Optimal Sampling for Single Parameter Sets

In this chapter, optimal sampling schemes Ω for the Tofts model are derived according to section 4.2 for the parameter sets pB and pM.

4.3 Methods 59 a) Ktrans [min−1] ve [ ] benign 0.2≤ Ktrans< 1.0 0.2≤ ve< 0.4 intermediate 1.0≤ Ktrans< 1.5 0.4≤ ve< 0.6 malignant Ktrans_{≥ 1.5 0.6≤ v} e< 0.9

Table 4.1: a) Ranges of physiologically realistic PK parameter values for different tissue types. b) Graphical illustration of a).

Constraints and Assumptions of Sampling in DCE MRI

To determine an OSS for the application to DCE MRI, the following constraints and assumptions need to be considered:

• There is a natural ordering of the sampling times ti, since contrast agent is admin-

istered only once: 0 < t₁ < t₂< ... < tN,

where N is the total number of samples.

• Imaging takes place during a time interval with a minimal time Δtmin needed to

acquire an image. For two adjacent time points this imposes the constraint:

ti₊₁− ti ≥ Δtmin.

• For Cartesian acquisitions, image contrast is mainly determined by the central k-

space lines. Therefore, it is assumed here that the actual imaging process occurs instantaneously during a single time point ti within the interval.

Sensitivity Functions

It is beneficial for accurate model fitting to sample data at time points ti at which the

sensitivity functions have a high magnitude. They are given by

ζK(t) = ∂C(t) ∂Ktrans, ζv(t) = ∂C(t) ∂ve , ζτ(t) = ∂C(t) ∂τ . (4.12)

Figure 4.2: Flow chart of the algorithm to determine optimal sampling schemes for a single parameter set (adapted from [Xie2008])

Here, ζK(t), ζv(t) and ζτ(t) are calculated using Mathematica 8 [Wolfram Research,

Inc., Champaign, Illinois, USA].

In figure 4.4 the sensitivity functions ζK(t), ζv(t) and ζτ(t) are plotted along with the

concentration time curve C(t) for each parameter set pB and pM to obtain an estimate

where sampling is relevant. The analytical solutions of the sensitivity functions are given in appendix B.

Algorithm to Determine the OSS Using D-Optimality

To determine the optimal sampling scheme, a search algorithm is implemented which is illustrated in the flow chart of figure 4.2.

The algorithm takes as input an a priori known parameter set p = (Ktrans, ve, τ ), the

total number of sampling points N , the total start time Tmin and total end time Tmax

of imaging, the search resolution r and the minimal distance between neighboring time point Δtmin. The final output of the algorithm is the optimal sampling scheme Ω(p, N )

for the parameter set p and N sampling points.

The inverse cost function Ψ = det( H) is to be maximized against all sampling schemes

within the given constraints to minimize the parameter variance. Scurrdescribes the cur-

rent sampling scheme and Ccurr the current value of the inverse cost function Ψ(Scurr).

4.3 Methods 61

successively adjusted. The first sampling point t₁ is adjusted with step size r between

Tmin and Tmax and with every step the current scheme Scurr(1) is updated to t₁. The

time t₁ is found at which the inverse cost function Ccurr = Ψ(Scurr(1)) is maximized.

Ω(1) = t₁ is set. For the next iteration Scurr(1) = t₁ is set fixed and Scurr(2) is adjusted

analogously, however keeping the distance Δtmin to Scurr(1). For all i > 1 it is pro-

ceeded analogously. The possible locations for tiare constrained by keeping the minimal

distance Δtmin to previously determined time points Scurr(j) with j < i.

Here, a search resolution of r = 1 s, Tmin = 0 min, Tmax = 8 min and a minimal

distance of Δtmin = 10 s is employed. The number of samples is varied from N = 6 to

N = 40.

Comparison of the Performance of the EDS and the OSS

To be able to compare different sampling schemes in terms of resulting parameter sta- bility, the following two measures are used.

(i) Theoretical parameter accuracy: The standard deviation σp_j of the parameter pj

can be theoretically estimated from the curvature of the χ2-function close to the minimum. Under the assumption that there is no correlation between the fit parameters, the variance σ2p_j of parameter pj is given by equation 4.6. Combined

with equation 4.5, this yields:

σp_j =  2(∂χ2 ∂p2_j)−1 =    _1/N i=1 1 σi (∂C(ti) ∂pj )2 (4.13)

In MRI, it can be assumed that the noise standard deviation σi = σ is the same at

all time points ti. Here, the second derivative is evaluated at the true underlying

parameter set p. The resulting standard deviations σK, σvand στof the parameters

Ktrans, veand τ are compared for the OSS and the EDS for N = 6, ..., 40 and the

two different noise standard deviations σ₁ = 0.01 and σ₂ = 0.05. The calculation is performed for pB and pM.

(ii) Measured fitting accuracy: The concentration time curve C(t) is generated with the underlying parameter set p = [Ktrans, ve, τ ]. Gaussian noise with a standard

deviation σ for each single measurement point C(ti) at sampling time ti is added,

resulting in the noisy concentration curve Cσ(t). Using a Levenberg-Marquardt

algorithm with the initial guess being the true parameter set p, the Tofts model

is fitted to Cσ(t). This process is repeated Nrep = 100 times, resulting in a set

{pf it} of resulting parameter sets. From this set, the relative means Kf it, vf it, τf it

and standard deviations σK_fit, σv_fit, στ_fit of the parameters Ktrans, ve and τ are

calculated. This calculation is performed for N = 6, ..., 40 and two different noise standard deviations σ₁ = 0.01 and σ₂ = 0.05. It is repeated for the parameter sets



Figure 4.3: Flow chart of the algorithm to determine optimal sampling schemes for a distribution of parameter sets (adapted from [Xie2008])

Additionally, the χ2-function of the parameter set pM for Ktransand veat the true value

τ = 1.5 min and of the parameters Ktrans and τ at the true value ve = 0.8 is plotted

for N = 7 and σ = 0.01 for the EDS and the OSS. This is done to visualize the varying

χ2-surface of different sampling schemes, leading to varying fitting stabilities.