Andreas Mittermeier^{1}, Birgit Ertl-Wagner^{2}, Jens Ricke^{1}, Olaf Dietrich^{1}, and Michael Ingrisch^{1}

We implemented a tracer-kinetic model within a Bayesian framework which infers full posterior probability distributions for parameter estimates. We validate our Bayesian model using a digital reference object and compare it to a standard non-linear least squares approach. Furthermore, we use this approach to obtain pharmacokinetic parameter distributions during the course of a therapy for breast cancer DCE-MRI data, and we demonstrate how Bayesian posterior distributions can be utilized to assess treatment response.

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is an imaging technique used to quantify microvascular tissue perfusion. By acquiring images during the progression of a contrast agent (CA) through a tissue of interest, a time-dependent CA concentration can be extracted. By fitting pharmacokinetic (PK) models to the concentration-time curves, quantitative tracer-kinetic parameters are obtained. The standard approach to infer parameters is a non-linear least squares (NLLS) analysis which yields point estimates. Bayesian statistics offers an alternative approach which infers full posterior probability distributions of each parameter, yielding additional information about their uncertainty.

Here, we implemented a Bayesian Toft's model (BTM) with the purpose i) to evaluate accuracy against a NLLS approach using a digital reference object, ii) to validate uncertainty estimates against a bootstrapped NLLS approach and iii) to demonstrate how Bayesian posterior probability distributions can be used to assess treatment response in a breast cancer DCE-MRI dataset.

**Validation** To validate the parameter estimates of the BTM, the QIBA_v6_Tofts^{1}
DRO, provided by the Quantitative Imaging Biomarkers Alliance
(QIBA), was used. The DRO contains tissue concentration
curves constructed with the standard Toft’s model (TM)^{2}
for an arterial input function (AIF) and 30 combinations of
$$$K^{trans}$$$ and $$$v_e$$$ (Fig. 1). For a more realistic setting,
complex Gaussian noise with standard deviation $$$\sigma$$$=0.2
relative to the baseline signal was added. The concentration-time
curves were fitted with the NLLS approach using the L-BFGS-B^{3}
algorithm via python’s *scipy*^{4} package. A
*bootstrapping*^{5} method was applied to assess the
uncertainty in parameter estimation.

A BTM
was implemented in *stan*^{6} which infers the parameter’s
posterior distribution via Bayes’ theorem $$P(\theta\mid D)={\frac
{P(D\mid \theta)\,P(\theta)}{P(D)}}$$ and *Markov Chain Monte Carlo*
(MCMC) methods
using the NUTS^{7} algorithm. For both methods, the percentage error
was calculated between estimated parameters and ground truth. Uncertainty $$$\sigma$$$ was inferred from half the width (17^{th}-83^{rd}
percentile) of bootstrap samples and posterior distributions. The
structural similarity index (SSIM)^{8} was calculated between
estimated and true maps, and between both uncertainty maps to
quantify differences.

**Application** A set of breast DCE-MRI data^{9} was used,
acquired from 10 patients before and during preoperative neoadjuvant
chemotherapy (NACT), provided by the Quantitative Imaging Network
(QIN). Patients were labeled according to their pathologic response –
3 pathologic complete responder (pCR) and 7 non-pCR. Concentration
curves averaged within regions of interest (ROI) were evaluated with
the BTM. Posterior distributions for $$$K^{trans}$$$ were compared
between visits and the change was assessed by means of Cohen’s *d*,
calculated as: $$d={\frac
{{\bar{x}}_{1}-{\bar{x}}_{2}}{{\sqrt{(\sigma_{1}^{2}+\sigma_{2}^{2})/2}}}}.$$
A univariate logistic regression (ULR) model was fit to the Cohen’s *d* values to obtain a prediction of response to NACT.

Fig. 2 shows the fitted median-$$$K^{trans}$$$ values of the DRO for the BTM (SSIM 95%) and point estimates for NLLS (SSIM 91%) on the left. The middle column displays the calculated percentage error. The uncertainty maps from posterior distributions and bootstrap samples are shown on the right, coinciding with a SSIM of 91%. Regions for $$$v_e$$$=0.01 are difficult to infer for both methods, indicated by the highest percentage error and uncertainty.

Fig. 3 shows the posterior distributions for $$$K^{trans}$$$ for
visit 1 (blue) and visit 2 (orange) for all patients. Cohen’s *d*
values are visualized in Fig. 4; light-gray represents non-pCR,
dark-gray pCR. The ULR analysis revealed an area under the receiver
operator curve (AUC) of 0.952.

In this study, we assessed posterior probability distributions of tracer-kinetic parameters obtained with a BTM against a standard NLLS approach. Validation with a DRO revealed high accuracy of BTM and NLLS approaches, indicated by strong similarity between estimated and ground truth maps. In addition, precision of estimates, assessed via the width of the posterior probability distributions and bootstrapping, respectively, was in very good agreement between both approaches.

The breast cancer DCE-MRI dataset revealed that the degree of
decrease in $$$K^{trans}$$$ gives information about the pathologic
response to NACT. The response, including measures of uncertainty of
the parameter estimates, was quantified with Cohen’s *d*,
calculated from the posterior distributions between visit 1 and 2.
ULR modeling indicated excellent prediction of response. Further
analysis could include a model selection step to decrease the
influence of systematic modeling errors on posterior distributions
and may further improve assessment of therapy response.

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9. Huang, Wei, Tudorica, Alina, Chui, Stephen, Kemmer, Kathleen, Naik, Arpana, Troxell, Megan, Holtorf, Megan. (2014). Variations of dynamic contrast-enhanced magnetic resonance imaging in evaluation of breast cancer therapy response: a multicenter data analysis challenge. The Cancer Imaging Archive. http://doi.org/10.7937/K9/TCIA.2014.A2N1IXOX