🧬 biology

PerfuRange: feasible-set identifiability analysis of component blood flow in dual-input perfusion using a groundtruth digital phantom

This paper introduces PerfuRange, a framework that quantifies the identifiability of component blood flow in dual-input perfusion by determining the feasible range of physiological targets compatible with specific acquisition and modeling constraints, revealing that good model fits often fail to guarantee correct physiological decomposition due to inherent limitations in data and model assumptions.

Original authors: Tomoki Saka

Published 2026-08-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tomoki Saka

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Medical imaging has long relied on a simple but powerful idea: by watching how a contrast dye moves through the body, doctors can map out how blood flows to organs. This process, known as perfusion imaging, turns a series of pictures into a story about life inside the tissue. For organs like the liver, which receive blood from two distinct sources—the heart via the arteries and the gut via the portal vein—the story is more complex. The liver acts as a mixing bowl where these two streams converge, and clinicians have long wanted to know exactly how much blood comes from each source. Separating these two contributions is crucial because diseases often affect one stream more than the other, and knowing the difference could guide better treatments. However, untangling two overlapping flows from a single measurement is a notoriously difficult puzzle, one where the answer often depends more on the assumptions made by the computer than on the data itself.

A researcher at Tokyo Denki University, Tomoki Saka, has developed a new way to look at this problem, not by trying to force a single answer, but by mapping out the entire range of possible answers that the data allows. He calls this method PerfuRange. Instead of asking a computer to pick the one "best" number for blood flow, PerfuRange asks a different question: given the measurements we have, and the rules we agree to follow about how blood behaves, what is the smallest and largest possible value for the blood flow? This approach treats the result not as a fixed point, but as a window of possibility. To test this, Saka built a perfect, digital simulation of a liver. In this virtual world, he knew the exact truth: he knew exactly how much blood came from the artery and how much from the vein, and he knew the exact shape of the dye's journey. He then ran the PerfuRange analysis on this known truth to see if the method could correctly identify the limits of what is actually knowable.

The results revealed a surprising fragility in how we currently interpret these scans. When the researchers allowed the computer to consider any physically possible shape for the blood flow, the range of answers was incredibly wide. Even with perfect data and no noise, the method could not rule out the possibility that one of the blood sources contributed nothing at all. In nearly half of the test cases, the "feasible set" included zero flow for the second source, meaning the data alone could not prove that source was active. The only thing that narrowed the range of answers was imposing strict assumptions about the shape of the flow curve. When the researchers forced the computer to choose from a specific list of pre-defined shapes, the range of answers became very tight and precise. But this precision was an illusion of the assumption, not the data.

The study showed that if the true shape of the blood flow did not match the list of shapes the computer was allowed to pick from, the method would still produce a very narrow, confident-looking answer, but that answer would be wrong. In tests where the computer was forced to guess the shape of a flow it had never seen before, it frequently produced narrow intervals that completely missed the true value. This happened even when the computer was allowed to pick from a list of shapes that shared the same basic statistical properties as the real flow. The computer would confidently declare a specific flow rate, but that declaration was entirely dependent on the fact that the real flow happened to look like one of the allowed shapes. If the real flow looked slightly different, the confidence evaporated, or worse, the computer would confidently give the wrong number.

Noise in the images, which is inevitable in real medical scans, made the problem even more stark. As the researchers added more simulated noise to the data, the range of possible answers grew wider and wider. At a noise level of just one Hounsfield unit—a standard measure of image grain—the method could no longer rule out zero flow for the second source in about half of the cases. The only way to get a narrow, precise answer in the presence of noise was to accept a very large range of possible solutions, which rendered the specific number useless for clinical decision-making. The study also found that small errors in measuring the strength of the input dye curves could completely skew the results. If the computer underestimated the strength of the dye entering from the portal vein by a small amount, it would overestimate the blood flow from that vein by a corresponding amount, yet the overall fit of the model to the data would look perfect. The computer had no way to know it was wrong because the error was hidden in the calibration, not the curve.

Perhaps the most instructive finding came from a test where the researchers extended the time they watched the dye. By watching the scan for 80 seconds instead of 58, they were able to confidently identify the total volume of blood in the liver. However, even with this extra time and information, they still could not determine how much of that volume came from the artery versus the vein. This proved that knowing the total amount of blood does not automatically solve the puzzle of where it came from. The extra time helped answer one question but left the specific source attribution completely unresolved.

The core message of this work is that the precision we see in medical imaging reports is often a reflection of the assumptions we feed into the computer, not a direct measurement of reality. A narrow, confident number for blood flow is only trustworthy if the true shape of the flow happens to match the limited set of shapes the computer was allowed to consider. If the real biology is slightly different, that confidence is misplaced. The study suggests that for dual-input perfusion, we cannot simply report a single number for blood flow from each source without explicitly stating the assumptions that made that number possible. The data alone, even when perfect, does not contain enough information to separate the two flows uniquely. The "answer" is not a hidden truth waiting to be found, but a range of possibilities constrained by the rules we choose to apply. This does not mean the technology is useless, but it does mean that the certainty we feel about a specific number must be tempered by an understanding of the model that produced it.

Technical Summary: PerfuRange: Feasible-Set Identifiability Analysis of Component Blood Flow in Dual-Input Perfusion

Problem Statement
Quantitative perfusion imaging in dual-supply organs (e.g., the liver) requires decomposing a single measured tissue concentration curve into two distinct component impulse responses (arterial and portal-venous). The fundamental inverse problem is underdetermined: one measured curve must support the recovery of two unknown response functions. While prior studies have established that uniqueness depends on specific model assumptions (e.g., Becker et al., 2005; Miyazaki et al., 2008; Shalom et al., 2024), there remains a lack of explicit frameworks to distinguish between precision derived from data constraints versus precision imposed by restrictive modeling assumptions. Furthermore, successful curve fitting does not guarantee that the individual component flows are identifiable, particularly in the presence of model misspecification or input calibration errors.

Methodology
The author introduces PerfuRange, a target-specific feasible-set framework for perfusion inverse problems. Rather than providing a single point estimate, PerfuRange computes the minimum and maximum values of a physiological target (e.g., component blood flow) over the set of all solutions compatible with:

The measured tissue curve and input functions.
A declared admissible response class (e.g., causality, non-negativity, monotonicity, or specific parametric families).
A residual/error budget (L2 norm).

The study utilizes a ground-truth digital phantom constructed independently of clinical data, featuring:

Inputs: An analytic gamma-variate arterial input ( $A_0$ ) and a dispersed, area-conserving portal input ( $A_1$ ).
Truth: 735 combinations of second-component blood volume, mean transit time (MTT), onset delay, and residue shapes (exponential, linear, box, and unseen holdouts like gamma, lognormal, and inverse-Gaussian distributions).
Experiments:
- Model-Light Analysis: Testing weak physiological constraints (causality, non-negativity, monotonicity) without specific parametric families.
- Structured Family Analysis: Testing finite candidate banks of response shapes against both represented and unseen (holdout) truth shapes.
- Dense-Grid Isolation: Separating parameter-grid alignment effects from response-shape misspecification by using off-grid truth parameters and refined analysis grids.
- Sensitivity Analyses: Assessing the impact of noise levels, input amplitude scaling, input timing shifts, and input waveform similarity.
- Positive-Identifiability Control: A prospectively frozen experiment testing whether extending the observation window (0–58s to 0–80s) could identify total blood volume without resolving component flows.

Key Contributions

PerfuRange Framework: A formal operational construction that maps acquisition parameters, admissible response classes, and error budgets to the complete feasible range of a physiological target. This makes the assumptions conditioning precision explicit.
Identifiability Hierarchy: The study demonstrates that identifiability is not a binary property of the dataset but a conditional property of the target, acquisition, and model class. It distinguishes between "data-supported identifiability" (narrow intervals across broad classes) and "model-conditioned precision" (narrow intervals resulting from restrictive assumptions).
Quantification of Ambiguity: The paper provides quantitative bounds on how much precision is lost when moving from represented truth shapes to unseen, physiologically admissible shapes, and how noise and input calibration errors degrade these bounds.

Key Results

Weak Constraints Are Insufficient: Even with exact inputs, oracle onset times, and strict physiological constraints (causality, non-negativity, monotonicity), the median feasible interval width for the second-component flow was 2.98 times the true flow, with the lower bound being zero in 40.1% of conditions.
Model-Conditioned Precision vs. Miscoverage:
- When the true response shape was represented in the candidate bank, coverage was near-perfect (99.9%).
- When the truth was an unseen shape (e.g., gamma-3, lognormal), support dropped to 84.1%, and coverage among supported cases fell to 44.1%.
- Crucially, 45.2% of "observable-narrow" intervals (width $\le$ 0.5 relative span) for unseen shapes missed the true value, compared to only 0.123% for represented shapes. This indicates that narrow intervals can be highly misleading if the response family is misspecified.
Noise and Calibration Limits:
- At a tissue-curve noise level of 1 HU, approximately 50.4% of cases allowed for zero second-component flow, rendering the interval uninformative.
- Input Amplitude: A scaling error in the second input ( $A_1$ ) by a factor $\beta$ results in a reciprocal scaling of the recovered flow ( $1/\beta$ ) with no change in the curve-fit residual. This ambiguity is invisible to goodness-of-fit metrics.
- Timing: Timing errors are absorbed by onset shifts until causal support limits are reached, at which point large flow errors occur.
Target-Specific Identifiability: Extending the observation window from 58s to 80s increased the identification rate of total blood volume from 0/64 to 24/64 cases (with 100% certified coverage). However, component flows remained unidentifiable in all 64 cases, demonstrating that additional data can resolve integrated quantities without resolving source attribution.
Portal-Dominant Regime: The model-dependence phenomenon persists in portal-dominant flow fractions ( $\ge$ 0.70). In this regime, unseen shapes often resulted in a lack of model support (visible warning) rather than a confident narrow wrong interval, though miscoverage remained high (48.5%) when support existed.

Significance
The paper argues that identifiability is target-specific, acquisition-dependent, and model-class-dependent. A narrow, precise component-flow estimate does not establish correct physiological decomposition; it may simply reflect the restrictiveness of the assumed response family.

The significance of PerfuRange lies in its ability to make the assumptions behind precision explicit. It suggests that reporting dual-input perfusion should not rely on point estimates alone but must include:

The target functional claimed to be identified.
The acquisition information used.
The response assumptions and noise model.
Diagnostics for model support and sensitivity to input calibration.

The study concludes that while dual-input perfusion is not inherently unidentifiable, the degree of identifiability must be stated alongside the specific assumptions under which it was obtained. Good fit or narrow model-conditioned intervals do not guarantee that the data independently support the physiological decomposition.

Technical Summary: PerfuRange: Feasible-Set Identifiability Analysis of Component Blood Flow in Dual-Input Perfusion

More like this