Modeling of the contrast-enhanced perfusion test in liver based on the multi-compartment flow in porous media

Abstract

The paper deals with modeling the liver perfusion intended to improve quantitative analysis of the tissue scans provided by the contrast-enhanced computed tomography (CT). For this purpose, we developed a model of dynamic transport of the contrast fluid through the hierarchies of the perfusion trees. Conceptually, computed time-space distributions of the so-called tissue density can be compared with the measured data obtained from CT; such a modeling feedback can be used for model parameter identification. The blood flow is characterized at several scales for which different models are used. Flows in upper hierarchies represented by larger branching vessels are described using simple 1D models based on the Bernoulli equation extended by correction terms to respect the local pressure losses. To describe flows in smaller vessels and in the tissue parenchyma, we propose a 3D continuum model of porous medium defined in terms of hierarchically matched compartments characterized by hydraulic permeabilities. The 1D models corresponding to the portal and hepatic veins are coupled with the 3D model through point sources, or sinks. The contrast fluid saturation is governed by transport equations adapted for the 1D and 3D flow models. The complex perfusion model has been implemented using the finite element and finite volume methods. We report numerical examples computed for anatomically relevant geometries of the liver organ and of the principal vascular trees. The simulated tissue density corresponding to the CT examination output reflects a pathology modeled as a localized permeability deficiency.

Appendix

1.1 Computing parameters of the perfusion model

Parameters ${\varvec{K}}^i$ and $G_j^i$ can be determined using the perfusion tree models $\mathcal {T}_g$ represented by line segments $\mathbf{s }$, each associated with a vessel of length $\ell$ and diameter d. Each vessel is labelled by the hierarchical parameter $\xi _0 \in ]0,1[$. We shall consider a finite number of disjoint intervals ${{\varDelta }}^h\xi _0 \subset ]0,1[$, thus $\sum _h {{\varDelta }}^h\xi _0 = 1$ which constitute the hierarchies h, see Sect. 2.1.

Fig. 21

Averaging scheme for computing tensor $K_{ij}$ associated with a given compartment, see formula (30)

1.2 The permeability tensor ${\varvec{K}}(x)$

Let $x \in {{\varOmega }}$ and $\mathcal {O}(x) \subset {{\varOmega }}$ be the averaging volume. By $I(x,{{\varDelta }}\xi _0)$ we denote the index set of all vessels k which intersect $\mathcal {O}(x)$, have the hierarchical parameter within the interval ${{\varDelta }}\xi _0$, and which belong to the same perfusion tree $\mathcal {T}_g$ (i.e. either the portal $g=P$, or hepatic $g=H$ veins). Further, by ${{\varDelta }}\mathbf{s }^k$ we denote the portion of line segment $\mathbf{s }^k$ intersecting $\mathcal {O}(x)$ whereby ${{\varDelta }}\ell ^k = \ell ^k |{{\varDelta }}\mathbf{s }^k|/|\mathbf{s }^k|$ denote the proportional length, see Fig. 21. According to the theoretical work Huyghe and Campen (1995), the effective permeability associated with all vessels in $I(x,{{\varDelta }}\xi _0)$ can be computed by the following formula:

$$\begin{aligned} K_{ij}(x) = \frac{\pi }{128 \mu |\mathcal {O}(x)| {{\varDelta }}\xi _0} \sum _{k \in I(x,{{\varDelta }}\xi _0)} \frac{(d^k)^4 {{\varDelta }}s_i^k {{\varDelta }}s_j^k}{{{\varDelta }}\ell ^k}\;. \end{aligned}$$

(30)

In the FE model, the permeability is defined element-wise, thus the averaging volumes $\mathcal {O}(x^e)$ are balls defined in the center $x^e$ of each element. Since we use the artificially generated trees, ${{\varDelta }}\ell ^k = |{{\varDelta }}\mathbf{s }^k|$. It is worthy of noting that this definition of ${\varvec{K}}(x)$ disregards connectivity of segments ${{\varDelta }}\mathbf{s }^k$ within $\mathcal {O}(x^e)$.

1.3 The local perfusion coefficients $G_j^i(x)$

In contrast with the permeability tensor, coefficients cannot be determined merely using the geometrical data describing the perfusion tree. A flow model is needed to establish fluxes in all segments of the tree. Coefficients $G_j^i$ are relevant to the decomposition of the tree $\mathcal {T}$ into compartments according to the hierarchical intervals ${{\varDelta }}\xi _0^i$. Let ${Q}^k$ be flux in the line segment $\mathbf{s }^k$. We consider such bifurcations belonging to $\mathcal {O}(x)$ which join vessels associated with different compartments i and j. Then by $Y_j^i$ we denote the index set of all vessels k connecting compartments i and j. Further we compute the flux $Q_j^i(x)$ between the two compartments and the average pressures ${\overline{p^i}}(x)$, ${\overline{p^j}}(x)$ by the following formulae:

$$\begin{aligned} {\overline{Q}}_j^i(x) = \frac{1}{|\mathcal {O}(x)|} \sum _{k \in Y_j^i} {Q}^k\;,\quad {\overline{p^i}}(x) = \frac{\sum _{k \in I(x,{{\varDelta }}\xi _0)} P^k V^k}{\sum _{k \in I(x,{{\varDelta }}\xi _0)} V^k}\;, \end{aligned}$$

(31)

where $V^k = \pi (d^k)^2\ell ^k/4$ is the volume of the vessel k and $P^k$ is the average pressure in the the same vessel. Then we can determine the local perfusion coefficients using the obvious formula (a tolerance $\epsilon > 0$ to be chosen):

$$\begin{aligned} G_j^i(x) = \left\{ \begin{array}{ll} \frac{{\overline{Q}}_j^i(x)}{|{\overline{p^i}}(x) - {\overline{p^j}}(x)|} &{} \text{ if } |{\overline{p^i}}(x) - {\overline{p^j}}(x)| \ge \epsilon \;,\\ 0 &{} \text{ otherwise }\;. \end{array} \right. \end{aligned}$$

(32)