1 Introduction

In aging societies, cardiovascular conditions such as aortic aneurysms and aortic dissections persist as life-threatening diseases. Moreover, congenital diseases such as hypoplastic left heart syndrome constitute an important issue for our society. In recent years, patient-specific simulations have become common in the biomedical engineering field. Several mathematical viewpoints are expected to be added and to play important roles in this context. For instance, geometrical characterization of blood vessels, which vary widely among individuals, provides useful information to medical sciences. Differences in blood vessel morphology give rise to different flow characteristics, which cause different stress distributions and outcomes. Therefore, characterization of these vessels’ respective morphologies represents an important clinical question. Our objective in this study is to understand possible mechanisms connecting geometrical characteristics and stress distributions through flow behaviors. The studies presented in this paper are parts of a CREST [1] framework supported by the Japan Science and Technology Agency in a strategic area for promoting collaboration between mathematical science and other scientific fields.

2 Numerical Methods and Results

2.1 Governing Equations

We adopted incompressible Navier–Stokes equations as governing equations.

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\displaystyle \partial u_i}{\displaystyle \partial t} + u_j \frac{\displaystyle \partial u_i}{\displaystyle \partial x_j}= -\frac{\displaystyle 1}{\displaystyle \rho }\frac{\displaystyle \partial p}{\displaystyle \partial x_i} +\nu \frac{\displaystyle \partial }{\displaystyle \partial x_j} \left( \frac{\displaystyle \partial u_i}{\displaystyle \partial x_j} +\frac{\displaystyle \partial u_j}{\displaystyle \partial x_i}\right) , \\ \frac{\displaystyle \partial u_j}{\displaystyle \partial x_j} = 0 \\ \end{array} \right. \;\; \text{ in } \;\; \Omega \times \left( 0,T\right) . \end{aligned}$$
(1)

In those equations, \(t, u_i \, (i=1,2,3),\, p,\, \rho ,\) and \(\nu \) respectively represent time, velocity, pressure, density, and the kinematic viscosity of blood. We assumed that blood can be regarded as a Newtonian fluid in large arteries. Several numerical results with different numerical methods are presented in the following subsections. Finite difference method is used in Sect. 2.2, applied for blood flows in a thoracic aorta and for flows in simple spiral tubes to examine torsion effects. Then, finite element method is applied in Sect. 2.3 where fluid structure interaction (FSI) is considered and some flow mechanisms in a configuration after Norwood surgery are examined.

2.2 Finite Difference Approximation

2.2.1 Visualization of Flows in a Thoracic Aorta

Effects of curvature on flows in curved tubes have been discussed extensively in earlier studies [2,3,4]. When a tube has curvature, centrifugal force acts in the opposite direction, depending on the axial component of the velocity. Subsequently, secondary flow occurs on the cross-section and forms a set of twin vortices called Dean’s vortices, thereby playing an important role in blood flow through the aortic arch where a strong curvature exists.

Figure 1 presents streamlines that can be visualized based on numerical results obtained through an earlier study [5]. We assumed a blood vessel as a rigid body and applied finite-difference method on a centerline-fitted curvilinear coordinate system, where the centerlines and cross-sections were extracted from patient-specific CT scans of patients with aortic aneurysms. Incompressible Navier–Stokes equations were solved numerically with a boundary condition for the inflow velocity profile given by a phase-contrast MRI measurement.

Figure 1a presents streamlines through the whole thoracic aorta at peak systolic phase. Circulation in the aneurysm is apparent. Figure 1b shows the Dean’s vortices on the aortic arch superimposed to the main axial flow. In Fig. 1c, a spiral flow is apparent in the descending aorta.

Fig. 1
figure 1

Instantaneous streamlines

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Helicity, \(\boldsymbol{u}\cdot \left( \nabla \times \boldsymbol{u}\right) \), represents swirling flow regions of opposite signs. Figure 2a depicts helicity isosurfaces of a positive and a negative values, which shows Dean’s vortices generated at the aortic arch and subsequently flowing down to the descending aorta. In Fig. 2b, an isosurface of the second largest eigenvalue \(\lambda _2\) of \(S^2 + \Omega ^2\), where S and \(\Omega \) respectively represent symmetric and antisymmetric parts of the velocity gradient tensor, also shows a swirling flow region [6]. Enstrophy, \(\left| \nabla \times \boldsymbol{u}\right| ^2\), exhibits the strength of vorticity in Fig. 2c. In Fig. 2b, c, colors of isosurfaces show \(\lambda _2\) values.

2.2.2 Effects of Torsion in Simple Spiral Tubes

We also examined the effects of torsion using a pulsating flow in simple spiral tubes, as shown in [5]. Torsion of a three-dimensional curve is defined through the Frenet–Serret formula shown below.

$$\begin{aligned} \frac{d}{ds} \left( \begin{array}{c} \boldsymbol{t}\\ \boldsymbol{n}\\ \boldsymbol{b}\\ \end{array} \right) = \left( \begin{array}{rrr} 0 &{} \chi &{} 0 \\ -\chi &{} 0 &{} \tau \\ 0 &{} -\tau &{} 0 \\ \end{array} \right) \left( \begin{array}{c} \boldsymbol{t}\\ \boldsymbol{n}\\ \boldsymbol{b}\\ \end{array} \right) . \end{aligned}$$
(2)

Therein, \(\chi \) and \(\tau \) respectively represent curvature and torsion, where \(\boldsymbol{t}\), \(\boldsymbol{n}\), and \(\boldsymbol{b}\) respectively denote the tangential, normal, and bi-normal vectors.

Fig. 2
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Several fluid dynamics quantities

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Fig. 3
figure 3

Secondary flows in a zero-torsion tube

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Fig. 4
figure 4

Secondary flows in a nonzero-torsion tube

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Figures 3 and 4 portray secondary flows, which are obtainable by subtracting the main axial flow from the total flow velocities at peak systolic, late systolic, and late diastolic phases, respectively, for zero-torsion and nonzero-torsion cases. When the torsion is zero, the secondary flow is invariably symmetric. However, when the torsion is not zero, merging phenomena occur; one large vortex persists in a diastolic phase. Such difference brings about differences in torque exerted on vessel walls.

2.3 Finite Element Approximation

2.3.1 Torsion Effects on Flows in the Thoracic Aorta

Next we consider fluid–structure interaction (FSI) to examine torsion effects using patient-specific morphologies [7]. Here, FSI analysis is handled with the Sequentially-Coupled Arterial FSI (SCAFSI) technique [8] because the class of an FSI problem here has temporally–periodic FSI dynamics. Fluid mechanics equations are solved using Space–Time Variational Multiscale (ST-VMS) method [9,10,11]. First, we carry out structural mechanics computation to assess arterial deformation under an observed blood pressure profile in a cardiac cycle. Then we apply fluid mechanics computation over a mesh that moves to follow the lumen as the artery deforms. These steps are iterated where the stress obtained in fluid mechanics computation is used for the next structural mechanics computation. To assess torsion effects, the torsion-free model geometry is generated by projecting the original centerline to its averaged plane of curvature, as presented in Fig. 5.

Fig. 5
figure 5

Projected and original shapes

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Figure 6 presents secondary flows. On the left-hand side (projected shape), symmetric Dean’s vortices are apparent, although they are not visible on the right-hand side (original shape), similarly to the simple spiral tubes in Fig. 4.

Fig. 6
figure 6

Secondary flows in projected and original shapes

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Next we compare the wall shear stresses (WSS) patterns corresponding to the projected and the original geometries to examine the influence of torsion. Figure 7 presents WSS at peak systolic phase. In the projected torsion-free shape, a high WSS region is apparent at the aortic arch, which results from the strong Dean’s twin vortices, although it is not apparent in the original shape with torsion there.

Fig. 7
figure 7

Wall shear stresses at peak systolic phase

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2.3.2 Flow Mechanism in Morphology After Norwood Surgery

This subsection presents examples of patient-specific blood flow simulations at an anastomosis site after Norwood surgery for hypoplastic left heart syndrome. Our target is the geometry surrounding an anastomosis site of the aortic arch and pulmonary artery after Norwood surgery, which is one step taken during surgeries for hypoplastic left heart syndrome. The target geometry was extracted from a CT scan with boundary conditions obtained from ultrasound measurements. Here, we again adopt the rigid body assumption, i.e., not considering fluid–structure interactions. The SUPG/PSPG stabilized finite element formulation is used, which is solved on P1/P1 elements.

Fig. 8
figure 8

Streamlines at an anastomosis site after Norwood surgery

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Fig. 9
figure 9

Front and back views of streamlines

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Figure 8a portrays instantaneous streamlines at the peak systolic phase, whereas Fig. 8b depicts the energy-dissipation distribution. Energy dissipation is a clinically important quantity because it imposes a load on the heart directly [12]. In Fig. 8b, high energy dissipation is apparent at the anastomosis site, which can be understood straightforwardly because the velocity is extremely high there. Although high energy dissipation is also apparent in the descending aorta, it cannot be qualified straightforwardly. This dissipation apparently derives from spiral flow there, which is generated at the aortic arch immediately after blood passes out of the thin anastomosis channel, as shown in Fig. 9. Here, a relation can be found between morphology and energy dissipation patterns through flow structures.

3 Conclusions

We have presented some relations between geometrical characteristics of blood vessels and flow behaviors. Those relations are expected to explain how and why vessel morphologies affect WSS distributions and energy dissipations. As described in Sect. 2.2, vessel curvature induces Dean’s vortices as a secondary flow by centrifugal force, thereby creating strong WSS there. Moreover, Dean’s vortices show different behaviors depending on the existence of torsion. In the example from a Norwood surgery morphology, an energy dissipation pattern on the descending aorta can be explained through flow structures. As a next step, predictions based on geometrical characteristics of blood vessels are expected to contribute to better risk assessments and surgery planning through mathematical modellings and numerical simulations.