Tracer kinetic modelling of tumour angiogenesis based on dynamic contrast-enhanced CT and MRI measurements

Abstract

Purpose

Technical developments in both magnetic resonance imaging (MRI) and computed tomography (CT) have helped to reduce scan times and expedited the development of dynamic contrast-enhanced (DCE) imaging techniques. Since the temporal change of the image signal following the administration of a diffusible, extracellular contrast agent (CA) is related to the local blood supply and the extravasation of the CA into the interstitial space, DCE imaging can be used to assess tissue microvasculature and microcirculation. It is the aim of this review to summarize the biophysical and tracer kinetic principles underlying this emerging imaging technique offering great potential for non-invasive characterization of tumour angiogenesis.

Methods

In the first part, the relevant contrast mechanisms are presented that form the basis to relate signal variations measured by serial CT and MRI to local tissue concentrations of the administered CA. In the second part, the concepts most widely used for tracer kinetic modelling of concentration-time courses derived from measured DCE image data sets are described in a consistent and unified manner to highlight their particular structure and assumptions as well as the relationships among them. Finally, the concepts presented are exemplified by the analysis of representative DCE data as well as discussed with respect to present and future applications in cancer diagnosis and therapy.

Results

Depending on the specific protocol used for the acquisition of DCE image data and the particular model applied for tracer kinetic analysis of the derived concentration-time courses, different aspects of tumour angiogenesis can be quantified in terms of well-defined physiological tissue parameters.

Conclusions

DCE imaging offers promising prospects for improved tumour diagnosis, individualization of cancer treatment as well as the evaluation of novel therapeutic concepts in preclinical and early-stage clinical trials.

Appendix: Convolution and generalized functions [103]

Given two functions f(t) and g(t) specified for t ≥ 0, the convolution (denoted by the symbol ⊗) is defined by

$$ f(t) \otimes g(t) = \int\limits_0^\infty {f\left( \tau \right) \cdot g\left( {t - \tau } \right)d\tau } . $$

(A1)

The convolution f ⊗ g exists if f and g are both integrable functions. In this case, the convolution product is also integrable. If ƒ and g are rapidly decreasing functions, then so is f ⊗ g.

The improper integral of the convolution f ⊗ g on the whole space is simply given by the product of the integrals over both functions

$$ \int\limits_0^\infty {f(t) \otimes g(t)dt = \int\limits_0^\infty {f(t)dt \cdot \int\limits_0^\infty {g(t)dt} } } $$

(A2)

as can be verified by means of Eq. A1 and applying Fubini’s theorem to exchange the order of integration.

The convolution of a continuous function f(t) with the discontinuous (‘generalized’) delta function δ(t) (representing an infinitely sharp peak at t = 0 bounding unit area) is

$$ f(t) \otimes \delta (t) = f(t) $$

(A3)

and with the discontinuous unit step function σ(t) (whose value is 0 for t < 0 and 1 for t > 0)

$$ f(t) \otimes \sigma (t) = \int\limits_0^t {f\left( \tau \right)d\tau } . $$

(A4)

The latter relation in combination with the associative and distributive property of the convolution product yields

$$ \int\limits_0^t {f\left( \tau \right) \otimes g\left( \tau \right)d\tau = \left[ {\left( {f(t) \otimes g(t)} \right.} \right] \otimes \sigma (t) = f(t) \otimes \left[ {g(t) \otimes \sigma (t)} \right] = f(t) \otimes \int\limits_0^t {g\left( \tau \right)d\tau } } $$

(A5)

and thus

$$ \int\limits_0^t {\left[ {f\left( \tau \right) - f\left( \tau \right) \otimes g\left( \tau \right)} \right]d\tau = f(t) \otimes \left[ {\sigma (t) - \int\limits_0^t {g\left( \tau \right)d\tau } } \right]} . $$

(A6)