Parameter estimation of perfusion models in dynamic contrast-enhanced imaging: a unified framework for model comparison

doi:10.1016/j.media.2016.07.008

Medical Image Analysis

Volume 35, January 2017, Pages 360-374

https://doi.org/10.1016/j.media.2016.07.008 Get rights and content

Highlights

•
We propose a mathematical framework to analyse models of perfusion.
•
We estimate parameters from dynamic imaging data.
•
We compare the most widely used compartment models of perfusion.
•
We apply our methodology to data acquired on patients with abdominal tumours.

Abstract

Patients follow-up in oncology is generally performed through the acquisition of dynamic sequences of contrast-enhanced images. Estimating parameters of appropriate models of contrast intake diffusion through tissues should help characterizing the tumour physiology. However, several models have been developed and no consensus exists on their clinical use. In this paper, we propose a unified framework to analyse models of perfusion and estimate their parameters in order to obtain reliable and relevant parametric images. After defining the biological context and the general form of perfusion models, we propose a methodological framework for model assessment in the context of parameter estimation from dynamic imaging data: global sensitivity analysis, structural and practical identifiability analysis, parameter estimation and model comparison. Then, we apply our methodology to five of the most widely used compartment models (Tofts model, extended Tofts model, two-compartment model, tissue-homogeneity model and distributed-parameters model) and illustrate the results by analysing the behaviour of these models when applied to data acquired on five patients with abdominal tumours.

Graphical abstract

Introduction

Providing earlier assessment of drugs efficiency is a major challenge for the improvement of patient care in oncology. Follow-up of patients presenting tumours is traditionally performed through various morphological measurements (number of tumours, size of tumours,...) (Shanbhogue et al., 2010). However, such measurements are known to reflect only partially the disease progression. Indeed, morphological assessment may be insensitive to or provide markedly delayed indications of the tumour response to treatment even when the therapeutic effect is substantial. It has been shown that the physiological response to a given therapy often precedes the evolution of its morphological descriptors (Li and Padhani, 2012). In this context, dynamic contrast-enhanced (DCE) imaging is a promising tool for the assessment of tissue differentiation based on its intrinsic nature (normal, tumour, necrotic...). The technique, based on the temporal analysis of the contrast intake curve extracted from the images after bolus injection of a contrast agent, aims at characterising tissue microcirculation and microvascularisation.

Functional imaging of the microcirculation, using either magnetic resonance imaging (MRI) (Sourbron and Buckley, 2012), computed tomography (CT) (Miles, Lee, Goh, Klotz, Cuenod, Bisdas, Groves, Hayball, Alonzi, Brunner, 2012, Ingrisch, Sourbron, 2013), ultrasound (Lassau et al., 2010), positron-emission tomography (Keiding, 2012) or single-photon emission CT (Zhang et al., 2012), is based on dynamic contrast imaging, although the diffusion kinetics depends on the type of contrast agent. In this paper, we focus on DCE-MRI and DCE-CT. In such cases, contrast agents are small inert molecules that propagate in extracellular spaces. The kinetics of contrast agent diffusion are similar for both modalities.

From contrast intake curves, simple parameters, such as time to peak or mean transit time (Našel et al., 2000), may be extracted to describe the change of contrast agent concentration and compute parametric images. However, more informative approaches exist, that aim to provide physiologically-based parameters. By associating the temporal variation of contrast intake to physiological parameters, the analysis of contrast intake curves provides a way to characterize the pharmacokinetics of the contrast agent. These parameters can be derived using the tracer kinetic theory (Brix et al., 2010). In this theory, two different concepts have been highlighted. The first one, the indicator dilution theory, is based on a convolution approach and does not rely on any assumption about the diffusion process (Meier and Zierler, 1954). The second approach, which will be considered in this paper, is based on pharmacokinetic compartment models, usually formulated as systems of coupled differential equations. They rely on the assumption that tissues can be represented as a set of interacting subcompartments within which an administered tracer can circulate with different dynamics that depend on each compartment properties. Compartment models are likely to provide a better biological insight, relying on several physiological parameters such as the local vascular permeability, blood flow, intravascular and extracellular volumes (Ingrisch and Sourbron, 2013). These parameters can be extracted from contrast intake curves in a given tissue region through model parameter estimation. However, several models have been developed in the literature and no consensus exists on their clinical use. Each type of model has long been confined to specific applicative domains and only recently efforts have been undertaken to review the full range of existing models (Sourbron, Buckley, 2012, Ingrisch, Sourbron, 2013, Brix, Griebel, Kiessling, Wenz, 2010). For instance, Sourbron and Buckley (2012) and Ingrisch and Sourbron (2013) present a classification that provides clearer insight into the links between the different models. Ingrisch and Sourbron (2013) discuss the difficult problem of model selection: they argue that, while the classical goodness-of-fit criteria such as the χ² value are the most widely used, they are inappropriate to compare models with different numbers of free parameters and should be replaced in that case by penalized criteria such as the Akaike information criterion (AIC). Some mathematical aspects of model analysis, and their systematic applications to a whole group of models, are nevertheless still lacking. Indeed, to obtain reliable and relevant estimates of parameters, one must ensure that the chosen model meets a certain number of requirements: plausibility of the underlying hypotheses and parameters biological meaning, parameter sensitivity and identifiability, solvability of the optimization problem associated to the estimation process, robustness to noise. With a few exceptions (Orton et al., 2007), these conditions are hardly examined; most studies focus on direct applications of the models without questioning their validity.

Therefore, complementary to other review papers that present and classify the models, the objectives of this paper are (i) to define the biological context and general form of perfusion models, (ii) to propose a generic framework for parametric estimation problem of perfusion models in the context of DCE-MRI or DCE-CT imaging through the definition of a unified mathematical framework providing tools to perform robust parameter estimation, sensitivity and identifiability analysis, (iii) to apply the framework to the most widely used compartment models, and finally (iv) to illustrate the results on abdominal DCE-CT images.

Section snippets

Data

DCE imaging consists in the acquisition of temporal sequences of 3D images. Several acquisition protocols can be selected, depending on the modality (CT or MRI) (Kambadakone, Sahani, 2009, Sourbron, 2010) and on the values of the temporal resolution (time delay between consecutive acquisitions) and the duration of acquisition. In this work, the parameters of reconstruction (the spatial resolution and the smoothing filter) were set following the recommendations of Romain et al. (2012). From

Description of extracellular models

The most general model is the Distributed Parameter (DP) model. First introduced by Sangren and Sheppard (1953), it makes no additional assumption on the two-compartment structure than those presented in section 2.2 and uses the four parameters: $θ_{D P} = (F_{p}, P S, v_{e}, v_{p})$ . The impulse residue function R(t) for the DP model is composed of a vascular transit phase R_vasc(t) and a parenchyma phase R_par(t): $R (t) = R_{v a s c} (t) + R_{p a r} (t - t_{1})$ with $t_{1} = \frac{v_{p}}{F_{p}}$ denoting the vascular transit time, i.e. the mean time for blood

Loss function and constraints

In the optimization problem described in Eq. (7), the goal is to minimize the cost function $L_{*},$ i.e. $L_{s q u a r e} (θ)$ in the least squares case, with respect to the constraints. Some of the constraints are common to all models (e.g. positivity of the parameters), and a maximum of three additional constraints may be added, depending on the involved parameters: $f_{1} (θ) = v_{e} - 1 \leq 0,$ $f_{2} (θ) = v_{p} - 1 \leq 0$ and $f_{3} (θ) = v_{e} + v_{p} - 1 \leq 0$ .

Given the non-convexity of objective functions associated to the five selected models,

Discussion

In this paper, we proposed a methodology for model assessment in the context of parametric estimation from DCE-CT or DCE-MRI imaging. This methodology gathers a set of mathematical and statistical tools that we recommend to systematically apply before using a model in a particular clinical context. Few other studies have focused on model comparison. Ingrisch and Sourbron (2013) discussed the problem of model selection and recommended the use of the AIC to choose, for each clinical application,

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As a result, the model-based concentration curves may match the noisy observed data, but the inferred parameters can be far from the actual values. It has further been shown that model parameters are correlated (Romain et al., 2017), which causes several distinct parameter combinations to produce concentration curves which may all well fit the observed data, therefore making the identification of the correct set of parameters difficult (Buckley, 2002; Ahearn et al., 2005). This has lead to more complex fitting algorithms being proposed but these have not yet seen widespread adoption (Kelm et al., 2009; Dikaios et al., 2017; Scannell et al., 2020a; Dikaios, 2020).
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View full text

Parameter estimation of perfusion models in dynamic contrast-enhanced imaging: a unified framework for model comparison

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Data

Description of extracellular models

Loss function and constraints

Discussion

Phys. Med.

Int. J. Radiat. Oncol. Biol. Phys.

Radiol Clin North Am

Eur J Radiol

J. Stat. Software

Phys Med Biol

A new look at the statistical model identification

Autom. Control, IEEE Trans.

Systems biology: parameter estimation for biochemical models

Febs J.

Comprehensive model for simultaneous MRI determination of perfusion and permeability using a blood-pool agent in rats rhabdomyosarcoma

Eur. radiol.

Dynamic contrast-enhanced CT of head and neck tumors: perfusion measurements using a distributed-parameter tracer kinetic model. initial results and comparison with deconvolution-based analysis

Phys. med. biol.

Regional blood flow, capillary permeability, and compartmental volumes: measurement with dynamic ct?initial experience

Radiol.

Tracer kinetic modelling of tumour angiogenesis based on dynamic contrast-enhanced ct and mri measurements

Eur. j. nuclear med. mol. imaging

Pharmacokinetic analysis of tissue microcirculation using nested models: multimodel inference and parameter identifiability

Med. phys.

Practical identifiability analysis of large environmental simulation models

Water Resour. Res.

Model selection and multimodel inference: a practical information-theoretic approach

The microvascular characteristics of cervical cancer: limitations of the modified-tofts tracer kinetic model for the analysis of DCE-MRI data

Proc Intl Soc Mag Reson Med

Interior methods for nonlinear optimization

SIAM rev.

A novel estimation method for physiological parameters in dynamic contrast-enhanced MRI: application of a distributed parameter model using fourier-domain calculations

Med. Imaging, IEEE Trans.

Quantitative tumor perfusion assessment with multidetector CT: Are measurements from two commercial software packages interchangeable? 1

Radiol.

Tracer-kinetic modeling of dynamic contrast-enhanced MRI and CT: a primer.

J Pharmacokinet Pharmacodyn

A model for capillary exchange

Am. J. Physiol-Legacy Content

Bringing physiology into PET of the liver.

J Nucl Med

Fundamentals of tracer kinetics for dynamic contrast-enhanced MRI

Journal of Magnetic Resonance Imaging

Dynamic contrast-enhanced CT imaging of hepatocellular carcinoma in cirrhosis: feasibility of a prolonged dual-phase imaging protocol with tracer kinetics modeling

Eur. radiol.

Hepatic metastases: In vivo assessment of perfusion parameters at dynamic contrast-enhanced MR imaging with dual-input two-compartment tracer kinetics mode

Radiol.

Tracer-kinetic models for measuring cerebral blood flow using externally detected radiotracers

J. Cereb. Blood Flow Metab.

Dynamic contrast-enhanced ultrasonography (DCE-US): a new tool for the early evaluation of antiangiogenic treatment

Targeted oncol.