Pharmacokinetic Models for Dynamic Contrast-Enhanced MRI Data

Parameter estimation for single compartment models is performed using literature-based or user-specified arterial input functions. The Levenburg-Marquardt algorithm does the heavy lifting.

dcemri.lm(conc, ...)

# S4 method for array
dcemri.lm(
  conc,
  time,
  img.mask,
  model = "extended",
  aif = NULL,
  control = minpack.lm::nls.lm.control(),
  user = NULL,
  guess = NULL,
  multicore = FALSE,
  verbose = FALSE,
  ...
)

Arguments

conc

is a multidimensional (1D-4D) array of contrast agent concentrations. The last dimension is assumed to be temporal, while the previous dimensions are assumed to be spatial.
...

Additional parameters to the function.
time

is a vector of acquisition times (in minutes) relative to injection of the contrast agent. Negative values should be used prior to the injection.
img.mask

is a (logical) multidimensional array that identifies the voxels to be analyzed. Has to have same dimension as conc minus temporal dimension.
model

is a character string that identifies the type of compartmental model to be used. Acceptable models include:

"weinmann"

Tofts & Kermode AIF convolved with single compartment model

"extended"

Weinmann model extended with additional vascular compartment (default)

"orton.exp"

Extended model using Orton's exponential AIF

"orton.cos"

Extended model using Orton's raised cosine AIF

"kety.orton.exp"

Kety model using Orton's exponential AIF

"kety.orton.cos"

Kety model using Orton's raised cosine AIF
aif

is a character string that identifies the parameters of the type of arterial input function (AIF) used with the above model. Acceptable values are:

  • tofts.kermode(default) for the weinmann and extended models

  • fritz.hansen for the weinmann and extended models

  • “empirical” for the weinmann and extended models

  • orton.exp(default) for the orton.exp and kety.orton.exp model

  • orton.cos(default) for the orton.cos and kety.orton.cos model.

  • user for the orton.exp and orton.cos model.

All AIF models set the parametric form and parameter values -- except user, where a set of user-defined parameter values are allowed, and empirical, where a vector of values that fully characterize the empirical AIF.
control

is a list of parameters used by nls.lm.control that are set by default, but may be customized by the user.
user

if aif="user" a list with the following parameters required: D, AB, muB, AG, muG. If aif="empirical" a vector of values that fully characterize the empirical AIF.
guess

is a vector of starting values for kinetic parameter estimation. The vector must have length = 3 (with names th0, th1 and th3) when the extended Kety model is used with the vascular parameter and length = 2 (with names th1 and th3) otherwise.
multicore

is a logical variable (default = FALSE) that allows parallel processing via parallel.
verbose

is a logical variable (default = FALSE) that allows text-based feedback during execution of the function.

Value

Parameter estimates and their standard errors are provided for the masked region of the multidimensional array. All multi-dimensional arrays are provided in nifti format. They include:

ktrans

Transfer rate from plasma to the extracellular, extravascular space (EES).

kep

Rate parameter for transport from the EES to plasma.

ve

Fractional occupancy by EES (the ratio between \(K^{trans}\) and \(k_{ep}\)).

vp

Fractional occupancy in the plasma space.

ktranserror

Standard error for \(K^{trans}\).

keperror

Standard error for \(k_{ep}\).

vperror

Standard error for \(v_p\).

The residual sum-of-squares is also provided, along with the original acquisition times (for plotting purposes).

Details

Compartmental models are the solution to the modified general rate equation (Kety 1951). The specific parametric models considered here include the basic Kety model $$C_t(t)=K^{trans}\left[C_p(t)\otimes\exp(-k_{ep}t)\right],$$ where \(\otimes\) is the convoluation operator, and the so-called extended Kety model $$C_t(t)=v_pC_p(t)+K^{trans}\left[C_p(t)\otimes\exp(-k_{ep}t)\right].$$ The arterial input function must be either literature-based (with fixed parameters) or the exponential AIF from Orton et al. (2008) with user-defined parameters.

Note

WARNING: when using the empirical AIF, a linear interpolation is used to upsample the AIF to a one-second sampling rate. This allows one to utilize a computationally efficient numeric method to perform the convolution. If the empirical AIF is sampled faster than one Hertz, then the upsampling operation will become a downsampling operation. This should not have any serious effect on the parameter estimates, but caution should be exercised if very fast sampling rates are used to obtain an empirical AIF.

References

Ahearn, T.S., Staff, R.T., Redpath, T.W. and Semple, S.I.K. (2005) The use of the Levenburg-Marquardt curve-fitting algorithm in pharmacokinetic modelling of DCE-MRI data, Physics in Medicine and Biology, 50, N85-N92.

Fritz-Hansen, T., Rostrup, E., Larsson, H.B.W, Sondergaard, L., Ring, P. and Henriksen, O. (1993) Measurement of the arterial concentration of Gd-DTPA using MRI: A step toward quantitative perfusion imaging, Magnetic Resonance in Medicine, 36, 225-231.

Orton, M.R., Collins, D.J., Walker-Samuel, S., d'Arcy, J.A., Hawkes, D.J., Atkinson, D. and Leach, M.O. (2007) Bayesian estimation of pharmacokinetic parameters for DCE-MRI with a robust treatment of enhancement onset time, Physics in Medicine and Biology 52, 2393-2408.

Orton, M.R., d'Arcy, J.A., Walker-Samuel, S., Hawkes, D.J., Atkinson, D., Collins, D.J. and Leach, M.O. (2008) Computationally efficient vascular input function models for quantitative kinetic modelling using DCE-MRI, Physics in Medicine and Biology 53, 1225-1239.

Tofts, P.S., Brix, G, Buckley, D.L., Evelhoch, J.L., Henderson, E., Knopp, M.V., Larsson, H.B.W., Lee, T.-Y., Mayr, N.A., Parker, G.J.M., Port, R.E., Taylor, J. and Weiskoff, R. (1999) Estimating kinetic parameters from dynamic contrast-enhanced \(T_1\)-weighted MRI of a diffusable tracer: Standardized quantities and symbols, Journal of Magnetic Resonance, 10, 223-232.

Tofts, P.S. and Kermode, A.G. (1984) Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts, Magnetic Resonance in Medicine, 17, 357-367.

Weinmann, H.J., Laniado, M. and Mutzel, W. (1984) Pharmacokinetics of Gd-DTPA/dimeglumine after intraveneous injection into healthy volunteers, Physiological Chemistry and Physics and Medical NMR, 16, 167-172.

See also

dcemri.bayes, dcemri.map, dcemri.spline, nls.lm

Author

Brandon Whitcher bwhitcher@gmail.com,
Volker Schmid volkerschmid@users.sourceforge.net

Examples


data("buckley")

## Empirical arterial input function
img <- array(t(breast$data), c(13,1,1,301))
time <- buckley$time.min
mask <- array(TRUE, dim(img)[1:3])

## Estimate kinetic parameters directly from Buckley's empirical AIF
fit1 <- dcemri.lm(img, time, mask, model="weinmann", aif="empirical",
                  user=buckley$input)
fit2 <- dcemri.lm(img, time, mask, model="extended", aif="empirical",
                  user=buckley$input)

## Set up breast data for dcemri
xi <- seq(5, 300, by=3)
img <- array(t(breast$data)[,xi], c(13,1,1,length(xi)))
time <- buckley$time.min[xi]
input <- buckley$input[xi]

## Generate AIF params using the orton.exp function from Buckley's AIF
(aifparams <- orton.exp.lm(time, input))
#> $AB
#> [1] 37.44682
#> 
#> $muB
#> [1] 3.920004
#> 
#> $AG
#> [1] 1.522074
#> 
#> $muG
#> [1] -0.009832244
#> 
#> $info
#> [1] 1
#> 
#> $message
#> [1] "Relative error in the sum of squares is at most `ftol'."
#> 
fit3 <- dcemri.lm(img, time, mask, model="orton.exp", aif="user",
                  user=aifparams)

## Scatterplot comparing true and estimated Ktrans values
plot(breast$ktrans, fit1$ktrans, xlim=c(0,0.75), ylim=c(0,0.75),
     xlab=expression(paste("True ", K^{trans})),
     ylab=expression(paste("Estimated ", K^{trans})))
points(breast$ktrans, fit2$ktrans, pch=2)
points(breast$ktrans, fit3$ktrans, pch=3)
abline(0, 1, lwd=1.5, col=2)
legend("bottomright", c("weinmann/empirical", "extended/empirical",
                        "orton.exp/user"), pch=1:3)

cbind(breast$ktrans, fit1$ktrans[,,1], fit2$ktrans[,,1], fit3$ktrans[,,1])
#>              [,1]      [,2]       [,3]       [,4]
#>  [1,] 0.145599052 0.1678452 0.16592766 0.55254849
#>  [2,] 0.218344559 0.2658134 0.24219283 0.50264588
#>  [3,] 0.250521815 0.3129245 0.27084876 0.48798627
#>  [4,] 0.268391926 0.3403433 0.28533489 0.48014076
#>  [5,] 0.279725126 0.3582433 0.29400920 0.47516094
#>  [6,] 0.009912791 0.7331255 0.01475737 0.09028424
#>  [7,] 0.146992146 0.2043296 0.15691602 0.47338401
#>  [8,] 0.328712966 0.3961533 0.36284044 0.49722427
#>  [9,] 0.387767110 0.4608153 0.43840557 0.50468090
#> [10,] 0.250521815 0.2491542 0.24915417 0.45785051
#> [11,] 0.250521815 0.2801023 0.25820501 0.47252745
#> [12,] 0.250521815 0.3476498 0.28871475 0.50510375
#> [13,] 0.250521815 0.3833826 0.31198211 0.52406079

## Scatterplot comparing true and estimated Ktrans values
plot(breast$vp, fit1$vp, type="n", xlim=c(0,0.15), ylim=c(0,0.15),
     xlab=expression(paste("True ", v[p])),
     ylab=expression(paste("Estimated ", v[p])))
points(breast$vp, fit2$vp, pch=2)
points(breast$vp, fit3$vp, pch=3)
abline(0, 1, lwd=1.5, col=2)
legend("bottomright", c("extended/empirical","orton.exp/user"), pch=2:3)

cbind(breast$vp, fit2$vp[,,1], fit3$vp[,,1])
#>         [,1]         [,2]         [,3]
#>  [1,] 0.0600 1.840405e-03 1.371417e-03
#>  [2,] 0.0600 1.754635e-02 1.710091e-02
#>  [3,] 0.0600 2.818906e-02 2.771863e-02
#>  [4,] 0.0600 3.486796e-02 3.447373e-02
#>  [5,] 0.0600 3.933510e-02 3.905019e-02
#>  [6,] 0.0600 5.451225e-02 5.395206e-02
#>  [7,] 0.0600 3.971422e-02 3.895468e-02
#>  [8,] 0.0600 1.899215e-02 1.873550e-02
#>  [9,] 0.0600 1.138390e-02 1.112868e-02
#> [10,] 0.0001 1.128819e-17 3.850360e-12
#> [11,] 0.0300 1.521879e-02 1.482669e-02
#> [12,] 0.0900 3.789190e-02 3.776697e-02
#> [13,] 0.1200 4.396546e-02 4.480785e-02