R/kinfitr_refPatlak.R

In mathesong/kinfitr: Kinetic Modelling of PET Time Activity Curves

#' Patlak Reference Tissue Model
#'
#' Function to fit the Patlak Reference Tissue Model of Patlak & Blasbert (1985) to data.
#'
#' @param t_tac Numeric vector of times for each frame in minutes. We use the time halfway through the frame as well as a
#' zero. If a time zero frame is not included, it will be added.
#' @param reftac Numeric vector of radioactivity concentrations in the reference tissue for each frame. We include zero at
#' time zero: if not included, it is added.
#' @param roitac Numeric vector of radioactivity concentrations in the target tissue for each frame. We include zero at time
#' zero: if not included, it is added.
#' @param tstarIncludedFrames The number of frames to be used in the regression model, i.e. the number of frames for which
#' the function is linear after pseudo-equilibrium is reached. This is a count from the end of the measurement, so a value of
#' 10 means that last 10 frames will be used. This value can be estimated using \code{refPatlak_tstar}.
#' @param weights Optional. Numeric vector of the weights assigned to each frame in the fitting. We include zero at time zero:
#' if not included, it is added. If not specified, uniform weights will be used.
#' @param dur Optional. Numeric vector of the time durations of the frames. If
#' not included, the integrals will be calculated using trapezoidal integration.
#' @param frameStartEnd Optional: This allows one to specify the beginning and final frame to use for modelling, e.g. c(1,20).
#' This is to assess time stability.
#'
#' @return A list with a data frame of the fitted parameters \code{out$par}, the model fit object \code{out$fit}, a dataframe
#' containing the TACs of the data \code{out$tacs}, a dataframe containing the TACs of the fitted values \code{out$fitvals},
#' a vector of the weights \code{out$weights}, and the specified tstarIncludedFrames value \code{out$tstarIncludedFrames}
#'
#' @examples
#' # Note: Reference region models, and irreversible binding models, should not
#' # be used for PBR28 - this is just to demonstrate function
#'
#' data(pbr28)
#'
#' t_tac <- pbr28$tacs[[2]]$Times / 60
#' reftac <- pbr28$tacs[[2]]$CBL
#' roitac <- pbr28$tacs[[2]]$STR
#' weights <- pbr28$tacs[[2]]$Weights
#'
#' fit <- refPatlak(t_tac, reftac, roitac, tstarIncludedFrames = 10, weights = weights)
#' @author Granville J Matheson, \email{mathesong@@gmail.com}
#'
#' @references Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. Journal of Cerebral Blood Flow & Metabolism. 1985 Dec 1;5(4):584-90.
#'
#' @export

refPatlak <- function(t_tac, reftac, roitac, tstarIncludedFrames,
                      weights = NULL, dur = NULL, frameStartEnd = NULL) {


  # Tidying

  tidyinput <- tidyinput_ref(t_tac, reftac, roitac, weights, frameStartEnd)

  if (!is.null(dur)) {
    tidyinput_dur <- tidyinput_ref(dur, reftac, roitac, weights, frameStartEnd)
    dur <- tidyinput_dur$t_tac
  }

  t_tac <- tidyinput$t_tac
  reftac <- tidyinput$reftac
  roitac <- tidyinput$roitac
  weights <- tidyinput$weights


  # Parameters

  if (!is.null(dur)) {

    patlak_roi <- roitac / reftac
    patlak_ref <- frame_cumsum(dur, reftac) / reftac

  } else {

    patlak_roi <- roitac / reftac
    patlak_ref <- pracma::cumtrapz(t_tac, reftac) / reftac

  }

  patlak_equil_roi <- tail(patlak_roi, tstarIncludedFrames)
  patlak_equil_ref <- tail(patlak_ref, tstarIncludedFrames)
  weights_equil <- tail(weights, tstarIncludedFrames)


  # Solution

  patlak_model <- lm(patlak_equil_roi ~ patlak_equil_ref, weights = weights_equil)

  # Output

  par <- as.data.frame(list(K = as.numeric(patlak_model$coefficients[2])))
  fit <- patlak_model

  tacs <- data.frame(Time = t_tac, Reference = reftac, Target = roitac )

  if (!is.null(dur)) {
    tacs$Duration <- dur
  }

  fitvals <- data.frame(Patlak_ROI = patlak_roi, Patlak_Ref = patlak_ref)

  out <- list(
    par = par, fit = fit, tacs = tacs, fitvals = fitvals, weights = weights,
    tstarIncludedFrames = tstarIncludedFrames, model = "refPatlak"
  )

  class(out) <- c("refPatlak", "kinfit")

  return(out)
}

#' Plot: Patlak Reference Tissue Model
#'
#' Function to visualise the fit of the refPatlak model to data.
#'
#' @param refpatlakout The output object of the refPatlak fitting procedure.
#' @param roiname Optional. The name of the Target Region to see it on the plot.
#'
#' @return A ggplot2 object of the plot.
#'
#' @examples
#' # Note: Reference region models, and irreversible binding models, should not
#' # be used for PBR28 - this is just to demonstrate function
#'
#' data(pbr28)
#'
#' t_tac <- pbr28$tacs[[2]]$Times / 60
#' reftac <- pbr28$tacs[[2]]$CBL
#' roitac <- pbr28$tacs[[2]]$STR
#' weights <- pbr28$tacs[[2]]$Weights
#'
#' fit <- refPatlak(t_tac, reftac, roitac, tstarIncludedFrames = 10, weights = weights)
#'
#' plot_refPatlakfit(fit)
#' @author Granville J Matheson, \email{mathesong@@gmail.com}
#'
#' @import ggplot2
#'
#' @export

plot_refPatlakfit <- function(refpatlakout, roiname = NULL) {
  plotdf <- data.frame(
    Weights = refpatlakout$weights,
    Patlak_ref = refpatlakout$fitvals$Patlak_Ref,
    Patlak_roi = refpatlakout$fitvals$Patlak_ROI,
    Equilibrium = as.character("Before")
  )


  plotdf$Equilibrium <- as.character(plotdf$Equilibrium)
  plotdf$Equilibrium [ (nrow(plotdf) - (refpatlakout$tstarIncludedFrames - 1)):nrow(plotdf)  ] <- "After"

  plotdf$Equilibrium <- forcats::fct_inorder(factor(plotdf$Equilibrium))

  myColors <- RColorBrewer::brewer.pal(3, "Set1")
  names(myColors) <- levels(plotdf$Equilibrium)
  colScale <- scale_colour_manual(name = paste0(roiname, "\nEquilibrium"), values = myColors)

  outplot <- ggplot(data = plotdf, aes(x = Patlak_ref, y = Patlak_roi, colour = Equilibrium)) +
    geom_point(aes(shape = "a", size = Weights)) +
    geom_abline(
      slope = as.numeric(refpatlakout$fit$coefficients[2]),
      intercept = as.numeric(refpatlakout$fit$coefficients[1])
    ) +
    xlab("Integ(C_Ref) / C_Ref") + ylab("C_Tissue / C_Ref") + colScale +
    guides(shape = "none", color = guide_legend(order = 1)) + scale_size(range = c(1, 3))

  return(outplot)
}


#' Tstar Finder: Patlak Reference Tissue Model
#'
#' Function to identify where t* is for the Patlak Reference Tissue Model.
#'
#'
#' @param t_tac Numeric vector of times for each frame in minutes. We use the time halfway through the frame as well as a
#' zero. If a time zero frame is not included, it will be added.
#' @param reftac Numeric vector of radioactivity concentrations in the reference tissue for each frame.
#' @param lowroi Numeric vector of radioactivity concentrations in a target tissue for each frame. This should be from a ROI with low binding.
#' @param medroi Numeric vector of radioactivity concentrations in a target tissue for each frame. This should be from a ROI with medium binding.
#' @param highroi Numeric vector of radioactivity concentrations in a target tissue for each frame. This should be from a ROI with high binding.
#' @param filename The name of the output image: filename_refPatlak.jpeg
#' @param frameStartEnd Optional: This allows one to specify the beginning and final frame to use for modelling, e.g. c(1,20).
#' This is to assess time stability.
#' @param gridbreaks Optional. The size of the grid in the plots. Default: 2.
#'
#' @return Saves a jpeg of the plots as filename_refPatlak.jpeg
#'
#' @examples
#' \dontrun{
#' refPatlak_tstar(t_tac, reftac, taclow, tacmed, tachigh, "demonstration")
#' }
#'
#' @author Granville J Matheson, \email{mathesong@@gmail.com}
#'
#' @import ggplot2
#'
#' @export

refPatlak_tstar <- function(t_tac, reftac, lowroi, medroi, highroi, filename = NULL, frameStartEnd = NULL, gridbreaks = 2) {
  frames <- length(reftac)
  lowroi_fit <- refPatlak(t_tac, reftac, lowroi, length(reftac), frameStartEnd = frameStartEnd)
  medroi_fit <- refPatlak(t_tac, reftac, medroi, length(reftac), frameStartEnd = frameStartEnd)
  highroi_fit <- refPatlak(t_tac, reftac, highroi, length(reftac), frameStartEnd = frameStartEnd)

  patlak_xlab <- "Integ(C_Ref) / C_Ref"
  patlak_ylab <- "C_Tissue / C_Ref"

  low_linplot <- qplot(lowroi_fit$fitvals$Patlak_Ref, lowroi_fit$fitvals$Patlak_ROI) + ggtitle("Low") + xlab(patlak_xlab) + ylab(patlak_ylab)
  med_linplot <- qplot(medroi_fit$fitvals$Patlak_Ref, medroi_fit$fitvals$Patlak_ROI) + ggtitle("Medium") + xlab(patlak_xlab) + ylab(patlak_ylab)
  high_linplot <- qplot(highroi_fit$fitvals$Patlak_Ref, highroi_fit$fitvals$Patlak_ROI) + ggtitle("High") + xlab(patlak_xlab) + ylab(patlak_ylab)

  tstarInclFrames <- 3:frames
  zeros <- rep(0, length(tstarInclFrames))

  r2_df <- data.frame(Frames = tstarInclFrames, Low = zeros, Medium = zeros, High = zeros)
  maxperc_df <- data.frame(Frames = tstarInclFrames, Time = t_tac[ tstarInclFrames ], Low = zeros, Medium = zeros, High = zeros)
  k_df <- data.frame(Frames = tstarInclFrames, Time = t_tac[ tstarInclFrames ], Low = zeros, Medium = zeros, High = zeros)

  for (i in 1:length(tstarInclFrames)) {
    lowfit <- refPatlak(t_tac, reftac, lowroi, tstarIncludedFrames = tstarInclFrames[i], frameStartEnd = frameStartEnd)
    medfit <- refPatlak(t_tac, reftac, medroi, tstarIncludedFrames = tstarInclFrames[i], frameStartEnd = frameStartEnd)
    highfit <- refPatlak(t_tac, reftac, highroi, tstarIncludedFrames = tstarInclFrames[i], frameStartEnd = frameStartEnd)

    r2_df$Low[i] <- summary(lowfit$fit)$r.squared
    r2_df$Medium[i] <- summary(medfit$fit)$r.squared
    r2_df$High[i] <- summary(highfit$fit)$r.squared

    maxperc_df$Low[i] <- maxpercres(lowfit)
    maxperc_df$Medium[i] <- maxpercres(medfit)
    maxperc_df$High[i] <- maxpercres(highfit)

    k_df$Low[i] <- lowfit$par$K
    k_df$Medium[i] <- medfit$par$K
    k_df$High[i] <- highfit$par$K
  }

  xlabel <- "Number of Included Frames"
  ylab_r2 <- expression(R^2)
  ylab_mp <- "Maximum Percentage Variance"


  # R Squared plots

  low_r2plot <- ggplot(r2_df, aes(x = Frames, y = Low)) + geom_point() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + coord_cartesian(ylim = c(0.99, 1)) + xlab(xlabel) + ylab(ylab_r2)
  med_r2plot <- ggplot(r2_df, aes(x = Frames, y = Medium)) + geom_point() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + coord_cartesian(ylim = c(0.99, 1)) + xlab(xlabel) + ylab(ylab_r2)
  high_r2plot <- ggplot(r2_df, aes(x = Frames, y = High)) + geom_point() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + coord_cartesian(ylim = c(0.99, 1)) + xlab(xlabel) + ylab(ylab_r2)

  # Max Percentage Variation Plots

  maxperc_df$inclmins <- rev(max(t_tac) - t_tac)[-c(1, 2)]
  maxperc_df$tstar <- rev(t_tac)[-c(1, 2)]

  low_mpplot <- ggplot(maxperc_df, aes(x = Frames, y = Low)) + geom_point() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + coord_cartesian(ylim = c(0, 20)) + xlab(xlabel) + ylab(ylab_mp) + annotate("text", x = 3, y = 20, label = "t* Minutes", colour = "red", size = 3, hjust = 0) + annotate("text", x = maxperc_df$Frames, y = maxperc_df$Low + 1.4, label = round(maxperc_df$tstar, 1), size = 3, colour = "red") + annotate("text", x = 3, y = 20 - 0.7, label = "Included Minutes", colour = "blue", size = 3, hjust = 0) + annotate("text", x = maxperc_df$Frames, y = maxperc_df$Low + 0.7, label = round(maxperc_df$inclmins, 1), size = 3, colour = "blue")
  med_mpplot <- ggplot(maxperc_df, aes(x = Frames, y = Medium)) + geom_point() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + coord_cartesian(ylim = c(0, 20)) + xlab(xlabel) + ylab(ylab_mp) + annotate("text", x = 3, y = 20, label = "t* Minutes", colour = "red", size = 3, hjust = 0) + annotate("text", x = maxperc_df$Frames, y = maxperc_df$Medium + 1.4, label = round(maxperc_df$tstar, 1), size = 3, colour = "red") + annotate("text", x = 3, y = 20 - 0.7, label = "Included Minutes", colour = "blue", size = 3, hjust = 0) + annotate("text", x = maxperc_df$Frames, y = maxperc_df$Medium + 0.7, label = round(maxperc_df$inclmins, 1), size = 3, colour = "blue")
  high_mpplot <- ggplot(maxperc_df, aes(x = Frames, y = High)) + geom_point() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + coord_cartesian(ylim = c(0, 20)) + xlab(xlabel) + ylab(ylab_mp) + annotate("text", x = 3, y = 20, label = "t* Minutes", colour = "red", size = 3, hjust = 0) + annotate("text", x = maxperc_df$Frames, y = maxperc_df$High + 1.4, label = round(maxperc_df$tstar, 1), size = 3, colour = "red") + annotate("text", x = 3, y = 20 - 0.7, label = "Included Minutes", colour = "blue", size = 3, hjust = 0) + annotate("text", x = maxperc_df$Frames, y = maxperc_df$High + 0.7, label = round(maxperc_df$inclmins, 1), size = 3, colour = "blue")


  # TAC Plot

  tacplotdf <- data.frame(cbind(Time = lowroi_fit$tacs$Time, Reference = lowroi_fit$tacs$Reference, Low = lowroi_fit$tacs$Target, Medium = medroi_fit$tacs$Target, High = highroi_fit$tacs$Target))
  tacplotdf <- tidyr::gather(tacplotdf, key = Region, value = Radioactivity, -Time)

  tacplotdf$Region <- forcats::fct_rev(forcats::fct_inorder(factor(tacplotdf$Region)))

  myColors <- RColorBrewer::brewer.pal(4, "Set1")
  names(myColors) <- levels(tacplotdf$Region)
  colScale <- scale_colour_manual(name = "Region", values = myColors)

  tacplot <- ggplot(tacplotdf, aes(x = Time, y = Radioactivity, colour = Region)) + geom_point() + geom_line() + colScale


  # K Plot

  kplotdf <- tidyr::gather(k_df, key = Region, value = K, -Frames, -Time)
  kplotdf$Region <- forcats::fct_rev(forcats::fct_inorder(factor(kplotdf$Region)))

  kplot <- ggplot(kplotdf, aes(x = Frames, y = K, colour = Region)) + geom_point() + geom_line() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + ylab(expression(K[i])) + colScale


  # Output

  linrow <- cowplot::plot_grid(low_linplot, med_linplot, high_linplot, nrow = 1)
  r2row <- cowplot::plot_grid(low_r2plot, med_r2plot, high_r2plot, nrow = 1)
  mprow <- cowplot::plot_grid(low_mpplot, med_mpplot, high_mpplot, nrow = 1)
  outrow <- cowplot::plot_grid(tacplot, kplot, rel_widths = c(2, 1))

  totalplot <- cowplot::plot_grid(linrow, r2row, mprow, outrow, nrow = 4)

  if (!is.null(filename)) {
    jpeg(filename = paste0(filename, "_refPatlak.jpeg"), width = 300, height = 400, units = "mm", res = 600)
    totalplot
    dev.off()
  }

  return(totalplot)
}