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Simulation of Dissolution Profiles
In bootf2: Simulation and Comparison of Dissolution Profiles
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#"
)
library(bootf2)
Introduction
Dissolution profiles are simulated either by mathematical models, or by
multivariate normal distribution based on function mvrnorm() from R package
MASS.
Mathematical Models
Two models are implemented at the moment: first-order model and Weibull
model (the default).
The first-order model is expressed as
$$f_t = f_\mathrm{max}\left(1 - e^{-k\cdot (t - t_\mathrm{lag})}\right),$$
and the Weibull model either as
$$f_t = f_\mathrm{max}\left(1 - e^{-\left(\frac{t - t_\mathrm{lag}}
{\mathrm{MDT}}\right)^\beta}\right),$$
or as
$$f_t = f_\mathrm{max}\left(1 - e^{-\frac{\left(t - t_\mathrm{lag}\right)^\beta}
{\alpha}}\right).$$
Obviously, the two expressions of Weibull model are mathematically equivalent
with $\alpha = \mathrm{MDT}^\beta$.
Parameter $f_\mathrm{max}$ is the maximum dissolution. In theory, it is 100,
but in practice, it might be slightly high than 100, or much less if the
dissolution is not complete. For immediate-release formulation, the lag time
$t_\mathrm{lag}$ is typically zero, but not necessary so for the extended
release formulation. Parameter $k$ is the first-order rate constant, and
$\alpha$ and $\beta$ are scale and shape parameters, and $\mathrm{MDT}$ is
the mean dissolution time.
Simulation based on mathematical models
Let $P_\mu$ be the set of population model parameters, i.e.,
$P_\mu = {f_\mathrm{max}, k, t_\mathrm{lag}}$ for the first-order
model and $P_\mu = {f_\mathrm{max}, t_\mathrm{lag}, \mathrm{MDT}, \beta}$
or $P_\mu = {f_\mathrm{max}, t_\mathrm{lag}, \alpha, \beta}$ for Weibull
model, given any time points tp, the dissolution profile $f_t$ will be
calculated according to the equations above with parameters $P_\mu$.
The generated profile is considered as population dissolution profile dp.
Individual dissolution profiles are generated with the same equation, but with
individual model parameters $P_i$, which is simulated according to exponential
error model
$$P_i = P_\mu \cdot e^{N\left(0,\, \sigma^2\right)},$$
where $N\left(0,\, \sigma^2\right)$ represents the normal distribution
with mean zero and variance $\sigma^2$ ($\sigma = \mathrm{CV}/100$)
Simulation based on multivariate normal distribution
Note that the population dissolution profile dp in the previous section
is generated by equation with population model parameters $P_\mu$. Sometime
we might have a mean dissolution profile dp, which is considered as population
profile, and we want to simulate many individual profiles such that the
calculated mean profile is equal to dp. In such case, multivariate normal
distribution approach will be used. Given any tp, dp, and the required
number of individual units, n.units, the function will simulate individual
profiles fulfil the above mentioned requirements.
Recommendation
Depending on the variability and the target profile, the run time of simulation
based on multivariate normal distribution might be very long, sometimes it might
even fail to finish. So in general, the modelling approach is recommended.
Usage
The complete list of arguments of the function is as follows:
sim.dp(tp, dp, dp.cv, model = c("Weibull", "first-order"),
model.par = NULL, seed = NULL, n.units = 12L, product,
max.disso = 100, ascending = FALSE, message = FALSE,
time.unit = c("min", "h"), plot = TRUE,
plot.max.unit = 36L)
The approach used to simulate individual dissolution profiles depends on if the
target mean dissolution profile dp is provided by the user or not.
- If
dp is not provided, then it will be calculated using tp, model, and
model.par. The parameters defined by model.par are considered as the
population parameter; consequently, the calculated dp is considered as the
targeted population profile. In addition, n.units sets of individual
model parameters will be simulated based on exponential error model, and
individual dissolution profiles will be generated using those individual
parameters. The mean of all simulated individual profiles, sim.mean,
included in one of the outputs data frames, sim.summary, will be similar,
but not identical, to dp. The difference between sim.mean and dp is
determined by the number of units and the variability of the model parameters.
In general, the larger the number of units and the lower of the variability,
the smaller the difference.
- If
dp is provided, then n.units of individual dissolution profiles will be
simulated using multivariate normal distribution. The mean of all simulated
individual profiles, sim.mean, will be identical to dp. In such case,
it is recommended that dp.cv, the CV at time points tp, should also be
provided by the user. If dp.cv is not provided, it will be generated
automatically such that the CV is between 10% and 20% for time points up to
10 min, between 1% and 3% for time points where the dissolution is more than
95%, between 3% and 5% for time points where the dissolution is between 90%
and 95%, and between 5% and 10% for the rest of time points. Whether the
dp.cv is provided or generated automatically, the CV calculated using all
individual profiles will be equal to dp.cv.
Minimum required arguments that must be provided by the user
If dp is provided by the user, logically, tp must also be provided, and the
approach based on multivariate normal distribution is used, as explained above.
If dp is not provided, tp could be missing, i.e., a simple function call
such as sim.dp() will simulate dissolution profiles. In such case, a default
tp will be generated depending on the time.unit: if the time.unit is
"min", then tp would be c(5, 10, 15, 20, 30, 45, 60); otherwise the tp
would be c(1, 2, 3, 4, 6, 8, 10, 12). The rest arguments are either optional
or required by the function but default values will be used.
Notes on function arguments
- Model parameter
model.par has to be specified as named list, e.g.,
list(fmax = 100, fmax.cv = 3, tlag = 0, tlag.cv = 0, k = 0.9, k.cv = 30).
The parameter fmax/k/tlag are used to generate dp, and the corresponding
.cv parts are used to generate individual model parameters. If model.par
is missing, it will be generated randomly.
- When multivariate normal distribution approach is used, depending on the
target profile and the variability, sometimes the simulated individual
profiles will decrease in the end. This could also happen to the real-life
profiles, especially for those products that are unstable in the dissolution
media. To force the profiles always increase with time, set option
ascending = TRUE. However, in that case, it is possible that the function
takes long time to run or even fails.
- Parameter
product is not really necessary for the simulation. so if
missing, it will be generated automatically with 3 letters and 4 digits. It
might be useful in the situations such as many simulation will be run and
output are pooled together for analysis.
- Read the function manual by
help("sim.dp") for more details on
each argument.
Examples
Simple case
For the most basic use, the function can be run without any user provided
arguments, e.g., sim.dp(). In such case, 12 units of individual dissolution
profiles will be simulated using Weibull model with a typical sampling time
points of 5, 10, 15, 20, 30, 45, and 60 min. A seed number will be randomly
generated, if not provided by the user, and included in the output for
reproducibility purpose.
# simulation
tmp1 <- sim.dp(seed = 1234)
The output is a list of 5 components:
sim.summary: A data frame with summary statistics of all individual
dissolution profiles.
tmp1$sim.summary
dp is the population mean profile obtained with the parameters in sim.info,
as explained in previous section. The rest columns with prefix sim are basic
descriptive statistics calculated from all simulated individual profiles.
sim.qt05, sim.qt25, ..., are 5%, 25%, .., quantiles.
sim.disso: A data frame with all individual dissolution profiles.
tmp1$sim.disso
sim.info: A data frame with information of the simulation.
tmp1$sim.info
model.par.ind: A data frame of individual model parameters that are used
to simulate the individual dissolution profiles if mathematical models are
chosen for the simulation.
tmp1$model.par.ind
sim.plot: A plot since the default argument plot is set to be TRUE.
tmp1$sim.plot
When the number of individual units is not large, all individual profiles will
be plotted, together with the population profile in green (as target profile),
and mean profile in blue calculated with simulated individual profiles. When
the number is small, there will be visible difference between the mean and the
target profile due to random error. When the number of units increase, the
difference will become smaller.
The argument plot.max.unit control how individual profile will be represented
in the plot. When the actual number of units is greater than the value of
plot.max.unit, the individual profile will be represented as boxplots at each
time points, as shown below.
# default plot.max.unit = 36
sim.dp(n.units = 100)$sim.plot
Example with model parameters
To obtain better controlled simulation, model parameters should be provided.
CV should be specified in percentage.
fo.par <- list(fmax = 100, fmax.cv = 3, k = 0.1, k.cv = 20,
tlag = 0, tlag.cv = 0)
fo.dat <- sim.dp(model = "first-order", model.par = fo.par, seed = 123)
fo.dat$sim.plot
fo.dat$sim.summary
Similarly, for Weibull model, the model parameter should be provided like the
example below. The order of the parameter does not matter, but the parameter
names have to be specified exactly as the following:
# with alpha = xx and alpha.cv = yy to replace beta/beta.cv if alternative
# expression of Weibull model is used.
mod.par <- list(fmax = 100, fmax.cv = 3, tlag = 0, tlag.cv = 0,
mdt = 20, mdt.cv = 25, beta = 2, beta.cv = 30)
Example with multivariate normal distribution
# target mean profile
dp <- c(39, 56, 67, 74, 83, 90, 94)
# CV at each time points
dp.cv <- c(19, 15, 10, 8, 8, 5, 3)
mvn.dat <- sim.dp(tp, dp = dp, dp.cv = dp.cv, seed = 1234)
mvn.dat$sim.summary
Notice that the the mean and CV of the simulated individual profile (
sim.mean and sim.cv) are equal to the target profile dp and dp.cv.
The plot look like this:
mvn.dat$sim.plot
Another example with missing dp.cv.
mvn.dat2 <- sim.dp(tp, dp = dp, seed = 123)
mvn.dat2$sim.summary
Notice that the dp.cv were generated automatically as explained in
[Notes on function arguments].
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knitr::opts_chunk$set( collapse = TRUE, comment = "#" )
library(bootf2)
Introduction
Dissolution profiles are simulated either by mathematical models, or by
multivariate normal distribution based on function mvrnorm() from R package
MASS.
Mathematical Models
Two models are implemented at the moment: first-order model and Weibull model (the default).
The first-order model is expressed as
$$f_t = f_\mathrm{max}\left(1 - e^{-k\cdot (t - t_\mathrm{lag})}\right),$$ and the Weibull model either as $$f_t = f_\mathrm{max}\left(1 - e^{-\left(\frac{t - t_\mathrm{lag}} {\mathrm{MDT}}\right)^\beta}\right),$$ or as $$f_t = f_\mathrm{max}\left(1 - e^{-\frac{\left(t - t_\mathrm{lag}\right)^\beta} {\alpha}}\right).$$ Obviously, the two expressions of Weibull model are mathematically equivalent with $\alpha = \mathrm{MDT}^\beta$.
Parameter $f_\mathrm{max}$ is the maximum dissolution. In theory, it is 100, but in practice, it might be slightly high than 100, or much less if the dissolution is not complete. For immediate-release formulation, the lag time $t_\mathrm{lag}$ is typically zero, but not necessary so for the extended release formulation. Parameter $k$ is the first-order rate constant, and $\alpha$ and $\beta$ are scale and shape parameters, and $\mathrm{MDT}$ is the mean dissolution time.
Simulation based on mathematical models
Let $P_\mu$ be the set of population model parameters, i.e.,
$P_\mu = {f_\mathrm{max}, k, t_\mathrm{lag}}$ for the first-order
model and $P_\mu = {f_\mathrm{max}, t_\mathrm{lag}, \mathrm{MDT}, \beta}$
or $P_\mu = {f_\mathrm{max}, t_\mathrm{lag}, \alpha, \beta}$ for Weibull
model, given any time points tp, the dissolution profile $f_t$ will be
calculated according to the equations above with parameters $P_\mu$.
The generated profile is considered as population dissolution profile dp.
Individual dissolution profiles are generated with the same equation, but with individual model parameters $P_i$, which is simulated according to exponential error model $$P_i = P_\mu \cdot e^{N\left(0,\, \sigma^2\right)},$$ where $N\left(0,\, \sigma^2\right)$ represents the normal distribution with mean zero and variance $\sigma^2$ ($\sigma = \mathrm{CV}/100$)
Simulation based on multivariate normal distribution
Note that the population dissolution profile dp in the previous section
is generated by equation with population model parameters $P_\mu$. Sometime
we might have a mean dissolution profile dp, which is considered as population
profile, and we want to simulate many individual profiles such that the
calculated mean profile is equal to dp. In such case, multivariate normal
distribution approach will be used. Given any tp, dp, and the required
number of individual units, n.units, the function will simulate individual
profiles fulfil the above mentioned requirements.
Recommendation
Depending on the variability and the target profile, the run time of simulation based on multivariate normal distribution might be very long, sometimes it might even fail to finish. So in general, the modelling approach is recommended.
Usage
The complete list of arguments of the function is as follows:
sim.dp(tp, dp, dp.cv, model = c("Weibull", "first-order"), model.par = NULL, seed = NULL, n.units = 12L, product, max.disso = 100, ascending = FALSE, message = FALSE, time.unit = c("min", "h"), plot = TRUE, plot.max.unit = 36L)
The approach used to simulate individual dissolution profiles depends on if the target mean dissolution profile dp is provided by the user or not.
- If
dpis not provided, then it will be calculated usingtp,model, andmodel.par. The parameters defined bymodel.parare considered as the population parameter; consequently, the calculateddpis considered as the targeted population profile. In addition,n.unitssets of individual model parameters will be simulated based on exponential error model, and individual dissolution profiles will be generated using those individual parameters. The mean of all simulated individual profiles,sim.mean, included in one of the outputs data frames,sim.summary, will be similar, but not identical, todp. The difference betweensim.meananddpis determined by the number of units and the variability of the model parameters. In general, the larger the number of units and the lower of the variability, the smaller the difference. - If
dpis provided, thenn.unitsof individual dissolution profiles will be simulated using multivariate normal distribution. The mean of all simulated individual profiles,sim.mean, will be identical todp. In such case, it is recommended thatdp.cv, the CV at time pointstp, should also be provided by the user. Ifdp.cvis not provided, it will be generated automatically such that the CV is between 10% and 20% for time points up to 10 min, between 1% and 3% for time points where the dissolution is more than 95%, between 3% and 5% for time points where the dissolution is between 90% and 95%, and between 5% and 10% for the rest of time points. Whether thedp.cvis provided or generated automatically, the CV calculated using all individual profiles will be equal todp.cv.
Minimum required arguments that must be provided by the user
If dp is provided by the user, logically, tp must also be provided, and the
approach based on multivariate normal distribution is used, as explained above.
If dp is not provided, tp could be missing, i.e., a simple function call
such as sim.dp() will simulate dissolution profiles. In such case, a default
tp will be generated depending on the time.unit: if the time.unit is
"min", then tp would be c(5, 10, 15, 20, 30, 45, 60); otherwise the tp
would be c(1, 2, 3, 4, 6, 8, 10, 12). The rest arguments are either optional
or required by the function but default values will be used.
Notes on function arguments
- Model parameter
model.parhas to be specified as named list, e.g.,list(fmax = 100, fmax.cv = 3, tlag = 0, tlag.cv = 0, k = 0.9, k.cv = 30). The parameterfmax/k/tlagare used to generatedp, and the corresponding.cvparts are used to generate individual model parameters. Ifmodel.paris missing, it will be generated randomly. - When multivariate normal distribution approach is used, depending on the
target profile and the variability, sometimes the simulated individual
profiles will decrease in the end. This could also happen to the real-life
profiles, especially for those products that are unstable in the dissolution
media. To force the profiles always increase with time, set option
ascending = TRUE. However, in that case, it is possible that the function takes long time to run or even fails. - Parameter
productis not really necessary for the simulation. so if missing, it will be generated automatically with 3 letters and 4 digits. It might be useful in the situations such as many simulation will be run and output are pooled together for analysis. - Read the function manual by
help("sim.dp")for more details on each argument.
Examples
Simple case
For the most basic use, the function can be run without any user provided
arguments, e.g., sim.dp(). In such case, 12 units of individual dissolution
profiles will be simulated using Weibull model with a typical sampling time
points of 5, 10, 15, 20, 30, 45, and 60 min. A seed number will be randomly
generated, if not provided by the user, and included in the output for
reproducibility purpose.
# simulation tmp1 <- sim.dp(seed = 1234)
The output is a list of 5 components:
sim.summary: A data frame with summary statistics of all individual dissolution profiles.
tmp1$sim.summary
dp is the population mean profile obtained with the parameters in sim.info,
as explained in previous section. The rest columns with prefix sim are basic
descriptive statistics calculated from all simulated individual profiles.
sim.qt05, sim.qt25, ..., are 5%, 25%, .., quantiles.
sim.disso: A data frame with all individual dissolution profiles.
tmp1$sim.disso
sim.info: A data frame with information of the simulation.
tmp1$sim.info
model.par.ind: A data frame of individual model parameters that are used to simulate the individual dissolution profiles if mathematical models are chosen for the simulation.
tmp1$model.par.ind
sim.plot: A plot since the default argumentplotis set to beTRUE.
tmp1$sim.plot
When the number of individual units is not large, all individual profiles will be plotted, together with the population profile in green (as target profile), and mean profile in blue calculated with simulated individual profiles. When the number is small, there will be visible difference between the mean and the target profile due to random error. When the number of units increase, the difference will become smaller.
The argument plot.max.unit control how individual profile will be represented
in the plot. When the actual number of units is greater than the value of
plot.max.unit, the individual profile will be represented as boxplots at each
time points, as shown below.
# default plot.max.unit = 36 sim.dp(n.units = 100)$sim.plot
Example with model parameters
To obtain better controlled simulation, model parameters should be provided. CV should be specified in percentage.
fo.par <- list(fmax = 100, fmax.cv = 3, k = 0.1, k.cv = 20, tlag = 0, tlag.cv = 0) fo.dat <- sim.dp(model = "first-order", model.par = fo.par, seed = 123) fo.dat$sim.plot fo.dat$sim.summary
Similarly, for Weibull model, the model parameter should be provided like the example below. The order of the parameter does not matter, but the parameter names have to be specified exactly as the following:
# with alpha = xx and alpha.cv = yy to replace beta/beta.cv if alternative # expression of Weibull model is used. mod.par <- list(fmax = 100, fmax.cv = 3, tlag = 0, tlag.cv = 0, mdt = 20, mdt.cv = 25, beta = 2, beta.cv = 30)
Example with multivariate normal distribution
# target mean profile dp <- c(39, 56, 67, 74, 83, 90, 94) # CV at each time points dp.cv <- c(19, 15, 10, 8, 8, 5, 3) mvn.dat <- sim.dp(tp, dp = dp, dp.cv = dp.cv, seed = 1234) mvn.dat$sim.summary
Notice that the the mean and CV of the simulated individual profile (
sim.mean and sim.cv) are equal to the target profile dp and dp.cv.
The plot look like this:
mvn.dat$sim.plot
Another example with missing dp.cv.
mvn.dat2 <- sim.dp(tp, dp = dp, seed = 123) mvn.dat2$sim.summary
Notice that the dp.cv were generated automatically as explained in
[Notes on function arguments].
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