given below:

  • Master Controller:

    $$\begin{array}{@{}rcl@{}} &&T(t)=K_{c}(Set Point(t)-C_{sat}(t)) \notag \\ &&+\frac{K_{c}}{\tau_{I}}\int^{t}_{0}(Set Point(t)-C_{sat}(t))dt \\ &&+K_{c}\tau_{D}\frac{d(Set Point(t)-C_{sat}(t))}{dt}+c_{s} \notag \\ \end{array} $$
    (12)
  • Slave Controller:

    $$\begin{array}{@{}rcl@{}} &&T_{c}(t)=K_{c}(Set Point(t)-T(t)) \notag \\ &&+\frac{K_{c}}{\tau_{I}}\int^{t}_{0}(Set Point(t)-T(t))dt \\ &&+K_{c}\tau_{D}\frac{d(Set Point(t)-T(t))}{dt}+c_s \end{array} $$
    (13)

Filtration Process

The main design equations of the cake filter is as follows (adapted from McCabe et al. [44]):

$$\begin{array}{@{}rcl@{}} &&\frac{dV(L_{1},L_{2},L_{3},t)}{dt} \notag \\ &&=\frac{A^{2}\Delta P}{\mu (\alpha(L_{1},L_{2},L_{3},t) c V(L_{1},L_{2},L_{3},t)+R_{m} A) } \end{array} $$
(14)

c and α can be found as follows:

$$ c=\frac{C_{F}}{1-(\frac{m_{F}}{m_{c}}-1)C_F/\rho_{s}} $$
(15)
$$ \alpha(L_{1},L_{2},L_{3},t)=\frac{150(1-\epsilon(L_{1},L_{2},L_{3},t))}{D_{p}(L_{1},L_{2},L_{3},t)\epsilon(L_{1},L_{2},L_{3},t)^{3}\rho_{s}} $$
(16)

Assuming that there are no solid particles present in the filtrate, the mass of wet cake deposited on the septum is given as follows:

$$\begin{array}{@{}rcl@{}} &&{} m_{F}(L_{1},L_{2},L_{3},t)= F(L_{1},L_{2},L_{3},t)V_{p} \notag \\ &&(L_{1},L_{2},L_{3},t)\rho_{s} N_a \end{array} $$
(17)

Drying Process

For the drying process, a model has been developed where the liquid is being evaporated from the solid surface (adapted from Mezhericher et al. [45]).

The change in particle diameter with time can be represented as follows:

$$ \frac{dD_{p}(L_{1},L_{2},L_{3},t)}{dt}=-\frac{m_{v}(L_{1},L_{2},L_{3},t)}{\rho_{l} 2 \pi D_{p}(L_{1},L_{2},L_{3},t)^{2}} $$
(18)

The temperature profile of particle can be given as shown below:

$$\begin{array}{@{}rcl@{}} &&h_{fg}m_{v}(L_{1},L_{2},L_{3},t)+ c_{ps} \rho_{avg}(L_{1},L_{2},L_{3},t) \notag \\ &&V_{p}(L_{1},L_{2},L_{3},t)\frac{dT_{p}(L_{1},L_{2},L_{3},t)}{dt} \\ &&=h(T_g-T_{p}(L_{1},L_{2},L_{3},t))2\pi D_{p}(L_{1},L_{2},L_{3},t)^{2} \notag \\ \end{array} $$
(19)

The evaporation rate can be calculated as follows:

$$\begin{array}{@{}rcl@{}} && m_{v}(L_{1},L_{2},L_{3},t)=k(L_{1},L_{2},L_{3},t)(x_p-x_{eql})2 \pi \notag\\ &&D_{p}(L_{1},L_{2},L_{3},t)^2 \end{array} $$
(20)

The heat and mass transfer coefficients are given below:

$$ h(L_{1},L_{2},L_{3},t)=\frac{Sh(L_{1},L_{2},L_{3},t)D_{v}}{D_{p}(L_{1},L_{2},L_{3},t)} $$
(21)
$$ k(L_{1},L_{2},L_{3},t)=\frac{Nu(L_{1},L_{2},L_{3},t) k_{g}}{D_{p}(L_{1},L_{2},L_{3},t)} $$
(22)

such that Nusselts number is given as follows:

$$\begin{array}{@{}rcl@{}} Nu(L_{1},L_{2},L_{3},t)=(2+0.6Re(L_{1},L_{2},L_{3},t)^{\frac{1}{2}}Pr^{\frac{1}{3}}) \notag \\ (1+(C_{pv}*(T_g-T_p)/h_{fg}))^{-0.7} \notag \\ \end{array} $$
(23)

and Sherwood number is given as shown in Eq. 26.

$$\begin{array}{@{}rcl@{}} Sh(L_{1},L_{2},L_{3},t)=(2+0.6Re(L_{1},L_{2},L_{3},t)^{\frac{1}{2}}Sc^{\frac{1}{3}}) \notag \\ (1+(Cpv*(T_g-T_p)/h_{fg}))^{-0.7} \notag \\ \end{array} $$
(24)

The outflow from dryer is given as shown below:

$$\begin{array}{@{}rcl@{}} mass_{out}(L_{1},L_{2},L_{3},t)=m_{F}(L_{1},L_{2},L_{3},t) \notag \\ -m_{v}(L_{1},L_{2},L_{3},t)*F(L_{1},L_{2},L_{3},t)*Na \notag \\ \end{array} $$
(25)

The heat transfer between the air and superheated steam can be represented as follows:

$$ U A_{s} (T_s-T_{gas})=M_{steam} C_{psteam} (T_s-T_{in-steam}) $$
(26)

The controller equation for drying can be given as follows:

$$\begin{array}{@{}rcl@{}} &&T_{s}(t)=K_{c}(Set Point(t)-T_{gas}(t)) \\ &&+\frac{K_{c}}{\tau_{I}}\int^{t}_{0}(Set Point(t)-T_{gas}(t))dt \\ &&+K_{c}\tau_{D}\frac{d(Set Point(t)-T_{gas}(t))}{dt}+c_{s} \\ \end{array} $$
(27)

Mixer

The mixing model has been assumed to be independent of size change based on previous work [46]. Hence the internal coordinates have been dropped from the population balance model. The PBM for the mixer can be written as follows:

$$\begin{array}{@{}rcl@{}} \frac{\partial}{\partial t}F(\textbf{z},t)+\frac{\partial}{\partial \textbf{z}}\left[F(\textbf{z},t)\frac{d\textbf{z}}{dt}\right] = 0 \end{array} $$
(28)

The above equation can be written in multidimensional form as shown below:

$$\begin{array}{@{}rcl@{}} \frac{\partial}{\partial t}F(n,x,y,t)+\frac{\partial}{\partial x}\left[F(n,x,y,t)\frac{dx}{dt}\right] \\ +\frac{\partial}{\partial y}\left[F(n,x,y,t)\frac{dy}{dt}\right] \\ =Inflow - Outflow \end{array} $$
(29)

The mass balance of a single component can be simplified according to the equation given below:

$$\begin{array}{@{}rcl@{}} &&\frac{\partial{F(n,x,y,t)}}{\partial{t}}= \frac{V_{f} [F_{n,x-1,y,t}-F_{n,x,y,t}]}{\Delta x} \\ &&+ \frac{V_{b} [F_{n,x+1,y,t}-F_{n,x,y,t}]}{\Delta x} \\ &&+ V_{r} \frac{[F_{n,x,y+1,t}+F_{n,x,y-1,t}-2F_{n,x,y,t}]}{\Delta y} \\ \end{array} $$
(30)

The properties of the final blend from the mixer output have been presented in terms of mean API composition (y API ) relative standard deviation (RSD).

$$ y_{API}=\frac{\sum^{y_{max}}_{y=1} {F(API,x_{max},y,t)}}{\sum^{n_{max}}_{n=1}{\sum^{y_{max}}_{y=1} {F(n,x_{max},y,t)}}} $$
(31)

The controller equation for drying can be given as follows:

$$\begin{array}{@{}rcl@{}} Excipient_{flowrate}(t) &=& K_{c}(Set Point(t)-y_{API}(t))\\ &&{} +\frac{K_{c}}{\tau_{I}}\int^{t}_{0}(Set Point(t)-y_{API}(t))dt\\ &&{} +K_{c}\tau_{D}\frac{d(Set Point(t)-y_{API}(t))}{dt}+c_{s}\\ \end{array} $$
(32)

Nomenclature

Symbol

A c

B 0

b

C

C sat C pw

D p

F

G 1

G 2

g 1

g 2

k g1

k g2

k b

L 1

L 2

L 3

M w

T

T c

T in

U

𝜖

ρ avg

ρ s

ΔP

μ

α

A

C F

c

m F

m c

m v

N a

R m

V

V p

A s

c ps

C psteam

k

h fg

h

T s

T p

T g

T insteam

mass out

M s

Nu

Pr

Re

Sh

U

x p

x eql

n

n max

t

V f

V b

V r

x

x max

y

y max

y API

y avg

y i

Description

Area of heat transfer

Primary nucleation term

Kinetic parameter for crystallization

Solute concentration in crystallization

Saturation concentration of solute

Specific heat constant for water

Crytal diameter

Particle density

Growth rate

Growth rate

Kinetic parameter for crystallization

Kinetic parameter for crystallization

Kinetic parameter for crystallization

Kinetic parameter for crystallization

Kinetic parameter for crystallization

Internal coordinate for length of solid

Internal coordinate for length of solid

Internal coordinate for length of liquid

Cooling water flowrate

Temperature (cooling schedule)

Temperature of cooling water

Inlet temperature of water

Overall heat transfer coefficient

porosity of cake

Average density of wet particles

Density of solid

Filter pressure difference

Fluid viscosity

Specific cake resistance

Filter surface area

concentration of solutes in slurry

Mass of solute deposited on filter

per unit volume of filtrate

Mass of wet cake

Mass of dry cake

Rate of evaporation during drying

Avogadro number

Filter medium resistance

Filtrate volume

Particle volume

Area of heat transfer

Specific heat capacity

Specific heat constant for steam

Mass transfer coefficient

Specific heat of evaporation

Heat transfer coefficient

Temperature of steam

Temperature of particle

Drying gas temperature

Inlet temperature of steam

Outlet flowrate of API crystals from dryer

Steam flowrate

Nusselts Number

Prandtl Number

Reynolds Number

Sherwood Number

Overall heat transfer coefficient

Liquid content of solid particle

Liquid content of solid particle at equillibrium

Counter for number of components

Maximum number of components

Time

Forward axial velocity

Backward axial velocity

Radial velocity

Spatial coordinate in axial direction

Maximum number of axial compartments

Spatial coordinate in axial direction

Maximum number of radial compartments

Fractional API composition at mixer outlet

Average spatial composition of component A

Composition of component A in ith compartment

Units

m 2

particles/m 3/s

Dimensionless

moles/m 3

moles/m 3

J/K

μm

particles/m 3

m/s

m/s

Dimensionless

Dimensionless

m/s

m/s

particles/m 3/s

m

m

m

kg/s

Kelvin

Kelvin

Kelvin

W/m 2K

Dimensionless

kg/m 3

kg/m 3

kPa

kg/ms

m/kg

m 2

moles/m 3

kg/m 3

kg

kg

kg/s

Dimensionless

1/m

m 3

m 3

m 2

J/kgK

J/K

m/s

J/kg

W/M 2 K

Kelvin

Kelvin

Kelvin

Kelvin

kg/s

kg/s

Dimensionless

Dimensionless

Dimensionless

Dimensionless

W/m 2K

Dimensionless

Dimensionless

Dimensionless

[−]

s

m/s

m/s

m/s

[−]

[−]

[−]

[−]

Dimensionless

moles/m 3

moles/m 3

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Sen, M., Singh, R. & Ramachandran, R. Simulation-Based Design of an Efficient Control System for the Continuous Purification and Processing of Active Pharmaceutical Ingredients. J Pharm Innov 9, 65–81 (2014). https://doi.org/10.1007/s12247-014-9173-6

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