R/kinfitr_loganplot.R

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

#' Logan Plot
#'
#' Function to fit the Logan Plot 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 tac 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 input Data frame containing the blood, plasma, and parent fraction
#'   concentrations over time.  This can be generated using the
#'   \code{blood_interp} function.
#' @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{Logan_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 inpshift Optional. The number of minutes by which to shift the timing
#'   of the input data frame forwards or backwards. If not specified, this will
#'   be set to 0. This can be fitted using 1TCM or 2TCM.
#' @param vB Optional. The blood volume fraction.  If not specified, this will
#'   be ignored and assumed to be 0%. If specified, it will be corrected for
#'   prior to parameter estimation using the following equation: \deqn{C_{T}(t)
#'   = \frac{C_{Measured}(t) - vB\times C_{B}(t)}{1-vB}}
#' @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$par},
#'   their percentage standard errors \code{out$par.se}, the model fit object
#'   \code{out$fit}, a
#'   dataframe containing the TACs of the data \code{out$tacs}, a dataframe
#'   containing the fitted values \code{out$fitvals}, the blood input data frame
#'   after time shifting \code{input}, a vector of the weights
#'   \code{out$weights}, the inpshift value used \code{inpshift}, the specified
#'   vB value \code{out$vB} and the specified tstarIncludedFrames value
#'   \code{out$tstarIncludedFrames}.
#'
#' @examples
#'
#' data(pbr28)
#'
#' t_tac <- pbr28$tacs[[2]]$Times / 60
#' tac <- pbr28$tacs[[2]]$FC
#' weights <- pbr28$tacs[[2]]$Weights
#' dur <- pbr28$tacs[[2]]$Duration/60
#'
#' input <- blood_interp(
#'   pbr28$procblood[[2]]$Time / 60, pbr28$procblood[[2]]$Cbl_dispcorr,
#'   pbr28$procblood[[2]]$Time / 60, pbr28$procblood[[2]]$Cpl_metabcorr,
#'   t_parentfrac = 1, parentfrac = 1
#' )
#'
#' fit1 <- Loganplot(t_tac, tac, input, 10, weights)
#' fit2 <- Loganplot(t_tac, tac, input, 10, weights, inpshift = 0.1, vB = 0.05)
#' fit3 <- Loganplot(t_tac, tac, input, 10, weights, inpshift = 0.1, vB = 0.05, dur = dur)
#' @author Granville J Matheson, \email{mathesong@@gmail.com}
#'
#' @references Logan J, Fowler JS, Volkow ND, Wolf AP, Dewey SL, Schlyer DJ,
#'   MacGregor RR, Hitzemann R, Bendriem B, Gatley SJ, Christman DR. Graphical
#'   Analysis of Reversible Radioligand Binding from Time-Activity Measurements
#'   Applied to N-11C-Methyl-(-)-Cocaine PET Studies in Human Subjects.
#'   Journal of Cerebral Blood Flow & Metabolism. 1990 Sep 1;10(5):740-7.
#'
#' @export

Loganplot <- function(t_tac, tac, input, tstarIncludedFrames, weights = NULL,
                      inpshift = 0, vB = 0, dur = NULL, frameStartEnd = NULL) {

  # Tidying

  tidyinput <- tidyinput_art(t_tac, tac, weights, frameStartEnd)

  if (!is.null(dur)) {
    tidyinput_dur <- tidyinput_art(dur, tac, weights, frameStartEnd)
    dur <- tidyinput_dur$t_tac
  }

  t_tac <- tidyinput$t_tac
  tac <- tidyinput$tac
  weights <- tidyinput$weights

  newvals <- shift_timings(
    t_tac = t_tac,
    tac = tac,
    input = input,
    inpshift = inpshift
  )

  t_tac <- newvals$t_tac
  tac <- newvals$tac

  t_inp <- newvals$input$Time
  blood <- newvals$input$Blood
  aif <- newvals$input$AIF

  # Parameters

  interptime <- newvals$input$Time
  i_tac <- pracma::interp1(t_tac, tac, interptime, method = "linear")

  # Blood Volume Correction (nothing happens if vB = 0)
  i_tac <- (i_tac - vB * blood) / (1 - vB)
  tac_uncor <- tac
  tac <- pracma::interp1(interptime, i_tac, t_tac, method = "linear")

  if (!is.null(dur)) {

    logan_roi <- frame_cumsum(dur, tac) / tac
    logan_plasma <- as.numeric(pracma::cumtrapz(interptime, aif)) / i_tac

    logan_plasma <- pracma::interp1(interptime, logan_plasma, t_tac, method = "linear")

  } else {

    logan_roi <- as.numeric(pracma::cumtrapz(interptime, i_tac) / i_tac)
    logan_plasma <- as.numeric((pracma::cumtrapz(interptime, aif)) / i_tac)

    logan_roi <- pracma::interp1(interptime, logan_roi, t_tac, method = "linear")
    logan_plasma <- pracma::interp1(interptime, logan_plasma, t_tac, method = "linear")

  }

  logan_equil_roi <- tail(logan_roi, tstarIncludedFrames)
  logan_equil_plasma <- tail(logan_plasma, tstarIncludedFrames)
  weights_equil <- tail(weights, tstarIncludedFrames)


  # Solution

  logan_model <- lm(logan_equil_roi ~ logan_equil_plasma, weights = weights_equil)

  # Output

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

  tacs <- data.frame(Time = t_tac, Target = tac, Target_uncor = tac_uncor) # uncorrected for blood volume

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

  fitvals <- data.frame(Logan_Plasma = logan_plasma, Logan_ROI = logan_roi)

  input <- newvals$input

  par.se <- par
  par.se[1,] <- purrr::map_dbl(names(coef(logan_model)), ~ get_se(logan_model, .x))[2]
  names(par.se) <- paste0(names(par.se), ".se")

  out <- list(
    par = par, par.se = par.se, fit = logan_model, tacs = tacs,
    fitvals = fitvals, input = input, weights = weights,
    inpshift = inpshift, vB = vB, tstarIncludedFrames = tstarIncludedFrames,
    model = "Logan"
  )

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

  return(out)
}


#' Plot: Logan Plot
#'
#' Function to visualise the fit of the Logan Plot model to data.
#'
#' @param loganout The output object of the Logan Plot 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
#'
#' data(pbr28)
#'
#' t_tac <- pbr28$tacs[[2]]$Times / 60
#' tac <- pbr28$tacs[[2]]$FC
#' weights <- pbr28$tacs[[2]]$Weights
#'
#' input <- blood_interp(
#'   pbr28$procblood[[2]]$Time / 60, pbr28$procblood[[2]]$Cbl_dispcorr,
#'   pbr28$procblood[[2]]$Time / 60, pbr28$procblood[[2]]$Cpl_metabcorr,
#'   t_parentfrac = 1, parentfrac = 1
#' )
#'
#' fit <- Loganplot(t_tac, tac, input, 10, weights)
#' plot_Loganfit(fit)
#' @author Granville J Matheson, \email{mathesong@@gmail.com}
#'
#' @import ggplot2
#'
#' @export


plot_Loganfit <- function(loganout, roiname = NULL) {
  plotdf <- data.frame(
    Weights = loganout$weights,
    Logan_Plasma = loganout$fitvals$Logan_Plasma,
    Logan_ROI = loganout$fitvals$Logan_ROI,
    Equilibrium = as.character("Before")
  )

  plotdf$Equilibrium <- as.character(plotdf$Equilibrium)
  plotdf$Equilibrium [ (nrow(plotdf) - (loganout$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, "\nLinear"),
                                  values = myColors)

  xlimits <- c(0, tail(plotdf$Logan_Plasma, 1))

  xlabel <- "Integ(C_Plasma)/C_Tissue"
  ylabel <- "Integ(C_Tissue)/C_Tissue"

  # xlabel <- expression(paste("", "", integral("", paste("0"), paste("", "t")),
  #                            "C", phantom()[{paste("Plasma")}], "",
  #                            phantom()/phantom(), "C", phantom()[{
  #                              paste("Tissue", "")}]))
  #
  # ylabel <- expression(paste("", "", integral("", paste("0"), paste("", "t")),
  #                            "C", phantom()[{paste("Tissue")}], "",
  #                             phantom()/phantom(), "C", phantom()[{
  #                             paste("Tissue", "")}]))

  outplot <- ggplot(data = plotdf, aes(x = Logan_Plasma, y = Logan_ROI, colour = Equilibrium)) +
    geom_point(aes(shape = "a", size = Weights)) +
    geom_abline(
      slope = as.numeric(loganout$fit$coefficients[2]),
      intercept = as.numeric(loganout$fit$coefficients[1])
    ) +
    xlab(xlabel) + ylab(ylabel) + xlim(xlimits) + colScale +
    guides(shape = "none", color = guide_legend(order = 1)) + scale_size(range = c(1, 3))

  return(outplot)
}


#' Tstar Finder: Logan Plot
#'
#' Function to identify where t* is for the Logan Plot.
#'
#'
#' @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 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 input Data frame containing the blood, plasma, and parent fraction
#'   concentrations over time.  This can be generated using the
#'   \code{blood_interp} function.
#' @param filename The name of the output image: filename_Logan.jpeg
#' @param inpshift Optional. The number of minutes by which to shift the timing
#'   of the input data frame forwards or backwards. If not specified, this will
#'   be set to 0. This can be fitted using 1TCM or 2TCM.
#' @param vB Optional. The blood volume fraction.  If not specified, this will
#'   be ignored and assumed to be 0%. If specified, it will be corrected for
#'   prior to parameter estimation using the following equation: \deqn{C_{T}(t)
#'   = \frac{C_{Measured}(t) - vB\times C_{B}(t)}{1-vB}}
#' @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_Logan.jpeg
#'
#' @examples
#' \dontrun{
#' Logan_tstar(t_tac, lowroi, medroi, highroi, input,
#'   filename = "demonstration",
#'   inpshift = onetcmout$par$inpshift, vB = 0.05, gridbreaks = 4
#' )
#' }
#'
#' @author Granville J Matheson, \email{mathesong@@gmail.com}
#'
#' @import ggplot2
#'
#' @export

Logan_tstar <- function(t_tac, lowroi, medroi, highroi, input, filename = NULL, inpshift = 0, vB = 0.05, frameStartEnd = NULL, gridbreaks = 2) {
  frames <- length(t_tac)
  lowroi_fit <- Loganplot(t_tac, lowroi, input, tstarIncludedFrames = frames, inpshift = inpshift, vB = vB, frameStartEnd = frameStartEnd)
  medroi_fit <- Loganplot(t_tac, medroi, input, tstarIncludedFrames = frames, inpshift = inpshift, vB = vB, frameStartEnd = frameStartEnd)
  highroi_fit <- Loganplot(t_tac, highroi, input, tstarIncludedFrames = frames, inpshift = inpshift, vB = vB, frameStartEnd = frameStartEnd)

  xlabel <- "Integ(C_Plasma)/C_Tissue"
  ylabel <- "Integ(C_Tissue)/C_Tissue"

  low_xlimits <- c(0, tail(lowroi_fit$fitvals$Logan_Plasma, 1))
  med_xlimits <- c(0, tail(medroi_fit$fitvals$Logan_Plasma, 1))
  high_xlimits <- c(0, tail(highroi_fit$fitvals$Logan_Plasma, 1))

  low_linplot <- qplot(lowroi_fit$fitvals$Logan_Plasma, lowroi_fit$fitvals$Logan_ROI) + ggtitle("Low") + xlab(xlabel) + ylab(ylabel) + xlim(low_xlimits)
  med_linplot <- qplot(medroi_fit$fitvals$Logan_Plasma, medroi_fit$fitvals$Logan_ROI) + ggtitle("Medium") + xlab(xlabel) + ylab(ylabel) + xlim(med_xlimits)
  high_linplot <- qplot(highroi_fit$fitvals$Logan_Plasma, highroi_fit$fitvals$Logan_ROI) + ggtitle("High") + xlab(xlabel) + ylab(ylabel) + xlim(high_xlimits)

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

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

    vt_df$Low[i] <- lowfit$par$Vt
    vt_df$Medium[i] <- medfit$par$Vt
    vt_df$High[i] <- highfit$par$Vt
  }

  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, 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


  # Vt Plot

  vtplotdf <- tidyr::gather(vt_df, key = Region, value = Vt, -Frames, -Time)
  vtplotdf$Region <- forcats::fct_rev(forcats::fct_inorder(factor(vtplotdf$Region)))

  vtplot <- ggplot(vtplotdf, aes(x = Frames, y = Vt, colour = Region)) + geom_point() + geom_line() + scale_x_continuous(breaks = seq(min(tstarInclFrames), max(tstarInclFrames), by = gridbreaks)) + ylab(expression(V[T])) + 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, vtplot, rel_widths = c(2, 1))

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

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

  return(totalplot)
}