Why the same average radiation dose can leave some cell nuclei untouched

In boron neutron capture therapy, a nucleus can receive no energy at all from the reaction products even when the surrounding tissue contains boron and is being irradiated. In one modeled condition, spatial variation in boron raised that zero-hit probability by 40.49% compared with a uniform distribution having the same average boron concentration (Fardela et al., 2026). The finding does not mean 40.49% more cancer cells survive. It means that, at the scale of a modeled cell nucleus, the same average can conceal a less even pattern of energy delivery.

That distinction matters because boron neutron capture therapy, or BNCT, depends on a remarkably local reaction. Boron-10 captures a neutron and produces an alpha particle and a lithium-7 nucleus. These charged particles travel only a few micrometers before depositing their energy—roughly the scale of a cell. That short range is the source of BNCT’s appeal: in principle, energy can be concentrated near boron-bearing tissue rather than spread uniformly through a larger volume. But it also means that where boron sits within and around a cell can affect whether the nucleus is struck at all.

The study by Ramacos Fardela, Roni Muslim and Tista Artu Indra Kusuma builds a statistical model to separate two kinds of randomness that can blur this microscopic picture. One comes from neutron captures happening by chance. The other comes from boron being unevenly distributed in space. The paper’s central insight is that these sources do not rise in step as neutron exposure increases. That creates a changing balance between event-to-event randomness and the structure of the boron field.

The Science

BNCT treatment planning often relies on quantities such as average boron concentration and absorbed dose. Absorbed dose is energy deposited per unit mass, and it is indispensable for comparing radiation exposures. But an average is not a map. If particles travel only micrometers, the arrangement of boron at that scale can influence whether a particular nucleus receives energy, even when two tissues share the same average concentration.

The authors represent boron concentration as a spatially correlated lognormal field. “Lognormal” means the modeled concentration is always positive and can vary substantially, while “spatially correlated” means nearby locations tend to have more similar values than distant ones. They vary both the amplitude of concentration differences and their correlation length—the distance over which boron-rich or boron-poor patches tend to persist. This lets the model distinguish small, fine-grained variations from broader patches without changing the mean boron level.

For each hypothetical boron arrangement, neutron captures occur as a Poisson process: a mathematical description of events that happen randomly at a rate set by local boron concentration and neutron fluence. Fluence is the number of neutrons crossing a unit area over the exposure. Because the local capture rate itself varies with the boron field, the overall process is a Cox process—a Poisson process whose intensity is random. In plain terms, capture events are random, and the map that determines where they are likely to occur is random too.

To estimate what a single capture deposits in a nucleus, the team used Geant4, a particle-transport simulation toolkit. They modeled a spherical nucleus four micrometers in radius in water-equivalent material, then simulated the alpha and lithium products from captures at different distances. The calculations yielded the mean deposited energy, a second moment that captures the spread of event energies, and the probability that an event deposits any positive energy in the nucleus.

The statistical model then splits total dose variance—the spread of outcomes—into two pieces. “Shot” variance is the randomness in captures and energy deposition when the boron arrangement is held fixed. Heterogeneity variance is the variation in average dose from one boron arrangement to another. The distinction is important: the second contribution applies to an ensemble of different boron layouts. It is not the same as repeating irradiation on one fixed cell with an unchanged boron distribution.

Reported crossover fluence

Label	Value
H=0.8; ξ=3 μm	1.5 ×10¹² cm⁻²

One reported model condition; not a clinical treatment threshold.

The supplied paper text does not include figure panels or captions, so the visual references here use only the numerical results explicitly reported in its abstract. The first chart shows one such anchor: for heterogeneity amplitude H=0.8 and correlation length ξ=3 micrometers, the two variance contributions become equal at a neutron fluence of about 1.50×10¹² per square centimeter. This is a model-specific crossover, not a treatment prescription.

What They Found

The clearest general result is about scaling. As fluence rises, the variance from stochastic neutron capture grows linearly. The variance from boron heterogeneity grows quadratically. The difference follows from what each source represents: more irradiation brings more random capture events, while a higher capture intensity magnifies the effect of differences between boron-rich and boron-poor field realizations.

The mean dose, by contrast, remains the same as in a homogeneous-boron model when the average concentration and transport response are held fixed. Heterogeneity changes the spread around that mean, rather than the ensemble average itself. This is a subtle but consequential result. Two models can agree on average absorbed dose and still disagree on how reliably individual nuclei receive energy.

At low fluence, capture randomness can dominate. As fluence increases, the heterogeneity term catches up and may overtake it. The point where the contributions are equal depends on correlation length as well as on the amplitude of boron variation. The abstract reports a crossover of approximately 1.50×10¹² cm⁻² for H=0.8 and ξ=3 μm. It does not establish that this value applies to other boron distributions, cell geometries, neutron spectra or treatment settings.

At high fluence, the ensemble relative dose variance approaches a plateau of 0.160 in the reported model. Relative variance expresses the spread in dose in relation to the square of the mean, allowing a comparison that is less dependent on the dose scale itself. The plateau arises because, as exposure increases, the capture-noise contribution becomes smaller relative to the squared mean, while the field-to-field contribution remains. It should not be read as irreducible variation on repeated treatment of one unchanged boron layout; for a fixed layout, the paper says only the conditional capture variance describes repeat-to-repeat fluctuations.

High-fluence relative variance plateau

Label	Value
High-fluence limit	0.16 relative variance

The plateau concerns an ensemble of boron fields, not repeated irradiation of one unchanged field.

The second striking result concerns zero-hit events. The model defines a hit as an alpha–lithium reaction event that deposits any positive energy in the nucleus. Averaging across random boron fields, the probability of receiving no such hit is greater than the corresponding probability for homogeneous boron at the same mean concentration. This follows mathematically because the zero-hit probability depends nonlinearly on the local capture intensity: variability in that intensity raises the average chance of zero events, even if the average intensity stays constant.

For ξ=6 μm and neutron fluence Φₙ=3×10¹¹ cm⁻², the zero-hit probability was 40.49% higher than for the homogeneous-boron reference. The authors’ comparison is relative, not a statement that 40.49% of nuclei receive zero energy. The abstract does not provide the absolute zero-hit probability for either case, so the percentage should be understood only as the reported increase over the reference.

Increase in zero-hit probability

Label	Value
ξ=6 μm; Φₙ=3×10¹¹ cm⁻²	40.49 % higher

• ξ=6 μm; Φₙ=3×10¹¹ cm⁻² 40.49 % higher

A zero hit counts no positive alpha–lithium energy deposition in the modeled nucleus; it is not cell survival.

That result is about physical energy deposition, not biology. A nucleus that receives no alpha–lithium energy from the modeled reaction products is not necessarily an undamaged cell, and a nucleus that is hit is not necessarily a dead one. The model omits other radiation components, including capture gamma rays, and does not calculate DNA damage, repair, cell death or tumor control. The authors explicitly caution against interpreting zero-hit probability as cell survival.

Why This Changes Things

The practical message is not that average boron concentration is useless. It remains central to estimating the average capture rate and mean dose. Rather, average concentration does not contain enough information to describe all the variability at cellular scales. If boron is clustered, the local probabilities of capture are uneven. Since the reaction products travel only a few micrometers, those spatial patterns can translate into variation in dose to nuclei.

This helps clarify why measuring boron at one scale may not settle questions at another. Tissue-level imaging can estimate broad concentration patterns, but it cannot necessarily resolve subcellular distributions. The paper’s framework identifies correlation length as a potentially meaningful quantity alongside mean concentration and variation amplitude. It offers a way to ask whether boron-rich patches are fine-grained enough to average out within a nucleus, or broad enough to leave some nuclei comparatively underexposed and others more exposed.

There is a broader lesson for radiation dosimetry: the mean dose and the distribution of dose are different objects. This study does not demonstrate that incorporating microscopic heterogeneity will improve clinical outcomes. It does show, in a defined physical model, why a treatment-planning approach that uses only a mean can miss part of the statistical story. That distinction could matter when future studies link realistic boron microdistributions to biological response measurements.

The results also underscore the value of separating sources of uncertainty rather than bundling them together. Capture randomness is an intrinsic feature of counting discrete events. Heterogeneity variance reflects differences in the material’s spatial structure across realizations. They respond differently to fluence, so a single uncertainty estimate can conceal which mechanism is driving variation and how that mechanism may change as exposure changes.

The charts summarize three reported values, not complete dose-response curves. The paper’s abstract provides the crossover, plateau and zero-hit comparison, but the supplied text does not include its full result tables or figure captions. That limits what can responsibly be plotted here: no intermediate fluence values, error bars, or numerical comparisons across multiple correlation lengths can be reconstructed from the provided material.

What’s Next

The model is deliberately controlled. It assumes a spherical nucleus in water-equivalent material, a spatially uniform neutron field, and fixed boron statistics as fluence changes. It treats boron heterogeneity as not altering neutron transport, and it excludes boron depletion during irradiation. Its particle-transport calculation focuses on electromagnetic transport of the alpha and lithium products; it does not include gamma dose or all other components of a clinical BNCT exposure.

Those choices make the statistical mechanism easier to isolate, but they also mark the distance between a model result and a treatment decision. Real tumors differ in shape, tissue composition, boron uptake and subcellular localization. Neutron spectra and exposure conditions also matter. Most importantly, dose statistics alone do not say how cells respond. Biological models and experimental validation would be needed before translating a zero-hit statistic or variance plateau into predictions about tumor control or side effects.

The next useful step is therefore not simply to collect more precise average boron measurements. It is to characterize how boron is distributed over the distances that matter to particle ranges and nuclear dimensions, then test whether those maps improve predictions of energy deposition and biological response. Better delivery agents and imaging approaches may help, but the paper does not establish that any existing measurement can yet supply all the microstructural detail its model uses.

What the study opens is a more discriminating question for BNCT: not only how much boron reaches a tumor, but how its spatial arrangement shapes the odds that individual cell nuclei receive the intended dose. The answer will need experiments and clinically realistic modeling. But by showing that two mechanisms of dose fluctuation scale differently—and that heterogeneity can raise the chance of a nucleus receiving no reaction-product energy—the authors make clear why the microscopic map may matter as much as the tissue-wide average.