Optimal Sampling for a Parameter Distribution

In the previous section optimal sampling schemes were derived for two fixed parameter sets p. However, in reality tumors are heterogeneous structures comprising of a distribu-

tion of parameter sets, which varies from tumor to tumor dependent on type and stage. Therefore, a more feasible approach is to calculate the OSS for a distribution δ of PK parameters sets pδ as opposed to single parameter sets.

Here, the distribution δ is chosen to consist of equally weighted typical benign, interme- diate and malignant values. This is for example a sensible choice if the clinical question of interest is the distinction between benign and malignant lesions. The assumed distri- bution δ is based on table 4.1: 0.2 min−1≤ Ktrans ≤ 2.5 min−1, 0.2 ≤ ve ≤ 1.0. The

onset time τ is varied between τ =1.5 min and 1.6 min. Each parameter is iterated with step size s = 0.01.

4.4 Results 63

the OSS of an a priori given distribution of PK parameters. For all parameter sets



pδ of the distribution δ, the OSS for N samples is calculated as described for a single

parameter set (figure 4.2). The time points of all resulting optimal sampling times {ti}

of each parameter set within pδ are recorded in a histogram DOSS. DOSS displays the

distribution of optimal sampling times throughout all parameter sets of pδ. The optimal

sampling scheme Ωδof the whole distribution is obtained by dividing the area under the

resulting histogram into N equal parts. Here, Ωδ is calculated for N = 10. N = 10 is

chosen since according to figure 4.6 and 4.7 a constant parameter uncertainty is reached for the OSS of both tested parameter sets and noise standard deviations, as will be seen in the results section section.

4.4 Results

The methods described above were used to calculate the following sensitivity functions, optimal sampling schemes and comparisons of the EDS and the OSS.

4.4.1 Optimal Sampling for Single Parameter Sets

Sensitivity Functions

In figure 4.4 a) and b) the sensitivity functions ζτ(t) (green), ζK(t) (blue) and ζv(t) (red)

and the concentration time curves C(t) (black) of the two parameter sets pB and pM are

displayed. Both parameter sets exhibit a similar behavior.

It can be seen that ζτ(t) has a peak of high magnitude with a maximum at a few

seconds after the onset time τ with a narrow total width of about 0.5 min. When compared to C(t), the peak of ζτ(t) is located at the upslope of the concentration time

curve. A smaller second peak starting at about 0.5 min after the onset time with a width of about 1 min can be seen for both parameters sets, being higher for pM. It correlates

with the time of the initial fast downslope of C(t) after the peak.

ζK(t) displays a peak starting at τ with a maximum at approximately 15 s after τ

with a narrow total width of 1.0 min-1.5 min. The amplitude of ζK(t) is small compared

to that of ζτ(t) and ζv(t). Relative to C(t), the peak of ζK occurs during the upslope

and the initial fast downslope after the peak of C(t).

Starting at τ , the magnitude of the peak of ζv(t) increases. From about 0.5 min on,

ζv(t) is the sensitivity function with the highest magnitude, lasting until the end of the

measurement. The relatively high maximum is found at about 1 min-1.5 min after τ . The decrease of ζv after the maximum with increasing time is slow, resulting in a high

Figure 4.4: Sensitivity functions of the parameter sets a) pB and b) pM. The following

4.4 Results 65

Figure 4.5: Optimal sampling schemes Ω(p, N ) in dependence of number of sampling

points N for parameter set a) pBand b) pM. c) Equidistant sampling schemes

OSS for Single Parameter Sets Using D-optimality

The output Ω(p, N ) of the OSS algorithm as a function of the number of samples N with N = 6 to N = 40 can be seen in figure 4.5 a) - b) for the parameter sets pB and pM. In

figure 4.5 c) equidistant schemes with N = 6 to N = 40 are also plotted for comparison. For pB and pM a similar behavior can be seen. For N = 6, three clusters of time

points can be distinguished. The first is located at approximately [τ, τ + 0.1 min], the second approximately at [τ +0.3 min, τ +0.6 min] and the third cluster at approximately [τ + 1.3 min, τ + 1.7 min]. With increasing N , the later clusters broaden and merge into a single cluster, for pB at N = 16 and for pM at N = 11. For higher N each new

sample is added to the next possible higher time within the constraint. When t = Tmax

is reached, the next sample points are placed at the baseline.

Comparison of the Performance of the EDS and the OSS

The results of the comparison of the parameter uncertainty of the EDS and the OSS using method (i) and (ii) are shown in figures 4.6 and 4.7. For the EDS (left) and OSS (right) the expected parameter standard deviations σk, σv and στ (figure 4.6) and the

mean and standard deviation of the fitting results (figure 4.7) of the parameters Ktrans (blue), ve (red) and τ (green) are plotted for pB (a)- d)) and pM (e)- h)), each for noise

standard deviation σ₁=0.01 and σ₂=0.05. The following behavior is observed:

• For the larger noise standard deviation σ2, the parameter standard deviations σK,

σv and στ (figure 4.6) and σK_fit, σv_fit and στ_fit (figure 4.7) are significantly higher

than for σ₁ for both, the EDS and the OSS. It can additionally be seen that for

σ₂, systematic fitting errors in Kf it, vf it and τf it increase, especially for small N

(figure 4.7).

• For small N, the parameter standard deviations σK, σv and στ (figure 4.6), σK_fit,

σv_fit and στ_fit and systematic errors of Kf it, vf it and τf it (figure 4.7) of the EDS

are large. For the OSS, they are significantly smaller. Only for a few single N , the EDS performs comparable, as can be seen for example in figure 4.6 a) for N = 11, where στ of the EDS has a comparable value to that of the OSS.

• For the OSS, the parameter standard deviations of all parameters approach an

approximately constant value σK , σv and στ after NOSS samples (figure 4.6). The

same behavior can be seen in figure 4.7 for σK_fit, σv_fit, στ_fit and Kf it, vf it and

τf it, approaching σK _fit, σv_fit, στ_fit and Kf it , vf it and τf it . The constant value

depends on the parameter set and the noise standard deviation. With increasing noise standard deviation the constant values increase. With increasing malignancy, the constant values decrease.

• With increasing N the parameter standard deviations (figure 4.6) and fitting mean

and standard deviations (figure 4.7) of the EDS decrease and approach that of the OSS. An exception to this behavior is pB for σ2 (figure 4.6 c)- d)), where σK and

4.4 Results 67

Figure 4.6: Comparison of the expected parameter standard deviation of the EDS (left) and OSS (right) estimated from the curvature of the χ2-function for parame- ter sets pBand pM, each for noise standard deviation σ1=0.01 and σ2=0.05.

Figure 4.7: Comparison of the fitting accuracy of the EDS(left) and the OSS(right): The mean and standard deviation of Nrep = 100 fitting results is plotted for

parameter sets pB and pM, each for noise standard deviation σ1=0.01 and

4.4 Results 69

Figure 4.8: 2-D χ2-function for pM, N = 7 and a) EDS for Ktrans and ve at true τ =

1.5 min, b) OSS Ktrans _{and v}

e at true τ = 1.5 min, c) EDS for Ktrans and τ

at true ve= 0.8, d) OSS for Ktrans and τ at true ve= 0.8. The OSS exhibits

steeper slopes than the EDS.

• The EDS displays for both methods (i) and (ii) a periodic ’zig-zag’-pattern, whilst

the OSS shows a comparably smooth curve. The amplitude of the zig-zag-pattern decreases with increasing N . A correlation between the pattern of Ktrans and τ exists.

• Ktrans_{is the parameter with the highest parameter standard deviation σ}

Kand the

highest mean error Kf it and standard deviation σK_fit of the fitting parameters for

the OSS and the EDS. ve is the parameter with the smallest parameter standard

deviation σv and the smallest mean error vf it and standard deviation σv_fit of the

fitting parameters for the OSS and the EDS. For the OSS, τ is a stable parameter, for the EDS it is stable only for large N .

In figure 4.8 an example of the χ2-function for the EDS (left) and the OSS (right) are shown for the parameter set pM and N = 7. Comparing a) and b) it can be seen that

for the OSS the valley around the true parameter set (Ktrans, ve) = (2.0 min−1, 0.8)

exhibits steeper slopes than for the EDS when going to other parameter pairs. This is especially evident for varying Ktrans, showing a long broaden valley for the EDS.

Comparing c) and d) a similar pattern can be seen. For the OSS the minimum area around the true parameter set (Ktrans, τ ) = (2.0 min−1, 1.5 min) shows a more narrow

and better distinct valley than for the EDS. The minimum area of the EDS displays a large plateau over a large range of values, especially for parameter sets with higher values than the true parameter set.

4.4.2 Optimal Sampling for a Parameter Distribution

The histogram DOSS (black) and the resulting optimal sampling scheme Ωδ (red) are

shown in figure 4.9. It can be seen that the optimal sampling times are distributed such that high temporal resolution is required for the first 2 minutes after contrast agent arrival, followed by a single sampling time at about 3 minutes after the onset time. In the distribution 3 peaks can be distinguished, centered around t₁ ≈ τ + 0.05 min,

t₂≈ τ + 0.7 min and t₂ ≈ τ + 1.6 min.