# 1 Introduction

This post introduces a new formal code block for writing models where there are no compartments. The block is named after the analogous NONMEM block called `$PRED`

. This functionality has always been possible with mrgsolve, but only now is there a code block dedicated to these models. Also, a relaxed set of data set constraints have been put in place when these types of models are invoked.

# 2 Example model

As a most-basic model, we look at the `pred1`

model in `modlib()`

The model code is ::: {.cell}

```
$PROB
An example model expressed in closed form
$PARAM B = -1, beta0 = 100, beta1 = 0.1
$OMEGA 2 0.3
$PRED
double beta0i = beta0 + ETA(1);
double beta1i = beta1*exp(ETA(2));
capture Y = beta0i + beta1i*B;
```

:::

This is a random-intercept, random slope linear model. Like other models in mrgsolve, you can write parameters (`$PARAM`

), and random effects (`$OMEGA`

). But the model is actually written in `$PRED`

.

When mrgsolve finds `$PRED`

, it will generate an error if it also finds `$MAIN`

, `$TABLE`

, or `$ODE`

. However, the code that gets entered into `$PRED`

would function exactly as if you put it in `$TABLE`

.

In the example model, the response is a function of the parameter `B`

, so we’ll generate an input data set with some values of `B`

```
# A tibble: 6 × 2
ID B
<dbl> <dbl>
1 1 0.0398
2 1 0.335
3 1 0.0310
4 1 0.182
5 1 1.42
6 1 0.530
```

Like other models, we can simulate from a population

# 3 PK/PD Model

Here is an implementation of a PK/PD model using `$PRED`

In this model

- Calculate
`CL`

as a function of`WT`

and a random effect - Derive
`AUC`

from`CL`

and`DOSE`

- The response (
`Y`

) is a calculated from`AUC`

and the Emax model parameters

To simulate, look at 50 subjects at each of 5 doses ::: {.cell}

```
data <-
expand.idata(DOSE = c(30,50,80,110,200),ID = 1:50) %>%
mutate(WT = exp(rnorm(n(),log(80),1)))
head(data)
```

```
ID DOSE WT
1 1 30 59.13254
2 2 50 317.32739
3 3 80 242.15746
4 4 110 170.78136
5 5 200 248.18054
6 6 30 51.22012
```

:::

```
ID time WT DOSE AUC Y
1 1 0 59.13254 30 231.90852 110.61330
2 2 0 317.32739 50 36.76051 85.27834
3 3 0 242.15746 80 36.54808 98.90407
4 4 0 170.78136 110 23.68354 80.29131
5 5 0 248.18054 200 331.10229 108.81926
6 6 0 51.22012 30 251.58373 116.09649
```

Plot the response (`Y`

) versus `AUC`

, colored by dose ::: {.cell}

```
library(ggplot2)
ggplot(out, aes(AUC,Y,col =factor(DOSE))) +
geom_point() +
scale_x_continuous(trans = "log", breaks = 10^seq(-4,4)) +
geom_smooth(aes(AUC,Y),se = FALSE,col="darkgrey") + theme_bw() +
scale_color_brewer(palette = "Set2", name = "") +
theme(legend.position = "top")
```

``geom_smooth()` using method = 'loess' and formula 'y ~ x'`

:::