1 Introduction

Coronary arteries, attached to the heart surface, supply blood to the heart. “Atherosclerotic narrowing” of coronary arteries [1] may lead to the heart attack or “sudden cardiac death” [2]. Such artery diseases are the leading cause of morbidity and mortality among adults today [3]. Patient-specific computational flow analysis of coronary arteries can help doctors explain pathology of atherosclerosis and make treatment decisions. It has been known for quite a while (see, for example, [4,5,6]) that the wall shear stress (WSS) plays a significant role in arterial diseases, and its quantification can help identify the high-probability regions of the disease. Possible connection between the atherosclerosis and the WSS has been reported in several papers [7, 8]. Patient-specific computational flow analysis of coronary arteries is appealing in quantifying atherosclerosis diseases based on hemodynamic factors. In this class of computational flow analysis, we use the Space–Time Variational Multiscale (ST-VMS) method [9,10,11], which serves as the core method, and the ST Isogeometric Analysis (ST-IGA) [9, 12, 13]. The ST-VMS and ST-IGA give us increased accuracy in the flow solution. The ST-IGA also gives us a smoother and more accurate representation of the lumen geometry and motion. With cubic NURBS in time, we obtain a continuous acceleration from the lumen-motion representation. The higher-order representation in time might introduce sudden, unrealistic lumen motion. Here we focus focus on making the motion representation anatomically realistic.

1.1 Moving-boundary problems

It was shown as early as 2004 [4,5,6] that there are significant differences in the WSS obtained from blood flow computations with the rigid- and flexible-artery models. This created a strong incentive to conduct blood flow computations with flexible-artery models. Furthermore, it was recognized as early as 2008 [14] that high-refinement mesh layers near the lumen boundary are essential in high-resolution representation of the boundary layers and in accurate calculation of the WSS. In computation of flows with moving boundaries and interfaces (MBI), including fluid–structure interaction (FSI), a moving-mesh method enables mesh-resolution control near the interface and, consequently, high-resolution representation of the boundary layer. Because the coronary arteries undergo large motions during the cardiac cycle, the moving-mesh method will also need to be able to handle large lumen motions. Both the ST [9, 15] and Arbitrary Lagrangian–Eulerian (ALE) [16] methods are moving-mesh methods, with the ALE being the earlier and more commonly used one.

The Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method [15, 17], the precursor of the ST-VMS, was introduced for FSI and MBI computations. Because the stabilization components of the original DSD/SST are the Streamline-Upwind/Petrov-Galerkin (SUPG) [18] and Pressure-Stabilizing/Petrov-Galerkin [15, 19] stabilizations, the method is now also called “ST-SUPS.” The VMS components of the ST-VMS are from the residual-based VMS method [20,21,22,23]. The ST-VMS has two more stabilization terms beyond those the ST-SUPS has. The ALE-VMS method [23, 24] is the VMS version of the ALE. It succeeded the ST-SUPS and ALE-SUPS [25] methods and preceded the ST-VMS.

In the computational flow analysis presented here, the ST framework provides higher-order accuracy in a general context. The VMS feature of the ST-VMS addresses computational challenges associated with multiscale nature of unsteady flow in the arteries. The moving-mesh feature of the ST framework enables high-resolution computation near the lumen boundary as the artery moves.

1.2 ST-IGA

The ST-IGA was introduced in [9]. It is the integration of the ST framework with isogeometric discretization [24, 26, 27]. First computations with the ST-VMS and ST-IGA were reported in [9] in a 2D context, with IGA basis functions in space for flow past an airfoil, and in both space and time for the advection equation. The stability and accuracy analysis given in [9] for the advection equation showed that using higher-order basis functions in time would be essential in getting full benefit out of using higher-order basis functions in space.

As pointed out in [9, 10] and demonstrated in [12, 28, 29], higher-order NURBS basis functions in time provide a more accurate representation of the motion of the solid surfaces and a mesh motion consistent with that. They also provide more efficiency in temporal representation of the motion and deformation of the volume meshes, and better efficiency in remeshing. The ST framework and NURBS in time also enable, with the “ST-C” method, extracting a continuous representation from the computed data and, in large-scale computations, efficient data compression [11, 30,31,32,33,34,35].

In the computational flow analysis presented here, the ST-IGA gives us increased accuracy in the flow solution and a smoother and more accurate representation of the lumen geometry and motion.

1.3 Motion representation

The ST-IGA with IGA basis functions in time provides a good framework for smooth motion representation. With cubic NURBS in time, the representation gives us continuous acceleration [12], which is key to obtaining a reasonable time-dependent behavior from the calculation of the fluid mechanics forces acting on the moving surface. The desirable features of the ST-IGA have been used in many 3D computations. The classes of problems solved are flapping-wing aerodynamics for an actual locust [12, 23, 28, 36], bioinspired MAVs [29, 37,38,39] and wing-clapping [40, 41], separation aerodynamics of spacecraft [42], aerodynamics of horizontal-axis [38, 39, 43, 44] and vertical-axis [45,46,47] wind-turbines, thermo-fluid analysis of ground vehicles and their tires [11, 31], thermo-fluid analysis of disk brakes [32], flow-driven string dynamics in turbomachinery [33,34,35], and flow analysis of turbocharger turbines [13, 48,49,50].

In the locust flapping-wing aerodynamics [12, 23, 28, 36], for example, the wing-motion data in discrete form, extracted from the high-speed video cameras of the wind tunnel, was least-squares projected to cubic NURBS representation in time. We use the same method here in obtaining the cubic NURBS representation of the lumen motion from the medical-image data in discrete form. With that, we have a continuous acceleration in the lumen-motion representation. However, because of the higher-order nature of the NURBS representation, sometimes we might see sudden unrealistic changes in time. This would degrade the flow solution quality. Here, we propose a remedy for that.

1.4 Penalty least-squares projection

Starting with the method of least-squares projection to cubic NURBS representation given in [12, 28], we introduce a penalty least-squares (PLS) projection method to make the lumen-motion representation anatomically realistic in patient-specific computational flow analysis of coronary arteries with time-dependent medical-image data. We supplement the method given in [12, 28] with two penalty terms, corresponding to the first and second time derivatives of the control-point trajectories. The penalty terms help us avoid the sudden unrealistic changes and obtain an anatomically realistic NURBS representation of the lumen motion. We, of course, still retain the smooth representation that comes with the NURBS basis functions.

1.5 Outline of the remaining sections

In Sect. 2, we provide the time-dependent anatomical model in terms of the lumen motion, and a mesh motion consistent that. Making the lumen motion anatomically realistic with the PLS projection method is described in Sect. 3. The fluid mechanics computation method and the computational conditions are described in Sect. 4. The results are presented in Sect. 5, and the concluding remarks are given in Sect. 6.

2 Time-dependent anatomical model

We obtain the lumen motion from the coronary artery centerline motion in [1], which is the step-populated and somewhat smoothed version of the centerline motion [51] extracted from MRI data. The cardiac cycle is \(T= 1~\mathrm {s}\). There are 400 steps in a cardiac cycle, making the time-step size \(2.5~\mathrm {ms}\). The lumen geometry, represented by the lumen boundary mesh, is generated as part of the lumen NURBS mesh. In generating the quadratic NURBS mesh for each time step, we first we create a quadrilateral control mesh for the inlet section and sweep it along the centerline (see [52, 53]). Figure 1 shows the lumen geometry at three instants.

Fig. 1
figure 1

Lumen geometry at three instants during the cardiac cycle

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Fig. 2
figure 2

Control mesh after the Laplacian smoothing. We identify two layered zones: external (purple) and internal (yellow). (Color figure online)

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Next we apply Laplacian smoothing [54] to improve the control mesh quality. Figure 2 shows the control mesh after the Laplacian smoothing. To have higher-resolution representation of the boundary layers, we increase the mesh refinement. We make the number of external-zone layers 4, with a progression factor of 1.8, and the number of internal-zone layers also 4, by evenly splitting the existing layers. The refined mesh has 44,928 control points and 42,315 elements. Figure 3 shows the control mesh after the refinement. Figure 4 shows the control mesh at \(t = 0\).

Fig. 3
figure 3

Control mesh after the refinement for higher-resolution representation of the boundary layers

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Fig. 4
figure 4

Control mesh at \(t = 0\)

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3 Anatomically realistic representation of the lumen motion

Representing the medical-image data given in discrete form with linear basis functions in time does not result in a smooth representation. The first and second derivatives of the control-point trajectories, corresponding to the velocity and acceleration, cannot be expected to be continuous. Figure 5 shows an example. A continuous acceleration is key to obtaining a reasonable time-dependent behavior from the calculation of the fluid mechanics forces acting on the moving arterial wall. We need a smoother representation of the lumen motion.

Fig. 5
figure 5

A control point “A” (top) and the time derivative of the x component of its position (bottom) in linear temporal representation. We see a jump at t = 0.12 s, and a kink at t = 0.87 s

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3.1 NURBS representation of the lumen motion

The ST-IGA with cubic NURBS in time gives us a representation with continuous acceleration [12, 28]. In addition, with quadratic NURBS in space, we have increased accuracy in the flow solution and a smoother and more accurate representation of the lumen geometry. Similar to what was done in [12, 28] for the locust flapping-wing aerodynamics, the medical-image data in discrete form is least-squares projected to cubic NURBS representation in time. As a consequence of the acceleration being continuous, the WSS will be continuous.

Because of the higher-order nature of the NURBS representation, sometimes we might see sudden unrealistic changes in time. We need to make the representation anatomically realistic.

3.2 Periodic data

The lumen motion is roughly periodic. We generate from that a periodic data set, where the first and last points of the cycle are colocated and have the desired continuity. Thus, a single cycle of lumen motion can be repeated to produce as many cardiac cycles as needed.

To obtain a periodic data set, after the least-squares projection, we extract one cardiac cycle. To maintain continuity, the control points corresponding to the knot at the beginning of the cardiac cycle are colocated with the control points corresponding to the knot at the end of the cardiac cycle (three control points correspond to a given knot). To obtain such repetition, we average those control points. Finally, we insert knots to extract a single cycle. See [28] for more on the process, including figures illustrating the averaging.

3.3 PLS projection

A spatial-control point \(\mathbf{x}_A\) will be represented as

$$\begin{aligned} {\mathbf {x}}_A =\sum \limits _{\alpha =0}^{n_{{\mathrm {ct}}}-1}N_{\alpha ,3}({\vartheta }) \mathbf {x}_{A}^\alpha , \end{aligned}$$
(1)

where \(\mathbf {x}_{A}^\alpha \) is the temporal-control point associated with the NURBS basis function \(N_{\alpha ,3}({\vartheta })\), with \({\vartheta }\) representing the NURBS parametric space. The basis functions are defined over the parametric space given by the open knot vector \(\{{\vartheta }_1,~\ldots ,~{\vartheta }_{n_{{\mathrm {kt}}}}\}\), where \(n_{{\mathrm {ct}}}\) and \(n_{{\mathrm {kt}}}\) are the number of temporal-control points and knots.

The PLS projection is derived from the functional

$$\begin{aligned} E_A(\mathbf {x}_A^\alpha )&=\sum \limits _{k=0}^{m}\omega _k\left\| \mathbf {x}_A(\tilde{{\vartheta }}_k) -\tilde{\mathbf {x}}_A(\tilde{{\vartheta }}_k)\right\| ^2 \nonumber \\&\quad +\lambda _1\sum \limits _{\alpha =1}^{n_{{\mathrm {ct}}}-1}\left\| D_{t} \mathbf {x}^\alpha _{A}\right\| ^2 +\lambda _2\sum \limits _{\alpha =2}^{n_{{\mathrm {ct}}}-1}\left\| D_{tt} \mathbf {x}^\alpha _{A}\right\| ^2, \end{aligned}$$
(2)

where

$$\begin{aligned} D_{t} \mathbf {x}^\alpha _{A}&=\mathbf {x}^\alpha _{A} - \mathbf {x}^{\alpha -1}_{A}, \end{aligned}$$
(3)
$$\begin{aligned} D_{tt} \mathbf {x}^\alpha _{A}&=\mathbf {x}^\alpha _{A} - 2 \mathbf {x}^{\alpha -1}_{A}+\mathbf {x}^{\alpha -2}_{A}, \end{aligned}$$
(4)

m is the number of time steps for the medical-image data, and \(\omega _k\), \(\lambda _1\) and \(\lambda _2\) are the PLS projection parameters. The parametric value \(\tilde{{\vartheta }}_k\) corresponds to the time step k. It is related to the temporal knot vector \(\{{\vartheta }_1,~\ldots ,~{\vartheta }_{n_{{\mathrm {kt}}}}\}\) by the formula \(\tilde{{\vartheta }}_k={\vartheta }_1+k\frac{{\vartheta }_{n_{{\mathrm {kt}}}}-{\vartheta }_1}{m}\) (\(k=0,~\ldots ,~m\)).

The first summation in Eq. (2) represents the discrete least-squares projection, with the option of having different weights for different time steps of the medical-image data. The second and third summations represent the penalty terms corresponding to the first and second time derivatives of the control-point trajectories.

3.4 Selection of the PLS projection parameters

When \(\lambda _1=0\), \(\lambda _2=0\), and \(\omega _k=1\), the PLS projection becomes just a least-squares projection, which we will label “LS” for the purpose of presenting the results. When \(\omega _k=1\) and only one of the parameters \(\lambda _1\) and \(\lambda _2\) is nonzero, Eq. (2) gives the penalty B-splines [55]. In the computations reported in this article, \(\lambda _1=1\), \(\lambda _2=1\), and \([\omega _k]\) = \(\begin{bmatrix} 1000,&1000,&1,&\ldots&1,&1000,&1000\end{bmatrix}\). The number of elements in time is 240, which represents the three cycles we use to extract the middle cycle and generate periodic data as described in Sect. 3.2.

Figure 6 shows the position and acceleration of a spatial-control point, from the medical-image data and from the LS and PLS projections.

Fig. 6
figure 6

Position and acceleration of a spatial-control point, from linear representation and cubic NURBS representation with LS and PLS projections

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4 Fluid mechanics computation

The arterial diameter D at the inflow is \(4.02~\mathrm {mm}\) and the average flow rate Q is \(1.08~\mathrm {cm^3/s}\). The average shear rate in the artery can be estimated from the laminar-flow approximation as \(\dot{\gamma } = 32Q/(\pi D^3) = 169~\mathrm {s}^{-1}\). As explained in [56, 57], the viscosity of the blood can be approximated as a constant when the shear rate is higher than 150 \(\mathrm {s}^{-1}\). Therefore we assume the blood to be Newtonian here. The density and kinematic viscosity are 1,060 kg/m\(^3\) and \(4.0{\times }10^{-6}\) \(\mathrm {m^2/s}\).

The governing equations are the Navier–Stokes equations of incompressible flow:

$$\begin{aligned} \rho \left( \frac{\partial {\mathbf {u}}}{\partial t} +{\mathbf {u}}\cdot \pmb {\nabla }{\mathbf {u}}- {\mathbf {f}}\right) -\pmb {\nabla }\cdot \pmb {\sigma }&= \mathbf{0}, \end{aligned}$$
(5)
$$\begin{aligned} \pmb {\nabla }\cdot {\mathbf {u}}&= 0, \end{aligned}$$
(6)

where \(\rho \), \({\mathbf {u}}\) and \({\mathbf {f}}\) are the density, velocity and the body force. The stress tensor is defined as \(\pmb {\sigma }({\mathbf {u}}, p) = -p{\mathbf {I}}+ 2 \mu \pmb {\varepsilon }({\mathbf {u}})\), where \(p\) is the pressure, \({\mathbf {I}}\) is the identity tensor, \(\mu = \rho \nu \) is the viscosity, \(\nu \) is the kinematic viscosity, and \(\pmb {\varepsilon }({\mathbf {u}}) = \left( \pmb {\nabla }{\mathbf {u}}+\left( \pmb {\nabla }{\mathbf {u}}\right) ^T\right) /2\) is the strain-rate tensor.

4.1 ST-VMS

The ST-SUPS and ST-VMS formulations (see for example [9,10,11, 15, 17, 58]) are written over a sequence of N ST slabs \(Q_n\), where \(Q_n\) is the slice of the ST domain between the time levels \(t_n\) and \(t_{n+1}\), and \(P_n\) is the lateral boundary of \(Q_n\). At each time step, the integrations are performed over \(Q_n\). The essential and natural boundary conditions are enforced over \((P_n)_\mathrm{{g}}\) and \((P_n)_\mathrm{{h}}\), the complementary subsets of the lateral boundary of the ST slab. The ST basis functions are continuous within a ST slab, but discontinuous from one ST slab to another. The notation \((\cdot )_n^-\) and \((\cdot )_n^+\) will denote the function values at \(t_n\) as approached from below and above. Each \(Q_n\) is decomposed into elements \(Q_n^e\), where \(e=1,2,\ldots ,(n_{\mathrm{el}})_n\). The subscript n used with \(n_{\mathrm{el}}\) is for the general case where the number of ST elements may change from one ST slab to another. The ST-VMS formulation is given as

$$\begin{aligned}&\int _{{Q_{n}}} {\mathbf {w}}^h \cdot \rho \left( \frac{\partial {\mathbf {u}}^h}{\partial t} + {\mathbf {u}}^h \cdot \pmb {\nabla }{\mathbf {u}}^h - {\mathbf {f}}^h \right) \mathrm{d}Q\nonumber \\&\qquad + \int _{{Q_{n}}} \pmb {\varepsilon }({\mathbf {w}}^h) : \pmb {\sigma }({\mathbf {u}}^h,p^h) \mathrm{d}{Q}-\int _{\left( P_{n}\right) _{\mathrm{h}}} {\mathbf {w}}^h \cdot {\mathbf{h}}^h \mathrm{d}{P}\nonumber \\&\qquad +\int _{{Q_{n}}} q^h \pmb {\nabla }\cdot {\mathbf {u}}^h \mathrm{d}{Q}+ \int _{\varOmega _{n}} ({\mathbf {w}}^h)_{n}^{+} \cdot \rho \left( ({\mathbf {u}}^h)_{n}^+ - ({\mathbf {u}}^h)_{n}^- \right) \mathrm{d}\varOmega \nonumber \\&\qquad + \sum _{e=1}^{(n_{\mathrm{el}})_n} \int _{{Q_{n}^{e}}} \frac{\tau _{\mathrm{SUPS}}}{\rho } \left[ \rho \left( \frac{\partial {\mathbf {w}}^h}{\partial t} + {\mathbf {u}}^h \cdot \pmb {\nabla }{\mathbf {w}}^h \right) \right. \nonumber \\&\quad \qquad \left. +\pmb {\nabla }q^h \right] \cdot {\mathbf {r}}_{\mathrm{M}}({\mathbf {u}}^h,p^h) \mathrm{d}{Q}\nonumber \\&\qquad + \sum _{e=1}^{(n_{\mathrm{el}})_n} \int _{{Q_{n}^{e}}} \nu _{\mathrm{LSIC}}\pmb {\nabla }\cdot {\mathbf {w}}^h \rho {r}_{\mathrm{C}}({\mathbf {u}}^h) \mathrm{d}{Q}\nonumber \\&\qquad -\sum _{e=1}^{(n_{\mathrm{el}})_n} \int _{{Q_{n}^{e}}} \tau _{\mathrm{SUPS}}{\mathbf {w}}^h \cdot \left( {\mathbf {r}}_{\mathrm{M}}({\mathbf {u}}^h,p^h) \cdot \pmb {\nabla }{\mathbf {u}}^h \right) \mathrm{d}{Q}\nonumber \\&\qquad -\sum _{e=1}^{(n_{\mathrm{el}})_n} \int _{{Q_{n}^{e}}} \frac{\tau _{\mathrm{SUPS}}^2}{\rho } {\mathbf {r}}_{\mathrm{M}}({\mathbf {u}}^h,p^h) \cdot \left( \pmb {\nabla }{\mathbf {w}}^h\right) \cdot {\mathbf {r}}_{\mathrm{M}}({\mathbf {u}}^h,p^h) \mathrm{d}{Q}\nonumber \\&\quad = 0. \end{aligned}$$
(7)

Here

$$\begin{aligned} {\mathbf {r}}_{\mathrm{M}}({\mathbf {u}}^h, p^h)&= \rho \left( \frac{\partial {\mathbf {u}}^h}{\partial t} + {\mathbf {u}}^h \cdot \pmb {\nabla }{\mathbf {u}}^h - {\mathbf {f}}^h \right) - \pmb {\nabla }\cdot \pmb {\sigma }({\mathbf {u}}^h, p^h), \end{aligned}$$
(8)
$$\begin{aligned} {r}_{\mathrm{C}}({\mathbf {u}}^h)&= \pmb {\nabla }\cdot {\mathbf {u}}^h \end{aligned}$$
(9)

are the residuals of the momentum equation and incompressibility constraint, and \(\tau _{\mathrm{SUPS}}\) and \(\nu _{\mathrm{LSIC}}\) are the stabilization parameters. There are many ways of defining these stabilization parameters (for examples, see [11, 17, 43, 58,59,60,61]). Here the stabilization parameters are those given by Eqs. (2.4)–(2.9) in [45]. We calculate the WSS by using the expression given by Eq. (5.119) in [23].

Fig. 7
figure 7

Volumetric flow rate

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4.2 Boundary and starting conditions

At the inflow boundary, we specify the velocity profile as a function of time. The velocity profile is generated by applying the Womersley solution [62, 63] with 400 Fourier coefficients. The corresponding time-dependent volumetric flow rate is shown in Fig. 7. The Reynolds number based on this flow rate varies from 45 to 300. The inflow data is least-squares projected to cubic NURBS representation in time. At the outflow the boundary condition is stress-free. The inflow and outflow boundaries are indicated in Fig. 8. We perform the flow computation with two different representations of the lumen motion: linear representation in time and cubic NURBS representation in time with PLS projection. We start the lumen motion suddenly at some instant in the cardiac cycle and expect that the solution will settle within one or two cardiac cycles. We compute for three cardiac cycles.

Fig. 8
figure 8

Inflow (red) and outflow (yellow) boundaries. (Color figure online)

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5 Results

First we compare cubic-NURBS lumen-motion representation with LS and PLS projections. After that we compare the flow computations based on lumen-motion with linear and PLS-projected cubic NURBS representations.

5.1 Cubic-NURBS lumen-motion representation with LS and PLS projections

Fig. 9
figure 9

Lumen volume and its first and second time derivatives, based on linear representation and cubic NURBS representation with LS and PLS projections

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Figure 9 shows the lumen volume and its first and second time derivatives, based on linear representation and cubic NURBS representation with LS and PLS projections. Cubic NURBS representation, as expected, with both LS and PLS projections, brings a smoothness that we cannot get from linear representation. For the first derivative, with the PLS projection we circumvent the sudden, unrealistic changes introduced by the higher-order representation at around 0.12 s. We make a similar observation for the second derivative, this time at instants: 0.12 s and 0.87 s.

Fig. 10
figure 10

Among all the spatial-control points on the lumen boundary, the maximum difference between the linear and PLS-projected cubic NURBS representations

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Figure 10 shows, among all the spatial-control points on the lumen boundary, the maximum difference between the linear and PLS-projected cubic NURBS representations. This deviation from the medical-image data is actually a small price for an anatomically realistic smooth representation of the lumen motion. The maximum difference is about 0.2 mm, occurring at around 0.12 s. Compared to the centerline length, which is over \(100~\mathrm {mm}\) (see Fig. 11 for the model length scales), the difference is about 0.2 %. It is also within the spatial resolution (\(0.45~\mathrm {mm}\)) of the images [51].

Fig. 11
figure 11

Model length scales

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5.2 Flow computations based on lumen-motion with linear and PLS-projected cubic NURBS representations

In both computations, the mesh motion is consistent with the lumen-motion representation. We use 5 ST slabs for each of the 80 elements of the cubic NURBS representation in time, and that makes the time-step size \(2.5{\times }10^{-3}~\mathrm {s}\). The number of nonlinear iterations per time step is 3, and number GMRES iterations per nonlinear iteration is 400.

Fig. 12
figure 12

Mass balance for the PLS-projected cubic NURBS representation

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Figure 12 shows the mass balance for the PLS-projected cubic NURBS representation. By checking the mass balance, we are following an old custom that goes back to early arterial FSI computations with the ST-SUPS (see, for example, [63,64,65,66]). The objective is to make sure that the number of GMRES iterations is high enough for the part of the equation system associated with the incompressibility constraint, with the convergence measured not only by the residual decay, but also by the mass balance. Here the derivative of the lumen volume is calculated from the boundary integral of the normal component of the mesh velocity.

Fig. 13
figure 13

Spatially-maximum WSS from flow computations based on lumen-motion with linear and PLS-projected cubic NURBS representations

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Smooth and anatomically realistic representation of the lumen motion makes the WSS calculation more reliable. Figure 13 shows the spatially-maximum WSS. As can be seen from the figure, with the PLS-projected cubic NURBS representation, we circumvent the sudden, unrealistic changes at around 0.12 s and 0.87 s. Figure 14 shows the spatial distribution of the WSS at \(t=0.1275~\mathrm {s}\). In the linear-representation case, the high-WSS distribution we see at the lower part of the artery is consistent with the spikes we see at around 0.12 s in Fig. 13.

Fig. 14
figure 14

Spatial distribution of the WSS (dyn/cm\(^2\)) at \(t=0.1275~\mathrm {s}\), from flow computations based on lumen motion with linear (left) and PLS-projected cubic NURBS (right) representations. The top end of the artery is the inflow end

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6 Concluding remarks

In making treatment decisions related to coronary arteries, patient-specific computational flow analysis with time-dependent medical-image data can play a significant role. A good core computational method, high-fidelity space and time discretizations, and an anatomically realistic representation of the lumen motion would make the computational analysis more reliable. To that end, we have proposed in this article a method for anatomically realistic representation of the lumen motion. The method is used in combination with the ST-VMS, which has a good track record as a core method, and the ST-IGA, which is a superior discretization method. The ST framework, in a general context, provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady flow in the artery. The moving-mesh feature of the ST framework enables high-resolution flow computation near the moving fluid–solid interfaces. The ST-IGA, with IGA basis functions in space, enables more accurate representation of the lumen geometry and increased accuracy in the flow solution. With IGA basis functions in time, it enables a smoother representation of the lumen motion and a mesh motion consistent with that. With cubic NURBS in time, we obtain a continuous acceleration from the lumen-motion representation, which is key to obtaining a reasonable time-dependent behavior from the calculation of the fluid mechanics forces acting on the moving surface. The method we propose here for obtaining from medical-image data in discrete form an anatomically realistic NURBS representation of the lumen motion circumvents the sudden, unrealistic changes that might be introduced by the higher-order representation. In the discrete projection from the medical-image data to the NURBS representation, we supplement the least-squares terms with two penalty terms, corresponding to the first and second time derivatives of the control-point trajectories. The PLS projection helps us avoid the sudden unrealistic changes. The test computations presented demonstrate that. In the test computations, cubic NURBS representation, as expected, brings a smoothness that we cannot get from linear representation. Beyond that, the PLS-projected cubic NURBS representation circumvents the sudden, unrealistic changes introduced by the higher-order representation. It is known that the WSS plays a significant role in arterial diseases, and its quantification can help identify the high-probability regions of the disease. Smooth and anatomically realistic representation of the coronary artery lumen motion makes the WSS calculation more reliable, and this has also been demonstrated with the test computations presented.