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![Page 1: Modelos matemáticos. As necessary, the food business operators responsible for the manufacture of the product shall conduct studies in accordance with.](https://reader035.fdocuments.us/reader035/viewer/2022070417/5665b4601a28abb57c90fb1c/html5/thumbnails/1.jpg)
Modelos matemáticosModelos matemáticos
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As
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An
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Art
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3
![Page 3: Modelos matemáticos. As necessary, the food business operators responsible for the manufacture of the product shall conduct studies in accordance with.](https://reader035.fdocuments.us/reader035/viewer/2022070417/5665b4601a28abb57c90fb1c/html5/thumbnails/3.jpg)
•W
hen
nece
ssary
on
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e b
asi
s of
the a
bovem
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tion
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stu
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s,
the f
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ay in
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de:
pre
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row
th o
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ctors
for
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-org
an
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e p
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uct
,
An
nex II
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Generalidades sobre los modelos matemáticos predictivos
Generalidades sobre los modelos matemáticos predictivos
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Every model is wrong. The question is, how much wrong still useful it can be. (Box and Draper)
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MICROBIOLOGÍA PREDICTIVA
Campo de estudio que combina elementos de microbiología,
matemáticas y estadística para desarrollar modelos que
describan y predigan matemáticamente el crecimiento o
muerte de los microorganismos, cuando se les somete a
condiciones medioambientales específicas (Whiting, 1995).
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Los modelos son descripciones simplificadasde la realidad
La realidad descrita por el modelo es una parte de la realidad total llamada espacio
modelo
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Los modelos deben reflejar lo que está pasando y deben ser capaces de predecir con precisión los estados presente y futuro de las cosas que
describen
Hay que ser conscientes de que un modelo no puede dar una representación total de la realidad.
Un modelo particular puede describir algún aspecto de forma muy adecuada mientras que
falla en la descripción de otro
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Suposiciones en modelizaciónEspacio Modelo: No se puede modelizar todo, hayque escoger la parte de la realidad que se quieremodelizar. A esto se le llama espacio modelo yno tiene conexión con el resto de la realidad
espacio modelorealidad
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Se define como todos los factores que juegan un papel en la determinación del fenómeno bajo estudio, los conocidos y no conocidos
Espacio modelo:
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Fenómeno: Los modelos se usan para describir relaciones entre variables dependiente e indepen-dientes.
V. dependiente
V. Independientes
Relación Fenómeno
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Para poder modelizar un fenómeno en un espacio modelo determinado es necesario entender la relación entre las variables depen-diente e independientes. Este ejercicio ayudará a elegir el modelo apropiado
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Microbiología predictiva
El objetivo de la microbiología predictiva esconseguir un Espacio Modelo para describir unFenómeno de forma matemática o probabilística
Espacio modelo
Fenómeno
Medioambiene TemperaturapHaw
Respuesta microbianaCrecimientoInactivación
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La microbiología predictiva no revela, generalmente, comportamientos inesperadosde los microorganismos.
La Microbiología predictiva cuantifica los efectos de la interacción entre dos o más factores y permite la interpolación decombinaciones de factores no comprobadosde forma explícita
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Clasificación de los modelos
Modelos de nivel primario:
Modelos de nivel secundario:
Superficie de respuesta
Modelo de Bigelow
Modelos de nivel terciario:Tejedor y Martínez
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Los modelos de nivel primario describencambios en el número de microorganismos u otras respuestas microbianas con el tiempo.
0
2
4
6
8
10
0 0.1 0.2 0.3 0.4 0.5
time (h)
conc
. (lo
g10
cfu/
ml)
0
2
4
6
8
10
0 0.1 0.2 0.3 0.4 0.5
time (h)
conc
. (lo
g10
cfu/
ml)
0
2
4
6
8
10
0 10 20 30
time (h)
conc
. (lo
g10
cfu/
ml)
0
2
4
6
8
10
0 10 20 30
time (h)
conc
. (lo
g10
cfu/
ml)
inactivación crecimiento
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Los modelos secundarios describen lasrespuestas de los parámetros de los modelosprimarios a los cambios en las condiciones medioambientales
5.6
6
6.4
6.8
1
2
3
4
5
-2.6
-2.2
-1.8
-1.4
-1
-2.6
-2.2
-1.8
-1.4
-1
Ln(
spec
.g.r
ate)
NaCl (%) pH
superficie de respuesta
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Los modelos terciarios son programas deordenador que transforman a los modelosprimarios y secundarios en herramientas de facil uso para los usuarios del modelo
Inactivación crecimiento
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Consideraciones en el desarrollo de un modelo
Precisión en el ajuste. Capacidad de predecir combinaciones de factores no probadas. Incorporación de todos los factores relevantes. Que tenga el mínimo número de parámetros. Especificación del término de error. Los parámetros deben tener un significadobiológico y valores realistas. Reparametrización si se mejoran las propiedades estadísticas.
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Termoresistencia y Modelos primarios de inactivación/supervivencia
01234567
116 118 120 122 124 126 128Temperatura (ºC)
Log
N
experimentalpredicho
Bacillus stearothermophilus
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Obtención de datos experimentales
A) Tratamiento térmico isotérmico
B) Tratamiento térmico no isotérmico
B.1) La temperatura de la muestra varía conel tiempo
B.2)La temperatura de la muestra varía con eltiempo y después permanece constante hasta la fase de enfriamiento
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Modelos de inactivación: Velocidad alta de muerte de los microorganismos por la acción deun agente activo
Modelos de supervivencia: Disminución de la carga microbiana de forma mas lenta y noimplica esterilidad comercial
Los modelos matemáticos son los mismos en ambos casos
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Capilares
Data logger
Baño calentamiento Baño enfriamiento
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Capilares
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Detalle termorresistómetro
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Modelos primarios
La modelización matemática comenzó en 1920con los cálculos de tiempo de destrucción térmica.Los valores D y Z se usaron con éxito para asegurar que los alimentos enlatados estabanlibres de riesgo de alteración por Cl. botulinumEstos modelos establecen la relación existenteentre el tiempo y la inactivación de un microorga-nismo a una temperatura dada.
A) Modelos logarítmicos
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Los datos experimentales para la obtención delos parámetros, D y Z, que definen la inactivaciónde los microorganismos se pueden analizar de diferentes maneras:
Dos regresiones lineales consecutivas
Una regresión no lineal en un solo paso
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DDTT
Tiempo de exposiciónTiempo de exposición
Lo
g.
sup
ervi
vien
tes
Lo
g.
sup
ervi
vien
tes
11
22
33
Curva de supervivenciaCurva de supervivencia Curva de supervivenciaCurva de supervivencia
Dos regresiones lineales
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dN
dtkN
LnN=LnNo-ktLnN=LnNo-kt
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D tk
T 2 303,
lgN=lgNo-(k/2,303)tlgN=lgNo-(k/2,303)t
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zz
TemperaturaTemperatura
Lo
g D
Lo
g D
TT
DDT1T1
DDT2T2
TT11 TT22
Curva de muerte térmicaCurva de muerte térmica Curva de muerte térmicaCurva de muerte térmica
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log( ) log( )D D
T T Z
2 1
1 2
1
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log logN No
D
t
R
T T
zR
1
10
Tratamiento isotérmicoTratamiento isotérmico Tratamiento isotérmicoTratamiento isotérmico
Una regresión no lineal
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SSQNo
N
No
Ni
m
f m
1
2
log log
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Tabla 1. Parámetros cinéticos predichos para dos cepas de Bacillus cereus
Temperature D value (min)
(ºC) AV TZ415 AV Z421
Linear Non-linear Linear Non-linear
859095100105
165a
3.90.70.940.170.220.06
ND
17.10.5a
4.040.080.950.02
0.2250.007ND
ND
4020a
1132.50.4
0.600.19
ND
393a
9.80.52.480.060.630.03
z (ºC) 8.10.3 7.970.10 8.00.6 8.40.2
ND not determined.a D value confidence interval (95%).
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Log (No/N) predicted
Log
(N
o/N
) ob
serv
ed
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
3
Log (No/N) predicted
Log
(N
o/N
) ob
serv
ed
0 0.5 1 1.5 2 2.50
1
2
3
4
Curvas de equivalencia
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0
5
10
15
20
25
-0.3 -0.2 -0.1 0.1 0.2 0.3
(Log Nexp - Log Ncal)F
req
uen
cy
0
0
5
10
15
20
25
30
35
-0.7 -0.46 -0.22 0.02 0.26 0.5
(Log Nexp - Log Ncal)
Fre
qu
ency
Residuos normalescon media cero
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D (min)
z (º
C)
1 1.5 2 2.5 3 3.5 4 4.5 5
7.5
7.9
8.3
8.7
9.1
95ºCAV Z421
90ºCAV TZ415
Regiones de confianza conjunta
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D (min)
Z (ºC)
1 2 3 4 56
8
10
12
14118 ºC
Z (ºC)
2 4 6 8 10 126
8
10
12
14 115 ºC
Efecto del pH sobre el valor D del
B. stearothermophilus en ensaladilla
D (min)
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D (min)
Z (ºC)
1 1,2 1,4 1,6 1,8 26
8
10
12
14121 ºC
D (min)
Z (ºC)
0 1 26
8
10
12
14125 ºC
Efecto del pH sobre el valor D del
B. stearothermophilus en ensaladilla
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Tiempo de exposiciónTiempo de exposición
Lo
g.
sup
ervi
vien
tes
Lo
g.
sup
ervi
vien
tes
11
22
33
Diferentes tipos de curvas de supervivencia
Hombro
Cola
Lineal
Concavidad hacia abajo
Concavidad hacia arriba
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Los hombros se han atribuido:
a la necesidad de mas de un evento dañino
a la necesidad de una activación de las esporas
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Teoría vitalistaTeoría vitalista
Presencia de colas
Distribución de termorresistencia
Teoría mecanicistaTeoría mecanicista
La termorresistenciadepende del ciclo celular en que serecoja
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Presencia de artefactos experimentalesMezcla de poblaciones
La curva de supervivenciaes una forma acumulativade distribución de eventos letales con el tiempo
Cada organismo individual o espora de una poblaciónmuere a un tiempo específico
Otras explicacionesOtras explicaciones
Nueva aproximaciónNueva aproximación
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0 8 16 24 32 40
Time (min)
85°C
90°C
95°C
100°C
S(t
) (N
/No
) AVTZ415 strain
0.00001
0.0001
0.001
0.01
0.1
1
Curvas con hombros
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n
a
t-
e S(t)
Función de supervivenciaFunción de supervivencia Función de supervivenciaFunción de supervivencia
a= Scalan= Forma
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El parámetro de forma “n” se puede considerar como un índice de comportamiento
Si n >1 describe una curva con hombroSi n < 1 describe una curva con colaSi n = 1 la curva de supervivencia sera lineal en coordenadas semilogarítmicas y se comportará como una reacción de primer orden
El parámetro de escala “a”se puede considerar como una constante de velocidad de reacción. Similar al Valor D
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0.00 3.20 6.40 9.60 12.80 16.00
0.00
0.20
0.40
0.60
0.80
1.0095°C
97.5°C
100°C
102.5°C
105°C
S(t
) (N
/No
)
AVZ421 strain
Curvas de supervivenciaCurvas de supervivencia Curvas de supervivenciaCurvas de supervivencia
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nate1-nn- tnaf(t)
Función de densidadFunción de densidad Función de densidadFunción de densidad
frecuencia de muertes por unidad de tiempo
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0.00 1.80 3.60 5.40 7.20 9.00
Tiempo (min)
0.00
0.09
0.18
0.27
0.36
0.45
AVZ421 strain
Fre
cuen
cia
(1/m
in)
Curva de distribuciónCurva de distribución Curva de distribuciónCurva de distribución
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-1n1a tc Función Gama
Medida de la resistencia térmicaMedida de la resistencia térmica Medida de la resistencia térmicaMedida de la resistencia térmica
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t N
(min) Nobs NW NB
0481216
1990000013266000836000034500001417000
1990000013710010762965037598611688864
2413098912433299640615833007221700671
Af - 1.10 1.20
Comparación entre el número supervivientes Comparación entre el número supervivientes
experimentales y predichosexperimentales y predichos
Comparación entre el número supervivientes Comparación entre el número supervivientes
experimentales y predichosexperimentales y predichos
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T Weibull distribution Bigelow model
(ºC) scale (a) shape (n) tc (min) D (min)95.097.5100.0102.5105.0
8.34.52.101.350.65
1.361.721.582.031.69
8.04.01.851.200.58
14 5a
5.9 1.52.5 0.51.5 0.5
0.76 0.18
z (ºC) (8.9) 8.1
Parámetros para la distribución Parámetros para la distribución
de Weibull y valor Dde Weibull y valor D
Parámetros para la distribución Parámetros para la distribución
de Weibull y valor Dde Weibull y valor D
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0.00 2.40 4.80 7.20 9.60 12.00
Tiempo ( min)
0.00001
0.0001
0.001
0.01
0.1
1
Fra
cció
n su
perv
ivie
ntes
Curva de supervivencia para Curva de supervivencia para BacillusBacilluspumilluspumillus en condiciones isotérmicas en condiciones isotérmicas
Curva de supervivencia para Curva de supervivencia para BacillusBacilluspumilluspumillus en condiciones isotérmicas en condiciones isotérmicas
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)(
t
tLnS
Función de supervivenciaFunción de supervivencia Función de supervivenciaFunción de supervivencia
n
a
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0.00 2.20 4.40 6.60 8.80 11.00
Tiempo( min )
-15
-12
-9
-6
-3
0
90 º C
a =5.47, n =0.32
Ln
frac
tion
of
surv
ivor
sCurva de supervivencia para Curva de supervivencia para BacillusBacillus
pumilluspumillus mediante Weibull en mediante Weibull en condiciones isotérmicascondiciones isotérmicas
Curva de supervivencia para Curva de supervivencia para BacillusBacilluspumilluspumillus mediante Weibull en mediante Weibull en
condiciones isotérmicascondiciones isotérmicas
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Ventajas de los métodos no isotérmicosVentajas de los métodos no isotérmicos Ventajas de los métodos no isotérmicosVentajas de los métodos no isotérmicos
Se obtiene una gran información de cada experimento
Se ahorra tiempo
Se ahorra material y costo en mano de hobra
Son mas cercanos a lo que en realidad pasa en un proceso industrial
Métodos no isotérmicosMétodos no isotérmicos Métodos no isotérmicosMétodos no isotérmicos
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Tratamiento no isotérmicoTratamiento no isotérmico Tratamiento no isotérmicoTratamiento no isotérmico
n
i z
TT
R
t
D
LogNNo
LogLogR
110
1
Ecuación 1
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az
TT
zTT
Dz
NNo
Log R
R
11010
10ln
0
0
a=Velocidad de calentamiento
Ecuación 2
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Cálculo de las regiones de confianza conjuntaCálculo de las regiones de confianza conjunta Cálculo de las regiones de confianza conjuntaCálculo de las regiones de confianza conjunta
SQ SSQp
m pF p m p
1 ( , )
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Temperature
(°C)
Isothermic heating D values (min) Non-isothermic heating D values (min)
118 9.03 10.49
121 3.08 4.38
125 0.93 1.37
z (°C) 7 7.90
No isotérmico con tramo isotérmico
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0
1
2
3
4
5
6
7
116 118 120 122 124 126 128
Temperatura (ºC)
Log
N experimentalpredicho
Bacillus stearothermophilus
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0
10
20
30
40
50
60
70
80
-0.4 -0.2 0 0.2 0.4
(Log Nexp - Log N cal)
Fre
cuen
cia
Distribución de residuos
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1 1.5 2 2.5 3 3.5 4
D (min)
6.0
6.5
7.0
7.5
8.0
8.5
9.0z
(°C
)
125 ºC 124 ºC 123 ºC 122 ºC
Regiones de confianza conjunta
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Temperature D (min)
(ºC) non-isothermal Isothermala
85
90
95
100
16.0
3.93
0.96
0.236
17.1 a
4.04 a
0.95 a
0.225 a
z ( C) 8.19 7.97 a
A fb 1.11
Bacillus cereus
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Activation rate a Inactivation rate bEquation 1 Equation 2
(ºC/min) (ºC/min) T (ºC) D (min) z (ºC) Afc T (ºC) D (min) z (ºC) Af
c
0.5 0.5
1.0
90
95
90
95
3.50
0.42
5.20
0.46
5.4
4.8
1.17
1.13
90
95
90
95
3.50
0.44
5.00
0.46
5.6
4.8
1.17
1.13
1.0 0.5
1.0
90
95
90
95
3.10
0.58
3.90
0.64
6.9
6.3
1.11
1.18
90
95
90
95
3.10
0.62
3.90
0.66
7.1
6.5
1.12
1.18
Bacillus cereus
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5.6
6
6.4
6.8
1
2
3
4
5
-2.6
-2.2
-1.8
-1.4
-1
-2.6
-2.2
-1.8
-1.4
-1
Modelos secundarios de inactivación
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Modelos secundarios
Tanto los parámetros que definen las curvas de Inactivación D ó z, como los que definen las curvasDe crecimiento , se ven afectados por factoresMediombientales pH, ClNa, aw, entre otros.
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Los modelos probabilísticos o matemáticos que relacionan las variables dependientes, parámetros cinéticos, con los factores medioambientales son los denominados modelos secundarios
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Modelos secundarios de inactivción
Modelo basado en la ecuación de Arrhenius (Davey, 1993)
Lnk = c0+(c1/T)+c2pH+c3(pH)2+
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Modelo basado en la ecuación de Bigelow (Mafart y Leguérinel, 1998)
LogD = LogD*-(1/zT)(T-T*)-(1/zpH)2(pH-pH*)2+
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Modelo cuadrático polinomial (Fernández y col., 1996)
LogD = c1+c2T+c3pH+c4(TpH)+c5T2+c6(pH)2 +
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Modelo básico (Fernández y col., 1996)
LogD = c1+c2T+c3pH+
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refref
pHTref TTR
EapHpH
tLn
11exp
),(
Curvas con colas o con hombros
Modelo basado en la distribución de
Frecuencia de Weibull (Fernández y col., 2001)
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VALIDACIÓN Y EVALUACIÓN DE LOS MODELOS
Con nuevos datos obtenidos de forma independiente
En condiciones reales de elaboración del alimento
A través de ciertos índices (Estadísticamente)
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ANÁLISIS DE LOS MODELOS
Indices estadísticos
Coeficiente de determinación
Estudio de los residuos
Datos influyentes
Multicolinealidad
Índices para evaluar modelos en microbiología de alimentos
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Coeficiente de determinación
Este coeficiente indica la proporción de variabilidadde las observaciones de la variable dependiente (lnK)explicada por el conjunto de las variables independientesconsideradas en cada caso.
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Estudio de los residuos
Los residuos se definen como la diferencia entre el valorobservado de la variable dependiente y el valor ajustadoen el modelo.
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Pruebas habituales para los residuos
Descriptivas básicas
Test de normalidad (Kolmogorov-Smirnov)
Linealidad, homocedasticidad y valores atípicos
Autocorrelación entre residuos consecutivos (Durbin-Watson)
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