Diffusion weighted imaging of prostate cancer xenografts: comparison of bayesian modeling and independent least squares fitting

Parisa Movahedi^{1}, Hanne Hakkarainen^{2}, Harri Merisaari^{1}, Heidi LiljenbĂ¤ck^{1}, Helena Virtanen^{1}, Hannu Juhani Aronen^{1}, Heikki Minn^{1}, Matti Poutanen^{1}, Anne Roivainen^{1}, Timo Liimatainen^{2}, and Ivan Jambor^{1}

One million PC-3 (Anticancer Inc., USA) human prostate cancer cells were inoculated subcutaneously in immunodeficient mice (n=11, HSD: Athymic Nude Foxn 1nu, Harlan Laboratories, Indianapolis, IN, USA). All animal handling was conducted in accordance with the local ethics committee for the use and care of laboratory animals and the institutional animal care policies, which fully meet the requirements as defined in the U.S. National Institutes of Health guidelines on animal experimentation. The mice were divided into 2 groups: 1. control group (n=10), 2. treatment group (n=9). The treatment group received Docetaxel (Docetaxel, Actavis, Espoo, Finland) given once a week for three weeks as i.p. injections. The dose was 15mg/kg. Tumor growth in both of the groups was followed by weekly MRI examinations performed using a 7T MR scanner (7T Pharmascan, Bruker GmbH, Ettlingen, Germany) and a 72 mm volume transmitter (Bruker GmbH) and 10 mm surface receiver coil (Bruker GmbH). Multislice T2-weighted anatomical images covering the whole tumor area were obtained (TR/TE 2500 ms/33 ms, field of view (FOV) = 30 × 30 mm2, matrix size 256 × 256, 15 slices) to localize a slice with maximum tumor diameter for DWI measurements. Diffusion weighted single shot spin-echo echo planar imaging was applied with the parameters: TR/TE 3750/25.3 (low b-value set) 3000/30 ms (high b-value set), FOV 3 × 1.5 cm2, matrix 128 ×64, slice thickness 1 mm, three orthogonal diffusion directions, and two different sets of b-values: low b-value set (15 b-values in total): 0, 2, 4, 6, 9, 12, 14, 18, 23, 25, 28, 50, 100, 300, 500 s/mm2, and high b-value set (12 b-values in total): 0, 100, 300, 500, 700, 900, 1100, 1300, 1500, 1700, 1900, 2000 s/mm2. For further analysis, the mean value of the signal from three directions was calculated. The following four mathematical functions/models were applied to the DWI signal obtained using low and high b-values:

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

2. Stretched exponential model (2): $$S(b)=S_{0}e^{-bADC^{\alpha}}$$ Eq. 2

3. Kurtosis model (3): $$S(b)=S_{0}(e^{-bADC_{k}+1/6b^{2}ADC_{k}^{2}K})$$ Eq. 3

4a. Bi-exponential model for low b-values (4):

$$S(b)=S_{0}(fe^{-bD_{p}}+(1-f)e^{-bD_{f}})$$ Eq. 4

4b. Bi-exponential model for high b-values (5):

$$S(b)=S_{0}(fe^{-bD_{f}}+(1-f)e^{-bD_{s}})$$ Eq. 5

The DWI signal decay of each individual voxel has been fitted using four mathematical models, as described above, to generate parametric maps of the parameters. The fitting procedure has been performed using the Levenberg–Marquardt algorithm in Python programming language and following multiple initialization values to prevent local minima in the fitting procedure. Furthermore, all of the parameters except of the Mono-exponential model’s parameter (ADCm) have been fitted by a Bayesian shrinkage method (6). The Bayesian shrinkage prior (BSP) model takes the estimated parameters from least squares fit as a prior distribution of the region of interest, jointly estimating the voxel-wise parameters based on the ROI distribution. The Markove chain algorithm utilizing Gibbs sampling with Metropolis-Hastings updates for each voxel parameters has been applied to each ROI to robustly and quickly converge to the stationary distribution of each parameter. The tumor area was manually delineated on T2-weighted anatomical images and the regions of interest (ROIs) were transferor to the corresponding parametric images. Corrected Akaike information criteria difference (AICc) (7) was used to evaluate fitting quality.

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7. Akaike H. Information theory as an extension of the maximum likelyhood principle. In: Petrov BN, Csaki F, editors. Second International Symposium on Information Theory. Budapest: Akademiai Kaido; 1973. p 267–281.

Box plots of voxel
values derived using independent least squares fitting (A, B, C, D, I, J, K, L)
and bayesian shrinkage method (F, G, H, M, N, O, P) for high b values data sets
of control group. ADCm parameter (A) was
estimated only using independent least squares fitting. The box extends from the 25th to 75th
percentiles while the error bars extend from minimal to maximal values.

Parametric maps using
independent least squares fitting, LSG fit, (A, B, D, F, H, J, L, N) and
Bayesian shrinkage method, BSP Fit, (C, E, G, I, K, M, O) for high b value data
sets of control group mice in week 3.
ADCm parameter (A) was estimated only using independent least squares
fitting.

Median percentage value
of root mean square error increase of the bayesian shrinkage method over
independent least squares fitting.

High b-value DWI data
sets: 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.

Low b-value DWI data
sets: 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.

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

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