Calculate 24-Motion Variables

A quick guide to calculating 24-motion variables

fMRI
motion
R
preprocessing
Published

March 20, 2018

Modified

March 24, 2019

This is a quick guide to calculate the 24 motion parameters from the original six motion parameters. These 24 motion parameters can be used as control variables (covariates of no interest; nuisance regressors) that is typically performed in fMRI first-level analyses, especially in resting-state and task-based functional connectivity analyses.



1 R Libraries

Code 
# data loading, manipulation, and writing
library(tidyverse)



2 Function

A helper function to plot the motion parameters.

Code 
# graph motion parameters and facet by motion type
graph_motion <- function(dataset, title) {
  require(reshape2)
  data_long <- melt(dataset, id.var = "Duration") %>%
    mutate(variable = gsub("_", " ", variable))
  figure <- ggplot(data_long, aes(x = Duration, y = value)) +
    geom_line(color = '#737373') +
    theme_minimal() +
    facet_wrap(~ variable, scales = "free_y", ncol = 3) +
    labs(title = title,
         y = NULL) +
    theme(plot.title = element_text(hjust = 0.5))
  print(figure)
}



3 Calculate 24-Motion Parameters

3.1 6-Rigid Body Motion Regressors

The first six motion parameters are the standard motion parameters and include:

  1. \(t_x\), translation in the x-direction
  2. \(t_y\), translation in the y-direction
  3. \(t_z\), translation in the z-direction
  4. \(\theta_x\), rotation around the x-axis
  5. \(\theta_y\), rotation around the y-axis
  6. \(\theta_z\), rotation around the z-axis
Code 
# load 6-rigid body motion parameters
# add duration
motion_file <- "data/data_demeaned.csv"
motion_demeaned <- read_csv(motion_file, col_names = F) %>%
  mutate(Duration = c(1:nrow(.)))

# rename columns
motion_variables <- c(paste0("Translation_", c("X", "Y", "Z")),
                      paste0("Rotation_", c("X", "Y", "Z")))
colnames(motion_demeaned) <- c(motion_variables, "Duration")

# graph motion parameters
graph_motion(motion_demeaned, "6-Rigid Body Motion Parameters")



3.2 12-Motion Regressors

The next six motion parameters are the first temporal derivatives of the original six rigid body motion parameters. Specifically, it is the motion minus the motion from the previous time point.

  1. \(t_{x_{TD1}}\), first temporal derivative of the translation in the x-direction
  2. \(t_{y_{TD1}}\), first temporal derivative of the translation in the y-direction
  3. \(t_{z_{TD1}}\), first temporal derivative of the translation in the z-direction
  4. \(\theta_{x_{TD1}}\) first temporal derivative of the rotation around the x-axis
  5. \(\theta_{y_{TD1}}\), first temporal derivative of the rotation around the y-axis
  6. \(\theta_{z_{TD1}}\), first temporal derivative of the rotation around the z-axis
Code 
# calculate first temporal derivative
motion_demeaned_td1 <- motion_demeaned %>%
  sapply(., FUN = function(x) c(NA, diff(x))) %>%
  as_tibble() %>%
  mutate(Duration = c(1:nrow(.)))
  
# rename columns
colnames(motion_demeaned_td1) <- c(paste0(motion_variables, "_TD1"), "Duration")

# graph motion parameters
graph_motion(motion_demeaned_td1, "First Temporal Derivatives of the 6-Rigid Body Motion Parameters")



3.3 18-Motion Regressors

The next six motion parameters are the original six rigid body motion parameters squared:

  1. \(t_x^2\), translation in the x-direction squared
  2. \(t_y^2\), translation in the y-direction squared
  3. \(t_z^2\), translation in the z-direction squared
  4. \(\theta_x^2\), rotation around the x-axis squared
  5. \(\theta_y^2\), rotation around the y-axis squared
  6. \(\theta_z^2\), rotation around the z-axis squared
Code 
# square 6-rigid body motion parameters
motion_demeaned_sq <- motion_demeaned^2 %>%
  as_tibble() %>%
  mutate(Duration = c(1:nrow(.)))
  
# rename columns
colnames(motion_demeaned_sq) <- c(paste0(motion_variables, "_Squared"), "Duration")

# graph motion parameters
graph_motion(motion_demeaned_sq, "6-Rigid Body Motion Parameters Squared")



3.4 24-Motion Regressors

The next six motion parameters are the first temporal derivatives squared.

  1. \(t_{x_{TD1}}^2\), first temporal derivative of the translation in the x-direction squared
  2. \(t_{y_{TD1}}^2\), first temporal derivative of the translation in the y-direction squared
  3. \(t_{z_{TD1}}^2\), first temporal derivative of the translation in the z-direction squared
  4. \(\theta_{x_{TD1}}^2\), first temporal derivative of the rotation around the x-axis squared
  5. \(\theta_{y_{TD1}}^2\) first temporal derivative of the rotation around the y-axis squared
  6. \(\theta_{z_{TD1}}^2\), first temporal derivative of the rotation around the z-axis squared
Code 
# square the first temporal derivative
motion_demeaned_td1_sq <- motion_demeaned_td1^2 %>%
  as_tibble() %>%
  mutate(Duration = 1:nrow(.)) 

# rename columns
colnames(motion_demeaned_td1_sq) <- c(paste0(motion_variables, "_TD1_Squared"), "Duration")

# plot motion.td1.sq into one figure
graph_motion(motion_demeaned_td1_sq, "First Temporal Derivatives of the 6-Rigid Body Motion Parameters Squared")



4 Full R Code Example

Code 
# function to create 24 motion variables =======================================
create_24_motion_variables <- function(dataset) {
  # mean-center (demean) ----
  # subtract each score from its respective column mean
  dataset_motion_demeaned <- scale(x = dataset, center = T, scale = F)

  # first temporal derivatives ----
  # subtract motion from prior motion time point
  dataset_motion_td1 <- sapply(dataset_motion_demeaned, FUN = function(x) c(NA, diff(x)))

  # combine the variables into one dataset
  dataset_motion <- data.frame(dataset_motion_demeaned, dataset_motion_td1)

  # squares ----
  # square 6-rigid body motion parameters and its temporal derivatives
  dataset_motion_squared <- dataset_motion^2

  # combine the variables into one dataset
  dataset_motion <- data.frame(dataset_motion, dataset_motion_squared)

  return(dataset_motion)
}

# load 6-rigid body motion parameters file =====================================
# assign to dataset_motion
dataset_motion <- read_csv(file.choose())

# create 24 motion variables using function ====================================
# assign output to dataset_motion_24
dataset_motion_24 <- create_24_motion_variables(dataset = dataset_motion)

# rename columns ===============================================================
# assign original motion variables
motion_variables <- c(
  "trans_x", "trans_y", "trans_z",
  "rot_x", "rot_y", "rot_z"
)

# rename columns
colnames(dataset_motion_24) <- c(
  motion_variables,
  paste0(motion_variables, "_td1"),
  paste0(motion_variables, "_sq"),
  paste0(motion_variables, "_td1_sq")
)

# export 24-motion variables ===================================================
write_csv(x = dataset_motion_24, file = "dataset-motion-24.csv")



Note: Similar logic can also be applied to cerebral spinal fluid (CSF) and white matter (WM) time series to obtain an additional 8 nuisance regressors (32 nuisance regressors in total).

5 References

Friston, K. J., Williams, S., Howard, R., Frackowiak, R. S. J., & Turner, R. (1996). Movement-related effects in fMRI time-series. Magnetic Resonance in Medicine, 35(3), 346–355. http://doi.org/10.1002/mrm.1910350312

Yan, C.-G., Cheung, B., Kelly, C., Colcombe, S., Craddock, R. C., Di Martino, A., … Milham, M. P. (2013). A comprehensive assessment of regional variation in the impact of head micromovements on functional connectomics. NeuroImage, 76, 183–201. http://doi.org/10.1016/j.neuroimage.2013.03.004