We developed an algorithm to convert a series of IVUS signal data into a fully volumetric 3D visualization (Figure 1). The entire protocol including 2D image reconstruction from IVUS signal data, border tracing, segmentation, sequential alignment of 2D IVUS slices, and intermediary slice generation was conducted in a single image processing platform. Volumetric visualization of 3D IVUS images was performed using ImageJ, open-source Java-based image processing software provided by the National Institutes of Health.

### IVUS signal data acquisition *in vivo*

An atherosclerotic Yucatan miniswine atheroma model (20 kg, Sinclair Research Center Inc., Columbia, MO) was used. The animal protocol was approved by the Institutional Animal Care and Use Committee of The University of Texas Health Science Center at Houston. Following anesthesia, the right femoral artery was exposed with groin incisions, and an arteriotomy was performed. A 5 F sheath was inserted into the femoral artery. A high frequency (20 MHz, 3.5 F) IVUS imaging catheter connected to a Volcano s5i IVUS Imaging System (Volcano Co, Rancho Cordova, CA) was utilized. The IVUS catheter was inserted through the arterial segment past the region of interest, and withdrawn using an automatic pullback device at a constant speed of 0.5 mm/s, while 2D IVUS images and signal data of the arterial segment were continuously recorded. A total of 256 scan lines with 1,024 sampling data per scan line were recorded (dynamic range of 40–60 dB) for each 2D IVUS slice. Since there was no curvature in the arterial segment evaluated, it was assumed that the direction of pullback of the IVUS catheter was parallel to the longitudinal direction of the artery.

The enveloped amplitude (i.e., acoustic backscatter) was utilized to reconstruct images in the polar coordinate, in which the x- and y- axes refer to the radial and circumferential directions, respectively. The reconstructed images in the polar coordinate were transformed to the Cartesian coordinate system for standard vascular imaging. A graphical user interface (GUI)-based image processing system was developed for interactive tracing and segmenting procedures under MATLAB (Mathworks Inc., Natick, MA) platform. The inner and outer arterial borders in each IVUS image were manually segmented. A series of signal data in the segmented ROIs was placed in tomographic sequence to create intermediary slices.

We collected a total of 25 original 2D IVUS slices of the arterial segment with a distance of 0.5 mm between slices to generate 10 intermediary slices between two adjacent slices resulting in a total of 265 cross-sectional images along the longitudinal direction. In order to interpolate IVUS signal data of the arterial segments in 3D space, we utilized the natural cubic spline interpolation considering the nonlinearity of both vascular structure geometry and acoustic backscatter information within the arterial wall.

### Intermediary IVUS slice generation using the shape-based nonlinear interpolation

Figure 2 demonstrates a schematic diagram of the intermediary IVUS slice generation algorithm using the shape-based nonlinear interpolation method. Intermediary slices between original 2D IVUS slices are generated utilizing both arterial geometry and acoustic backscatter information using vascular geometry information in the three neighboring slices (Step 1). We applied the natural cubic spline interpolation method to the segmented ROIs of 2D IVUS data sequentially placed along the longitudinal direction. The contours delineating arterial wall structure in the three neighboring 2D IVUS slices were utilized to calculate inner and outer arterial borders in intermediary slices along the 256 scan lines. Next, nonlinear acoustic backscatter interpolation was performed (Step 2). The segmented ROI of the arterial wall in each IVUS slice contains acoustic backscatter profile. This information was utilized to interpolate acoustic backscatter distribution within the ROIs in the consequent intermediary slices using the natural cubic interpolation.

Figure 3A demonstrates variables, functions and conditions utilized in the shape-based nonlinear interpolation method. Acoustic backscatter information within the segmented ROI was defined as **A** = {(*r*_{m}, *φ*_{n}, *I*(*r*_{m}, *φ*_{n}))} = {(*r*_{m}, *φ*_{n}, *I*_{mn})} along the scan lines bounded by the inner arterial structure border contour **M** = {(*r*_{p}(*φ*_{n}), *φ*_{n})} = {(*r*_{pn}, *φ*_{n})} and the outer border contour **N** = {(*r*_{q}(*φ*_{n}), *φ*_{n})} = {(*r*_{qn}, *φ*_{n})} in each 2D IVUS image in the polar coordinate system. In this configuration, the radial distance *r* is defined as a function of *φ* which is the angular position of the scan line. There are a total of 256 scan lines (n) with 1,024 sampling data (m) per scan line.

Detailed information of the shape-based nonlinear interpolation algorithm is described in Figure 3B. First, the original 2D IVUS slices were registered in the global coordinate and the centroid of the arterial structure on each slice was aligned to the centerline. The inner boundary point on the *n*^{th} scan line in the *j*^{th} slice was defined *r*_{
pn,j
} = (*x*_{
j
}, *r*_{
j
})^{T}. In order to calculate the coefficients of the polynorms that constitute boundary splines of the *n*^{th} scan line along the longitudinal direction using *j + 1* number of slices, the boundary points data were defined as (*x*_{
1,
}*r*_{
1
}),…, (*x*_{
j+1,
}*r*_{
j+1
}).

For any *i* ∈ {*1,…, n*}, we set *z*_{
i
} *= s″*(*x*_{
i
}) and Δ *x*_{
i
} *= x*_{
i+1
}*- x*_{
i
}. For all *j* ∈ {*1,…, n - 2*} and *x* ∈ [*x*_{
j
}*, x*_{
j+1
}], the second derivative of a cubic spine function *s*_{
j
}*″*(*x*) is the line connecting (*x*_{
j,
}*z*_{
j
}) and (*x*_{
j+1,
}*z*_{
j+1
}). The equation is given by

{{s}^{\u2033}}_{j}\left(x\right)=\frac{x-{x}_{j}}{\Delta {x}_{j}}{z}_{j+1}-\frac{x-{x}_{j+1}}{\Delta {x}_{j}}{z}_{j}

(1)

By integrating (1), we have

{{s}^{\prime}}_{j}\left(x\right)=\frac{1}{2}{\left(x-{x}_{j}\right)}^{2}\frac{{z}_{j+1}}{\Delta {x}_{j}}-\frac{1}{2}{\left(x-{x}_{j+1}\right)}^{2}\frac{{z}_{j}}{\Delta {x}_{j}}+{B}_{j}

(2)

For (*A*_{
j
}*, B*_{
j
}) ∈ **R**^{2}, we have

{s}_{j}\left(x\right)=\frac{{\left({x}_{j+1}-x\right)}^{3}}{6\Delta {x}_{j}}{z}_{j}+\frac{{\left(x-{x}_{j}\right)}^{3}}{6\Delta {x}_{j}}{z}_{j+1}+{A}_{j}+{B}_{j}\left(x-{x}_{j}\right)

(3)

Given with *s*(*x*_{
j
}) = *r*_{
j
} and *s*(*x*_{
j+1
}) = *r*_{
j+1
}, the coefficients of the polynorms can be calculated as

{A}_{j}={r}_{j}-\frac{{z}_{j}{\left(\Delta {x}_{j}\right)}^{2}}{6}\phantom{\rule{0.12em}{0ex}}\mathit{and}\phantom{\rule{0.12em}{0ex}}{B}_{j}=\frac{\Delta {r}_{j}}{\Delta {x}_{j}}-\frac{\Delta {z}_{j}\Delta {x}_{j}}{6}

(4)

The continuity of *s″* in *x*_{
1
} and *x*_{
n
} implies that

For all *j* ∈ {*2,…, n – 1*}, the continuity of *s′* in *x*_{
j
} implies that

\frac{\Delta {x}_{j-1}}{6}{z}_{j-1}+\left(\frac{\Delta {x}_{j}}{3}+\frac{\Delta {x}_{j-1}}{3}\right){z}_{j}+\frac{\Delta {x}_{j}}{6}{z}_{j+1}=\frac{\Delta {r}_{j}}{\Delta {x}_{j}}-\frac{\Delta {r}_{j-1}}{\Delta {x}_{j-1}}

(6)

A system of linear equations (6) represented by a tridiagonal matrix can be employed to calculate the spline interpolation functions in equation (3) to define the inner boundary curvature between the slices. Likewise, the outer boundary curvature was determined.

Next, we created the same number of acoustic backscatter data points between the two borders with an arterial wall thickness *d*_{n} = dist(*r*_{qn} - *r*_{pn}) on each scan line. Following the inner and outer boundaries defined on the intermediary slices as above, the acoustic backscatter within the ROI in the intermediary slices were determined across 100 data points for each scan line using the same natural cubic interpolation algorithm.

### Validation studies of the shape-based nonlinear interpolation

Our shape-based nonlinear interpolation is developed to create intermediary slices incorporating both geometric interpolation and grayscale interpolation. In order to validate our shape-based nonlinear interpolation algorithm, we utilized a virtual vascular phantom containing cross-sectional images with varying shapes, wall thicknesses and grayscale values (Figure 4). Three cross-sectional images (#1, #5, #9) were utilized as input data to calculate three intermediary slices between each pair. Detailed dimensional information of the three original input slice images are described in Figure 4. We compared geometry and grayscale information of the intermediary slices created by the shape-based nonlinear interpolation with the original cross-sectional images at the corresponding location of the phantom. Next, we evaluated differences in image quality between the conventional pixel-based interpolation and the shape-based nonlinear interpolation methods using both the phantom data and the femoral artery IVUS data.

### Volumetric 3D IVUS visualization of the femoral artery

We created 10 intermediary slices for every pair of 25 original IVUS slices using the conventional pixel-based interpolation and our shape-based nonlinear interpolation methods. This permitted evaluation of a total of 265 IVUS images along the longitudinal direction of the femoral artery. Image stocks containing 265 IVUS slice images created from each interpolation method were utilized for volumetric 3D IVUS visualization. The stacks were imported to the volume viewer in ImageJ. In order to better demonstrate the difference between the two interpolation methods, we displayed both inner and outer surfaces of the arterial segment.

In order to demonstrate the practical applicability of our shape-based nonlinear interpolation method to 3D IVUS reconstruction along curved pullback trajectories (e.g., coronary arteries), we created a curved trajectory (s-shaped with one third height with respect to the arterial segment length) and incorporated the same IVUS data into the trajectory.