Date Published: October 9, 2014
Publisher: Public Library of Science
Author(s): Elias Kellner, Peter Gall, Matthias Günther, Marco Reisert, Irina Mader, Roman Fleysher, Valerij G. Kiselev, Timothy W. Secomb.
Evaluation of blood supply of different organs relies on labeling blood with a suitable tracer. The tracer kinetics is linear: Tracer concentration at an observation site is a linear response to an input somewhere upstream the arterial flow. The corresponding impulse response functions are currently treated empirically without incorporating the relation to the vascular morphology of an organ. In this work we address this relation for the first time. We demonstrate that the form of the response function in the entire arterial tree is reduced to that of individual vessel segments under approximation of good blood mixing at vessel bifurcations. The resulting expression simplifies significantly when the geometric scaling of the vascular tree is taken into account. This suggests a new way to access the vascular morphology in vivo using experimentally determined response functions. However, it is an ill-posed inverse problem as demonstrated by an example using measured arterial spin labeling in large brain arteries. We further analyze transport in individual vessel segments and demonstrate that experimentally accessible tracer concentration in vessel segments depends on the measurement principle. Explicit expressions for the response functions are obtained for the major middle part of the arterial tree in which the blood flow in individual vessel segments can be treated as laminar. When applied to the analysis of regional cerebral blood flow measurements for which the necessary arterial input is evaluated in the carotid arteries, present theory predicts about 20% underestimation, which is in agreement with recent experimental data.
Blood flow is a process of fundamental physiological importance. Today, several imaging modalities of positron emission tomography (PET), magnetic resonance imaging (MRI) and computer assisted tomography (CT) are available for its assessment using different kinds of tracers. Modeling the tracer kinetics in the vasculature is a pivotal component of these methods. The time-dependent tracer concentration in blood is described by the indicator dilution theory introduced by Stewart in his pioneering work in 1893 . The theory has been validated and refined, in particular in Refs – and states the general linear dependence of the local tracer concentration on its input somewhere upstream. In observable terms, the tracer transport manifests itself as delay and dispersion of tracer concentration in the blood stream, which is described by the impulse response function h(t). In the present context, we refer to h(t) as the transport function.
Results are formulated in terms of the transport function, h(t), establishing the linear relation of the tracer concentration time course, , at a location b somewhere in the vasculature to the concentration at an upstream location a:(1)where ⊗ is used as a short notation for the time convolution in what follows. Figure 1 illustrates the model of the arterial tree used in this work. The generation number n = 1 is assigned to the stem vessel, n = 2 to the daughter vessels etc. The arterial tree ends at capillaries that form a mesh network rather than a tree and are outside the scope of the present model.
Two findings are reported here. First, we have shown that experimentally measured tracer concentration in a vessel segment depends on the measurement type and we derived an explicit new form of the transport function for the case of blood collected at an end of a straight cylindrical vessel. Second, we have shown that the approximation of good blood mixing at vessel bifurcations results in a drastically simplified description of the entire arterial tree with explicit results for the case of its self-similar structure. Arterial spin labeling measurements in large brain arteries support the developed model although are not sufficient for an unambiguous verification. Application of this model to cerebral blood flow measurements with a distal arterial input function suggests a 20% underestimation of blood flow, which is in agreement with a recent comparison of cerebral perfusion evaluations with MRI and PET.