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Flow-Dependent Fragility of Velocity-Normalized Doppler Intracranial Pressure Estimation: Mathematical Derivation and Empirical Validation Flow-Dependent Fragility in Velocity-Normalized Doppler Intracranial Pressure Estimation Corresponding author: Lorenzo Querci; lorenzo.querci@ospedaleniguarda.it Article type: Original Work - Clinical Investigation Body word count: Abstract word count: Figures: 4 Tables: 2 Online supplementary files: 4 Running title: Flow-dependent fragility of Doppler-derived ICP Details Page Compliance with instructions. This manuscript is submitted as Original Work and has been prepared in accordance with the February 2025 Neurocritical Care Instructions for Authors. Author contributions. Authorship and approval. All listed authors meet authorship requirements, accept responsibility for the work, and approved the submitted version. Originality. The manuscript has not been published previously and is not under consideration by another journal. Ethics and consent. The mathematical component did not involve new participants. Empirical validation used a publicly available, de-identified dataset. The original study was approved by the University of Cambridge research ethics board (REC 15/LO/1918), with written consent from participants' next of kin. Conflicts of interest. The authors declare no financial or non-financial conflicts of interest relevant to this work. Reporting guideline. A STROBE checklist adapted to the empirical validation component is included at the end of the manuscript. Funding. No funds, grants, or other support were received for this study. Data availability. The validation dataset is publicly available as S1 Data with Robba et al., PLOS Medicine 2017 (doi:10.1371/journal.pmed.1002356). Code availability. The complete R analysis, including a one-command pipeline, all mathematical and statistical analyses, validation checks, and figure generation, is included as supplementary material. Supplementary material. A separate Online Supplementary Appendix reports additional mathematical derivations, alternative measurement-error and covariance assumptions, LOPO recalibration, normalized and spline-based analyses, PaCO2-adjusted analyses, within-patient analyses, negative-control analyses, and supplementary tables and figures. AI-assisted technology. AI-assisted technologies (OpenAI ChatGPT/Codex, GPT-5) were used for language review, code drafting and debugging. The authors independently verified all analyses, citations, interpretations, figures, and final text and retain full responsibility for the work. Abstract Background/Objective: Velocity-normalized Doppler estimators of intracranial pressure (ICP) have shown only moderate accuracy for non-invasive ICP estimation. We hypothesized that part of this limitation arises from the mathematical structure of these estimators, in which mean flow velocity (FVm) appears in the denominator. Under this hypothesis, identical Doppler measurement uncertainty should produce progressively greater uncertainty in estimated ICP as FVm decreases. We tested this hypothesis mathematically and empirically. Methods: We first used sensitivity analysis and uncertainty propagation to quantify how identical Doppler measurement errors translate into uncertainty (the expected variability of the estimated ICP caused solely by measurement error) in velocity-normalized estimators, using the Czosnyka formula as an exemplar. We tested this prediction in 445 paired invasive and non-invasive ICP observations from 64 patients using FVm quintile-based differences and mixed-effects analyses. ONSD-based analyses, whose mathematical formulation does not include FVm normalization, served as negative controls. Results: Empirically, mean absolute error (MAE) of the Czosnyka estimator increased from 6.90 mmHg in the highest FVm quintile to 10.55 mmHg in the lowest, while root mean squared error (RMSE) increased from 8.94 to 13.58 mmHg. Lowest-minus-highest differences were 3.65 mmHg for MAE (95% CI 1.03–6.27; p=0.005) and 4.64 mmHg for RMSE (95% CI 1.77–7.31; p=0.002). Each 10-cm/s reduction in FVm was associated with a 1.18-mmHg increase in absolute error (95% CI 0.64–1.72; p<0.001). The observed low-flow error gradient was consistent with the mathematical prediction of increasing uncertainty at lower FVm. As expected for a negative-control estimator that does not normalize by FVm, ONSD-based models showed no comparable error gradient. Conclusions: The mathematical structure of velocity-normalized estimators predicts state-dependent fragility, and the predicted low-flow error gradient was observed empirically. Rather than assuming that all Doppler-derived nICP estimates are equally reliable, clinicians should recognize that confidence in an individual estimate depends on mean flow velocity, as lower FVm amplifies the impact of Doppler measurement uncertainty. Global accuracy metrics may therefore obscure clinically relevant state-dependent reliability. Keywords: intracranial pressure; multimodal monitoring; transcranial Doppler; uncertainty propagation; mean flow velocity; prediction error Introduction Invasive intracranial pressure (ICP) monitoring remains the gold standard for ICP measurement and management when clinically indicated [1,2], but non-invasive approaches are attractive when invasive monitoring is unavailable, delayed, or not justified [3,4]. Transcranial Doppler (TCD) is repeatable at the bedside and provides immediate information on cerebral hemodynamics.[5–7] Three principal methodological approaches to non-invasive ICP estimation can be identified: (i) velocity-normalized algebraic formulations (e.g., Czosnyka/Schmidt formula) [8,9], (ii) pulsatility-based estimators [10,11], and (iii) data-driven or machine-learning models [12]. Velocity-normalized and pulsatility-based estimators are among the best-characterized approaches because of their physiological interpretability and simplicity.[13] The Czosnyka/Schmidt approach estimates cerebral perfusion pressure from arterial pressure and middle cerebral artery flow velocities and then derives non-invasive ICP (nICP).[9,14] Its commonly used algebraic form is , where mABP is mean arterial pressure, FVd is diastolic flow velocity, and FVm is mean flow velocity. Validation studies have reported clinically useful association or discrimination together with heterogeneous absolute accuracy and broad limits of agreement.[15–18] Disagreement between invasive and non-invasive ICP estimates is usually attributed to technical variability, physiological heterogeneity, or imperfect calibration.[2,4,5] However, the equation itself may introduce an additional source of vulnerability because FVm appears in the denominator. This is a mathematical rather than physiological phenomenon: as the denominator becomes smaller, an identical Doppler measurement error produces a progressively larger uncertainty of the estimated ICP. We hypothesized that velocity normalization can produce greater uncertainty at lower mean flow velocity. We first developed this hypothesis mathematically for the Czosnyka estimator. We next tested whether the predicted low-flow error gradient could be detectable in the open Robba et al. dataset [5]. The primary objective was to determine whether this mathematically predicted state-dependent fragility was also observable in clinical data. Methods Sensitivity analysis Several Doppler-based approaches estimate ICP or related hemodynamic quantities using velocity-normalized formulations. Every Doppler measurement is affected by some degree of measurement uncertainty. Our question was therefore: if the measured values (mABP, FVs, or FVd) are slightly wrong, how much uncertainty will this introduce into the estimated ICP? Because flow velocity appears in the denominator, the mathematical propagation of Doppler measurement uncertainty may depend on mean flow velocity. We therefore examined whether decreasing FVm increases the estimator’s sensitivity to measurement uncertainty. The Czosnyka estimator was the primary exemplar because it is extensively studied and physiologically interpretable. A purely mathematical extension to a pulsatility-index estimator was provided in Online Supplementary Appendix S1.3; PI was not tested empirically. The Czosnyka estimator was expressed as a function of mABP, FVd, and FVm. Because FVm is conventionally derived from the Doppler waveform as , FVd contributes to both the numerator and denominator of the estimator. Substitution avoids treating FVd and FVm as independent inputs: To quantify how measurement errors affect the estimated ICP, we first quantified how much the estimated ICP changes in response to a small error in each measured variable (mABP, FVs, and FVd). Mathematically, these sensitivities correspond to the partial derivatives of the estimator with respect to each input variable.[19] Full derivations are provided in Online Supplementary Appendix S1.1. The resulting sensitivities are: Both velocity derivatives contain inverse powers of FVm. For a fixed velocity ratio , their magnitudes simplify to and . Uncertainty propagation, Fragility Index and threshold interpretation To quantify how measurement uncertainty propagates through velocity-normalized ICP estimators, we used first-order uncertainty propagation (delta method).[20] This approach approximates the variance of a function from the local sensitivities of the estimator and the errors of its input variables.[20] The delta method estimates how errors in the measured variables translates into uncertainty in the estimated ICP. For a vector of primitive measurements with measurement-error covariance matrix Σ, first-order delta-method propagation gives . Full covariance formulation was reported in the Online Supplementary Appendix S1.2. We defined the Fragility Index (FI) as: . The FI therefore quantifies the expected uncertainty of the estimated ICP arising from measurement error. It depends on the measured values because the same measurement error does not always produce the same error in the estimated ICP. For illustration, we assumed zero-centered approximately normal estimation error around the point estimate. For a clinical threshold T, the probability associated with the uncertainty model is . This is a decision-confidence illustration rather than a calibrated clinical probability because systematic bias and true repeatability variances were unavailable. The primary mathematical illustration used SD(mABP)=5 mmHg, SD(FVs)=SD(FVd)=3 cm/s, a correlation of 0.5 between FVs and FVd errors, and no covariance with mABP. Proportional uncertainty and alternative covariance assumptions were examined as robustness analyses in the Online Supplementary S1.4, Online Supplementary Table S1 and Online Supplementary Figure S1. Empirical validation dataset Empirical validation used data from the prospective observational study by Robba et al.[5] The original study enrolled adults with acute brain injury requiring invasive ICP monitoring at Addenbrooke's Hospital, Cambridge, from January 1 to November 1, 2016. It was approved by the University of Cambridge research ethics board (REC 15/LO/1918), and written consent was obtained from participants' next of kin. The present study is a post hoc secondary analysis of the publicly available de-identified dataset accompanying the original publication. The dataset contained repeated paired invasive ICP, mABP, middle cerebral artery velocities, optic nerve sheath diameter (ONSD), and PaCO2 measurements. TCD values were averaged across bilateral recordings. Acquisition details are reported in the source publication.[5] Empirical outcomes and statistical analysis For each observation, Czosnyka nICP was recalculated as . Error was invasive ICP minus nICP; absolute error, squared error, mean absolute error (MAE), root mean squared error (RMSE), median absolute error, and bias were calculated. FVm was divided into quintiles using the observed distribution. The primary descriptive contrast compared the lowest and highest quintiles. The continuous association was estimated with a linear mixed model of absolute error as a function of a 10-cm/s reduction in FVm, with patient-specific random intercept. To account for repeated measurements within patients, patient-cluster bootstrap with 5,000 resamples generated confidence intervals for quintile-specific MAE and RMSE and for lowest-minus-highest differences. The negative control was the published [5] ONSD equation, , which does not contain FVm in the denominator. Its absolute errors were analyzed across the same FVm quintiles and with the same mixed model. LOPO recalibration, normalization by overall model error, spline models, PaCO2 adjustment, patient fixed effects, random slopes, extended negative controls, secondary correlation analyses, and detailed bootstrap procedures are reported in the Online Supplementary Appendix S2, and S4. Analyses were performed in R 4.4.1 using lme4, lmerTest, clubSandwich, splines, ggplot2, and patchwork. Two-sided p<0.05 was considered statistically significant. The complete R analysis package, including a one-command pipeline, all mathematical and statistical analyses, validation checks, and figure generation, is included as supplementary material (Online Supplementary Appendix S5). Results Mathematical prediction Rewriting the estimator in terms of mABP, FVs, and FVd showed that velocity sensitivities scale inversely (hyperbolic) with FVm for a fixed velocity ratio (Figure 1A). Under the illustrative constant-absolute-error model, FI increased progressively as FVm decreased (Figure 1B). The clinical implication is conditional rather than deterministic. With mABP fixed at 90 mmHg and the FVd/FVm ratio fixed to produce an nICP of 24 mmHg, the illustrative FI was 7.65 mmHg at FVm 20 cm/s, 4.24 mmHg at 40 cm/s, 3.23 mmHg at 60 cm/s, and 2.80 mmHg at 80 cm/s. The corresponding modeled probabilities of exceeding a threshold of 22 mmHg were 0.60, 0.68, 0.73, and 0.76 (Figure 2). Thus, the same point estimate can imply different decision confidence when the propagated uncertainty differs. Dataset and global performance The public dataset contained 445 complete paired observations from 64 patients, with no missing values in the variables used for the main analysis. Mean FVm was 67.2 12.7 cm/s. Overall Czosnyka MAE was 8.25 mmHg, RMSE was 11.01 mmHg, and the estimator showed a small mean overestimation of invasive ICP (bias = -1.06 mmHg, calculated as invasive ICP minus estimated ICP). (Table 1). Validation across FVm quintiles The observed FVm distribution and quintile boundaries are shown in Table 2. The lowest FVm quintile contained 93 observations with mean FVm 49.2 cm/s; the highest contained 82 observations with mean FVm 84.4 cm/s. Czosnyka MAE decreased from 10.55 mmHg in the lowest quintile to 6.90 mmHg in the highest (Figure 3A), while RMSE decreased from 13.58 to 8.94 mmHg (Table 2). The patient-cluster bootstrap lowest-minus-highest difference was 3.65 mmHg for MAE (95% CI 1.03-6.27; p=0.005) and 4.64 mmHg for RMSE (95% CI 1.77-7.31; p=0.002). Median absolute error was also numerically greater in the lowest quintile (9.78 vs 5.92 mmHg), although its bootstrap difference was not significant (3.86 mmHg; 95% CI -1.59 to 6.42; p=0.132). In the continuous mixed-effects model, each 10-cm/s reduction in FVm was associated with a 1.18-mmHg increase in absolute Czosnyka error (95% CI 0.64-1.72; p<0.001) as shown in Figure 3B. Negative controls The corresponding continuous effect was 0.07 mmHg per 10-cm/s lower FVm for ONSD (95% CI -0.17 to 0.31; p=0.558). Its lowest-minus-highest MAE difference was 0.26 mmHg (95% CI -0.90 to 1.34; p=0.628) (Figure 4, and Online Supplementary Table S3). The observed gradient was therefore not observed for the negative-control model, which does not contain FVm in the denominator. Robustness analyses The principal pattern persisted after PaCO2 adjustment, patient fixed-effects analysis, random-slope modeling, normalization, and LOPO recalibration. The LOPO Czosnyka effect was 0.47 mmHg per 10-cm/s lower FVm (95% CI 0.12-0.82; p=0.009). Curvature was supported for the published equation (p=0.007), but not after LOPO recalibration (p=0.588), providing no reproducible threshold. Proportional measurement error attenuated the theoretical low flow FI gradient. Full results are reported in Online Supplementary Appendices S3-S4, Online Supplementary Table S2, and Online Supplementary Figure S2. Discussion The principal clinical message is that a velocity-normalized Doppler intracranial pressure estimation value should not always be interpreted with the same level of confidence. When mean flow velocity is lower, the same Doppler measurement error is expected to produce greater uncertainty in the estimated ICP. The mathematical derivation explains why such behavior is expected when Doppler error contains an absolute component. A velocity ratio can remain unchanged when its components scale proportionally, producing the same nICP point estimate, while the effect of an absolute perturbation becomes larger at lower signal magnitude. FI makes that distinction explicit by attaching a model-based propagated uncertainty to the point estimate. The threshold illustration shows the conceptual implication: same estimate does not necessarily mean same confidence. Clinically, two patients may have the same estimated nICP of 24 mmHg but different levels of confidence in that estimate. A patient with preserved mean flow velocity is expected to have a more reliable numerical estimate than a patient with markedly reduced flow velocity, even if the point estimate is identical. FI should be interpreted cautiously. It is not an observed patient characteristic, a replacement for prediction error, or a calibrated probability of true ICP. Its magnitude depends on the covariance matrix assigned to mABP, FVs, and FVd. The low-flow gradient was marked under constant absolute velocity error but much smaller under relative error. Before FI can be quantitatively calibrated for bedside use, the measurement-error structure of transcranial Doppler itself needs to be characterized. In particular, future repeatability studies should determine the magnitude, scaling (absolute versus proportional), and correlation structure of measurement errors in mABP, FVs, and FVd, ideally using replicate acquisitions, raw spectral envelopes, and reader-level variance components. The negative-control analysis strengthens, but does not prove, the structural interpretation. If low FVm simply marked globally more difficult or severely ill observations, one might expect ONSD error to increase in parallel; it did not. The broader TCD literature also shows why global association, and individual accuracy should not be conflated. Our findings complement rather than contradict previous validation studies. For example, Czosnyka et al. emphasized tracking cerebral perfusion pressure changes rather than estimating absolute ICP [9], while subsequent studies reported useful discrimination despite broad agreement or systematic bias.[5,15–17] This supports interpreting non-invasive TCD estimates as clinically informative but with state-dependent numerical reliability, consistent with current multimodal approaches. [6,7,17] Flow-dependent uncertainty provides an additional reason to retain that multimodal approach. Although the empirical analyses focused on Czosnyka nICP, the same denominator-driven mechanism may extend to other velocity-normalized estimators. Its exact form depends on the estimator, and on which measured quantities are treated as primitive inputs, as illustrated mathematically for PI in the Online Supplement. The results do not define a clinical FVm cutoff below which Czosnyka nICP should be discarded. Curvature was observed for the published equation but was not reproduced after LOPO recalibration. However, the available data contained relatively few very-low FVm observations, and the observed velocity range largely corresponded to the approximately linear portion of the theoretical sensitivity curve. These findings do not suggest that the Czosnyka estimator should be abandoned at low FVm. Rather, they indicate that low FVm should be interpreted as a condition in which numerical estimates deserve greater caution and should be integrated more carefully with the overall clinical assessment. TCD-derived nICP remains an adjunct and should not delay invasive monitoring when indicated.[1,6,7] Limitations The empirical validation was post hoc and used one heterogeneous open dataset. External validation in cohorts with different diagnoses, devices, operators, and flow distributions is required. FVm is both inside the Czosnyka equation and the variable used to examine effect modification, creating mathematical coupling. The theoretical derivation, continuous analysis, negative controls, within-patient models, and recalibration support the interpretation but cannot establish causality. The negative-control equations were developed in the source dataset. LOPO prediction reduces in-sample optimism but is not equivalent to independent external validation. The public file did not contain raw waveforms, replicate acquisitions, operator identifiers, treatment detail, or empirical measurement-error covariance matrices. FI scenarios are therefore illustrative and cannot quantify how much of the observed prediction error arose from measurement uncertainty. Very-low FVm measurements were underrepresented in the dataset. As a result, the observed velocity range may not fully capture the region where the mathematical model predicts the greatest amplification of Doppler measurement uncertainty. Conclusions The Czosnyka estimator is mathematically expected to become more fragile at lower mean flow velocity. The predicted low flow error gradient was empirically detectable, whereas ONSD negative control did not show the same pattern. Global accuracy metrics may therefore obscure state-dependent estimator reliability. FI provides a transparent framework for uncertainty-aware interpretation, but quantitative clinical implementation requires external validation and direct measurement of acquisition error. When FVm is low, clinicians should give greater weight to the full clinical picture, neuroimaging, trends, and the indication for invasive monitoring, because the same Doppler measurement uncertainty can produce a larger error in the estimated ICP. Acknowledgments The authors thank the investigators of the original Robba et al. study for making the individual-level data publicly available. Supplementary Material The Online Supplementary Appendix contains extended derivations and robustness analyses, including alternative measurement-error structures, covariance assumptions, LOPO recalibration, normalized error, splines, PaCO2 adjustment, patient fixed effects, extended negative controls, and detailed bootstrap results. References 1. Carney N, Totten AM, O’Reilly C, Ullman JS, Hawryluk GWJ, Bell MJ, et al. Guidelines for the Management of Severe Traumatic Brain Injury, Fourth Edition. NEUROSURGERY. 2017;80:6–15. https://doi.org/10.1227/NEU.0000000000001432 2. Nag DS, Sahu S, Swain A, Kant S. Intracranial pressure monitoring: Gold standard and recent innovations. WJCC. 2019;7:1535–53. https://doi.org/10.12998/wjcc.v7.i13.1535 3. Picetti E, Biasucci DG, Gouvea Bogossian E, Brasil S, Cardim D, Czosnyka M, et al. Non-invasive intracranial pressure estimation in the intensive care unit: narrative review of methods and clinical applications. Intensive Care Med. 2026;52:747–64. https://doi.org/10.1007/s00134-026-08420-7 4. Robba C, Picetti E, Bertuccio A, Rynkowski CB, Hawryluk GWJ, Bogossian EG, et al. Non-invasive ICP monitoring when invasive systems are available in the care of acute brain injured patients: a clinical approach. J Anesth Analg Crit Care. 2026;6:27. https://doi.org/10.1186/s44158-026-00356-0 5. Robba C, Cardim D, Tajsic T, Pietersen J, Bulman M, Donnelly J, et al. Ultrasound non-invasive measurement of intracranial pressure in neurointensive care: A prospective observational study. Schreiber M, editor. PLoS Med. 2017;14:e1002356. https://doi.org/10.1371/journal.pmed.1002356 6. Robba C, Czosnyka M, Smielewski P, others. Non-invasive assessment of intracranial pressure. Intensive Care Medicine. 2020;46:1270–83. https://doi.org/10.1007/s00134-020-06046-4 7. Sharma R, Tsikvadze M, Peel J, Howard L, Kapoor N, Freeman WD. Multimodal monitoring: practical recommendations (dos and don’ts) in challenging situations and uncertainty. Front Neurol. 2023;14:1135406. https://doi.org/10.3389/fneur.2023.1135406 8. Ursino M, Lodi CA. A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. Journal of Applied Physiology. 1988;82:1256–69. 9. Czosnyka M, Matta BF, Smielewski P, Kirkpatrick PJ, Pickard JD. Cerebral perfusion pressure in head-injured patients: a noninvasive assessment using transcranial Doppler ultrasonography. Journal of Neurosurgery. 1998;88:802–8. https://doi.org/10.3171/jns.1998.88.5.0802 10. Bellner J, Romner B, Reinstrup P, Kristiansson K-A, Ryding E, Brandt L. Transcranial Doppler sonography pulsatility index (PI) reflects intracranial pressure (ICP). Surgical Neurology. 2004;62:45–51. https://doi.org/10.1016/j.surneu.2003.12.007 11. Behrens A, Lenfeldt N, Ambarki K, Malm J, Eklund A. Transcranial Doppler pulsatility index: not an accurate method to assess intracranial pressure. Neurosurgery. 2010;66:1050–7. https://doi.org/10.1227/01.NEU.0000369620.43519.46 12. Kim DJ, Kasprowicz M, Carrera E, Castellani G, Zweifel C, Lavinio A, et al. Noninvasive intracranial pressure estimation using a data-driven approach. Journal of Neurotrauma. 2012;29:988–96. https://doi.org/10.1089/neu.2011.1978 13. Cardim D, Robba C, Bohdanowicz M, Donnelly J, Cabella B, Liu X, et al. Non-invasive Monitoring of Intracranial Pressure Using Transcranial Doppler Ultrasonography: Is It Possible? Neurocrit Care. 2016;25:473–91. https://doi.org/10.1007/s12028-016-0258-6 14. Schmidt B, Klingelhöfer J, Schwarze JJ, Sander D, Wittich I. Noninvasive prediction of intracranial pressure using transcranial Doppler ultrasonography. Journal of Neurology. 1997;244:497–500. 15. Brandi G, Béchir M, Sailer S, Haberthür C, Stocker R, Stover JF. Transcranial color-coded duplex sonography allows to assess cerebral perfusion pressure noninvasively following severe traumatic brain injury. Acta Neurochir. 2010;152:965–72. https://doi.org/10.1007/s00701-010-0643-4 16. Rasulo FA, Bertuetti R, Robba C, Lusenti F, Cantoni A, Bernini M, et al. The accuracy of transcranial Doppler in excluding intracranial hypertension following acute brain injury: a multicenter prospective pilot study. Crit Care. 2017;21:44. https://doi.org/10.1186/s13054-017-1632-2 17. Robba C, Pozzebon S, Moro B, Vincent J-L, Creteur J, Taccone FS. Multimodal non-invasive assessment of intracranial hypertension: an observational study. Crit Care. 2020;24:379. https://doi.org/10.1186/s13054-020-03105-z 18. Zweifel C, Czosnyka M, Carrera E, De Riva N, Pickard JD, Smielewski P. Reliability of the Blood Flow Velocity Pulsatility Index for Assessment of Intracranial and Cerebral Perfusion Pressures in Head-Injured Patients. Neurosurgery. 2012;71:853–61. https://doi.org/10.1227/NEU.0b013e3182675b42 19. Evaluation of measurement data — Guide to the expression of uncertainty in measurement [Internet]. 2008 [cited 2026 June 19]. https://doi.org/10.59161/JCGM100-2008E 20. Oehlert GW. A Note on the Delta Method. The American Statistician. 1992;46:27–9. https://doi.org/10.1080/00031305.1992.10475842 Figure Legends FIGURE 1. Mathematical origin of flow-dependent fragility. (A) Absolute local sensitivities to systolic flow velocity (FVs) and diastolic flow velocity (FVd) increase as mean flow velocity (FVm) decreases for the illustrative mean arterial blood pressure (mABP) of 90 mmHg and a fixed FVd/FVm ratio of 0.6. (B) The Fragility Index (FI), defined as the propagated standard deviation of estimated intracranial pressure under the specified measurement-error assumptions (mABP SD = 5 mmHg; FVs and FVd SD = 3 cm/s each; correlation = 0.5), increases toward lower FVm. In both panels, shading indicates values below the 20th percentile of FVm in the validation dataset, and the vertical dotted line marks that percentile (58.3 cm/s). FIGURE 2. Clinical interpretation of propagated uncertainty. All panels show the same non-invasive intracranial pressure (nICP) point estimate (24 mmHg) and clinical threshold (22 mmHg), but different Fragility Index (FI) values associated with mean flow velocities (FVm) of 20, 40, 60, and 80 cm/s. The dashed line marks the threshold, the solid vertical line marks the point estimate, and the shaded area represents probability mass above the threshold under the illustrative zero-centered normal error model. These are model-based uncertainty illustrations, not calibrated clinical probabilities. FIGURE 3. Empirical validation in the Robba et al. dataset. (A) Mean absolute Czosnyka prediction error across quintiles of mean flow velocity (FVm); error bars represent patient-cluster bootstrap 95% confidence intervals. (B) Continuous mixed-effects model relating absolute Czosnyka prediction error to FVm, with a 95% confidence band. Lower FVm was associated with progressively greater prediction error. FIGURE 4. Negative-control analysis across mean flow velocity (FVm) quintiles. Mean absolute error of the Czosnyka estimator and optic nerve sheath diameter (ONSD) estimator is shown with patient-cluster bootstrap 95% confidence intervals. Czosnyka error increased at lower FVm, whereas ONSD error remained relatively stable. 1 STROBE Checklist STROBE items adapted to the empirical validation component using a publicly available prospective observational dataset. STROBE Statement—Checklist of items that should be included in reports of cohort studies Item No Recommendation Page No / location Title and abstract 1 (a) Indicate the study’s design with a commonly used term in the title or the abstract p. 4 (Abstract, Methods: empirical analysis of 445 paired observations from the Robba et al. dataset); the source design is explicitly identified as prospective observational on p. 10. (b) Provide in the abstract an informative and balanced summary of what was done and what was found pp. 4–5 (structured Abstract: Background/Objective, Methods, Results, and Conclusions). Introduction Background/rationale 2 Explain the scientific background and rationale for the investigation being reported pp. 6–7 (Introduction: prior evidence, methodological problem, and rationale for investigating denominator-driven fragility). Objectives 3 State specific objectives, including any prespecified hypotheses p. 7 (Introduction, final paragraph: prespecified hypothesis and primary objective). Methods Study design 4 Present key elements of study design early in the paper pp. 7–10 (Methods); empirical component described under “Empirical validation dataset” on p. 10. Setting 5 Describe the setting, locations, and relevant dates, including periods of recruitment, exposure, follow-up, and data collection p. 10 (Addenbrooke’s Hospital, Cambridge; recruitment January 1 to November 1, 2016; public de-identified dataset). Participants 6 (a) Give the eligibility criteria, and the sources and methods of selection of participants. Describe methods of follow-up p. 10 (adults with acute brain injury requiring invasive ICP monitoring; source and selection from Robba et al. 1; repeated paired observations). Full original eligibility and follow-up procedures are in reference 1. (b) For matched studies, give matching criteria and number of exposed and unexposed Not applicable: this was not a matched cohort analysis. Variables 7 Clearly define all outcomes, exposures, predictors, potential confounders, and effect modifiers. Give diagnostic criteria, if applicable pp. 9–10 (FI and uncertainty model; invasive ICP, nICP, FVm, error metrics, ONSD negative control, and PaCO2); detailed robustness covariates in Online Supplement S2–S4. Data sources/ measurement 8* For each variable of interest, give sources of data and details of methods of assessment (measurement). Describe comparability of assessment methods if there is more than one group p. 10 (public dataset; paired invasive ICP, mABP, MCA velocities, ONSD, and PaCO2; bilateral TCD values averaged; acquisition methods referenced to Robba et al. 1); estimator calculations on p. 10. Bias 9 Describe any efforts to address potential sources of bias pp. 9–11 (primitive-input derivation, covariance-aware propagation, patient random intercepts, patient-cluster bootstrap, and negative control); p. 13 and Online Supplement S1–S4 (alternative assumptions, recalibration, within-patient and adjusted analyses). Study size 10 Explain how the study size was arrived at p. 12 and Table 1: all 445 complete paired observations from 64 patients available in the public dataset were analyzed; no prospective sample-size calculation was performed for this secondary analysis. Quantitative variables 11 Explain how quantitative variables were handled in the analyses. If applicable, describe which groupings were chosen and why p. 10 (FVm analyzed continuously per 10-cm/s reduction and in quintiles; primary contrast lowest vs highest quintile); p. 12 and Table 2 (observed quintile boundaries); p. 15 (reason for retaining a continuous model and not defining a cutoff). Statistical methods 12 (a) Describe all statistical methods, including those used to control for confounding pp. 10–11 (error definitions, mixed-effects model, patient-cluster bootstrap with 5,000 resamples, negative-control model, software and two-sided significance threshold); Online Supplement S2–S4 (PaCO2-adjusted, fixed-effect, random-slope, spline and LOPO analyses). (b) Describe any methods used to examine subgroups and interactions pp. 10–11 and Online Supplement S2–S4 (FVm quintile comparisons, spline models, estimator comparisons, within-patient models and random slopes). (c) Explain how missing data were addressed p. 12 and Table 1: no missing values in variables used for the main analysis (0 missing main-analysis observations). (d) If applicable, explain how loss to follow-up was addressed Not applicable: this was a secondary analysis of repeated paired measurements in a public dataset, with no study follow-up or loss-to-follow-up analysis. (e) Describe any sensitivity analyses pp. 9–10 and p. 13; Online Supplement S1–S4 (alternative measurement-error and covariance assumptions, proportional-error model, PaCO2 adjustment, patient fixed effects, random slopes, splines, LOPO recalibration, normalization and extended negative controls). Results Participants 13* (a) Report numbers of individuals at each stage of study—eg numbers potentially eligible, examined for eligibility, confirmed eligible, included in the study, completing follow-up, and analysed p. 12 (64 patients; 445 complete paired observations included and analysed). Original cohort screening/enrolment counts are reported in the source publication 1. (b) Give reasons for non-participation at each stage Not applicable to the secondary dataset analysis: all available complete observations were included. Reasons for non-participation in the original cohort are reported in reference 1. (c) Consider use of a flow diagram Not used: a flow diagram was not considered necessary because the analysis included all 445 available complete paired observations. Descriptive data 14* (a) Give characteristics of study participants (eg demographic, clinical, social) and information on exposures and potential confounders p. 12; Tables 1 and 2 (number of patients and observations, FVm distribution, exposure strata and model-performance summaries). Original clinical/demographic characteristics are available in reference 1. (b) Indicate number of participants with missing data for each variable of interest p. 12 and Table 1 (no missing values; 0 missing main-analysis observations). (c) Summarise follow-up time (eg, average and total amount) Not applicable: the analysis concerns repeated paired measurements and prediction error, not time-to-event outcomes or follow-up duration. Outcome data 15* Report numbers of outcome events or summary measures over time pp. 12–13; Tables 1 and 2; Figures 3–4 (MAE, RMSE, median absolute error, bias, quintile-specific summaries and continuous-model estimates). Main results 16 (a) Give unadjusted estimates and, if applicable, confounder-adjusted estimates and their precision (eg, 95% confidence interval). Make clear which confounders were adjusted for and why they were included pp. 12–13; Tables 1 and 2 (effect estimates, bootstrap 95% CIs and p values). Primary model was unadjusted apart from patient random intercepts; PaCO2-adjusted and other robustness models are reported on p. 13 and in Online Supplement S2–S4. (b) Report category boundaries when continuous variables were categorized p. 10 (FVm quintiles defined from the observed distribution); p. 12 and Table 2 (quintile sizes and mean FVm values: Q1 49.2, Q2 61.8, Q3 68.3, Q4 73.9, Q5 84.4 cm/s). (c) If relevant, consider translating estimates of relative risk into absolute risk for a meaningful time period Not applicable: no relative-risk estimate or time-based risk measure was reported. Other analyses 17 Report other analyses done—eg analyses of subgroups and interactions, and sensitivity analyses p. 13; Online Supplement S1–S4, Tables S1–S3 and Figure S2 (alternative error structures, PaCO2 adjustment, fixed effects, random slopes, spline models, normalization, LOPO recalibration and negative controls). Discussion Key results 18 Summarise key results with reference to study objectives pp. 13–16 (Discussion, especially first paragraph) and pp. 16–17 (Conclusions), explicitly related to the prespecified objective. Limitations 19 Discuss limitations of the study, taking into account sources of potential bias or imprecision. Discuss both direction and magnitude of any potential bias pp. 13–16, especially p. 16 (Limitations): mathematical coupling, post-hoc single-dataset validation, in-sample negative controls, absent raw/repeatability data, possible autocorrelation and skewed errors; likely direction/implications of bias are discussed. Interpretation 20 Give a cautious overall interpretation of results considering objectives, limitations, multiplicity of analyses, results from similar studies, and other relevant evidence pp. 13–16 (Discussion): cautious interpretation considering mathematical coupling, alternative error assumptions, negative controls, robustness analyses, prior validation studies and the absence of a clinical cutoff. Generalisability 21 Discuss the generalisability (external validity) of the study results p. 16 (Limitations: external validation required across different diagnoses, devices, operators and flow distributions); pp. 16–17 (Conclusions). Other information Funding 22 Give the source of funding and the role of the funders for the present study and, if applicable, for the original study on which the present article is based p. 2 (Funding: no funds, grants or other support were received; therefore there was no funder role). Funding of the original Robba et al. cohort is reported in reference 1. *Give information separately for exposed and unexposed groups. Note: An Explanation and Elaboration article discusses each checklist item and gives methodological background and published examples of transparent reporting. The STROBE checklist is best used in conjunction with this article (freely available on the Web sites of PLoS Medicine at http://www.plosmedicine.org/, Annals of Internal Medicine at http://www.annals.org/, and Epidemiology at http://www.epidem.com/). Information on the STROBE Initiative is available at http://www.strobe-statement.org.