## Introduction

This briefly describes the syntax used to define models that
`rxode2`

will translate into R-callable compiled code. It
also describes the communication of variables between `R`

and
the `rxode2`

modeling specification.

## Creating rxode2 models

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

- Inside a
`rxode2({})`

block statements:

```
library(rxode2)
#> rxode2 2.1.3.9000 using 2 threads (see ?getRxThreads)
#> no cache: create with `rxCreateCache()`
```

```
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$simulationModel
#> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
#> rxode2 2.1.3.9000 model named rx_3f85bebc6957ed055084d54955f3965f model (✔ ready).
#> x$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$simulationIniModel
#> using C compiler: ‘gcc (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0’
#> rxode2 2.1.3.9000 model named rx_f9f57facf9bbd48c86d945f784b14d3a model (✔ ready).
#> x$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.

## Syntax

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:

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:

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.

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`

:

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:

### 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`

)

## Note

The ODE specification mini-language is parsed with the help of the open source tool , Plevyak (2015).

## Example

```
f <- function() {
ini({
})
model({
# An rxode2 model specification (this line is a comment).
if(comed==0) { # concomitant medication (con-med)?
F <- 1.0 # full bioavailability w.o. con-med
} else {
F <- 0.80 # 20% reduced bioavailability
}
C2 <- centr/V2 # concentration in the central compartment
C3 <- peri/V3 # concentration in the peripheral compartment
# ODE describing the PK and PD
d/dt(depot) <- -KA*depot
d/dt(centr) <- F*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
})
}
```

## Interface and data handling between R and the generated C code

Users specify which variables are the dynamic system’s state
variables via the `d/dt(identifier)`

operator as part of the
model specification, and which are model parameters via the
`params=`

argument in `rxode2`

`solve()`

method:

```
m1 <- rxode2(model = ode, modName = "m1")
# model parameters -- a named vector is required
theta <-
c(KA=0.29, CL=18.6, V2=40.2, Q=10.5, V3=297, Kin=1, Kout=1, EC50=200)
# state variables and their amounts at time 0 (the use of names is
# encouraged, but not required)
inits <- c(depot=0, centr=0, peri=0, eff=1)
# qd1 is an eventTable specification with a set of dosing and sampling
# records (code not shown here)
solve(theta, event = qd1, inits = inits)
```

The values of these variables at pre-specified time points are saved
during model fitting/integration and returned as part of the fitted
values (see the function `et()`

, to define a set of time
points when to capture the values of these variables) and returned as
part of the modeling output.

The ODE specification mini-language is parsed with the help of the
open source tool *DParser*, Plevyak (2015).