---
title: "Case study: how long must PSA be followed to estimate regrowth?"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Case study: how long must PSA be followed to estimate regrowth?}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

```{r setup}
library(pkident)
```

A treatment-sensitive PSA component declines while a resistant component
grows: the ODE form of the biexponential decline-and-regrowth model used for
PSA and tumour size (for example Stein et al., Clin Cancer Res 2008).

```{r}
m <- pkpd_model(
  "d/dt(S) = -d*S
   d/dt(R) = g*R
   psa = S + R",
  outputs = "psa", initial = list(S = "s0", R = "r0")
)
```

The parameter values below are **illustrative**, not estimates from data:
PSA 20 ng/mL at the start of treatment, 2.5% from the resistant component,
sensitive half-life about 3.5 weeks (`d` = 0.8 per month), resistant doubling
time about 4.6 months (`g` = 0.15 per month). PSA reaches its nadir at about
5.6 months.

```{r}
values <- c(s0 = 19.5, r0 = 0.5, d = 0.8, g = 0.15)
error <- residual_error(add = 0.05, prop = 0.15)
```

## Structure

From PSA alone, all four parameters are locally identifiable:

```{r}
structural_identifiability(m)
```

## Length of observation

One patient sampled monthly from the start of treatment, for 3 to 24 months:

```{r}
lengths <- c(3, 6, 9, 12, 18, 24)
candidates <- setNames(lapply(lengths, function(L) experiment(seq(0, L, 1))),
                       paste0(lengths, " months"))
cmp <- compare_designs(m, candidates, values = values, error = error, structural = FALSE)
cmp
```

```{r, fig.width = 6, fig.height = 4}
p <- as_table(cmp, "parameters")
g <- p[p$parameter == "g", ]
d <- p[p$parameter == "d", ]
plot(lengths, g$rse_percent, log = "y", type = "b", pch = 19, ylim = c(3, 6000),
     xlab = "Months of follow-up", ylab = "Expected RSE (%)")
lines(lengths, d$rse_percent, type = "b", pch = 1, lty = 2)
abline(h = 30, col = "grey50")
abline(v = 5.6, col = "grey50", lty = 3)
legend("topright", c("regrowth rate g", "decline rate d", "RSE 30%", "nadir"),
       lty = c(1, 2, 1, 3), pch = c(19, 1, NA, NA), col = c(1, 1, "grey50", "grey50"), bty = "n")
```

The decline rate is determined by 6 months. The regrowth rate is not
determined until PSA has been followed well past the nadir: its expected RSE
falls below 30% only at about 12 months, twice the time to nadir.

## Do the expected RSEs hold?

The expected RSEs come from a local, linear approximation. For 12 and 24
months, 1,000 patients were simulated and each was fitted by maximum
likelihood (`case-studies/psa-regrowth.R` in the source repository):

```{r}
chk <- read.csv(system.file("extdata", "psa-simulation-check.csv", package = "pkident"))
knitr::kable(chk[, c("months", "parameter", "expected_rse", "empirical_rse", "ratio")], digits = 2)
```

The spread of the estimates is within 6% of the expected RSE.