Bolus Arrival Time

In fig. 4.5, the error in K^trans estimation is plotted as a function of BAT. The error is more prominent for negative BAT, i.e. the tissue response occurs after the onset of the AIF. Within a delay window of± 1s the absolute error is less than 3%. At 2 seconds negative shift the error is 6%. It is also evident that the optimal BAT does not necessarily happen at∆BAT = 0. In fig. 4.6, the optimal∆BAT is plotted as a function ofK^trans. Keeping the∆BAT inside the shaded area ensures that the absolute relative error stays below 2% (fig. 4.6a), and 2, 5, and 10% (fig. 4.6b).

Chapter 4. Results 35

Fig. 4.1: Plot of transfer functions and estimation of K ^trans for all models with true K ^trans = 7 ml/100g/min , v _e = 20 ml/100g and v _p = 2 ml/100g . Estimation of K ^trans converges at t approximately 10s for ETM and 2CXM. There is little frequency dependence when the analysis is done with TM. The vertical lines indicate the pole frequencies ! _p (a) and 2! ₀ (b and c) for 2CXM.

32

Ole Gunnar Johansen

Fp= 20

(a) (b)

(c)

Fig. 4.1: Plot of transfer functions and estimation of K

^trans

for all models with true K

^trans

= 7 ml/100g/min, v

e

= 20 ml/100g and v

p

= 2 ml/100g. Estimation of K

^trans

converges at t approximately 10s for ETM and 2CXM. There is little frequency dependence when the analysis is done with TM. The vertical lines indicate the pole frequencies !

p

(a) and 2!

0

(b and c) for 2CXM.

32

Ole Gunnar Johansen

Fp= 40

(a) (b)

(c)

Fig. 4.1: Plot of transfer functions and estimation of K

^trans

for all models with true K

^trans

= 7 ml/100g/min, v

e

= 20 ml/100g and v

p

= 2 ml/100g. Estimation of K

^trans

converges at t approximately 10s for ETM and 2CXM. There is little frequency dependence when the analysis is done with TM. The vertical lines indicate the pole frequencies !

_p

(a) and 2!

₀

(b and c) for 2CXM.

32

Ole Gunnar Johansen

Fp= 80

(a) (b)

(c)

Fig. 4.1: Plot of transfer functions and estimation of K

^trans

for all models with true K

^trans

= 7 ml/100g/min, v

_e

= 20 ml/100g and v

_p

= 2 ml/100g. Estimation of K

^trans

converges at t approximately 10s for ETM and 2CXM. There is little frequency dependence when the analysis is done with TM. The vertical lines indicate the pole frequencies !

p

(a) and 2!

0

(b and c) for 2CXM.

32

Fig. 4.2: Plot of transfer functions and estimation of K^trans for all models with true K^trans = 7 ml/100g/min,ve= 20 ml/100gand vp= 2 ml/100g. Estimation ofK^trans converges at ∆t approximately 10s for ETM and 2CXM. There is little frequency dependence when the analysis is done with TM. The vertical lines indicate the pole frequenciesω_p (a) and 2ω_p (b and c) for 2CXM.

36 4.3. AIF Dispersion and Bolus Arrival Time

−90

−80

−70

−60

−50

−40

−30

−20

−10

20log10|H(ω)|

K^trans= 3.0 K^trans= 7.0 K^trans= 10.0 K^trans= 13.0

ve= 5.0 ve= 10.0 ve= 20.0 ve= 30.0

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ Frequency (Hz)

−90

−80

−70

−60

−50

−40

−30

−20

−10

20log10|H(ω)|

vp= 1.0 vp= 1.5 vp= 2.0 vp= 3.0

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ Frequency (Hz)

Fp= 20.0 Fp= 40.0 Fp= 80.0 Fp= 100.0

Fig. 4.3: Plots of the transfer function for the 2CXM as a function of frequency. The input parameters are K^trans = 7 ml/100g/min, v_e = 20 ml/100g, v_p = 2 ml/100g, andF_p = 20 ml/100g/min if not otherwise indicated.

0 2 4 6 8 10

Fig. 4.4: Changing the width of the AIF to emulate varying degree of AIF BD between point of measurement and tissue voxels. The original (unchanged) AIF had BD = 13.2 s. (a) shows the resulting AIFs. Sub figure (b) and (d) show the error in estimation ofK^trans andF_p respectively. In sub figure (b), the top graph shows the estimation for TM, middle for ETM, and bottom for 2CXM.

Sub figure (c) shows the Fourier transform of the AIFs. The black solid line is FT of the 2CXM tissue response forF_p = 40 ml/100g/min. Steady state of theF_p estimation occurred at about 1 Hz, or at about 55dB attenuation of the AIFs.

38 4.3. AIF Dispersion and Bolus Arrival Time

in BAT of the fitting AIF compared to the nominal AIF for TM (top), ETM (middle) and 2CXM (bottom). in BAT of the fitting AIF compared to the nominal AIF for TM (top), ETM (middle) and 2CXM (bottom). in BAT of the fitting AIF compared to the nominal AIF for TM (top), ETM (middle) and 2CXM (bottom).

Change in arrival time (s)

−50

Change in arrival time (s)

−50

Change in arrival time (s) 0

500

(c)

Fig. 4.5: The error in the K^trans (a), v_e (b) and v_p (c) estimate as a function of change in BAT of AIF relative to BAT of tissue response for TM (top), ETM (middle) and 2CXM (bottom).

Chapter 4. Results 39

Fig. 4.6: The figures show the percentage error in theK^transestimate for a 40⇥40

phantom. The true value ofK^transincreases fromK^trans= 4 ml/100g/minto

K^trans= 14 ml/100g/minfrom top to bottom, and the standard deviation in

the noise increases from = 0to = 0.1from left to right. The sampling

frequency is t = 1s, and the other parameters were ve = 20 ml/100g,

vp= 2 ml/100g, andFp= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

Fig. 4.7

38

(a) TM (b) ETM (c) 2CXM

Fig. 4.6: The figures show the percentage error in the K^trans estimate for a 40⇥40 phantom. The true value ofK^trans increases fromK^trans = 4 ml/100g/minto K^trans = 14 ml/100g/minfrom top to bottom, and the standard deviation in the noise increases from = 0 to = 0.1 from left to right. The sampling frequency is t = 1s, and the other parameters were v_e = 20 ml/100g, v_p= 2 ml/100g, andF_p= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

Fig. 4.7

38

(a) TM (b) ETM (c) 2CXM

Fig. 4.6: The figures show the percentage error in the K^trans estimate for a 40⇥40 phantom. The true value ofK^trans increases fromK^trans = 4 ml/100g/min to K^trans= 14 ml/100g/min from top to bottom, and the standard deviation in the noise increases from = 0 to = 0.1 from left to right. The sampling frequency is t = 1s, and the other parameters were v_e = 20 ml/100g,

Fig. 4.6: The black lines indicate at which ∆BAT the best parameter estimation is found as a function ofK^trans. The shaded area shows the range within which the absolute relative error remains below 2% (a), and below 2, 5 and 10%

going outward (b). The red lines (figure (a)) indicate that illegal values of one of the parameters is reached, or that the absolute error would decline for a further increase or decrease in∆BAT. The latter is considered an obvious breakdown of the model, and it is assumed that no such situation will be encountered, or measures preventing it have been installed.

40 4.4. Effect of Noise

4.4 Effect of Noise

Fig. 4.7 shows the error of the estimate in K^trans for all the models as functions of SNR. The values have been truncated at absolute error of 20%.

SNR was varied between 10 and 100. The figure also shows the no-noise situation (SNR=∞).

Interestingly, error in TM based modelling showed little dependence on noise, but with increasing error for decreasing values ofK^trans. The 2CXM gave overall good estimate for all K^trans, however better for low K^trans. For K^trans below ∼10 and SNR above∼40, the error remained below 10%.

The ETM showed little dependence on both SNR and value of K^trans, however, decreasing Fp would yield larger relative errors than those indicated here (see fig. 4.2b).

The 2CXM exhibited the highest degree of noise sensitivity, especially at high values K^trans, and gave an overall better estimation of K^trans in the region where SNR>40 and K^trans <10 ml/100g/min.

4.5 Clinical Data

Fig. 4.8 shows histograms of the selected models for both patients, both showing a large fraction of voxels chosen for both ETM and 2CXM.

Table 4.1, shows the parametric estimations for patient 16, and the concentration time curves, together with the pixel positions are shown in fig. 4.9. Corresponding information for patient 19 is summarized in table 4.2, and fig. 4.10.

Note that the tumor is in both cases segmented out, and is here shown in black.

Chapter 4. Results 41

Ol eGu nnar Joh an sen

(a) TM (b) ETM

(c) 2CX M Fig .4.6 :Th efi gures sho w the per cen tag eer ror in the K trans estim ate for a4 0 ⇥ 40 pha ntom .Th etru eva lue of K trans incr eases from K trans =4 ml / 100g / mi n to K trans = 14 ml / 100g / mi n from top to bott om ,an dth est and ard dev iatio nin the nois ein crea ses from =0 to =0 . 1 from left to righ t. Th esa mp ling freq uen cy is t =1 s, and the oth er para met ers wer e v e = 20 ml / 100g ,

v p =2 ml / 100g ,a nd F p = 40 ml / 100g / mi n .P ixel swi th abso lute erro r> 20% hav eb een ex clud ed .

38

(a) TM (b) ETM

0

(c) 2CXM

Fig. 4.6: The figures show the percentage error in the K ^trans estimate for a 40 ⇥ 40 phantom. The true value of K ^trans increases from K ^trans = 4 ml/100g/min to K ^trans = 14 ml/100g/min from top to bottom, and the standard deviation in the noise increases from = 0 to = 0.1 from left to right. The sampling frequency is t = 1 s, and the other parameters were v _e = 20 ml/100g ,

Fig. 4.6: The figures show the percentage error in the K ^trans estimate for a 40 ⇥ 40 phantom. The true value of K ^trans increases from K ^trans = 4 ml/100g/min to K ^trans = 14 ml/100g/min from top to bottom, and the standard deviation in the noise increases from = 0 to = 0.1 from left to right. The sampling frequency is t = 1s, and the other parameters were v _e = 20 ml/100g, v _p = 2 ml/100g, and F _p = 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

Fig. 4.6: The figures show the percentage error in the K ^trans estimate for a 40 ⇥ 40 phantom. The true value of K ^trans increases from K ^trans = 4 ml/100g/min to K ^trans = 14 ml/100g/min from top to bottom, and the standard deviation in the noise increases from = 0 to = 0.1 from left to right. The sampling frequency is t = 1 s, and the other parameters were v _e = 20 ml/100g, v _p = 2 ml/100g , and F _p = 40 ml/100g/min . Pixels with absolute error > 20%

Fig. 4.6: The figures show the percentage error in the K ^trans estimate for a 40 ⇥ 40 phantom. The true value of K ^trans increases from K ^trans = 4 ml/100g/min to K ^trans = 14 ml/100g/min from top to bottom, and the standard deviation in the noise increases from = 0 to = 0.1 from left to right. The sampling frequency is t = 1 s, and the other parameters were v _e = 20 ml/100g , v _p = 2 ml/100g, and F _p = 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

Fig. 4.6: The figures show the percentage error in the K ^trans estimate for a 40 ⇥ 40 phantom. The true value of K ^trans increases from K ^trans = 4 ml/100g/min to K ^trans = 14 ml/100g/min from top to bottom, and the standard deviation in the noise increases from = 0 to = 0.1 from left to right. The sampling frequency is t = 1 s, and the other parameters were v _e = 20 ml/100g ,

Fig. 4.6: The figures show the percentage error in theK^trans estimate for a 40⇥40 phantom. The true value ofK^transincreases fromK^trans= 4 ml/100g/minto K^trans= 14 ml/100g/minfrom top to bottom, and the standard deviation in the noise increases from = 0to = 0.1from left to right. The sampling frequency is t = 1s, and the other parameters were ve = 20 ml/100g, vp= 2 ml/100g, andFp= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

38 Ole Gunnar Johansen

(a) TM (b) ETM

(c) 2CXM

Fig. 4.6: The figures show the percentage error in theK^trans estimate for a 40⇥40 phantom. The true value ofK^transincreases fromK^trans= 4 ml/100g/minto K^trans= 14 ml/100g/minfrom top to bottom, and the standard deviation in the noise increases from = 0to = 0.1from left to right. The sampling frequency is t = 1s, and the other parameters were ve = 20 ml/100g, vp= 2 ml/100g, andFp= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

38 Ole Gunnar Johansen

(a) TM (b) ETM

(c) 2CXM

Fig. 4.6: The figures show the percentage error in theK^trans estimate for a 40⇥40 phantom. The true value ofK^transincreases fromK^trans= 4 ml/100g/minto K^trans= 14 ml/100g/minfrom top to bottom, and the standard deviation in the noise increases from = 0to = 0.1from left to right. The sampling frequency is t = 1s, and the other parameters were ve = 20 ml/100g, vp= 2 ml/100g, andFp= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

38 Ole Gunnar Johansen

(a) TM (b) ETM

(c) 2CXM

Fig. 4.6: The figures show the percentage error in theK^trans estimate for a 40⇥40 phantom. The true value ofK^transincreases fromK^trans= 4 ml/100g/minto K^trans= 14 ml/100g/minfrom top to bottom, and the standard deviation in the noise increases from = 0to = 0.1from left to right. The sampling frequency is t = 1s, and the other parameters wereve = 20 ml/100g, vp= 2 ml/100g, andFp= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

38 Ole Gunnar Johansen

(a) TM (b) ETM

(c) 2CXM

Fig. 4.6: The figures show the percentage error in theK^trans estimate for a 40⇥40 phantom. The true value ofK^transincreases fromK^trans= 4 ml/100g/minto K^trans= 14 ml/100g/minfrom top to bottom, and the standard deviation in the noise increases from = 0to = 0.1from left to right. The sampling frequency is t = 1s, and the other parameters were ve = 20 ml/100g, vp= 2 ml/100g, andFp= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded.

38

Ole Gunnar Johansen

(a) TM (b) ETM

(c) 2CXM

Fig. 4.6: The figures show the percentage error in the K^trans estimate for a 40⇥40 phantom. The true value ofK^transincreases fromK^trans= 4 ml/100g/minto K^trans= 14 ml/100g/minfrom top to bottom, and the standard deviation in the noise increases from = 0to = 0.1from left to right. The sampling frequency is t = 1s, and the other parameters were v_e = 20 ml/100g, v_p= 2 ml/100g, andF_p= 40 ml/100g/min. Pixels with absolute error > 20%

have been excluded. NaN

Fig. 4.7: Percentage error in the K^trans estimate for a phantom measuring 40×40 voxels. The true value of K^trans increases from K^trans = 4 ml/100g/min toK^trans = 14 ml/100g/min from top to bottom, and noise increases from SNR = 100to SNR = 10from left to right. In the column on the far left, no noise was added. The sampling frequency is∆t= 1s, and the other parameters were ve = 20 ml/100g, vp = 2 ml/100g, and Fp = 40 ml/100g/min. Pixels with absolute error > 20% have been excluded.

42 4.5. Clinical Data

Noise VVp Patlak TM ETM 2CXM 0

10 20 30 40 50

%

(a) Patient 16

Noise VVp Patlak TM ETM 2CXM 0

10 20 30 40 50

%

(b) Patient 19

Fig. 4.8: Histogram of the percentage of selected models for patient 16 and 19. These patients were selected due to their high values of ETM and 2CXM selection ratios. The total number of voxels are 22013 and 5283 for patient 16 and 19 respectively.

Table 4.1: The parametric estimation done by nordicICE’s incremental model. ROIs were selected randomly for each model inside the tumor. The ROI positions are indicated in fig. 4.9. Patient 16, scan 1, slice 2.

TM ETM 2CXM

ROI K^trans k_ep v_e ROI K^trans k_ep v_e v_p ROI K^trans k_ep v_e v_p F_p

1 1.2 46 2.8 4 1.0 32 2.8 0.53 9 4.1 5 47.8 9.98 24

2 0.3 42 0.0 5 0.0 0.0 0.0 0.0 10 4.1 41 5.6 5.25 19

3 0.8 27 2.8 6 0.2 15 2.8 0.0 11 8.3 59 8.4 5.78 36

7 3.7 35 11.3 1.05 12 13 49 16.9 14.7 47

8 0.0 0.0 0.0 29.4 13 0.8 20 25.3 8.4 34

av. 0.8 38 1.9 av. 1.0 16 3.38 6.20 av. 6.1 35 21 8.8 32

0 50 100 150 Imaging time (s) 1000

1500

Signalintensity(arb.units)

ROI #1 ROI #2

ROI #3

(a) TM

0 50 100 150

Imaging time (s) 0

2000 4000

Signalintensity(arb.units)

ROI #4 ROI #5 ROI #6

ROI #7 ROI #8

(b) ETM

0 50 100 150

Imaging time (s) 1000

2000

Signalintensity(arb.units)

ROI #9 ROI #10 ROI #11

ROI #12 ROI #13

(c) 2CXM

Fig. 4.9: Signal intensity time curves for patient 16, scan 1, slice 2. The single voxel ROIs are all located within the tumor volume, and the figure indicate which model is selected by nICE in the different ROIs. The tumor volume is displayed in black in order to better visualize the ROI placements.

44 4.5. Clinical Data

0 50 100 150

Imaging time (s) 1000

2000

Signalintensity(arb.units)

ROI #1 ROI #2 ROI #3

ROI #4 ROI #5

(a) ETM

0 50 100 150

Imaging time (s) 1000

2000 3000

Signalintensity(arb.units)

ROI #6 ROI #7

ROI #8 ROI #9

(b) 2CXM

Fig. 4.10: Signal intensity time curves for patient 19, scan 1, slice 9. The single voxel ROIs are all located within the tumor volume, and the figure indicate which model is selected by nICE in the different ROIs. The tumor volume is displayed in black in order to better visualize the ROI placements.

Table 4.2: The parametric estimation done by nordicICE’s incremental model. ROIs were selected randomly for each model inside the tumor. The ROIs are 1 pixel in size, and are indicated in fig. 4.10. Patient 19, scan 1, slice 9.

ETM 2CXM

ROI K^trans kep ve vp ROI K^trans kep ve vp Fp

1 6.3 12 52 0.8 6 14 28 31 5.9 56

2 4.1 9.6 45 0.6 7 14 25 35 16 36

3 4.1 59 7.0 3.0 8 15 42 24 14 98

4 2.2 13 17 1.5 9 17 34 31 7.4 43

5 9.5 28 35 2.7

av. 5.4 24 31 1.7 av. 15 32 30 11 58

46

Discussion 5

The use of DCE-MRI both in research and in clinic is limited by lack of stan-dardization of both acquisition protocol, and the modelling approach. This makes qualitative evaluation of DCE parameters difficult to compare between studies, even between examinations of the same patient. Some of these factors are possible to standardize, however some prove more difficult.

This work attempts to identify, and give some insight into what to look out for, if and when standardization is attempted. The results indicate that the choice of kinetic model, how well the AIF can be measured, and the sampling rate of the signal, all play important roles in the complex task of accurate and robust DCE-MRI parameter estimation.

Most of the considerations in this work were made on noise free data, and are as such not representative for real clinical data where noise is always present.

This was, however, done deliberately to separate the effects of model choice and sampling conditions from the effect of image noise, under the assumption that the effect of noise is independent of the other effects, and that noise it will only add uncertainty to the model fit and model determination.

It has also been assumed that the underlying physiology is described sufficiently by the 2CXM. One goal of this work has been to determine and quantify under which conditions a simpler model can describe the physiology without significant loss of precision in the parameter estimates and physiological assumptions. Logically, in the fast flow limit Fp → ∞, the 2CXM approaches the ETM [24], as the TM and ETM both assume zero mean transit time (MTT) of the bolus through the capillary plasma compartment. The validity of using 2CXM as the "ground truth" relies on the assumption that the CA extraction fraction K^trans/Fp <<1; i.e. the fraction of the CA extracted during first pass to be relatively low, so that K^trans << Fp. With K^trans = 2−7 ml/100g/min, this assumption is always met.

The max flow used in the simulations was Fp = 80 ml/100g/min, which, with vp ' 2 ml/100g, corresponds to MTT = vp/Fp ' 1.5 s, requiring a sampling frequency of .0.75 s. The sampling frequency of the AIF used in signal generation constituted the lowest temporal resolution obtainable in the generated signal. The input AIF had a temporal resolution of 0.1 s and was therefore well within the required frequency range.

Since the 2CXM was used as input "gold standard" model, the use of this model in the analysis always yielded more accurate estimates of all kinetic parameters given sufficient sampling frequency and adequate SNR. Fitting the data to the simpler models tested, resulted in systematic errors due to non-zero MTT of the plasma flow (i.e. Fp <∞).

5.1 Study Limitations

Compartmentalization of CA in the tissue can have a profound effects on the relaxation properties induced by the CA in tissue. Compartmentalization restricts water protons to access the CA, and the fast water exchange assumption made for

48 5.2. Temporal Resolution

eq. (2.2.1) no longer holds. More advanced models accounting for water exchange effects have been proposed [31], but these have not been considered in the current project.

Any scientific work that is reliant on measurements, either physical or simulated, is prone to uncertainties. In computer simulations, the uncertainty is downward bounded to the floating point precision of the computer. On modern computers this precision is in the order of10⁻³⁸. Computer round-off errors can become significant if multiple iterations are needed for the calculation of a value. It was generally assumed in this work that other insecurities were of higher significance.

The application of the simulation study on the clinical dataset was limited to two sample datasets from an ongoing DCE-MRI study in patients with primary brain tumors. The small sample size makes these results only indicative, and therefore no hard conclusions can be drawn. Further studies on the matter should be conducted in the future.

The limited clinical validation was due to time limitations.

5.2 Temporal Resolution

Any non-zero capillary MTT will result in a broadening of the first pass peak of the tissue response compared to the AIF, and without any flow component in the model, this broadening will be interpreted as a leakage from the vascular space into the EES. The TM interprets therefore MTT > 0 as an increase in extracellular volume (fig. A.1a) and an artificial K^trans, whereas the ETM reports an increase in K^trans and extracellular volume, and a decrease in the capillary volume compared to the ground truth.

The impulse response function of the 2CXM contains two analytical pole frequencies. The highest of the two dictates a sampling frequency above which

unbiased estimation of all the parameters are done in the noise free data. The cutoff corresponded closely to 1/MTT = Fp/vp which is the component in the 2CXM with the highest frequency. For accurate estimation of K^trans, the lower

unbiased estimation of all the parameters are done in the noise free data. The cutoff corresponded closely to 1/MTT = Fp/vp which is the component in the 2CXM with the highest frequency. For accurate estimation of K^trans, the lower

In document DCE-MRI Pharmacokinetic Model Optimization and Implications for Brain Cancer Imaging (sider 47-65)