Date: 2020-01-06

# RxODE ODE solving syntax

#### 2020-01-06

## Introduction

This briefly describes the syntax used to define models that `RxODE` will translate into R-callable compiled code. It also describes the communication of variables between `R` and the `RxODE` modeling specification.

## Example

       # An RxODE 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;

## Syntax

An `RxODE` 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` 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 (`lag(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 RxODE data specification is found in RxODE events vignette.
- 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)`
- 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 `<-`.

Additionally, assignment can be done with the `~` operator, which causes RxODE to use the variable/expression while solving but suppress output to either the matrix or data-frame returned in R. The suppression works with simple assignments and time-derivative assignments. All other variable/assignments do not produce output, but are used for the solving.

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, however, no character nor integer expressions are currently supported.

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`). Notice that the modulo operator `%` is currently unsupported.

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

There are certain variable names that are in the `RxODE` event tables. To avoid confusion, the following event table-related items cannot be used in a RxODE model specification:

- `rate`
- `amt`
- `ss`
- `addl`
- `evid`
- `ii`
- `dvid`

The `RxODE` 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 `RxODE` 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 `solve()` or be in the supplied event data-set.

Sometimes RxODE generates variables that are fed back to RxODE. Similarly, nlmixr 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.

## 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 `RxODE` `solve()` method:

    m1 <- RxODE(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 `eventTable`, in particular its member function `add.sampling` function 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).

## Bugs and/or deficiencies

- The modulo operator `%` is currently unsupported.
- `else if` is not currently supported
- Conditional statements inside a single statement like `ifelse` are not currently supported.
- Logical expressions within a statement are not supported, that is `(cmt==2)*1+(cmt==3)*4` is not supported