A Smarter 3D Formula for Measuring Carotid Plaque: A Step Toward More Accurate Stroke Risk Assessment

Table of Contents

• Key Points
• Why This Research Matters
• How Carotid Plaque Severity Is Currently Measured
• Study Methods
• The Hemi-Ellipsoid Volume Formula
• Testing the Formula on Irregular Plaques
• Testing the Formula on Tubular Plaques
• Three-Dimensional Computer Models
• Validation in Real Patients on Follow-Up
• Key Findings
• Clinical Implications for Patients
• Limitations
• Recommendations
• Source Information
• Frequently Asked Questions

Key Points

• A new 3D formula estimates carotid plaque volume from standard ultrasound images, using circumferential length, sagittal length, and mean thickness.
• Using mean thickness, not maximum thickness, was essential to avoid overestimating volume of irregular plaques.
• In mathematical simulations, the formula accurately estimated volume for smooth, irregular, tubelike, and computer-generated 3D plaque shapes.
• In 115 stroke or TIA patients followed over five years, plaque volume tracked progression better than traditional diameter or area stenosis measurements.
• The approach requires no new equipment and may help detect hidden plaque growth that percent narrowing misses.

Why This Research Matters

Carotid atherosclerotic plaques (cholesterol and fatty deposits that build up in the walls of the carotid arteries, the main blood vessels in the neck that supply the brain) increase the risk of both stroke and coronary heart disease. The authors of this study note that early detection of these plaques before they cause symptoms is crucial for stroke prevention.

The stakes are high. According to the article, 60–80% of patients with carotid stenosis (narrowing of the artery due to plaque) have asymptomatic plaques — meaning they have no symptoms — with either mild stenosis (less than 30% narrowing of the artery diameter) or moderate stenosis (30–69% narrowing). These patients require ongoing monitoring of plaque severity until the plaques become symptomatic or dangerous.

This study, published in Scientific Reports, tackles a fundamental limitation of current practice: measuring plaque only by how much it narrows the artery (a one-dimensional or two-dimensional view) ignores the fact that plaques grow in all three dimensions over time. The researchers set out to develop and validate a practical formula that estimates true plaque volume from standard ultrasound images.

How Carotid Plaque Severity Is Currently Measured

Following the landmark European Carotid Surgery Trial (ECST) and the North American Symptomatic Carotid Endarterectomy Trial (NASCET), doctors have assessed carotid plaque severity using one-dimensional diameter-stenosis measurements — that is, the percentage by which the plaque narrows the vessel diameter. These include the ECST, NASCET, and Common Carotid methods, which were originally developed using angiographic (X-ray dye) findings.

The stenosis diameter (how narrow the artery opening is) helps determine the surgical approach for symptomatic severe carotid stenosis. Surgeons can choose between two procedures:

• Carotid endarterectomy (CEA) — surgical removal of the plaque
• Carotid artery stenting (CAS) — placing a mesh tube (stent) to open the narrowed artery

The rationale for focusing on diameter is that blood flow decreases to the fourth power of the diameter. In plain terms, a small reduction in vessel diameter causes a dramatic reduction in blood flow. This effect only becomes important, however, after the vessel has completed a process called compensatory positive remodeling, in which the artery expands outward to accommodate the growing plaque and temporarily preserve blood flow.

Here lies the problem the authors identify: carotid plaques have a three-dimensional shape from the moment they are first detected, with circumferential (around the vessel), sagittal (along the vessel length), and luminally protruding thickness (how far the plaque sticks into the channel). They progress in contour and size in all three directions over time. One-dimensional diameter and two-dimensional area stenosis measurements, taken from axial (cross-sectional) and sagittal (lengthwise) ultrasound images, do not reflect this three-dimensional progression of severity and atherosclerotic burden (the total amount of plaque a patient carries).

Recently, two-dimensional axial and sagittal images from carotid duplex ultrasonography (CDU) — an ultrasound test that combines B-mode imaging with Doppler blood-flow measurement — have been widely used to evaluate plaque morphology and severity. But practical tools to evaluate and follow up three-dimensional plaque volume using these two-dimensional images have been lacking. This study aimed to fill that gap.

Study Methods

The researchers pursued three parallel lines of investigation: mathematical simulation, computer-generated 3D model validation, and a real-world clinical study of patients.

For the clinical portion, the authors included 115 of 1,265 patients with transient ischemic attacks (TIAs, often called "ministrokes") or ischemic strokes (strokes caused by blocked blood vessels) who were enrolled in the Stroke Blood Sample Registry of Chungnam National University Hospital (CNUH) in South Korea between June 1, 2013, and May 28, 2015. These patients were followed for more than five years after their initial carotid ultrasound, until July 31, 2025.

The research team reviewed cardiovascular risk factors and fasting blood test results obtained within 24 hours of hospital admission for these patients. Plaque severity at follow-up was compared with baseline using three indices simultaneously: volume, diameter, and area. They recorded how many plaques decreased, increased, or remained unchanged in severity according to each measurement method. For each patient, total atherosclerotic burden was calculated as the sum of all plaque severity measurements. The researchers then compared the mean changes in total burden between initial and follow-up ultrasound tests, and categorized patients according to whether their summed plaque volume, diameter, or area increased, decreased, or stayed stable.

Ultrasound images were acquired from both common carotid arteries (CCAs) and proximal internal carotid arteries (ICAs) using one of three ultrasound systems: ALOKA Prosound SSD α-10 (Hitachi Aloka Medical Ltd.), Acuson X-300 (Siemens Medical Solutions), or Acuson Redwood (Siemens Medical Solutions). A carotid atherosclerotic plaque was defined as intimal thickening (thickening of the artery's inner lining) of at least 1.2 mm on sagittal and/or axial CDU images. Before each follow-up ultrasound, the researchers reviewed the patient's prior plaque images and captured new images at the same anatomical sites. They measured the most stenotic (most narrowed) portion of each plaque and evaluated changes at that exact site on follow-up. Plaques showing complete occlusion (full blockage) or carotid stent placement at either the initial or follow-up examination were excluded, because their extent could not be measured reliably.

Plaque length, diameter, and area were directly measured from the longitudinal and axial CDU images using MicroDicom DICOM Viewer software. Three-dimensional plaque models were built and analyzed using Rhinoceros 3D software (version 8.17, Robert McNeel & Associates). All statistical analyses were performed using IBM SPSS Statistics for Windows, version 29.0, with statistical significance set at p < 0.05.

The Hemi-Ellipsoid Volume Formula

The core idea of this study is elegantly simple. A carotid plaque, when viewed in cross-section, curves along the circular vessel wall. The researchers reasoned that if you "unfold" the tubular vessel lumen (the open channel inside the artery) onto a flat plane, the plaque takes on the shape of a hemi-ellipsoid — essentially half of a 3D ellipse, similar to an egg cut lengthwise.

To estimate plaque volume, the researchers measured three values from standard axial and sagittal ultrasound images:

1. Circumferential length (c_p) — the distance the plaque extends around the circular vessel wall
2. Sagittal length (l_p) — the distance the plaque extends along the vessel length
3. Mean thickness (t_p(mean)) — the average of the maximum and minimum thickness of the plaque, measured at right angles from the luminal surface (the inner surface facing the blood) to the vessel wall

The plaque volume was then calculated using the hemi-ellipsoid volume formula:

Plaque volume = (π × c_p × l_p × t_p(mean)) / 6

For comparison, the researchers also calculated the two traditional measures:

• Diameter stenosis = (1 − [stenosis diameter / lumen diameter]) × 100%, where the stenosis diameter and lumen (open channel) diameter are measured at the most stenotic site
• Area stenosis = (1 − [plaque area / lumen area]) × 100%, measured at the same site on the axial image

The authors also noted a useful mathematical shortcut for plaques that encircle the entire axial lumen (wrap all the way around the vessel): the circumferential length equals π times the vessel luminal diameter. This allows the formula to be computed from just two measurements in such cases.

Testing the Formula on Irregular Plaques

Real plaques are rarely perfect smooth shapes. They can have irregular lumpy surfaces, varying thickness, and partial involvement of the vessel circumference. To test whether the hemi-ellipsoid formula still works in these situations, the researchers created four simulated plaque shapes composed of alternating hemispheres (half-spheres) and hemi-ellipsoids on a flat plane:

• Shape 1: Six adjacent hemispheres, all with the same width, length, and thickness diameters
• Shape 2: Six oblate hemi-ellipsoids (flattened, like a squashed egg) with the same width and thickness as a hemisphere but twice the sagittal length
• Shape 3: Four hemispheres alternating with four oblate hemi-ellipsoids that have twice the length
• Shape 4: Four identical hemispheres alternating with four prolate hemispheres (elongated, like a rugby ball) that have twice the thickness

The actual volume of each component shape was calculated using standard sphere and ellipsoid volume formulas, and these were summed to obtain the true volume of the whole simulated plaque. The "logical" volume was then calculated using the hemi-ellipsoid formula applied to the total width, length, and thickness of each composite shape.

The results were striking:

• Shapes 1, 2, and 3: The estimated (logical) volumes matched the actual volumes exactly — in each case, both calculated to 0.5πd³ (Shape 1) or πd³ (Shapes 2 and 3), where d_hs is a virtual diameter length unit. This confirms that the formula accurately predicts true volume for plaques with uniform thickness, even when they are made of multiple bumps.
• Shape 4: Using the maximum thickness (which was twice the minimum thickness) caused marked overestimation — the formula gave a logical volume of 8/3πd³ (about 2.67πd³) versus an actual volume of πd³.
• Shape 4 with mean thickness: When the researchers substituted the mean of the maximum and minimum thickness (0.5 + 1.0 d_hs averaged), the logical volume matched the actual volume exactly at πd³.

This finding established a key principle: for plaques with irregular thickness, the mean (average) of the maximum and minimum thickness must be used, not the maximum thickness alone.

The researchers then confirmed this principle using real patient data. Because plaque volume cannot be directly measured from ultrasound images, they instead compared the logical sagittal plaque area (the 2D area of the plaque as seen along the vessel length) against the actual area traced directly on sagittal CDU images of 117 plaques with irregular thickness from the enrolled patients. The results:

• Actual sagittal area: 38.3 ± 19.4 mm²
• Logical area using mean thickness: 37.9 ± 19.4 mm² (p = 0.186 — not statistically different from the actual area, meaning the estimate was accurate)
• Logical area using maximum thickness: 50.1 ± 29.1 mm² (p < 0.001 — significantly different, confirming overestimation)

These results, combined with the 3D simulation findings, support using mean rather than maximum thickness for volume estimation in plaques with irregular contours.

Testing the Formula on Tubular Plaques

Some plaques involve the entire axial vessel lumen — meaning they wrap completely around the inside of the artery, forming a tube-like ring. To verify the formula for this situation, the researchers simulated a tubular plaque measuring 7 mm in diameter, 7 mm in length, with a 1.5 mm wall thickness, and with the proximal and distal ends cut at a 45° bevel (a slanted angle like a picture frame corner).

The actual volume was calculated by a complex process: removing the non-plaque luminal and beveled-region volumes from the original vessel column volume. The estimated volume was calculated using the hemi-ellipsoid formula. The two values were nearly identical:

• Actual volume: 51.80 mm³
• Estimated volume: 51.76 mm³

When the bevel angle was changed, the actual volume changed as expected: decreasing the angle to 30° reduced the volume to 38.07 mm³, while increasing it to 60° raised the volume to 57.04 mm³. Based on this analysis, the authors assumed a 45° bevel when applying the formula to plaques that encircle the vessel lumen in clinical practice.

Three-Dimensional Computer Models

To further validate the hemi-ellipsoid formula, the research team used Rhinoceros 3D software to construct two irregularly shaped three-dimensional plaque objects inside a cylindrical vessel model with a diameter of 10 mm and a height of 15 mm. Using subdivision surface (SubD) tools, they reshaped the luminal surfaces into curved, irregular geometries to mimic realistic plaques. They then used the Make2D command to obtain two-dimensional longitudinal and planar images, along with the length, width, and maximum and minimum thickness of each plaque.

The actual volume of each plaque was calculated precisely using the software's Volume command, and this was compared with the theoretical volume estimated by the hemi-ellipsoid formula using the measured length, width, and mean of the maximum and minimum thickness values. The first plaque measured 13.92 mm × 6.47 mm. The authors present the comparison between actual and theoretical volumes for both irregular geometries in the original article's Figure 5, demonstrating how the formula performed against the gold-standard computer-calculated volumes.

Validation in Real Patients on Follow-Up

The ultimate test of any measurement tool is whether it works in real patients over time. The clinical portion of the study included 115 patients from the CNUH Stroke Blood Sample Registry who had experienced TIAs or ischemic strokes and had been followed for more than five years after their initial carotid ultrasound.

Each patient's plaque severity was assessed at baseline and follow-up using all three measurement tools: plaque volume (via the hemi-ellipsoid formula), diameter stenosis, and area stenosis. The total atherosclerotic burden — the sum of all plaque severity measurements for each patient — was calculated for each method, and the mean changes between initial and follow-up ultrasound were compared.

In addition, the researchers used linear regression models to predict the severity of each atherosclerotic burden measurement tool (volume, diameter, and area) using clinical and laboratory cardiovascular risk factors. Each model was evaluated using the correlation coefficient (r), the determination coefficient (R²), and the p-value from analysis of variance. Dummy variables (categorical coding) were used for the following reference groups: male sex, previous history of hypertension, diabetes, smoking, alcohol consumption, and atrial fibrillation (an irregular heart rhythm that increases stroke risk).

Although the specific regression numbers appear in the full published figures, the authors' central conclusion from this patient analysis is clear: the hemi-ellipsoid volume evaluation was a better tool for assessing severity and atherosclerotic burden of carotid plaques on follow-up than either one-dimensional diameter stenosis or two-dimensional area stenosis.

Key Findings

The study produced several distinct findings that support the new measurement approach:

1. The hemi-ellipsoid volume formula is mathematically sound. For smooth plaques with uniform thickness, the formula estimated volumes exactly, with no measurable error in any of the first three simulated shapes.
2. Mean thickness is essential for irregular plaques. Using maximum thickness overestimated volume in the irregular simulation (8/3πd³ vs. πd³, about 167% of the true value), while mean thickness produced an exact match. The same pattern appeared in real patient data: mean thickness yielded a sagittal area estimate statistically indistinguishable from the measured area (37.9 vs. 38.3 mm², p = 0.186), whereas maximum thickness significantly overestimated it (50.1 mm², p < 0.001).
3. The formula handles tube-shaped plaques. For a simulated plaque encircling the entire vessel lumen, the estimated volume (51.76 mm³) was within 0.08% of the actual volume (51.80 mm³).
4. Volume beats diameter and area for monitoring. In the patient cohort, volume-based evaluation better reflected plaque progression and severity over more than five years of follow-up than traditional one- and two-dimensional measurements.
5. The method is practical. It uses only measurements already available from standard axial and sagittal carotid duplex ultrasound images, requiring no new equipment or specialized techniques.

Clinical Implications for Patients

For patients with carotid atherosclerosis, this research has several meaningful implications:

Better risk monitoring. Traditionally, doctors have relied on how much a plaque narrows the artery diameter (percent stenosis) to decide when to intervene. This study suggests that measuring actual plaque volume may give a more accurate picture of disease progression. A plaque can grow substantially in volume while the degree of stenosis changes very little, because the artery wall expands outward (positive remodeling) to accommodate the growth. Volume measurement captures this hidden progression that diameter measurement misses.

Earlier detection of worsening disease. Because volume better reflects true atherosclerotic burden, patients whose plaques are growing may be identified earlier. This could allow earlier adjustments to medications (such as statins, which lower cholesterol and stabilize plaque), lifestyle changes, or closer monitoring schedules.

A practical, low-cost tool. The hemi-ellipsoid formula runs on measurements already obtainable from standard carotid duplex ultrasound — a widely available, noninvasive, radiation-free imaging test. No expensive 3D ultrasound machines or CT/MRI scans are required to implement this approach. The only addition is a straightforward calculation using three numbers: circumferential length, sagittal length, and mean thickness.

Better understanding of total disease burden. Instead of focusing on a single plaque in isolation, the study's approach of summing plaque volumes across both carotid arteries provides a patient-level measure of total atherosclerotic burden. This holistic view could help doctors assess overall cardiovascular risk more comprehensively. As the authors note, carotid plaques increase the risk of both stroke and coronary heart disease, so tracking total carotid plaque burden may offer insight into systemic atherosclerosis (plaque buildup throughout the body's arteries).

Limitations

It is important to understand what this study does and does not prove. The authors acknowledge several inherent limitations:

• Indirect validation. Plaque volume cannot be directly measured from ultrasound images in living patients. The researchers validated the formula using mathematical simulations, computer-generated 3D models, and surrogate measurements (sagittal plaque area) rather than direct volumetric comparison in patients.
• Simulation assumptions. The simulations assume plaques conform to idealized geometric shapes (hemispheres, hemi-ellipsoids, tubes with beveled ends). Real plaques may have even more complex geometry, with calcification, ulceration, or intraplaque hemorrhage (bleeding inside the plaque).
• Single-center, retrospective design. The clinical validation involved patients from one hospital in South Korea, enrolled retrospectively, with the requirement for informed consent waived due to the study design. Results may not generalize to all populations.
• Selected patient group. The patient cohort consisted of individuals who had already experienced TIAs or ischemic strokes — a higher-risk group than the general population or patients with asymptomatic carotid plaques.
• The 45° bevel assumption. For plaques encircling the vessel lumen, the formula assumes a specific geometric relationship that may not hold in every case.
• Statistical limitations of surrogate endpoints. The study measured plaque severity indices, not clinical outcomes (such as actual stroke rates). Whether volume-based monitoring ultimately prevents more strokes than traditional measurement remains to be demonstrated in future prospective studies.

Recommendations

Based on this research, patients and clinicians may consider the following practical steps:

1. Ask about plaque volume. If you are being monitored for carotid artery disease, ask your doctor whether plaque volume measurement is available alongside the standard percent-stenosis measurement. Not all centers will have adopted this approach yet, but awareness of its value is growing.
2. Keep consistent follow-up imaging. The study's method relies on comparing images taken at the same anatomical site over time. Attending regular follow-up ultrasound appointments at the same imaging center increases the reliability of progression assessment.
3. Manage cardiovascular risk factors. Even with improved measurement tools, the fundamentals of treatment remain unchanged. Controlling blood pressure, managing diabetes, lowering LDL cholesterol (the "bad" cholesterol), quitting smoking, moderating alcohol, and treating atrial fibrillation are all known to reduce stroke risk. The study specifically accounted for these factors in its regression models, confirming their importance.
4. Understand what "mild" or "moderate" stenosis means. If your doctor tells you that you have mild (under 30%) or moderate (30–69%) stenosis, remember that 60–80% of patients with such findings have no symptoms. The plaque may still be growing in volume even if the stenosis percentage looks stable — another reason this volume-based approach matters.
5. Seek specialized centers for complex cases. For plaques with irregular surfaces or involving the entire vessel lumen, the choice of measurement technique becomes especially important. Centers familiar with volumetric assessment may provide a more accurate picture of disease burden.

Frequently Asked Questions

What is carotid plaque and why does it matter?

Carotid plaques are cholesterol and fatty deposits that build up in the walls of the carotid arteries, the main blood vessels in the neck that supply the brain. They increase the risk of stroke and coronary heart disease. According to the study, 60–80% of patients with carotid narrowing have no symptoms, making careful monitoring of plaque severity important for stroke prevention.

How was carotid plaque severity traditionally measured?

Traditionally, doctors measured how much a plaque narrows the artery diameter using one-dimensional or two-dimensional methods such as diameter stenosis and area stenosis. These were based on older clinical trial methods. However, plaques grow in all three dimensions over time, so these measurements may not capture hidden progression, especially when the artery expands outward to accommodate the growing plaque.

What is the new hemi-ellipsoid formula for plaque volume?

The formula estimates carotid plaque volume by treating the plaque like half of a three-dimensional egg shape, called a hemi-ellipsoid. It uses three ultrasound measurements: the distance the plaque extends around the vessel wall, the distance it extends along the vessel length, and the mean of its maximum and minimum thickness. These are measured from standard carotid ultrasound images.

How accurate is the new formula for irregular or tube-shaped plaques?

Among researchers, the formula was mathematically exact for smooth, irregular, and tube-shaped simulated plaques when mean thickness was used. For irregular plaques, using maximum thickness overestimated volume, while mean thickness gave accurate results. For a simulated plaque encircling the entire vessel lumen, the estimated volume was within 0.08% of the actual volume.

Does measuring plaque volume require special equipment?

No. The method uses images from standard carotid duplex ultrasound, which is widely available, noninvasive, and does not use radiation. Doctors already measure plaque dimensions on these images. The new approach only adds a straightforward calculation using three numbers: circumferential length, sagittal length, and mean thickness. No expensive 3D ultrasound machines, CT, or MRI are needed.

What does a plaque volume measurement mean for monitoring stroke risk?

In a study of 115 patients who had experienced ministrokes or ischemic strokes, plaque volume measured with this formula tracked disease progression over more than five years better than traditional diameter or area measurements. That could allow earlier identification of growing plaques and earlier adjustments to medications or lifestyle changes, although more research is needed to prove long-term stroke prevention.

I have mild carotid stenosis on ultrasound – should I get a second opinion to check my plaque volume for more accurate stroke risk?

Standard carotid ultrasound reports often give only percent stenosis, which measures how much the artery is narrowed. But plaque grows in three dimensions, and the artery can expand outward to hide that growth, so stenosis may look stable while plaque volume increases. Plaque volume measurement using a hemi-ellipsoid formula from standard ultrasound images has been shown in research to better track plaque progression over time. A second opinion that includes plaque volume assessment could give a fuller picture of your total plaque burden and whether your disease is truly changing. No evidence yet proves this method prevents more strokes. Diagnostic Detectives Network provides independent expert second opinions.

Source Information

Original article title: Hemi-ellipsoid formula enables accurate assessment of carotid plaque volume and atherosclerotic burden

Authors: Jei Kim, To Jeong, and Jeeyeon Kim

Affiliation: Department of Anatomy and Neurology, College of Medicine and Hospital, Chungnam National University, Daejeon, South Korea (Kim, Jei and Kim, Jeeyeon); ARATORE Studio, Seoul, South Korea (Jeong)

Journal: Scientific Reports (2026), volume 16, article 5138. Published online at Nature.com. DOI: https://doi.org/10.1038/s41598-026-35182-5

Study approval: Approved by the Institutional Review Board of Chungnam National University Hospital (CNUH-IRB-2013-01-019 for the registry; CNUH-IRB-2025-02-023 for this study). All procedures complied with the ethical standards of the 1975 Declaration of Helsinki.

This patient-friendly article is based on peer-reviewed research. It is intended to help patients and caregivers understand the study's findings and is not a substitute for professional medical advice. Always discuss your individual condition and treatment options with your healthcare provider.