Diffusion weighted imaging of prostate cancer: mathematical modeling of signal obtained using low b values

Harri Merisaari^{1}, Parisa Movahedi^{1}, Ileana Montoya^{1}, Jussi Toivonen^{1}, Marko Pesola^{1}, Pekka Taimen^{2}, Peter BostrĂ¶m^{2}, Tapio Pahikkala^{1}, Hannu Juhani Aronen^{1}, and Ivan Jambor^{1}

Eighty-one patients with histologically confirmed PCa underwent two
MR examinations on the same day performed using a 3T MR scanner (Ingenuity
PET/MR, Philips, Cleveland, USA). The
DWI was performed using a single shot SE-EPI sequence, monopolar
diffusion gradient scheme, and the following parameters: TR/TE 1394/44
ms, FOV 250x250 mm2, acquisition matrix size 124x124, reconstruction
matrix size 256x256, slice thickness 5.0 mm, no intersection gaps, diffusion
gradient timing (Δ) 21.204 ms, diffusion
gradient duration (δ) 6.600 m, SENSE factor of 2, partial-Fourier acquisition
0.69, SPAIR fat suppression, NSA 2, b values 0, 2, 4, 6, 9, 12, 14, 18, 23, 28,
50, 100, 300, 500 s/mm^{2},
acquisition time 3 minutes 45 seconds. The mean signal intensity of squared
shaped ROI (4.89x4.89x5.00 mm^{3}), placed in the center of PCa area, peripheral
zone (PZ), and central gland (CG), was fitted. The IVIM biexponential equation
(Eq. 1) was fitted using the following five different fitting methods:

$$S(b)=S_{0}(fe^{-bD_{p}}+(1-f)e^{-bD_{f}})$$ Eq. 1

1. “Full method”: All four parameters (S_{0},
D_{p}, D_{f}, f) were derived using least square fitting method, in-house written C++ code, utilizing Broyden–Fletcher–Goldfarb–Shanno
(BFGS) algorithm (2) in dlib library (3).

2. “Segmented method”: In the
first step, the monoexponential equation (Eq. 2) was used to derive D_{f}
parameter value by fitting signal intensities in the range 100 - 500 s/mm^{2}.
$$S(b)=S_{0}e^{-bD_{f}}$$ Eq. 2

In the second step, the D_{f}
parameter value from the first
step was inserted into the biexponential equation and the remaining three
parameters (S_{0}, D_{p}, f) were fitted.

3. “Over-segmented method” (4, 5): The first step
consisted of fitting signal intensities of
b values equal to or higher than 100 s/mm^{2} with the monoexponential model (Eq.2) identically to
the first step of the “segmented method”.
In the second step, the extrapolated signal of the fitted
monoexponential model was used to estimate f according to the equation 3:

$$f=(S_{0}-intercept)/S_{0}$$ Eq.3

, where the intercept is the S_{0 }estimated from the eq. 2

In the last step the D_{f} and f parameter values from the first and
second step, respectively, were inserted into the biexponential equation (Eq.1)
and the remaining 2 parameters (S_{0}, D_{p}) were fitted.

4. “Semi-continuous multi-exponential method ” (6-8): This fitting method
utilizes Non-negative least Squares (NNLS). The assumption is that the decay
curve is composed of multiple mono exponential components each with different
fraction of contribution to the decay curve extracted by NNLS. Accordingly,
arbitrarily number of coefficient between 0.1 and 1000 μm^{2}/ms were
chosen to derive the diffusion distribution spectrum of the signal decay curve.
Fraction f was determined from the spectrum by calculating the ratio of the
integral between 10-100 μm2/ms and the total integral.
$$S(b)=\sum_i^N f_{(ADC_{i})}.e^{-bADC_{i}}$$ Eq.4

5. “Simplified IVIM Model” (9): The biexponential function was modeled using delta function (Dirac delta function, d). Diffusion signal decay reduces to a mono-exponential function for all non-zero b-values: $$S(b)=S_{0}(f\delta(b)+(1-f)e^{-bD_{f}})$$ Eq. 5

In addition to the IVIM fitting methods the following mathematical models/functions were fitted:

1. Monoexponential model (10): $$S(b)=S_{0}(e^{-bADC_{m}})$$ Eq. 6

2. Kurtosis model (11): $$S(b)=S_{0}(e^{-bADC_{k}+\frac{1}{6}b^{2}ADC_{k}^{2}K})$$ Eq. 7

3. Stretched exponential model (12, 13): $$S(b)=S_{0}(e^{-(bADC_{s})^{\alpha}})$$ Eq. 8

The fitting quality was evaluated using corrected Akaike information criteria difference (ΔAICc) (14) while the repeatability of the fitted parameters was evaluated using coefficient of repeatability (CR) and Intraclass Correlation Coefficient (ICC) values (15), specifically ICC(3,1). Receiver operating characteristic curve (ROC) analysis was used to evaluate ability of the fitted parameters (17 parameters in total) to discriminate PCa with Gleason score of 3+3 from those with Gleason score of >3+3. Spearman correlation coefficient (ρ) values were calculated between the fitted parameters and the Gleason score groups (n=3).

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Selection of preferred model in different groups, each comparing two models. Percentage of ROIs described better by the first model of the comparison is shown in the table.

¶ - Corrected Akaike information criteria

Coefficient of repeatability (CR), ICC(3,1), area under the curve values (AUC), and spearman correlation coefficient values (ρ). 95% confidence intervals are shown in brackets. PCa= regions of interest placed in prostate cancer lesions; PZ= regions of interest placed in normal tissue of the peripheral zone, CG= regions of interest placed in normal tissue of the central gland; *=p value < 0.05; **=p value < 0.01; ***=p value < 0.001

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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