The algorithms used for analysis of dynamic MRI data are detailed here. The following algorithms are covered:

- Automatic identification of the Arterial Input Function (AIF) for brain perfusion.
- Calculation of Brain Perfusion parameters (CBF, CBV, MTT, TTP).
- Calculation of DCE-MRI parameters
(K
^{trans}, v_{e}, v_{p}).

The automatic AIF finder uses a two-stage process to find those pixels in the input image(s) that have characteristics that are most like an arterial input function.

The pixels are ranked in order of largest change, and those pixels that show the largest change are retained after this first sweep. The number of pixels retained is user-configurable

peak intensity * initial slope / tPeak

Thus, the scoring system favours pixels where there is a large change in signal intensity, with a rapid change immediately after the bolus injection, and with an early peak in intensity. The top-scoring pixels are retained as those pixels that are used for the arterial input function. The number of pixels retained after this second stage is user-configurableAfter identification, as above, the [Gd]/time course of each of these AIF pixels is averaged to form the final AIF.

Deconvolution of the arterial input function from the tissue response function
follows the method in
Ostergaard L. *et al.* "High Resolution Measurement of Cerebral Blood
Flow Using Intravascular Tracer Bolus Passages, I: Mathematical Approach
and Statistical Analysis". *Magn. Reson. Med.* **36:**715-728 (1996).
The method is summarised below, and is repeated for every spatial pixel
location in the image. First, the AIF matrix, **A**, is formed following
Eq. [14] of Ostergaard 1996. Singular value decomposition is then used to
construct **A**^{-1} as
**V**.**W**.**U**^{T}.
The diagonal elements of **W** below a user-set threshold percentage (the SVD
threshold) of the maximum value are set to zero.

- The pre-contrast signal intensity for a pixel is averaged, and this average is used, in combination with the pulse sequence parameters and relaxivity, to calculate the post-contrast [Gd] time course for the tissue in that pixel.
- The residue function, R(t), for the pixel is found from Eq. [18] of Ostergaard 1996.
- The areas under the AIF curve and the tissue response curves are found
using the trapezium rule, and the cerebral blood volume (CBV) is calculated as:
CBV=∫[Gd]

_{tissue}d*t*/ ∫[Gd]_{artery}d*t* - The tissue mean transit time (MTT) is found from the residue function
for that tissue:
MTT=Σ (R

where Δt is the time between images; and max(R) is the maximum value of the residue function. (R_{i-1}- R_{i}). (i-0.5).Δt / max(R))_{i-1}- R_{i}) represents the fraction of blood that has a transit time of (i-0.5).Δt - Finally, the cerebral blood flow (CBV) is calculated as:
CBF = CBV / MTT

- The time-to-peak (TTP) is simply computed as the time at which the maximum change in concentration occurs, minus the time of arrival in the contrast in the feeding artery (set by the user).

Calculation of dynamic contrast-enhanced parameters follows the general approach
described in Tofts PS *et al.* "Estimating Kinetic Parameters from
Dynamic Contrast-Enhanced T1-weighted MRI of a Diffusible Tracer:
Standardized Quantities and Symbols" *JMRI* **10:** 223-232
(1999). Refinements to the algorithms are taken from M.A. Horsfield and
B. Morgan "Algorithms for Calculation of Kinetic Parameters from
T1-Weighted Dynamic Contrast-Enhanced MRI". JMRI **20:** 723-729 (2004).

All signal intensities are converted to [Gd] values by averaging
pre-contrast R_{1}, and assuming that:

R_{1} = R_{1}pre + ρ.[Gd]

In the standard Tofts model, the tissue [Gd], C_{t}(t) is related to the plasma [Gd],
C_{p}(t) by:

C_{t}(t) =
K^{trans}∫_{0}^{t} C_{p}(τ)
exp(-K^{trans}(t-τ) / v_{e}) dτ

An alternative model ("Tofts with vp term") takes account of the contribution that the plasma
volume makes to the signal intensity in the tissue.
C_{p}(t) by:

C_{t}(t) =
K^{trans}∫_{0}^{t} C_{p}(τ)
exp(-K^{trans}(t-τ) / v_{e}) dτ + C_{p}(t)v_{p}

The iterative Levenburg-Marquardt method is used to perform the
deconvolution, solving these expressions for K^{trans},
v_{e} and (optionally) v_{p}. The solution that minimises the
summed squared difference between the measured tissue concentration/time
curve and the modelled tissue concentration/time according to the above
expressions, is obtained by iteratively making adjustments to the variable parameters.

The two-compartment exchange mode (2CXM) is that
described by Sourbron. A
bi-exponential residue function is deconvolved from the tissue response. From the parameters of
the bi-exponential form, the parameters F_{p} (flow), PS (permeability-surface area
product), v_{e} and v_{p} can be calculated.