Date: 2019-11-11

# Introduction to RxODE

#### 2019-11-11

# Introduction

`RxODE` is an R package that facilitates simulation with ODE models in R. It is designed with pharmacometrics models in mind, but can be applied more generally to any ODE model.

# Description of RxODE illustrated through an example

The model equations can be specified through a text string, a model file or an R expression. Both differential and algebraic equations are permitted. Differential equations are specified by `d/dt(var_name) =`. Each equation can be separated by a semicolon.

To load `RxODE` package and compile the model:

    library(RxODE)
    library(units)
    #> udunits system database from /usr/share/udunits

    mod1 <-RxODE({
        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;
    });

## Specify ODE parameters and initial conditions

Model parameters can be defined as named vectors. Names of parameters in the vector must be a superset of parameters in the ODE model, and the order of parameters within the vector is not important.

    theta <-
       c(KA=2.94E-01, CL=1.86E+01, V2=4.02E+01, # central
         Q=1.05E+01,  V3=2.97E+02,              # peripheral
         Kin=1, Kout=1, EC50=200)               # effects

Initial conditions (ICs) are defined through a vector as well. If the elements are not specified, the initial condition for the compartment is assumed to be zero.

    inits <- c(eff=1);

## Specify Dosing and sampling in RxODE

`RxODE` provides a simple and very flexible way to specify dosing and sampling through functions that generate an event table. First, an empty event table is generated through the “eventTable()” function:

    ev <- eventTable(amount.units='mg', time.units='hours')

Next, use the `add.dosing()` and `add.sampling()` functions of the `EventTable` object to specify the dosing (amounts, frequency and/or times, etc.) and observation times at which to sample the state of the system. These functions can be called multiple times to specify more complex dosing or sampling regiments. Here, these functions are used to specify 10mg BID dosing for 5 days, followed by 20mg QD dosing for 5 days:

    ev$add.dosing(dose=10000, nbr.doses=10, dosing.interval=12)
    ev$add.dosing(dose=20000, nbr.doses=5, start.time=120, dosing.interval=24)
    ev$add.sampling(0:240)

If you wish you can also do this with the `mattigr` pipe operator `%>%`

    ev <- eventTable(amount.units="mg", time.units="hours") %>%
        add.dosing(dose=10000, nbr.doses=10, dosing.interval=12) %>%
        add.dosing(dose=20000, nbr.doses=5, start.time=120,dosing.interval=24) %>%
        add.sampling(0:240);

The functions `get.dosing()` and `get.sampling()` can be used to retrieve information from the event table.

    head(ev$get.dosing())
    #>   id low time high       cmt   amt rate ii addl evid ss dur
    #> 1  1  NA    0   NA (default) 10000    0 12    9    1  0   0
    #> 2  1  NA  120   NA (default) 20000    0 24    4    1  0   0

    head(ev$get.sampling())
    #>   id low time high   cmt amt rate ii addl evid ss dur
    #> 1  1  NA    0   NA (obs)  NA   NA NA   NA    0 NA  NA
    #> 2  1  NA    1   NA (obs)  NA   NA NA   NA    0 NA  NA
    #> 3  1  NA    2   NA (obs)  NA   NA NA   NA    0 NA  NA
    #> 4  1  NA    3   NA (obs)  NA   NA NA   NA    0 NA  NA
    #> 5  1  NA    4   NA (obs)  NA   NA NA   NA    0 NA  NA
    #> 6  1  NA    5   NA (obs)  NA   NA NA   NA    0 NA  NA

You may notice that these are similar to NONMEM event tables; If you are more familiar with NONMEM data and events you could use them directly with the event table function `et`

    ev  <- et(amountUnits="mg", timeUnits="hours") %>%
        et(amt=10000, addl=9,ii=12,cmt="depot") %>%
        et(time=120, amt=2000, addl=4, ii=14, cmt="depot") %>%
        et(0:240) # Assumes sampling when there is no dosing information

You can see from the above code, you can dose to the compartment named in the RxODE model. This slight deviation from NONMEM can reduce the need for compartment renumbering.

These events can also be combined and expanded (to multi-subject events and complex regimens) with `rbind`, `c`, `seq`, and `rep`. For more information about creating complex dosing regimens using RxODE see the RxODE events vignette.

# Solving ODEs

The ODE can now be solved by calling the model object’s `run` or `solve` function. Simulation results for all variables in the model are stored in the output matrix x.

    x <- mod1$solve(theta, ev, inits);
    knitr::kable(head(x))

time | C2 | C3 | depot | centr | peri | eff
--- | --- | --- | --- | --- | --- | ---
0 | 0.00000 | 0.0000000 | 10000.000 | 0.000 | 0.0000 | 1.000000
1 | 44.37555 | 0.9198298 | 7452.765 | 1783.897 | 273.1895 | 1.084664
2 | 54.88296 | 2.6729825 | 5554.370 | 2206.295 | 793.8758 | 1.180825
3 | 51.90343 | 4.4564927 | 4139.542 | 2086.518 | 1323.5783 | 1.228914
4 | 44.49738 | 5.9807076 | 3085.103 | 1788.795 | 1776.2702 | 1.234610
5 | 36.48434 | 7.1774981 | 2299.255 | 1466.670 | 2131.7169 | 1.214742

You can also solve this and create a RxODE data frame:

    x <- mod1 %>% solve(theta, ev, inits);
    x
    #> ▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂ Solved RxODE object ▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂
    #> ── Parameters (x$params): ─────────────────────────────────────────────────
    #>
    #>      V2      V3      KA      CL       Q     Kin    Kout    EC50
    #>  40.200 297.000   0.294  18.600  10.500   1.000   1.000 200.000
    #> ── Initial Conditions (x$inits): ──────────────────────────────────────────
    #> depot centr  peri   eff
    #>     0     0     0     1
    #> ── First part of data (object): ───────────────────────────────────────────
    #> # A tibble: 241 x 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  44.4 0.920  7453. 1784.  273.  1.08
    #> 3     2  54.9 2.67   5554. 2206.  794.  1.18
    #> 4     3  51.9 4.46   4140. 2087. 1324.  1.23
    #> 5     4  44.5 5.98   3085. 1789. 1776.  1.23
    #> 6     5  36.5 7.18   2299. 1467. 2132.  1.21
    #> # … with 235 more rows
    #> ▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂

This returns a modified data frame. You can see the compartment values in the plot below:

    library(ggplot2)
    plot(x,C2) + ylab("Central Concentration") + xlab("Time");

Or,

    plot(x,eff)  + ylab("Effect") + xlab("Time");

Note that the labels are automatically labeled with the units from the initial event table. This is because RxODE uses the `units` package to handle units and `ggforce` to handle the plotting.