Estimate the Apparent Diffusion Coefficient (ADC)

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Description

Estimation of apparent diffusion coefficient (ADC) values, using a single exponential function, is achieved through nonlinear optimization.

Usage

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adc.lm(signal, b, guess, nprint=0)
ADC.fast(dwi, bvalues, dwi.mask, verbose=FALSE)

Arguments

signal

Signal intensity vector as a function of b-values.

b,bvalues

Diffusion weightings (b-values).

guess

Initial values of S_0 and D.

nprint

is an integer, that enables controlled printing of iterates if it is positive. In this case, estimates of par are printed at the beginning of the first iteration and every nprint iterations thereafter and immediately prior to return. If nprint is not positive, no tracing information on the progress of the optimization is produced.

dwi

Multidimensional array of diffusion-weighted images.

dwi.mask

Logical array that defines the voxels to be analyzed.

verbose

Additional information will be printed when verbose=TRUE.

Details

The adc.lm function estimates parameters for a vector of observed MR signal intensities using the following relationship

S(b) = S_0 \exp{-bD},

where S_0 is the baseline signal intensity and D is the apparent diffusion coefficient (ADC). It requires the routine nls.lm that applies the Levenberg-Marquardt algorithm. Note, low b-values (<50 or <100 depending on who you read) should be avoided in the parameter estimation because they do not represent information about the diffusion of water in tissue.

The ADC.fast function rearranges the assumed multidimensional (2D or 3D) structure of the DWI data into a single matrix to take advantage of internal R functions instead of loops, and called adc.lm.

Value

A list structure is produced with estimates of S_0, D and information about the convergence of the nonlinear optimization routine.

Author(s)

Brandon Whitcher

References

Buxton, R.B. (2002) Introduction to Functional Magnetic Resonance Imaging: Principles & Techniques, Cambridge University Press: Cambridge, UK.

Callahan, P.T. (2006) Principles of Nuclear Magnetic Resonance Microscopy, Clarendon Press: Oxford, UK.

Koh, D.-M. and Collins, D.J. (2007) Diffusion-Weighted MRI in the Body: Applications and Challenges in Oncology, American Journal of Roentgenology, 188, 1622-1635.

See Also

nls.lm

Examples

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S0 <- 10
b <- c(0,50,400,800)  # units?
D <- 0.7e-3           # mm^2 / s (normal white matter)

## Signal intensities based on the (simplified) Bloch-Torry equation
dwi <- function(S0, b, D) {
  S0 * exp(-b*D)
}

set.seed(1234)
signal <- array(dwi(S0, b, D) + rnorm(length(b), sd=.15),
                c(rep(1,3), length(b)))
ADC <- ADC.fast(signal, b, array(TRUE, rep(1,3)))
unlist(ADC) # text output

par(mfrow=c(1,1)) # graphical output
plot(b, signal, xlab="b-value", ylab="Signal intensity")
lines(seq(0,800,10), dwi(S0, seq(0,800,10), D), lwd=2, col=1)
lines(seq(0,800,10), dwi(ADC$S0, seq(0,800,10), ADC$D), lwd=2, col=2)