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

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
)

## 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 RxODE Syntax below. 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. 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 RxODE DLL for the model is created in the current directory named rx_????_platform, for example rx_129f8f97fb94a87ca49ca8dafe691e1e_i386.dll A file name or connection object where the ODE-based model specification resides. Only one of model or filename may be specified. 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. is a boolean indicating if the executable should be compiled with verbose debugging information turned on. boolean indicating if RxODE will calculate the Jacobain according to the specified ODEs. boolean indicating if RxODE will calculate the sensitivities according to the specified ODEs. boolean indicating if RxODE will remove all LHS variables when calculating sensitivities. Package name for pre-compiled binaries. ignored arguments. The method to calculate the linCmt() solutions Calculate inductive linearization matrices and compile with inductive linearization support. When TRUE be verbose with the linear compartmental model

## Value

An object (environment) of class RxODE (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.modelVarsa 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(RxODEobject, ...),
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), RxODE 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), RxODE 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).e

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 RxODE 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 RxODE
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 RxODE 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 a character string or in a text file, see Section RxODE Syntax below for coding details. An internal RxODE compilation manager object translates the ODE system into C, compiles it, and dynamically loads the object code into the current R session. The call to RxODE produces an object of class RxODE which consists of a list-like structure (environment) with various member functions (see Section Value below).

For evaluating RxODE models, two types of inputs may be provided: a required set of time points for querying the state of the ODE system and an optional set of doses (input amounts). These inputs are combined into a single event table object created with the function eventTable() or et().

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/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 RxODE data specification is found in RxODE 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.

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, 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).

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

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 rxSolve() or be in the supplied event data-set.

There are certain variable names that are in the RxODE 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 RxODE code:

• cmt

• dvid

• addl

• ss

• rate

• id

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

• evid

• ii

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.

### 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.

## 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, http://dparser.sourceforge.net. Web. 12 Oct. 2015.

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

## Author

Melissa Hallow, Wenping Wang and Matthew Fidler

## Examples

# \donttest{
# Step 1 - Create a model specification
ode <- "
# 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;
"

m1 <- RxODE(model = ode)
#>
print(m1)
#> $state: depot, centr, peri, eff #>$params: V2, V3, KA, CL, Q, Kin, Kout, EC50
#> $lhs: C2, C3 # Step 2 - Create the model input as an EventTable, # including dosing and observation (sampling) events # QD (once daily) dosing for 5 days. qd <- eventTable(amount.units = "ug", time.units = "hours") qd$add.dosing(dose = 10000, nbr.doses = 5, dosing.interval = 24)

# Sample the system hourly during the first day, every 8 hours
# then after

qd$add.sampling(0:24) qd$add.sampling(seq(from = 24 + 8, to = 5 * 24, by = 8))

# Step 3 - set starting parameter estimates and initial
# values of the state

theta <-
c(
KA = .291, CL = 18.6,
V2 = 40.2, Q = 10.5, V3 = 297.0,
Kin = 1.0, Kout = 1.0, EC50 = 200.0
)

# init state variable
inits <- c(0, 0, 0, 1)
# Step 4 - Fit the model to the data

qd.cp <- m1$solve(theta, events = qd, inits) #> Warning: Assumed order of inputs: depot, centr, peri, eff head(qd.cp) #> time C2 C3 depot centr peri eff #> [1,] 0 0.00000 0.0000000 10000.000 0.000 0.0000 1.000000 #> [2,] 1 43.99334 0.9113641 7475.157 1768.532 270.6751 1.083968 #> [3,] 2 54.50866 2.6510696 5587.797 2191.248 787.3677 1.179529 #> [4,] 3 51.65163 4.4243597 4176.966 2076.396 1314.0348 1.227523 #> [5,] 4 44.37513 5.9432612 3122.347 1783.880 1765.1486 1.233503 #> [6,] 5 36.46382 7.1389804 2334.004 1465.845 2120.2772 1.214084 # This returns a matrix. Note that you can also # solve using name initial values. For example: inits <- c(eff = 1) qd.cp <- solve(m1, theta, events = qd, inits) print(qd.cp) #> ▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂ Solved RxODE object ▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂ #> ── Parameters ($params): ───────────────────────────────────────────────────────
#>      V2      V3      KA      CL       Q     Kin    Kout    EC50
#>  40.200 297.000   0.291  18.600  10.500   1.000   1.000 200.000
#> ── Initial Conditions (\$inits): ────────────────────────────────────────────────
#> depot centr  peri   eff
#>     0     0     0     1
#> ── First part of data (object): ────────────────────────────────────────────────
#> # A tibble: 37 × 7
#>    time    C2    C3  depot centr  peri   eff
#>     [h] <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl>
#> 1     0   0   0     10000     0     0   1
#> 2     1  44.0 0.911  7475. 1769.  271.  1.08
#> 3     2  54.5 2.65   5588. 2191.  787.  1.18
#> 4     3  51.7 4.42   4177. 2076. 1314.  1.23
#> 5     4  44.4 5.94   3122. 1784. 1765.  1.23
#> 6     5  36.5 7.14   2334. 1466. 2120.  1.21
#> # … with 31 more rows
#> ▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂

plot(qd.cp)

# }