Create an ODE-based model specification

Create a dynamic ODE-based model object suitably for translation into fast C code

Usage

rxode2(
  model,
  modName = basename(wd),
  wd = getwd(),
  filename = NULL,
  extraC = NULL,
  debug = FALSE,
  calcJac = NULL,
  calcSens = NULL,
  collapseModel = FALSE,
  package = NULL,
  ...,
  linCmtSens = c("linCmtA", "linCmtB", "linCmtC"),
  indLin = FALSE,
  verbose = FALSE,
  fullPrint = getOption("rxode2.fullPrint", FALSE),
  envir = parent.frame()
)

RxODE(
  model,
  modName = basename(wd),
  wd = getwd(),
  filename = NULL,
  extraC = NULL,
  debug = FALSE,
  calcJac = NULL,
  calcSens = NULL,
  collapseModel = FALSE,
  package = NULL,
  ...,
  linCmtSens = c("linCmtA", "linCmtB", "linCmtC"),
  indLin = FALSE,
  verbose = FALSE,
  fullPrint = getOption("rxode2.fullPrint", FALSE),
  envir = parent.frame()
)

rxode(
  model,
  modName = basename(wd),
  wd = getwd(),
  filename = NULL,
  extraC = NULL,
  debug = FALSE,
  calcJac = NULL,
  calcSens = NULL,
  collapseModel = FALSE,
  package = NULL,
  ...,
  linCmtSens = c("linCmtA", "linCmtB", "linCmtC"),
  indLin = FALSE,
  verbose = FALSE,
  fullPrint = getOption("rxode2.fullPrint", FALSE),
  envir = parent.frame()
)

Arguments

model

This is the ODE model specification. It can be:

• a string containing the set of ordinary differential equations (ODE) and other expressions defining the changes in the dynamic system.

• a file name where the ODE system equation is contained

An ODE expression enclosed in \{\}

(see also the filename argument). For details, see the sections “Details” and rxode2 Syntax below.

modName

a string to be used as the model name. This string is used for naming various aspects of the computations, including generating C symbol names, dynamic libraries, etc. Therefore, it is necessary that modName consists of simple ASCII alphanumeric characters starting with a letter.

wd

character string with a working directory where to create a subdirectory according to modName. When specified, a subdirectory named after the “modName.d” will be created and populated with a C file, a dynamic loading library, plus various other working files. If missing, the files are created (and removed) in the temporary directory, and the rxode2 DLL for the model is created in the current directory named rx_????_platform, for example rx_129f8f97fb94a87ca49ca8dafe691e1e_i386.dll

filename

A file name or connection object where the ODE-based model specification resides. Only one of model or filename may be specified.

extraC

Extra c code to include in the model. This can be useful to specify functions in the model. These C functions should usually take double precision arguments, and return double precision values.

debug

is a boolean indicating if the executable should be compiled with verbose debugging information turned on.

calcJac

boolean indicating if rxode2 will calculate the Jacobain according to the specified ODEs.

calcSens

boolean indicating if rxode2 will calculate the sensitivities according to the specified ODEs.

collapseModel

boolean indicating if rxode2 will remove all LHS variables when calculating sensitivities.

package

Package name for pre-compiled binaries.

...

ignored arguments.

linCmtSens

The method to calculate the linCmt() solutions

indLin

Calculate inductive linearization matrices and compile with inductive linearization support.

verbose

When TRUE be verbose with the linear compartmental model

fullPrint

When using printf within the model, if TRUE print on every step (except ME/indLin), otherwise when FALSE print only when calculating the d/dt

envir

is the environment to look for R user functions (defaults to parent environment)

Value

An object (environment) of class rxode2 (see Chambers and Temple Lang (2001)) consisting of the following list of strings and functions:

* `model` a character string holding the source model specification.
* `get.modelVars`a function that returns a list with 3 character
    vectors, `params`, `state`, and `lhs` of variable names used in the model
    specification. These will be output when the model is computed (i.e., the ODE solved by integration).

  * `solve`{this function solves (integrates) the ODE. This
      is done by passing the code to [rxSolve()].
      This is as if you called `rxSolve(rxode2object, ...)`,
      but returns a matrix instead of a rxSolve object.

      `params`: a numeric named vector with values for every parameter
      in the ODE system; the names must correspond to the parameter
      identifiers used in the ODE specification;

      `events`: an `eventTable` object describing the
      input (e.g., doses) to the dynamic system and observation
      sampling time points (see  [eventTable()]);

      `inits`: a vector of initial values of the state variables
      (e.g., amounts in each compartment), and the order in this vector
      must be the same as the state variables (e.g., PK/PD compartments);


      `stiff`: a logical (`TRUE` by default) indicating whether
      the ODE system is stiff or not.

      For stiff ODE systems (`stiff = TRUE`), `rxode2` uses
      the LSODA (Livermore Solver for Ordinary Differential Equations)
      Fortran package, which implements an automatic method switching
      for stiff and non-stiff problems along the integration interval,
      authored by Hindmarsh and Petzold (2003).

      For non-stiff systems (`stiff = FALSE`), `rxode2` uses `DOP853`,
      an explicit Runge-Kutta method of order 8(5, 3) of Dormand and Prince
      as implemented in C by Hairer and Wanner (1993).

      `trans_abs`: a logical (`FALSE` by default) indicating
      whether to fit a transit absorption term
      (TODO: need further documentation and example);

      `atol`: a numeric absolute tolerance (1e-08 by default);

      `rtol`: a numeric relative tolerance (1e-06 by default).

      The output of \dQuote{solve} is a matrix with as many rows as there
      are sampled time points and as many columns as system variables
      (as defined by the ODEs and additional assignments in the rxode2 model
          code).}

  * `isValid` a function that (naively) checks for model validity,
      namely that the C object code reflects the latest model
      specification.
  * `version` a string with the version of the `rxode2`
      object (not the package).
  * `dynLoad` a function with one `force = FALSE` argument
      that dynamically loads the object code if needed.
  * `dynUnload` a function with no argument that unloads
      the model object code.
  * `delete` removes all created model files, including C and DLL files.
      The model object is no longer valid and should be removed, e.g.,
      `rm(m1)`.
  * `run` deprecated, use `solve`.
  * `get.index` deprecated.
  * `getObj` internal (not user callable) function.

Details

The Rx in the name rxode2 is meant to suggest the abbreviation Rx for a medical prescription, and thus to suggest the package emphasis on pharmacometrics modeling, including pharmacokinetics (PK), pharmacodynamics (PD), disease progression, drug-disease modeling, etc.

The ODE-based model specification may be coded inside four places:

• Inside a rxode2({}) block statements:

library(rxode2)
mod <- rxode2({
  # simple assignment
  C2 <- centr/V2

  # time-derivative assignment
  d/dt(centr) <- F*KA*depot - CL*C2 - Q*C2 + Q*C3;
})

## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’

• Inside a rxode2("") string statement:

mod <- rxode2("
  # simple assignment
  C2 <- centr/V2

  # time-derivative assignment
  d/dt(centr) <- F*KA*depot - CL*C2 - Q*C2 + Q*C3;
")

## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’

• In a file name to be loaded by rxode2:

writeLines("
  # simple assignment
  C2 <- centr/V2

  # time-derivative assignment
  d/dt(centr) <- F*KA*depot - CL*C2 - Q*C2 + Q*C3;
", "modelFile.rxode2")
mod <- rxode2(filename='modelFile.rxode2')
unlink("modelFile.rxode2")

• In a model function which can be parsed by rxode2:

mod <- function() {
  model({
    # simple assignment
    C2 <- centr/V2

    # time-derivative assignment
    d/dt(centr) <- F*KA*depot - CL*C2 - Q*C2 + Q*C3;
  })
}

mod <- rxode2(mod) # or simply mod() if the model is at the end of the function

# These model functions often have residual components and initial
# (`ini({})`) conditions attached as well. For example the
# theophylline model can be written as:

one.compartment <- function() {
  ini({
    tka <- 0.45 # Log Ka
    tcl <- 1 # Log Cl
    tv <- 3.45    # Log V
    eta.ka ~ 0.6
    eta.cl ~ 0.3
    eta.v ~ 0.1
    add.sd <- 0.7
  })
  model({
    ka <- exp(tka + eta.ka)
    cl <- exp(tcl + eta.cl)
    v <- exp(tv + eta.v)
    d/dt(depot) = -ka * depot
    d/dt(center) = ka * depot - cl / v * center
    cp = center / v
    cp ~ add(add.sd)
  })
}

# after parsing the model
mod <- one.compartment()

For the block statement, character string or text file an internal rxode2 compilation manager translates the ODE system into C, compiles it and loads it into the R session. The call to rxode2 produces an object of class rxode2 which consists of a list-like structure (environment) with various member functions.

For the last type of model (a model function), a call to rxode2 creates a parsed rxode2 ui that can be translated to the rxode2 compilation model.

mod$simulationModel

## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’

## rxode2 2.0.14.9000 model named rx_f8a38012c74774242195a2bca079f175 model (ready). 
## x$state: depot, center
## x$stateExtra: cp
## x$params: tka, tcl, tv, add.sd, eta.ka, eta.cl, eta.v, rxerr.cp
## x$lhs: ka, cl, v, cp, ipredSim, sim

# or 
mod$simulationIniModel

## using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’

## rxode2 2.0.14.9000 model named rx_8367e8f5e6c5a00454c7193fc896bf87 model (ready). 
## x$state: depot, center
## x$stateExtra: cp
## x$params: tka, tcl, tv, add.sd, eta.ka, eta.cl, eta.v, rxerr.cp
## x$lhs: ka, cl, v, cp, ipredSim, sim

This is the same type of function required for nlmixr2 estimation and can be extended and modified by model piping. For this reason will be focused on in the documentation.

This basic model specification consists of one or more statements optionally terminated by semi-colons ; and optional comments (comments are delimited by # and an end-of-line).

A block of statements is a set of statements delimited by curly braces, { ... }.

Statements can be either assignments, conditional if/else if/else, while loops (can be exited by break), special statements, or printing statements (for debugging/testing).

Assignment statements can be:

• simple assignments, where the left hand is an identifier (i.e., variable)

• special time-derivative assignments, where the left hand specifies the change of the amount in the corresponding state variable (compartment) with respect to time e.g., d/dt(depot):

• special initial-condition assignments where the left hand specifies the compartment of the initial condition being specified, e.g. depot(0) = 0

• special model event changes including bioavailability (f(depot)=1), lag time (alag(depot)=0), modeled rate (rate(depot)=2) and modeled duration (dur(depot)=2). An example of these model features and the event specification for the modeled infusions the rxode2 data specification is found in rxode2 events vignette.

• special change point syntax, or model times. These model times are specified by mtime(var)=time

• special Jacobian-derivative assignments, where the left hand specifies the change in the compartment ode with respect to a variable. For example, if d/dt(y) = dy, then a Jacobian for this compartment can be specified as df(y)/dy(dy) = 1. There may be some advantage to obtaining the solution or specifying the Jacobian for very stiff ODE systems. However, for the few stiff systems we tried with LSODA, this actually slightly slowed down the solving.

Note that assignment can be done by =, <- or ~.

When assigning with the ~ operator, the simple assignments and time-derivative assignments will not be output. Note that with the rxode2 model functions assignment with ~ can also be overloaded with a residual distribution specification.

Special statements can be:

• Compartment declaration statements, which can change the default dosing compartment and the assumed compartment number(s) as well as add extra compartment names at the end (useful for multiple-endpoint nlmixr models); These are specified by cmt(compartmentName)

• Parameter declaration statements, which can make sure the input parameters are in a certain order instead of ordering the parameters by the order they are parsed. This is useful for keeping the parameter order the same when using 2 different ODE models. These are specified by param(par1, par2,...)

An example model is shown below:

   # simple assignment
   C2 <- centr/V2

   # time-derivative assignment
   d/dt(centr) <- F*KA*depot - CL*C2 - Q*C2 + Q*C3;

Expressions in assignment and if statements can be numeric or logical.

Numeric expressions can include the following numeric operators +, -, *, /, ^ and those mathematical functions defined in the C or the R math libraries (e.g., fabs, exp, log, sin, abs).

You may also access the R’s functions in the R math libraries, like lgammafn for the log gamma function.

The rxode2 syntax is case-sensitive, i.e., ABC is different than abc, Abc, ABc, etc.

Identifiers

Like R, Identifiers (variable names) may consist of one or more alphanumeric, underscore _ or period . characters, but the first character cannot be a digit or underscore _.

Identifiers in a model specification can refer to:

• State variables in the dynamic system (e.g., compartments in a pharmacokinetics model).

• Implied input variable, t (time), tlast (last time point), and podo (oral dose, in the undocumented case of absorption transit models).

• Special constants like pi or R’s predefined constants.

• Model parameters (e.g., ka rate of absorption, CL clearance, etc.)

• Others, as created by assignments as part of the model specification; these are referred as LHS (left-hand side) variable.

Currently, the rxode2 modeling language only recognizes system state variables and “parameters”, thus, any values that need to be passed from R to the ODE model (e.g., age) should be either passed in the params argument of the integrator function rxSolve() or be in the supplied event data-set.

There are certain variable names that are in the rxode2 event tables. To avoid confusion, the following event table-related items cannot be assigned, or used as a state but can be accessed in the rxode2 code:

• cmt

• dvid

• addl

• ss

• rate

• id

However the following variables are cannot be used in a model specification:

• evid

• ii

Sometimes rxode2 generates variables that are fed back to rxode2. Similarly, nlmixr2 generates some variables that are used in nlmixr estimation and simulation. These variables start with the either the rx or nlmixr prefixes. To avoid any problems, it is suggested to not use these variables starting with either the rx or nlmixr prefixes.

Logical Operators

Logical operators support the standard R operators ==, != >= <= > and <. Like R these can be in if() or while() statements, ifelse() expressions. Additionally they can be in a standard assignment. For instance, the following is valid:

cov1 = covm*(sexf == "female") + covm*(sexf != "female")

Notice that you can also use character expressions in comparisons. This convenience comes at a cost since character comparisons are slower than numeric expressions. Unlike R, as.numeric or as.integer for these logical statements is not only not needed, but will cause an syntax error if you try to use the function.

Supported functions

All the supported functions in rxode2 can be seen with the rxSupportedFuns().

A brief description of the built-in functions are in the following table:

Function	Description	Aliases
gamma(x)	The Gamma function	gammafn
lgamma(x)	Natural logarithm of absolute value of gamma function	digamma
digamma(x)	First derivative of lgamma
trigamma(x)	Second derivative of lgamma
tetragamma(x)	Third derivative of lgamma
pentagamma(x)	Fourth derivative of lgamma
psigamma(x, deriv)	n-th derivative of Psi, the digamma function, which is the derivative of lgammafn. In other words, digamma(x) is the same as psigamma(x,0), trigamma(x) == psigamma(x,1), etc.
cospi(x)	cos(pi*x)
sinpi(x)	sin(pi*x)
tanpi(x)	tan(pi*x)
beta(a, b)	Beta function
lbeta(a, b)	log Beta function
bessel_i(x, nu, expo)	Bessel function type I with index nu	expo==1 is unscaled expo==2 is scaled by exp(-x)
bessel_j(x, nu)	Bessel function type J with index nu
bessel_k(x, ku, expo)	Bessel function type K with index nu	expo==1 is unscaled expo==2 is scaled by exp(x)
bessel_y(x, nu)	Bessel function type Y with index nu
R_pow(x, y)	x^y
R_pow_di(x, I)	x^y	y is an integer
log1pmx	log(1+x) - x
log1pexp	log(1+exp(x))
expm1(x)	exp(x)-1
lgamma1p(x)	log(gamma(x+1))
sign(x)	Compute the signum function where sign(x) is 1, 0 -1
fsign(x, y)	abs(x)*sign(y)
fprec(x, digits)	x rounded to digits (after the decimal point, used by signif()
fround(x, digits)	Round, used by R’s round()
ftrunc(x)	Truncated towards zero
abs(x)	absolute value of x	fabs
sin(x)	sine of x
cos(x)	cos of x
tan(x)	tan of x
factorial(x)	factorial of x
lfactorial(x)	log(factorial(x))
log10(x)	log base 10
log2(x)	log base 2
pnorm(x)	Normal CDF of x	normcdf, phi
qnorm(x)	Normal pdf of x	norminv
probit(x, low=0, hi=1)	Probit (normal pdf) of x transforming into a range
probitInv(q, low=0, hi=1)	Inverse probit of x transforming into a range
acos(x)	Inverse cosine
asin(x)	Inverse sine
atan(x)	Inverse tangent
atan2(a, b)	Four quadrant inverse tangent
sinh(x)	Hyperbolic sine
cosh(x)	Hyperbolic cosine
tanh(x)	Hyperbolic tangent
floor(x)	Downward rounding
ceil(x)	Upward rounding
logit(x, low=0, hi=1)	Logit transformation of x transforming into a range
expit(x, low=0, hi=1)	expit transofmration in range	invLogit, logitInv
gammaq(a, z)	Normalized incomplete gamma from boost
gammaqInv(a, q)	Normalized incomplete gamma inverse from boost
ifelse(cond, trueValue, falseValue)	if else function
gammap(a, z)	Normalized lower incomplete gamma from boost
gammapInv(a, p)	Inverse of Normalized lower incomplete gamma from boost
gammapInva(x, p)	Inverse of Normalized lower incomplete gamma from boost
rxnorm(x)	Generate one deviate of from a normal distribution for each observation scale
rxnormV(x)	Generate one deviate from low discrepancy normal for each observation
rxcauchy	Generate one deviate from the cauchy distribution for each observation
rxchisq	Generate one deviate from the chisq distribution for each observation
rxexp	Generate one deviate from the exponential distribution for each observation
rxf	Generate one deviate from low discrepancy normal for each observation
rxgamma	Generate one deviate from the gamma distribution for each observation
rxbeta	Generate one deviate from the beta distribution for each observation
rxgeom	Generate one deviate from the geometric distribution for each observation
rxpois	Generate one deviate from the poission distribution for each observation
rxt	Generate one deviate from the t distribtuion for each observation
tad() or tad(x)	Time after dose (tad()) or time after dose for a compartment tad(cmt)
tafd() or tafd(x)	Time after first dose (tafd()) or time after first dose for a compartment tafd(cmt)
dosenum()	Dose Number
tlast() or tlast(cmt)	Time of Last dose; This takes into consideration any lag time, so if there is a dose at time 3 and a lag of 1, the time of last dose would be 4. tlast(cmt) calculates the time since last dose of a compartment
tfirst() or tfirst(cmt)	Time since first dose or time since first dose of a compartment
prod(…)	product of terms; This uses PreciseSums so the product will not have as much floating point errors (though it will take longer)
sum(…)	sum of terms; This uses PreciseSums so the product will not have as much floating point errors (though it will take longer)
max(…)	maximum of a group of numbers
min(…)	Min of a group of numbers
lag(parameter, number=1)	Get the lag of an input parameter; You can specify a number of lagged observations
lead(parameter, number=2)	Get the lead of an input parameter; You can specify a number of lead observation
diff(par, number=1)	Get the difference between the current parameter and the last parameter; Can change the parameter number
first(par)	Get the first value of an input parameter
last(par)	Get the last value of an input parameter
transit()	The transit compartment psuedo function
is.na()	Determine if a value is NA
is.nan()	Determine if a value is NaN
is.infinite()	Check to see if the value is infinite
rinorm(x)	Generate one deviate of from a normal distribution for each individual
rinormV(x)	Generate one deviate from low discrepancy normal for each individual
ricauchy	Generate one deviate from the cauchy distribution for each individual
richisq	Generate one deviate from the chisq distribution for each individual
riexp	Generate one deviate from the exponential distribution for each individual
rif	Generate one deviate from low discrepancy normal for each individual
rigamma	Generate one deviate from the gamma distribution for each individual
ribeta	Generate one deviate from the beta distribution for each individual
rigeom	Generate one deviate from the geometric distribution for each individual
ropois	Generate one deviate from the poission distribution for each individual
rit	Generate one deviate from the t distribtuion for each individual
simeps	Simulate EPS from possibly truncated sigma matrix. Will take sigma matrix from the current study. Simulated at the very last moment.
simeta	Simulate ETA from possibly truncated omega matrix. Will take the omega matrix from the current study. Simulated at the initilization of the ODE system or the intialization of lhs

Note that lag(cmt) = is equivalent to alag(cmt) = and not the same as = lag(wt)

Reserved keywords

There are a few reserved keywords in a rxode2 model. They are in the following table:

Reserved Name	Meaning	Alias
time	solver time	t
podo	In Transit compartment models, last dose amount
tlast	Time of Last dose
M_E	Exp(1)
M_LOG2E	log2(e)
M_LOG10E	log10(e)
M_LN2	log(2)
M_LN10	log(10)
M_PI	pi
M_PI_2	pi/2
M_PI_4	pi/4
M_1_PI	1/pi
M_2_PI	2/pi
M_2_SQRTPI	2/sqrt(pi)
M_SQRT2	sqrt(2)
M_SQRT1_2	1/sqrt(2)
M_SQRT_3	sqrt(3)
M_SQRT_32	sqrt(32)
M_LOG10_2	Log10(2)
M_2PI	2*pi
M_SQRT_PI	sqrt(pi)
M_1_SQRT_2PI	1/(sqrt(2*pi))
M_LN_SQRT_PI	log(sqrt(pi))
M_LN_SQRT_2PI	log(sqrt(2*pi))
M_LN_SQRT_PId2	log(sqrt(pi/2))
pi	pi
NA	R’s NA value
NaN	Not a Number Value
Inf	Infinite Value
newind	1: First record of individual; 2: Subsequent record of individual	NEWIND
rxFlag	Flag for what part of the rxode2 model is being run; 1: ddt; 2: jac; 3: ini; 4: F; 5: lag; 6: rate; 7: dur; 8: mtime; 9: matrix exponential; 10: inductive linearization; 11: lhs

Note that rxFlag will always output 11 or calc_lhs since that is where the final variables are calculated, though you can tweak or test certain parts of rxode2 by using this flag.

Residual functions when using rxode2 functions

In addition to ~ hiding output for certain types of output, it also is used to specify a residual output or endpoint when the input is an rxode2 model function (that includes the residual in the model({}) block).

These specifications are of the form:

var ~ add(add.sd)

Indicating the variable var is the variable that represents the individual central tendencies of the model and it also represents the compartment specification in the data-set.

You can also change the compartment name using the | syntax, that is:

var ~ add(add.sd) | cmt

In the above case var represents the central tendency and cmt represents the compartment or dvid specification.

Transformations

For normal and related distributions, you can apply the transformation on both sides by using some keywords/functions to apply these transformations.

Transformation	rxode2/nlmixr2 code
Box-Cox	+boxCox(lambda)
Yeo-Johnson	+yeoJohnson(lambda)
logit-normal	+logitNorm(logit.sd, low, hi)
probit-normal	+probitNorm(probid.sd, low, hi)
log-normal	+lnorm(lnorm.sd)

By default for the likelihood for all of these transformations is calculated on the untransformed scale.

For bounded variables like logit-normal or probit-normal the low and high values are defaulted to 0 and 1 if missing.

For models where you wish to have a proportional model on one of these transformation you can replace the standard deviation with NA

To allow for more transformations, lnorm(), probitNorm() and logitNorm() can be combined the variance stabilizing yeoJohnson() transformation.

Normal and t-related distributions

For the normal and t-related distributions, we wanted to keep the ability to use skewed distributions additive and proportional in the t/cauchy-space, so these distributions are specified differently in comparison to the other supported distributions within nlmixr2:

Distribution	How to Add	Example
Normal (log-likelihood)	+dnorm()	cc ~ add(add.sd) + dnorm()
T-distribution	+dt(df)	cc ~a dd(add.sd) + dt(df)
Cauchy (t with df=1)	+dcauchy()	cc ~ add(add.sd) + dcauchy()

Note that with the normal and t-related distributions nlmixr2 will calculate cwres and npde under the normal assumption to help assess the goodness of the fit of the model.

Also note that the +dnorm() is mostly for testing purposes and will slow down the estimation procedure in nlmixr2. We suggest not adding it (except for explicit testing). When there are multiple endpoint models that mix non-normal and normal distributions, the whole problem is shifted to a log-likelihood method for estimation in nlmixr2.

Notes on additive + proportional models

There are two different ways to specify additive and proportional models, which we will call combined1 and combined2, the same way that Monolix calls the two distributions (to avoid between software differences in naming).

The first, combined1, assumes that the additive and proportional differences are on the standard deviation scale, or:

y=f+(a+b* f^c)*err

The second, combined2, assumes that the additive and proportional differences are combined on a variance scale:

y=f+[sqrt(a^2+b^2 *f^(2c))]*err

The default in nlmixr2/rxode2 if not otherwise specified is combined2 since it mirrors how adding 2 normal distributions in statistics will add their variances (not the standard deviations). However, the combined1 can describe the data possibly even better than combined2 so both are possible options in rxode2/nlmixr2.

Distributions of known likelihoods

For residuals that are not related to normal, t-distribution or cauchy, often the residual specification is of the form:

cmt ~ dbeta(alpha, beta)

Where the compartment specification is on the left handed side of the specification.

For generalized likelihood you can specify:

ll(cmt) ~ llik specification

Ordinal likelihoods

Finally, ordinal likelihoods/simulations can be specified in 2 ways. The first is:

err ~ c(p0, p1, p2)

Here err represents the compartment and p0 is the probability of being in a specific category:

Category	Probability
1	p0
2	p1
3	p2
4	1-p0-p1-p2

It is up to the model to ensure that the sum of the p values are less than 1. Additionally you can write an arbitrary number of categories in the ordinal model described above.

It seems a little off that p0 is the probability for category 1 and sometimes scores are in non-whole numbers. This can be modeled as follows:

err ~ c(p0=0, p1=1, p2=2, 3)

Here the numeric categories are specified explicitly, and the probabilities remain the same:

Category	Probability
0	p0
1	p1
2	p2
3	1-p0-p1-p2

General table of supported residual distributions

In general all the that are supported are in the following table (available in rxode2::rxResidualError)

Error model	Functional Form	Transformation	code	addProp	lhs
constant		None	var ~ add(add.sd)		response variable
proportional		None	var ~ prop(prop.sd)		response variable
power		None	var ~ pow(pow.sd, exponent)		response variable
additive+proportional	combined1	None	var ~ add(add.sd) + prop(prop.sd) + combined1()	addProp=1	response variable
additive+proportional	combined2	None	var ~ add(add.sd) + prop(prop.sd) + combined2()	addProp=2	response variable
additive+power	combined1	None	var ~ add(add.sd) + pow(pow.sd, exponent) + combined1()	addProp=1	response variable
additive+power	combined2	None	var ~ add(add.sd) + pow(pow.sd, exponent) + combined2()	addProp=2	response variable
constant		log	var ~ lnorm(add.sd)		response variable
proportional		log	var ~ lnorm(NA) + prop(prop.sd)		response variable
power		log	var ~ lnorm(NA) + pow(pow.sd, exponent)		response variable
additive+proportional	combined1	log	var ~ lnorm(add.sd) + prop(prop.sd) + combined1()	addProp=1	response variable
additive+proportional	combined2	log	var ~ lnorm(add.sd) + prop(prop.sd) + combined2()	addProp=2	response variable
additive+power	combined1	log	var ~ lnorm(add.sd) + pow(pow.sd, exponent) + combined1()	addProp=1	response variable
additive+power	combined2	log	var ~ lnorm(add.sd) + pow(pow.sd, exponent) + combined2()	addProp=2	response variable
constant		boxCox	var ~ boxCox(lambda) + add(add.sd)		response variable
proportional		boxCox	var ~ boxCox(lambda) + prop(prop.sd)		response variable
power		boxCox	var ~ boxCox(lambda) + pow(pow.sd, exponent)		response variable
additive+proportional	combined1	boxCox	var ~ boxCox(lambda) + add(add.sd) + prop(prop.sd) + combined1()	addProp=1	response variable
additive+proportional	combined2	boxCox	var ~ boxCox(lambda) + add(add.sd) + prop(prop.sd) + combined2()	addProp=2	response variable
additive+power	combined1	boxCox	var ~ boxCox(lambda) + add(add.sd) + pow(pop.sd, exponent) + combined1()	addProp=1	response variable
additive+power	combined2	boxCox	var ~ boxCox(lambda) + add(add.sd) + pow(pop.sd, exponent) + combined2()	addProp=2	response variable
constant		yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd)		response variable
proportional		yeoJohnson	var ~ yeoJohnson(lambda) + prop(prop.sd)		response variable
power		yeoJohnson	var ~ yeoJohnson(lambda) + pow(pow.sd, exponent)		response variable
additive+proportional	combined1	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + prop(prop.sd) + combined1()	addProp=1	response variable
additive+proportional	combined2	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + prop(prop.sd) + combined2()	addProp=2	response variable
additive+power	combined1	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + pow(pop.sd, exponent) + combined1()	addProp=1	response variable
additive+power	combined2	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + pow(pop.sd, exponent) + combined2()	addProp=2	response variable
constant		logit	var ~ logitNorm(logit.sd)		response variable
proportional		logit	var ~ logitNorm(NA) + prop(prop.sd)		response variable
power		logit	var ~ logitNorm(NA) + pow(pow.sd, exponent)		response variable
additive+proportional	combined1	logit	var ~ logitNorm(logit.sd) + prop(prop.sd)	addProp=1	response variable
additive+proportional	combined2	logit	var ~ logitNorm(logit.sd) + prop(prop.sd)	addProp=2	response variable
additive+power	combined1	logit	var ~ logitNorm(logit.sd) + pow(pow.sd, exponent)	addProp=1	response variable
additive+power	combined2	logit	var ~ logitNorm(logit.sd) + pow(pow.sd, exponent)	addProp=2	response variable
additive		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd)		response variable
proportional		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(NA) + prop(prop.sd)		response variable
power		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(NA) + pow(pow.sd, exponent)		response variable
additive+proportional	combined1	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + prop(prop.sd)	addProp=1	response variable
additive+proportional	combined2	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + prop(prop.sd)	addProp=2	response variable
additive+power	combined1	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + pow(pow.sd, exponent)	addProp=1	response variable
additive+power	combined2	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + pow(pow.sd, exponent)	addProp=2	response variable
constant		logit	var ~ probitNorm(probit.sd)		response variable
proportional		probit	var ~ probitNorm(NA) + prop(prop.sd)		response variable
power		probit	var ~ probitNorm(NA) + pow(pow.sd, exponent)		response variable
additive+proportional	combined1	probit	var ~ probitNorm(probit.sd) + prop(prop.sd) + combined1()	addProp=1	response variable
additive+proportional	combined2	probit	var ~ probitNorm(probit.sd) + prop(prop.sd) + combined2()	addProp=2	response variable
additive+power	combined1	probit	var ~ probitNorm(probit.sd) + pow(pow.sd, exponent) + combined1()	addProp=1	response variable
additive+power	combined2	probit	var ~ probitNorm(probit.sd) + pow(pow.sd, exponent) + combined2()	addProp=2	response variable
additive		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd)		response variable
proportional		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(NA) + prop(prop.sd)		response variable
power		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(NA) + pow(pow.sd, exponent)		response variable
additive+proportional	combined1	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + prop(prop.sd) + combined1()	addProp=1	response variable
additive+proportional	combined2	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + prop(prop.sd) + combined2()	addProp=2	response variable
additive+power	combined1	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + pow(pow.sd, exponent) + combined1()	addProp=1	response variable
additive+power	combined2	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + pow(pow.sd, exponent) + combined2()	addProp=2	response variable
constant+t		None	var ~ add(add.sd) + dt(df)		response variable
proportional+t		None	var ~ prop(prop.sd) + dt(df)		response variable
power+t		None	var ~ pow(pow.sd, exponent) + dt(df)		response variable
additive+proportional+t	combined1	None	var ~ add(add.sd) + prop(prop.sd) + dt(df) + combined1()	addProp=1	response variable
additive+proportional+t	combined2	None	var ~ add(add.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	None	var ~ add(add.sd) + pow(pow.sd, exponent) + dt(df) +combined1()	addProp=1	response variable
additive+power+t	combined2	None	var ~ add(add.sd) + pow(pow.sd, exponent) + dt(df) +combined2()	addProp=2	response variable
constant+t		log	var ~ lnorm(add.sd) + dt(df)		response variable
proportional+t		log	var ~ lnorm(NA) + prop(prop.sd) + dt(df)		response variable
power+t		log	var ~ lnorm(NA) + pow(pow.sd, exponent) + dt(df)		response variable
additive+proportional+t	combined1	log	var ~ lnorm(add.sd) + prop(prop.sd) + dt(df) +combined1()	addProp=1	response variable
additive+proportional+t	combined2	log	var ~ lnorm(add.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	log	var ~ lnorm(add.sd) + pow(pow.sd, exponent) + dt(df) + combined1()	addProp=1	response variable
additive+power+t	combined2	log	var ~ lnorm(add.sd) + pow(pow.sd, exponent) + dt(df) + combined2()	addProp=2	response variable
constant+t		boxCox	var ~ boxCox(lambda) + add(add.sd)+dt(df)		response variable
proportional+t		boxCox	var ~ boxCox(lambda) + prop(prop.sd)+dt(df)		response variable
power+t		boxCox	var ~ boxCox(lambda) + pow(pow.sd, exponent)+dt(df)		response variable
additive+proportional+t	combined1	boxCox	var ~ boxCox(lambda) + add(add.sd) + prop(prop.sd) + dt(df) + combined1()	addProp=1	response variable
additive+proportional+t	combined2	boxCox	var ~ boxCox(lambda) + add(add.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	boxCox	var ~ boxCox(lambda) + add(add.sd) + pow(pop.sd, exponent) + dt(df) + combined1()	addProp=1	response variable
additive+power+t	combined2	boxCox	var ~ boxCox(lambda) + add(add.sd) + pow(pop.sd, exponent) + dt(df) + combined2()	addProp=2	response variable
constant+t		yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + dt(df)		response variable
proportional+t		yeoJohnson	var ~ yeoJohnson(lambda) + prop(prop.sd) + dt(df)		response variable
power+t		yeoJohnson	var ~ yeoJohnson(lambda) + pow(pow.sd, exponent) + dt(df)		response variable
additive+proportional+t	combined1	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + prop(prop.sd) + dt(df) + combined1()	addProp=1	response variable
additive+proportional+t	combined2	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + pow(pop.sd, exponent) + dt(df) + combined1()	addProp=1	response variable
additive+power+t	combined2	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + pow(pop.sd, exponent) + dt(df) + combined2()	addProp=2	response variable
constant+t		logit	var ~ logitNorm(logit.sd)+dt(df)		response variable
proportional+t		logit	var ~ logitNorm(NA) + prop(prop.sd)+dt(df)		response variable
power+t		logit	var ~ logitNorm(NA) + pow(pow.sd, exponent) + dt(df)		response variable
additive+proportional+t	combined1	logit	var ~ logitNorm(logit.sd) + prop(prop.sd) + dt(df) + combined1()	addProp=1	response variable
additive+proportional+t	combined2	logit	var ~ logitNorm(logit.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	logit	var ~ logitNorm(logit.sd) + pow(pow.sd, exponent) + dt(df) + combined1()	addProp=1	response variable
additive+power+t	combined2	logit	var ~ logitNorm(logit.sd) + pow(pow.sd, exponent) + dt(df) + combined2()	addProp=2	response variable
additive+t		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + dt(df)		response variable
proportional+t		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(NA) + prop(prop.sd) + dt(df)		response variable
power+t		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(NA) + pow(pow.sd, exponent) + dt(df)		response variable
additive+proportional+t	combined1	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + prop(prop.sd) + dt(df) + combined1()	addProp=1	response variable
additive+proportional+t	combined2	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + pow(pow.sd, exponent) + dt(df) + combined1()	addProp=1	response variable
additive+power+t	combined2	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + pow(pow.sd, exponent) + dt(df) + combined2()	addProp=2	response variable
constant+t		logit	var ~ probitNorm(probit.sd) + dt(df)		response variable
proportional+t		probit	var ~ probitNorm(NA) + prop(prop.sd) + dt(df)		response variable
power+t		probit	var ~ probitNorm(NA) + pow(pow.sd, exponent) + dt(df)		response variable
additive+proportional+t	combined1	probit	var ~ probitNorm(probit.sd) + prop(prop.sd) + dt(df) + combined1()	addProp=1	response variable
additive+proportional+t	combined2	probit	var ~ probitNorm(probit.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	probit	var ~ probitNorm(probit.sd) + pow(pow.sd, exponent) + dt(df) + combined1()	addProp=1	response variable
additive+power+t	combined2	probit	var ~ probitNorm(probit.sd) + pow(pow.sd, exponent) + dt(df) + combined2()	addProp=2	response variable
additive+t		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + dt(df)		response variable
proportional+t		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(NA) + prop(prop.sd) + dt(df)		response variable
power+t		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(NA) + pow(pow.sd, exponent) + dt(df)		response variable
additive+proportional+t	combined1	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + prop(prop.sd) + dt(df) + combined1()	addProp=1	response variable
additive+proportional+t	combined2	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + prop(prop.sd) + dt(df) + combined2()	addProp=2	response variable
additive+power+t	combined1	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + pow(pow.sd, exponent) + dt(df) + combined1()	addProp=1	response variable
additive+power+t	combined2	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + pow(pow.sd, exponent) + dt(df) +combined2()	addProp=2	response variable
constant+cauchy		None	var ~ add(add.sd) + dcauchy()		response variable
proportional+cauchy		None	var ~ prop(prop.sd) + dcauchy()		response variable
power+cauchy		None	var ~ pow(pow.sd, exponent) + dcauchy()		response variable
additive+proportional+cauchy	combined1	None	var ~ add(add.sd) + prop(prop.sd) + dcauchy() + combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	None	var ~ add(add.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	None	var ~ add(add.sd) + pow(pow.sd, exponent) + dcauchy() +combined1()	addProp=1	response variable
additive+power+cauchy	combined2	None	var ~ add(add.sd) + pow(pow.sd, exponent) + dcauchy() +combined2()	addProp=2	response variable
constant+cauchy		log	var ~ lnorm(add.sd) + dcauchy()		response variable
proportional+cauchy		log	var ~ lnorm(NA) + prop(prop.sd) + dcauchy()		response variable
power+cauchy		log	var ~ lnorm(NA) + pow(pow.sd, exponent) + dcauchy()		response variable
additive+proportional+cauchy	combined1	log	var ~ lnorm(add.sd) + prop(prop.sd) + dcauchy() +combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	log	var ~ lnorm(add.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	log	var ~ lnorm(add.sd) + pow(pow.sd, exponent) + dcauchy() + combined1()	addProp=1	response variable
additive+power+cauchy	combined2	log	var ~ lnorm(add.sd) + pow(pow.sd, exponent) + dcauchy() + combined2()	addProp=2	response variable
constant+cauchy		boxCox	var ~ boxCox(lambda) + add(add.sd)+dcauchy()		response variable
proportional+cauchy		boxCox	var ~ boxCox(lambda) + prop(prop.sd)+dcauchy()		response variable
power+cauchy		boxCox	var ~ boxCox(lambda) + pow(pow.sd, exponent)+dcauchy()		response variable
additive+proportional+cauchy	combined1	boxCox	var ~ boxCox(lambda) + add(add.sd) + prop(prop.sd) + dcauchy() + combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	boxCox	var ~ boxCox(lambda) + add(add.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	boxCox	var ~ boxCox(lambda) + add(add.sd) + pow(pop.sd, exponent) + dcauchy() + combined1()	addProp=1	response variable
additive+power+cauchy	combined2	boxCox	var ~ boxCox(lambda) + add(add.sd) + pow(pop.sd, exponent) + dcauchy() + combined2()	addProp=2	response variable
constant+cauchy		yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + dcauchy()		response variable
proportional+cauchy		yeoJohnson	var ~ yeoJohnson(lambda) + prop(prop.sd) + dcauchy()		response variable
power+cauchy		yeoJohnson	var ~ yeoJohnson(lambda) + pow(pow.sd, exponent) + dcauchy()		response variable
additive+proportional+cauchy	combined1	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + prop(prop.sd) + dcauchy() + combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + pow(pop.sd, exponent) + dcauchy() + combined1()	addProp=1	response variable
additive+power+cauchy	combined2	yeoJohnson	var ~ yeoJohnson(lambda) + add(add.sd) + pow(pop.sd, exponent) + dcauchy() + combined2()	addProp=2	response variable
constant+cauchy		logit	var ~ logitNorm(logit.sd)+dcauchy()		response variable
proportional+cauchy		logit	var ~ logitNorm(NA) + prop(prop.sd)+dcauchy()		response variable
power+cauchy		logit	var ~ logitNorm(NA) + pow(pow.sd, exponent) + dcauchy()		response variable
additive+proportional+cauchy	combined1	logit	var ~ logitNorm(logit.sd) + prop(prop.sd) + dcauchy() + combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	logit	var ~ logitNorm(logit.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	logit	var ~ logitNorm(logit.sd) + pow(pow.sd, exponent) + dcauchy() + combined1()	addProp=1	response variable
additive+power+cauchy	combined2	logit	var ~ logitNorm(logit.sd) + pow(pow.sd, exponent) + dcauchy() + combined2()	addProp=2	response variable
additive+cauchy		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + dcauchy()		response variable
proportional+cauchy		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(NA) + prop(prop.sd) + dcauchy()		response variable
power+cauchy		yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(NA) + pow(pow.sd, exponent) + dcauchy()		response variable
additive+proportional+cauchy	combined1	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + prop(prop.sd) + dcauchy() + combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + pow(pow.sd, exponent) + dcauchy() + combined1()	addProp=1	response variable
additive+power+cauchy	combined2	yeoJohnson(logit())	var ~ yeoJohnson(lambda) + logitNorm(logit.sd) + pow(pow.sd, exponent) + dcauchy() + combined2()	addProp=2	response variable
constant+cauchy		logit	var ~ probitNorm(probit.sd) + dcauchy()		response variable
proportional+cauchy		probit	var ~ probitNorm(NA) + prop(prop.sd) + dcauchy()		response variable
power+cauchy		probit	var ~ probitNorm(NA) + pow(pow.sd, exponent) + dcauchy()		response variable
additive+proportional+cauchy	combined1	probit	var ~ probitNorm(probit.sd) + prop(prop.sd) + dcauchy() + combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	probit	var ~ probitNorm(probit.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	probit	var ~ probitNorm(probit.sd) + pow(pow.sd, exponent) + dcauchy() + combined1()	addProp=1	response variable
additive+power+cauchy	combined2	probit	var ~ probitNorm(probit.sd) + pow(pow.sd, exponent) + dcauchy() + combined2()	addProp=2	response variable
additive+cauchy		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + dcauchy()		response variable
proportional+cauchy		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(NA) + prop(prop.sd) + dcauchy()		response variable
power+cauchy		yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(NA) + pow(pow.sd, exponent) + dcauchy()		response variable
additive+proportional+cauchy	combined1	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + prop(prop.sd) + dcauchy() + combined1()	addProp=1	response variable
additive+proportional+cauchy	combined2	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + prop(prop.sd) + dcauchy() + combined2()	addProp=2	response variable
additive+power+cauchy	combined1	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + pow(pow.sd, exponent) + dcauchy() + combined1()	addProp=1	response variable
additive+power+cauchy	combined2	yeoJohnson(probit())	var ~ yeoJohnson(lambda) + probitNorm(probit.sd) + pow(pow.sd, exponent) + dcauchy() +combined2()	addProp=2	response variable
poission		none	cmt ~ dpois(lamba)		compartment specification
binomial		none	cmt ~ dbinom(n, p)		compartment specification
beta		none	cmt ~ dbeta(alpha, beta)		compartment specification
chisq		none	cmt ~ dchisq(nu)		compartment specification
exponential		none	cmt ~ dexp(r)		compartment specification
uniform		none	cmt ~ dunif(a, b)		compartment specification
weibull		none	cmt ~ dweibull(a, b)		compartment specification
gamma		none	cmt ~ dgamma(a, b)		compartment specification
geometric		none	cmt ~ dgeom(a)		compartment specification
negative binomial form #1		none	cmt ~ dnbinom(n, p)		compartment specification
negative binomial form #2		none	cmt ~ dnbinomMu(size, mu)		compartment specification
ordinal probability		none	cmt ~ c(p0=0, p1=1, p2=2, 3)		compartment specification
log-likelihood		none	ll(cmt) ~ log likelihood expression		likelihood + compartment expression

Creating rxode2 models

NA

References

Chamber, J. M. and Temple Lang, D. (2001) Object Oriented Programming in R. R News, Vol. 1, No. 3, September 2001. https://cran.r-project.org/doc/Rnews/Rnews_2001-3.pdf.

Hindmarsh, A. C. ODEPACK, A Systematized Collection of ODE Solvers. Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pp. 55-64.

Petzold, L. R. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148.

Hairer, E., Norsett, S. P., and Wanner, G. Solving ordinary differential equations I, nonstiff problems. 2nd edition, Springer Series in Computational Mathematics, Springer-Verlag (1993).

Plevyak, J. dparser, https://dparser.sourceforge.net/. Web. 12 Oct. 2015.

See also

eventTable(), et(), add.sampling(), add.dosing()

Author

Melissa Hallow, Wenping Wang and Matthew Fidler

Examples

# \donttest{

mod <- function() {
  ini({
    KA   <- .291
    CL   <- 18.6
    V2   <- 40.2
    Q    <- 10.5
    V3   <- 297.0
    Kin  <- 1.0
    Kout <- 1.0
    EC50 <- 200.0
  })
  model({
    # A 4-compartment model, 3 PK and a PD (effect) compartment
    # (notice state variable names 'depot', 'centr', 'peri', 'eff')
    C2 <- centr/V2
    C3 <- peri/V3
    d/dt(depot) <- -KA*depot;
    d/dt(centr) <- KA*depot - CL*C2 - Q*C2 + Q*C3;
    d/dt(peri)  <-                    Q*C2 - Q*C3;
    d/dt(eff)   <- Kin - Kout*(1-C2/(EC50+C2))*eff;
    eff(0)      <- 1
  })
}

m1 <- rxode2(mod)
#>  
#>  
print(m1)
#>  ── rxode2-based free-form 4-cmt ODE model ────────────────────────────────────── 
#>  ── Initalization: ──  
#> Fixed Effects ($theta): 
#>      KA      CL      V2       Q      V3     Kin    Kout    EC50 
#>   0.291  18.600  40.200  10.500 297.000   1.000   1.000 200.000 
#> 
#> States ($state or $stateDf): 
#>   Compartment Number Compartment Name
#> 1                  1            depot
#> 2                  2            centr
#> 3                  3             peri
#> 4                  4              eff
#>  ── Model (Normalized Syntax): ── 
#> function() {
#>     ini({
#>         KA <- 0.291
#>         CL <- 18.6
#>         V2 <- 40.2
#>         Q <- 10.5
#>         V3 <- 297
#>         Kin <- 1
#>         Kout <- 1
#>         EC50 <- 200
#>     })
#>     model({
#>         C2 <- centr/V2
#>         C3 <- peri/V3
#>         d/dt(depot) <- -KA * depot
#>         d/dt(centr) <- KA * depot - CL * C2 - Q * C2 + Q * C3
#>         d/dt(peri) <- Q * C2 - Q * C3
#>         d/dt(eff) <- Kin - Kout * (1 - C2/(EC50 + C2)) * eff
#>         eff(0) <- 1
#>     })
#> }

# Step 2 - Create the model input as an EventTable,
# including dosing and observation (sampling) events

# QD (once daily) dosing for 5 days.

qd <- et(amountUnits = "ug", timeUnits = "hours") %>%
  et(amt = 10000, addl = 4, ii = 24)

# Sample the system hourly during the first day, every 8 hours
# then after
qd <- qd %>% et(0:24) %>%
  et(from = 24 + 8, to = 5 * 24, by = 8)

# Step 3 - solve the system

qd.cp <- rxSolve(m1, qd)
#>  
#>  
#> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’

head(qd.cp)
#>    time       C2        C3     depot    centr      peri      eff
#> 1 0 [h]  0.00000 0.0000000 10000.000    0.000    0.0000 1.000000
#> 2 1 [h] 43.99334 0.9113641  7475.157 1768.532  270.6751 1.083968
#> 3 2 [h] 54.50866 2.6510696  5587.797 2191.248  787.3677 1.179529
#> 4 3 [h] 51.65163 4.4243597  4176.966 2076.396 1314.0348 1.227523
#> 5 4 [h] 44.37513 5.9432612  3122.347 1783.880 1765.1486 1.233503
#> 6 5 [h] 36.46382 7.1389804  2334.004 1465.845 2120.2772 1.214084

# }