9  Replicación en R de caso de estudio: Sovany et al. (2016)

Development of pellets for oral lysozyme delivery by using a quality by design approach

10 Artículo de estudio

Librerías requeridas: FrF2, dplyr, flextable

El artículo de Sovany et al. (2016), tiene como propósito identificar cuáles parámetros críticos de proceso influyen en los atributos críticos de calidad de una forma farmacéutica multiparticulada que contiene lisozima, una enzima con propiedades antibacterianas que actua contra bacterias Gram-positivas en infecciones gastrointestinales, con el fin de determinar cuáles de los factores evaluados (Ver Tabla 16.1) son realmente relevantes para la preservación de la actividad de esta proteína.

Variables

Nivel (-1)

Nivel (1)

Velocidad del impeller (IMP)

500 rpm

1000 rpm

Adición de líquido (LIQ)

5 mL/min

10 mL/min

Velocidad del extrusor (EXT)

70 rpm

120 rpm

Velocidad del esferonizador (SPEED)

1000 rpm

2000 rpm

Tiempo de esferonización(TIME)

15 min

30 min

Tabla 10.1: Tabla 1. Factores y niveles considerados por Sovany et al. (2016)

Las matrices del diseño en unidades codificadas y en unidades naturales se presentan en la Tabla 16.2. En dicho trabajo, se planteó inicialmente un diseño factorial \(2^6\) no replicado. Sin embargo, el factor temperatura fue descartado antes del desarrollo del experimento, por lo que realmente se realizó un diseño factorial completo \(2^5\) no replicado. Las variables respuesta evaluadas fueron la actividad del principio activo (ACT), la dureza (HARD) y la geometría de los gránulos (ROUND). Las variables respuesta evaluadas fueron dureza y la geometría de los gránulos, así como la actividad del principio activo.

Nota

Véase “Guía para la implementación de diseños experimentales en la industria farmacéutica” para una explicación detallada de los cálculos presentados.

11 Construcción de diseño factorial completo \(2^5\) no replicado

A continuación se presenta la réplica del artículo presentado por Sovany et al. (2016)

Código 
# Variables respuesta

ACT = c(68.39378,69.43005,96.89119,70.46632,
        71.50259,79.79275,84.45596,67.87565,
        70.46632,75.12953,71.50259,83.93782,
        77.72021,81.34715,65.80311,47.15026,
        72.53886,62.6943,86.5285,56.47668,
        67.87565,72.53886,78.75648,74.09326,
        75.64767,77.20207,94.81865,50.7772,
        80.82902,84.97409,62.6943,66.83938)

HARD = c(9.18,4.99,2.42,5.29,5.94,4.45,5.71,8.87,
         6.39,3.87,3.01,5.36,3.86,3.48,3.14,6.57,
         6.39,4.79,2.55,7.71,6.82,3.48,2.42,4.90,
         6.48,3.27,3.16,6.00,4.24,5.41,3.78,7.18)

ROUND = c(1.27,1.37,1.30,1.40,1.20,1.39,1.34,1.38,
          1.16,1.27,1.22,1.23,1.17,1.28,1.34,1.27,
          1.25,1.40,1.20,1.39,1.22,1.42,1.29,1.40,
          1.14,1.29,1.27,1.29,1.31,1.39,1.32,1.35)

# Diseño de variables codificadas

dis1_cod <- data.frame(
  TRAT = paste("TRAT",seq(1,32)),
  IMP = c(rep(c(-1,1),16)),
  LIQ = c(rep(c(rep(-1,2), rep(1,2)), 8)),
  EXT = c(rep(c(rep(-1,4), rep(1,4)), 4)),
  SPEED = c(rep(c(rep(-1,8), rep(1,8)), 2)),
  TIME = c(rep(-1,16), rep(1,16)), ACT, HARD, ROUND)

# Diseño de variables naturales

dis1_nat <- data.frame(
  TRAT = paste("TRAT",seq(1,32)),
  IMP = c(rep(c(500,1500),16)),
  LIQ = c(rep(c(rep(5,2), rep(10,2)), 8)),
  EXT = c(rep(c(rep(70,4), rep(120,4)), 4)),
  SPEED = c(rep(c(rep(1000,8), rep(2000,8)), 2)),
  TIME = c(rep(15,16), rep(30,16)), ACT, HARD, ROUND)

IMP(c)

LIQ(c)

EXT(c)

SPEED(c)

TIME(c)

IMP(n)

LIQ(n)

EXT(n)

SPEED(n)

TIMEn)

ACT

HARD

ROUND

-1

-1

-1

-1

-1

500

5

70

1,000

15

68.39378

9.18

1.27

1

-1

-1

-1

-1

1,500

5

70

1,000

15

69.43005

4.99

1.37

-1

1

-1

-1

-1

500

10

70

1,000

15

96.89119

2.42

1.30

1

1

-1

-1

-1

1,500

10

70

1,000

15

70.46632

5.29

1.40

-1

-1

1

-1

-1

500

5

120

1,000

15

71.50259

5.94

1.20

1

-1

1

-1

-1

1,500

5

120

1,000

15

79.79275

4.45

1.39

-1

1

1

-1

-1

500

10

120

1,000

15

84.45596

5.71

1.34

1

1

1

-1

-1

1,500

10

120

1,000

15

67.87565

8.87

1.38

-1

-1

-1

1

-1

500

5

70

2,000

15

70.46632

6.39

1.16

1

-1

-1

1

-1

1,500

5

70

2,000

15

75.12953

3.87

1.27

-1

1

-1

1

-1

500

10

70

2,000

15

71.50259

3.01

1.22

1

1

-1

1

-1

1,500

10

70

2,000

15

83.93782

5.36

1.23

-1

-1

1

1

-1

500

5

120

2,000

15

77.72021

3.86

1.17

1

-1

1

1

-1

1,500

5

120

2,000

15

81.34715

3.48

1.28

-1

1

1

1

-1

500

10

120

2,000

15

65.80311

3.14

1.34

1

1

1

1

-1

1,500

10

120

2,000

15

47.15026

6.57

1.27

-1

-1

-1

-1

1

500

5

70

1,000

30

72.53886

6.39

1.25

1

-1

-1

-1

1

1,500

5

70

1,000

30

62.69430

4.79

1.40

-1

1

-1

-1

1

500

10

70

1,000

30

86.52850

2.55

1.20

1

1

-1

-1

1

1,500

10

70

1,000

30

56.47668

7.71

1.39

-1

-1

1

-1

1

500

5

120

1,000

30

67.87565

6.82

1.22

1

-1

1

-1

1

1,500

5

120

1,000

30

72.53886

3.48

1.42

-1

1

1

-1

1

500

10

120

1,000

30

78.75648

2.42

1.29

1

1

1

-1

1

1,500

10

120

1,000

30

74.09326

4.90

1.40

-1

-1

-1

1

1

500

5

70

2,000

30

75.64767

6.48

1.14

1

-1

-1

1

1

1,500

5

70

2,000

30

77.20207

3.27

1.29

-1

1

-1

1

1

500

10

70

2,000

30

94.81865

3.16

1.27

1

1

-1

1

1

1,500

10

70

2,000

30

50.77720

6.00

1.29

-1

-1

1

1

1

500

5

120

2,000

30

80.82902

4.24

1.31

1

-1

1

1

1

1,500

5

120

2,000

30

84.97409

5.41

1.39

-1

1

1

1

1

500

10

120

2,000

30

62.69430

3.78

1.32

1

1

1

1

1

1,500

10

120

2,000

30

66.83938

7.18

1.35

(c): notación codificada (n): notación natural

Tabla 11.1: Tabla 2. Diseño factorial \(2^{5-1}\) no replicado

12 ACT

12.1 Modelo de regresión con interacciones hasta tres factores

Código 
# Actividad (%) 
mod1 <- lm(ACT~IMP + LIQ + EXT + SPEED + TIME +
                 IMP:LIQ + IMP:EXT + IMP:SPEED + IMP:TIME +
                 LIQ:EXT + LIQ:SPEED + LIQ:TIME +
                 EXT:SPEED + EXT:TIME + SPEED:TIME + 
                 IMP:LIQ:EXT + IMP:LIQ:SPEED + IMP:LIQ:TIME + IMP:EXT:SPEED + 
                 IMP:EXT:TIME + IMP:SPEED:TIME +
                 LIQ:EXT:SPEED + LIQ:EXT:TIME + LIQ:SPEED:TIME + EXT:SPEED:TIME
                 , data=dis1_cod)
summary(mod1)

Call:
lm.default(formula = ACT ~ IMP + LIQ + EXT + SPEED + TIME + IMP:LIQ + 
    IMP:EXT + IMP:SPEED + IMP:TIME + LIQ:EXT + LIQ:SPEED + LIQ:TIME + 
    EXT:SPEED + EXT:TIME + SPEED:TIME + IMP:LIQ:EXT + IMP:LIQ:SPEED + 
    IMP:LIQ:TIME + IMP:EXT:SPEED + IMP:EXT:TIME + IMP:SPEED:TIME + 
    LIQ:EXT:SPEED + LIQ:EXT:TIME + LIQ:SPEED:TIME + EXT:SPEED:TIME, 
    data = dis1_cod)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.4197 -3.4650  0.0648  3.3355  8.5492 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)    73.34845    1.71836  42.685 1.11e-08 ***
IMP            -3.30311    1.71836  -1.922   0.1029    
LIQ            -0.90674    1.71836  -0.528   0.6167    
EXT            -0.58290    1.71836  -0.339   0.7460    
SPEED          -0.42098    1.71836  -0.245   0.8146    
TIME           -0.51813    1.71836  -0.302   0.7732    
IMP:LIQ        -4.43653    1.71836  -2.582   0.0417 *  
IMP:EXT         2.36399    1.71836   1.376   0.2181    
IMP:SPEED       1.29534    1.71836   0.754   0.4795    
IMP:TIME       -1.32772    1.71836  -0.773   0.4691    
LIQ:EXT        -3.40026    1.71836  -1.979   0.0952 .  
LIQ:SPEED      -4.08031    1.71836  -2.375   0.0552 .  
LIQ:TIME       -0.55052    1.71836  -0.320   0.7595    
EXT:SPEED      -1.42487    1.71836  -0.829   0.4387    
EXT:TIME        1.32772    1.71836   0.773   0.4691    
SPEED:TIME      1.81347    1.71836   1.055   0.3319    
IMP:LIQ:EXT     0.90674    1.71836   0.528   0.6167    
IMP:LIQ:SPEED   0.68005    1.71836   0.396   0.7060    
IMP:LIQ:TIME   -0.25907    1.71836  -0.151   0.8851    
IMP:EXT:SPEED  -1.19819    1.71836  -0.697   0.5117    
IMP:EXT:TIME    3.30311    1.71836   1.922   0.1029    
IMP:SPEED:TIME -0.93912    1.71836  -0.547   0.6044    
LIQ:EXT:SPEED  -1.91062    1.71836  -1.112   0.3087    
LIQ:EXT:TIME    1.87824    1.71836   1.093   0.3163    
LIQ:SPEED:TIME  0.09715    1.71836   0.057   0.9567    
EXT:SPEED:TIME  0.29145    1.71836   0.170   0.8709    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 9.72 on 6 degrees of freedom
Multiple R-squared:  0.8478,    Adjusted R-squared:  0.2136 
F-statistic: 1.337 on 25 and 6 DF,  p-value: 0.3834

12.2 Gráfico de Daniel

Código 
mod2 <- lm(ACT~IMP*LIQ*EXT*SPEED*TIME, data=dis1_cod)
DanielPlot(mod2)

13 HARD

13.1 Modelo de regresión con interacciones hasta tres factores

Código 
# Actividad (%) 
mod3 <- lm(HARD~IMP + LIQ + EXT + SPEED + TIME +
                 IMP:LIQ + IMP:EXT + IMP:SPEED + IMP:TIME +
                 LIQ:EXT + LIQ:SPEED + LIQ:TIME +
                 EXT:SPEED + EXT:TIME + SPEED:TIME + 
                 IMP:LIQ:EXT + IMP:LIQ:SPEED + IMP:LIQ:TIME + IMP:EXT:SPEED + 
                 IMP:EXT:TIME + IMP:SPEED:TIME +
                 LIQ:EXT:SPEED + LIQ:EXT:TIME + LIQ:SPEED:TIME + EXT:SPEED:TIME
                 , data=dis1_cod)
summary(mod3)

Call:
lm.default(formula = HARD ~ IMP + LIQ + EXT + SPEED + TIME + 
    IMP:LIQ + IMP:EXT + IMP:SPEED + IMP:TIME + LIQ:EXT + LIQ:SPEED + 
    LIQ:TIME + EXT:SPEED + EXT:TIME + SPEED:TIME + IMP:LIQ:EXT + 
    IMP:LIQ:SPEED + IMP:LIQ:TIME + IMP:EXT:SPEED + IMP:EXT:TIME + 
    IMP:SPEED:TIME + LIQ:EXT:SPEED + LIQ:EXT:TIME + LIQ:SPEED:TIME + 
    EXT:SPEED:TIME, data = dis1_cod)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.86875 -0.27109  0.03969  0.31078  0.78937 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)     5.03469    0.18663  26.977 1.71e-07 ***
IMP             0.31656    0.18663   1.696 0.140768    
LIQ            -0.15531    0.18663  -0.832 0.437151    
EXT            -0.01906    0.18663  -0.102 0.921971    
SPEED          -0.33469    0.18663  -1.793 0.123078    
TIME           -0.12344    0.18663  -0.661 0.532906    
IMP:LIQ         1.28906    0.18663   6.907 0.000455 ***
IMP:EXT         0.21031    0.18663   1.127 0.302817    
IMP:SPEED       0.12594    0.18663   0.675 0.524930    
IMP:TIME        0.11469    0.18663   0.615 0.561428    
LIQ:EXT         0.46094    0.18663   2.470 0.048468 *  
LIQ:SPEED       0.23031    0.18663   1.234 0.263312    
LIQ:TIME       -0.04344    0.18663  -0.233 0.823690    
EXT:SPEED       0.02656    0.18663   0.142 0.891478    
EXT:TIME       -0.11344    0.18663  -0.608 0.565578    
SPEED:TIME      0.36344    0.18663   1.947 0.099415 .  
IMP:LIQ:EXT    -0.25719    0.18663  -1.378 0.217361    
IMP:LIQ:SPEED  -0.22906    0.18663  -1.227 0.265644    
IMP:LIQ:TIME    0.01469    0.18663   0.079 0.939830    
IMP:EXT:SPEED   0.29969    0.18663   1.606 0.159436    
IMP:EXT:TIME   -0.17781    0.18663  -0.953 0.377493    
IMP:SPEED:TIME -0.03219    0.18663  -0.172 0.868738    
LIQ:EXT:SPEED  -0.07594    0.18663  -0.407 0.698201    
LIQ:EXT:TIME   -0.47094    0.18663  -2.523 0.045078 *  
LIQ:SPEED:TIME  0.05844    0.18663   0.313 0.764783    
EXT:SPEED:TIME  0.31844    0.18663   1.706 0.138829    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.056 on 6 degrees of freedom
Multiple R-squared:  0.9343,    Adjusted R-squared:  0.6607 
F-statistic: 3.415 on 25 and 6 DF,  p-value: 0.06532

13.2 Gráfico de Daniel

Código 
mod4 <- lm(HARD~IMP*LIQ*EXT*SPEED*TIME, data=dis1_cod)
DanielPlot(mod4)

14 ROUND

14.1 Modelo de regresión con interacciones hasta tres factores

Código 
# Actividad (%) 
mod5 <- lm(ROUND~IMP + LIQ + EXT + SPEED + TIME +
                 IMP:LIQ + IMP:EXT + IMP:SPEED + IMP:TIME +
                 LIQ:EXT + LIQ:SPEED + LIQ:TIME +
                 EXT:SPEED + EXT:TIME + SPEED:TIME + 
                 IMP:LIQ:EXT + IMP:LIQ:SPEED + IMP:LIQ:TIME + IMP:EXT:SPEED + 
                 IMP:EXT:TIME + IMP:SPEED:TIME +
                 LIQ:EXT:SPEED + LIQ:EXT:TIME + LIQ:SPEED:TIME + EXT:SPEED:TIME
                 , data=dis1_cod)
summary(mod5)

Call:
lm.default(formula = ROUND ~ IMP + LIQ + EXT + SPEED + TIME + 
    IMP:LIQ + IMP:EXT + IMP:SPEED + IMP:TIME + LIQ:EXT + LIQ:SPEED + 
    LIQ:TIME + EXT:SPEED + EXT:TIME + SPEED:TIME + IMP:LIQ:EXT + 
    IMP:LIQ:SPEED + IMP:LIQ:TIME + IMP:EXT:SPEED + IMP:EXT:TIME + 
    IMP:SPEED:TIME + LIQ:EXT:SPEED + LIQ:EXT:TIME + LIQ:SPEED:TIME + 
    EXT:SPEED:TIME, data = dis1_cod)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.03188 -0.01094  0.00000  0.01094  0.03188 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)     1.297500   0.006409 202.436  9.8e-13 ***
IMP             0.047500   0.006409   7.411  0.00031 ***
LIQ             0.014375   0.006409   2.243  0.06609 .  
EXT             0.019375   0.006409   3.023  0.02331 *  
SPEED          -0.028750   0.006409  -4.486  0.00417 ** 
TIME            0.010625   0.006409   1.658  0.14845    
IMP:LIQ        -0.020625   0.006409  -3.218  0.01818 *  
IMP:EXT        -0.004375   0.006409  -0.683  0.52034    
IMP:SPEED      -0.020000   0.006409  -3.120  0.02057 *  
IMP:TIME        0.010625   0.006409   1.658  0.14845    
LIQ:EXT         0.005000   0.006409   0.780  0.46498    
LIQ:SPEED       0.003125   0.006409   0.488  0.64316    
LIQ:TIME       -0.008750   0.006409  -1.365  0.22117    
EXT:SPEED       0.015625   0.006409   2.438  0.05062 .  
EXT:TIME        0.010000   0.006409   1.560  0.16973    
SPEED:TIME      0.015625   0.006409   2.438  0.05062 .  
IMP:LIQ:EXT    -0.008750   0.006409  -1.365  0.22117    
IMP:LIQ:SPEED  -0.008125   0.006409  -1.268  0.25189    
IMP:LIQ:TIME    0.006250   0.006409   0.975  0.36716    
IMP:EXT:SPEED  -0.004375   0.006409  -0.683  0.52034    
IMP:EXT:TIME   -0.001250   0.006409  -0.195  0.85181    
IMP:SPEED:TIME -0.003125   0.006409  -0.488  0.64316    
LIQ:EXT:SPEED  -0.006250   0.006409  -0.975  0.36716    
LIQ:EXT:TIME   -0.008125   0.006409  -1.268  0.25189    
LIQ:SPEED:TIME  0.003750   0.006409   0.585  0.57982    
EXT:SPEED:TIME  0.002500   0.006409   0.390  0.70997    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.03626 on 6 degrees of freedom
Multiple R-squared:  0.959, Adjusted R-squared:  0.7884 
F-statistic:  5.62 on 25 and 6 DF,  p-value: 0.01965

14.2 Gráfico de Daniel

Código 
mod6 <- lm(ROUND~IMP*LIQ*EXT*SPEED*TIME, data=dis1_cod)
DanielPlot(mod6)

Sovany, Tamas, Kitti Csordas, András Kelemen, Geza Regdon Jr, y Klara Pintye-Hodi. 2016. «Development of pellets for oral lysozyme delivery by using a quality by design approach». Chemical Engineering Research and Design 106: 92-100.
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