In silico simulation of hepatic arteries: An open-source algorithm for efficient synthetic data generation

Abstract

Background: In silico testing of novel image reconstruction and quantitative algorithms designed for interventional imaging requires realistic high-resolution modeling of arterial trees with contrast dynamics. Furthermore, data synthesis for training of deep learning algorithms requires that an arterial tree generation algorithm be computationally efficient and sufficiently random.

Purpose: The purpose of this paper is to provide a method for anatomically and physiologically motivated, computationally efficient, random hepatic arterial tree generation.

Methods: The vessel generation algorithm uses a constrained constructive optimization approach with a volume minimization-based cost function. The optimization is constrained by the Couinaud liver classification system to assure a main feeding artery to each Couinaud segment. An intersection check is included to guarantee non-intersecting vasculature and cubic polynomial fits are used to optimize bifurcation angles and to generate smoothly curved segments. Furthermore, an approach to simulate contrast dynamics and respiratory and cardiac motion is also presented.

Results: The proposed algorithm can generate a synthetic hepatic arterial tree with 40 000 branches in 11 s. The high-resolution arterial trees have realistic morphological features such as branching angles (MAD with Murray's law $=1.2\pm 1.2^o$ ), radii (median Murray deviation $=0.08$ ), and smoothly curved, non-intersecting vessels. Furthermore, the algorithm assures a main feeding artery to each Couinaud segment and is random (variability = 0.98 ± 0.01).

Conclusions: This method facilitates the generation of large datasets of high-resolution, unique hepatic angiograms for the training of deep learning algorithms and initial testing of novel 3D reconstruction and quantitative algorithms designed for interventional imaging.

Keywords: data synthesis; hepatic arterial trees; interventional imaging simulation.

Figure 1:

Graphical depiction of the vessel tree formation by attaching new endpoints, where ⁎ represents endpoints in set $\mathit{P}$ and ${\overrightarrow{\mathit{p}}}_{\mathit{k}}$ is the new endpoint being attached to the tree. The points ${\overrightarrow{\mathit{s}}}_i$ and ${\overrightarrow{\mathit{e}}}_i$ are the start and end points of the replaced vessel branch.

Figure 2:

Graphical depiction of a cubic polynomial fit for branch ${\mathit{c}}_i$ with A) tangent vectors ${\overrightarrow{\mathit{k}}}_{\mathit{s}}$ and ${\overrightarrow{\mathit{k}}}_{\mathit{e}}$ at the start and end points. B) The starting point of the polynomial fit ${\overrightarrow{\mathit{s}}}_{\mathit{i}}^{\mathit{\text{'}}}$ is determined by finding the intersection of ${\overrightarrow{\mathit{k}}}_{\mathit{s}}$ with the edge of the sibling vessel segment. C) The final curved branch with theoretically ideal bifurcation angle.

Figure 3:

A) Hepatic arterial tree generated using straight segments. B) Hepatic arterial tree using cubic polynomial fits to generate smoothly curved segments and realistic bifurcations angles.

Figure 4:

Probability of a branch intersection vs radii of smaller intersecting branch. Bar height is the mean result for 1,000 vessel trees with 10k endpoints and whiskers represent standard deviation.

Figure 5:

A) Simulated hepatic arterial tree (100k endpoints) with color coded Couinaud segments. B) Endpoint distribution for each Couinaud segments main feeding artery. Histograms calculated from 1000 vessel trees with 60k endpoints. Bar plots represent mean values.

Figure 6:

Synthetic hepatic arterial trees generated with a fixed random seed, intersection check and 10k, 50k and 100k endpoints.

Figure 7:

Simulation time (left y-axis) as a function of the number of endpoints simulated. The corresponding radii of the terminal vessel segments for various number of endpoints is shown on the right y-axis. Note, terminal vessel radii were measured from a tree with a proper hepatic radius of 2.0 mm and a $\gamma =2/3$. Error bars represent standard deviation.

Figure 8:

A) User-defined concentration-time curve, B) measured proximal and distal time intensity curves from C) simulated contrast dynamics in synthetic hepatic arteries and D) a real hepatic angiogram showing early, middle and late stage contrast enhancement.

Figure 9:

Synthetic hepatic angiograms at A) max expiration, B) 50% inspiration and C) max inspiration. Callout box in A) visually demonstrates image noise texture.