The Biologically Effective Particle Number Ratio (BPNR): a new framework to quantify the therapeutic window in SFRT and other modalities

TL;DR

Spatially Fractionated Radiation Therapy (SFRT) presents a challenge for assessing therapeutic gain due to heterogeneous dose patterns. The authors introduce the Biologically Effective Particle Number Ratio (BPNR) and its relative form RBPNR as outcome-driven, MU-based metrics that quantify the therapeutic window using TCP and NTCP endpoints rather than spatial dose descriptors. The core definition, $BPNR_{50,50} = N_{NTCP_{50}} / N_{TCP_{50}}$, enables direct, model-independent comparisons across modalities, with RBPNR comparing test versus reference configurations. A preclinical proton minibeam radiotherapy (pMBRT) example demonstrates a widened therapeutic window (e.g., $BPNR_{50,50} > 1.93$ for pMBRT and $RBPNR_{50,50} > 1.46$), illustrating the framework’s translational potential for SFRT, FLASH, high-LET, and combination therapies.

Abstract

Spatially Fractionated Radiation Therapy (SFRT) produces highly heterogeneous dose distributions, for which conventional metrics such as peak, valley, or average dose can yield ambiguous or inconsistent estimates of therapeutic window changes. These dose descriptors are not uniquely linked to biological outcome, complicating comparisons between modalities. We introduce the Biologically Effective Particle Number Ratio (BPNR), a model-independent and outcome-based framework that quantifies therapeutic window changes using directly measurable delivery quantities rather than spatial dose metrics. BPNR is defined as the ratio of total particle numbers (proportional to monitor units, MU) required to reach specified levels of tumor control probability (TCP) and normal tissue complication probability (NTCP). The relative BPNR (RBPNR) compares therapeutic windows across modalities and avoids ambiguities of spatial dose averaging by relying solely on biological endpoints. As an illustration, we apply the framework to preclinical proton minibeam radiotherapy (pMBRT) versus conventional proton therapy. The absence of normal tissue complications at the highest tested MU in the pMBRT arm yields an RBPNR > 1.4, indicating a widened therapeutic window. The BPNR/RBPNR methodology provides a concise, experimentally grounded, and modality-agnostic way to quantify therapeutic window changes, with potential applications across SFRT, FLASH, high-LET radiation, and combination therapies.

Figures (2)

• Figure 1: Illustration of the BPNR concept. Each row represents a treatment modality: a reference plan (top) and a test plan (bottom). For each modality, $\mathrm{NTCP}_\mathrm{50\%}$ and $\mathrm{TCP}_\mathrm{50\%}$ response curves are measured as a function of total particle number $N$ (or equivalently, monitor units, MU). The spatial dose distribution is preserved for each of the two plans, and by scaling the plan MU, full response curves can be established. $\mathrm{BPNR}_{50,50}$ is extracted for each plan, and their ratio quantifies the relative change in the therapeutic window between the test and reference modalities. A $\mathrm{RBPNR}_{50,50}$$>$ 1 thus represents an increase in the therapeutic window relative to the reference modality.
• Figure 2: Illustrative TCP and NTCP response curves for conventional proton therapy (CONV treatment plan, top panel) and proton minibeam radiotherapy (pMBRT treatment plan, bottom panel), shown as a function of delivered monitor units. The abscissa is normalized to the MU needed for achieving ${\mathrm{TCP}_{50}}$ for the respective treatment plan. $\mathrm{BPNR}_{50,50}$ values (black intervals) are derived from the ${\mathrm{TCP}_{50}}$ and ${\mathrm{NTCP}_{50}}$ points for the CONV and the pMBRT plan. In the pMBRT case, the absence of normal tissue complications up to the maximum MU provides only a lower bound for $N_{\mathrm{NTCP}_{50}}$ and thus for the $\mathrm{BPNR}_{50,50}$ (indicated by the grey arrow in the lower panel).