# Browsing by Author "Howle, Laurens E"

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Item Open Access An Econophysics Approach to Short Time-Scale Dynamics of the Equities Markets(2017) Swingler, Ashleigh JaneFinancial markets have evolved drastically over the last decade due to the advent of high frequency trading and ubiquitous influence of algorithmic trading. Analyzing the equities markets has become an extremely data intensive and noisy undertaking. This work explores the information content of equity order book data outside of the inside price. First, an object-oriented library is presented to efficiently construct and maintain the order books of individual securities, by parsing and processing NASDAQ TotalView-ITCH data files. This library is part of the NASDAQ Order Processing software suite developed through this research effort. A framework for forecasting the returns of stock symbols that combines vector autoregression and principal component analysis is presented to determine if additional order book data, such as the volume of canceled orders and deleted orders, affects the price dynamics of stocks. Although the resulting model did not provide an adequate methodology for reliably forecasting prices, it was determined that information in the order book beyond returns and order volume should be included in market dynamics. This research also presents a novel visualization technique for viewing market dynamics and limit order book structure.

Item Open Access CFD Optimization of Small Gas Ejectors Used in Navy Diving Systems(2012) Cornman, Jacob KennethOptimization of small gas ejectors is typically completed by selecting a single set of operating conditions and optimizing the geometry for the specified conditions. The U.S. Navy is interested in utilizing a small gas ejector design in multiple diving systems with varying operational conditions. This thesis is directed at developing a Quasi Newton-Raphson Multivariate Optimization method using Computational Fluid Dynamics (CFD) to evaluate finite difference approximations. These approximations are then used as inputs to the gradient vector and the Hessian matrix of the standard Newton-Raphson multivariate optimization method. This optimization method was shown to be timely enough for use in the design phase of a multiple parameter system.

CFD investigation of the level curves of the simulation cost function hypersurface verified the success of the method presented at optimizing each independent parameter. Additional CFD simulations were used to investigate the ejector performance for operational conditions deviating from the operational conditions used during optimization. A correlation was developed for selecting the optimum throat diameter, and corresponding maximum efficiency, as functions of the input conditions only. Experimental models were manufactured using fused deposition modeling and evaluated with good agreement to the CFD simulation results.

Item Embargo Innovations in Decompression Sickness Prediction and Adaptive Ascent Algorithms(2023) Di Muro, GianlucaDecompression Sickness (DCS) is a potentially serious medical condition which can occur in humans when there is a decrease in ambient pressure. While it is generally accepted that DCS is initiated by the formation and growth of inert gas bubbles in the body, the mechanisms of its various forms are not completely understood. Complicating matters, divers often face challenges in adhering to predetermined safe ascent paths due to unpredictable environmental conditions. Therefore, the challenge of improving dive safety is twofold: 1) enhancing the accuracy of models in predicting DCS risk for a given dive profile; 2) developing algorithms, recommending safe ascent profiles, and capable of adapting in real time to new unforeseen diving conditions. This dissertation addresses both problems in the context of diving applications.First, we examine how the DCS risk is partitioned in air decompression dives to identify which portion of the dive is the most challenging. Our findings show that most of the risk might be accrued at surface, or during the ascent phase, depending on the specific mission parameters. Subsequently, we conducted a comprehensive investigation into DCS models incorporating inter-tissue perfusion dynamics. We proposed a novel algorithm to optimize these models efficiently. Our results determined that a model neglecting the coupling of faster tissue to slower tissues outperformed all other models on O2 surface decompression dive profiles. We further conducted experiments with various compartment tissue connections, involving diffusion phenomena and introducing delayed dynamics, while also exploring different risk functions. By adopting the Akaike Information Criterion, we found that the best performing model on the training set was BQE22AXT4, a four-compartment model featuring a risk threshold term only in the fourth compartment. Conversely, the classical Linear-Exponential model demonstrated superior performance on the extrapolation set. Finally, we introduce a groundbreaking real-time algorithm that delivers a secure and time optimized ascent path capable of adapting to unanticipated conditions. Our approach harnesses the power of advanced machine learning techniques and backward optimal control. Through our comprehensive analysis, we demonstrate that this innovative methodology attains a safety level on par with precomputed NAVY tables, while offering the added advantage of dynamic adaptation in response to unexpected events.

Item Open Access Multinomial Probabilistic Modeling for Decompression Sickness using Gas Content and Bubble Volume Models(2020) King, AmyDecompression sickness (DCS) is a condition resulting from the decrease in ambient pressure, both in hyperbaric and hypobaric environments. When ambient pressure is decreased, inert gas dissolved in the body’s tissues can become supersaturation and form bubbles, the physiological precursor to DCS. The signs and symptoms of DCS range from mild joint pain or rash to serious cardiopulmonary or neurological dysfunction. DCS is treated with recompression therapy, in which a patient is recompressed in a hyperbaric chamber and then decompressed following a treatment schedule specific to their symptoms.

This work focuses on DCS arising from hyperbaric exposures, and specifically underwater diving. DCS is a risk faced by U.S. Navy divers during underwater missions. To gain insight into the level of risk posed by a particular time-depth profile, the Navy uses probabilistic decompression models. These models quantify the probability of DCS occurring using survival analysis and a gas content or bubble volume model to define risk. Current probabilistic models make the binomial prediction of the probability of DCS occurring and not occurring.

Risk is a two-faceted entity, in which both the probability of injury and severity of injury contribute to the level of risk posed by an activity. Many Department of Defense organizations use two-dimensional risk assessment matrices to consider both the probability and severity of injury to manage operational risk. However, the U.S. Navy’s probabilistic decompression models only predict the probability of injury and do not provide divers with any information regarding potential DCS symptom severity.

The goal of this work was to investigate the efficacy of a variety of multinomial probabilistic models which simultaneously predict the probability and severity of DCS injury. We first conducted an analysis on the BIG292 calibration dataset to uncover the source of the bimodal trend in DCS symptom onset times. We concluded that neither of the two peaks alone correspond to DCS symptoms or DCS resulting from a particular dive type, and rather are a product of dive trial medical surveillance protocol. Because the dataset’s bimodal symptom onset time behavior is not related to the illness itself, it is not necessary for a probabilistic model to reproduce this trend.

Next, we developed 20 multinomial probabilistic models, testing the effectiveness of gas content versus bubble volume models to calculate risk, the justification of various gas content model parameters, the impact of using Type A/B versus Type I/II symptom severity splitting methods to define mild/serious DCS cases, and the influence of treating marginal DCS cases as separate, hierarchical events versus considering them non-events during model calibration.

The multinomial models presented in Chapters 3, 4, and 5 are able to accurately predict the incidence and severity of DCS as observed in the calibration data within 95% confidence. We find that trinomial models, which predict the probabilities of mild, serious, and no DCS, do perform better on the calibration dataset then their binomial counterparts. Multinomial models that predict the probabilities of marginal DCS are not able to accurately replicate the onset or distribution of marginal DCS cases in the calibration dataset, and the use of marginal cases during model calibration affects the model’s ability to accurately predict other types of DCS as well. The multinomial bubble volume models tested in this work were not able to achieve an optimal parameter set, and experienced model failure when predicting zero risk for a subset of dives.

We do not recommend the further use of multinomial models that predict the probability of marginal DCS separately from other symptom types. The multinomial bubble volume models should be reoptimized with a different algorithm and/or an alternative bubble nucleation criterion. To determine which multinomial probabilistic model presented here is optimal for U.S. Navy dive planning, all models should be evaluated on data extraneous to the datasets used for model calibration.

Item Open Access On the Advancement of Probabilistic Models of Decompression Sickness(2016) Hada, Ethan AlexanderThe work presented in this dissertation is focused on applying engineering methods to develop and explore probabilistic survival models for the prediction of decompression sickness in US NAVY divers. Mathematical modeling, computational model development, and numerical optimization techniques were employed to formulate and evaluate the predictive quality of models fitted to empirical data. In Chapters 1 and 2 we present general background information relevant to the development of probabilistic models applied to predicting the incidence of decompression sickness. The remainder of the dissertation introduces techniques developed in an effort to improve the predictive quality of probabilistic decompression models and to reduce the difficulty of model parameter optimization.

The first project explored seventeen variations of the hazard function using a well-perfused parallel compartment model. Models were parametrically optimized using the maximum likelihood technique. Model performance was evaluated using both classical statistical methods and model selection techniques based on information theory. Optimized model parameters were overall similar to those of previously published Results indicated that a novel hazard function definition that included both ambient pressure scaling and individually fitted compartment exponent scaling terms.

We developed ten pharmacokinetic compartmental models that included explicit delay mechanics to determine if predictive quality could be improved through the inclusion of material transfer lags. A fitted discrete delay parameter augmented the inflow to the compartment systems from the environment. Based on the observation that symptoms are often reported after risk accumulation begins for many of our models, we hypothesized that the inclusion of delays might improve correlation between the model predictions and observed data. Model selection techniques identified two models as having the best overall performance, but comparison to the best performing model without delay and model selection using our best identified no delay pharmacokinetic model both indicated that the delay mechanism was not statistically justified and did not substantially improve model predictions.

Our final investigation explored parameter bounding techniques to identify parameter regions for which statistical model failure will not occur. When a model predicts a no probability of a diver experiencing decompression sickness for an exposure that is known to produce symptoms, statistical model failure occurs. Using a metric related to the instantaneous risk, we successfully identify regions where model failure will not occur and identify the boundaries of the region using a root bounding technique. Several models are used to demonstrate the techniques, which may be employed to reduce the difficulty of model optimization for future investigations.

Item Open Access Probabilistic Modeling of Decompression Sickness, Comparative Hydrodynamics of Cetacean Flippers, Optimization of CT/MRI Protocols and Evaluation of Modified Angiocatheters: Engineering Methods Applied to a Diverse Assemblage of Projects(2010) Weber, Paul WilliamThe intent of the work discussed in this dissertation is to apply the engineering methods of theory/modeling, numerics/computation, and experimentation to a diverse assemblage of projects. Several projects are discussed: probabilistic modeling of decompression sickness, comparative hydrodynamics of cetacean flippers, optimization of CT/MRI protocols, evaluation of modified catheters, rudder cavitation, and modeling of mass transfer in amphibian cone outer segments.

The first project discussed is the probabilistic modeling of decompression sickness (DCS). This project involved developing a system for evaluating the success of decompression models in predicting DCS probability from empirical data. Model parameters were estimated using maximum likelihood techniques, and exact integrals of risk functions and tissue kinetics transition times were derived. Agreement with previously published results was excellent including maximum likelihood values within one log-likelihood unit of previous results and improvements by re-optimization, mean predicted DCS incidents within 1.4% of observed DCS, and time of DCS occurrence prediction. Alternative optimization and homogeneous parallel processing techniques yielded faster model optimization times. The next portion of this project involved investigating the nature and utility of marginal decompression sickness (DCS) events in fitting probabilistic decompression models to experimental dive trial data. Three null models were developed and compared to a known decompression model that was optimized on dive trial data containing only marginal DCS and no-DCS events. It was found that although marginal DCS events are related to exposure to decompression, empirical dive data containing marginal and full DCS outcomes are not combinable under a single DCS model; therefore, marginal DCS should be counted as no-DCS events when optimizing probabilistic DCS models with binomial likelihood functions. The final portion of this project involved the exploration of a multinomial DCS model. Two separate models based on the exponential-exponential/linear-exponential framework were developed: a trinomial model, which is able to predict the probabilities of mild, serious and no-DCS simultaneously, and a tetranomial model, which is able to predict the probabilities of mild, serious, marginal and no-DCS simultaneously. The trinomial DCS model was found to be qualitatively better than the tetranomial model, for reasons found earlier concerning the utility of marginal DCS events in DCS modeling.

The next project discussed is comparative hydrodynamics of cetacean flippers. Cetacean flippers may be viewed as being analogous to modern engineered hydrofoils, which have hydrodynamic properties such as lift coefficient, drag coefficient and associated efficiency. The hydrodynamics of cetacean flippers have not previously been rigorously examined and thus their performance properties are unknown. By conducting water tunnel testing using scale models of cetacean flippers derived via computed tomography (CT) scans, as well as computational fluid dynamic (CFD) simulations, a baseline work is presented to describe the hydrodynamic properties of several cetacean flippers. It was found that flippers of similar planform shape had similar hydrodynamic performance properties. Furthermore, one group of flippers of planform shape similar to modern swept wings was found to have lift coefficients that increased with angle of attack nonlinearly, which was caused by the onset of vortex-dominated lift. Drag coefficient versus angle of attack curves were found to be less dependent on planform shape. Larger cetacean flippers were found to have degraded performance at a Re of 250,000 compared to flippers of smaller odontocetes, while performance of larger and smaller cetacean flippers was similar at a swim speed of 2 m/s. Idealization of the planforms of cetacean flippers was found to capture the relevant hydrodynamic effects of the real flippers, although unintended consequences such as the lift curve slope changing from linear to nonlinear were sometimes observed. A numerical study of an idealized model of the humpback whale flipper showed that the leading-edge tubercles delay stall compared to a baseline (no tubercle) flipper because larger portions of the flow remaining attached at higher angles of attack.

The third project discussed is optimization of CT/MRI protocols. In order to optimize contrast material administration protocols for Computed Tomography (CT) and Magnetic Resonance Imaging (MRI), a custom-built physiologic flow phantom was constructed to model flow in the human body. This flow phantom was used to evaluate the effect of varying volumes, rates, and types of contrast material, use of a saline chase, and cardiac output on aortic enhancement characteristics. For CT, reducing the volume of contrast material decreased duration peak enhancement and reduced the maximum value of peak enhancement. Increasing the rate of contrast media administration increased peak enhancement and decreased duration of peak enhancement. Use of a saline chase resulted in an increase in peak enhancement. Peak aortic enhancement increased when reduced cardiac output was simulated. For MRI, when the same volume of contrast material was injected at the same rate, the type of contrast material used has a significant effect on the greatest peak signal intensity and duration peak signal intensity. A higher injection rate of saline chaser is more advantageous than a larger volume of saline chaser to increase the peak aortic signal intensity using low contrast material doses. Furthermore, for higher volumes of contrast material, the effect of increasing the volume of saline chaser makes almost no difference while increasing the rate of injection makes a significant difference. When a saline chaser with a high injection rate is used, the dose of the contrast material may be reduced by 25-50% and more than 86% of the non-reduced dose peak aortic enhancement will be attained.

The next project discussed is evaluation of modified angiocatheters. In this study, a standard peripheral end hole angiocatheter was compared to those modified with side holes or side slits by using experimental techniques to qualitatively compare the contrast material exit jets, and by using numeric techniques to provide flow visualization and quantitative comparisons. A Schlieren imaging system was used to visualize the angiocatheter exit jet fluid dynamics at two different flow rates, and a commercial computational fluid dynamics (CFD) package was used to calculate numeric results for various catheter orientations and vessel diameters. Experimental images showed that modifying standard peripheral intravenous angiocatheters with side holes or side slits qualitatively changed the overall flow field and caused the exiting jet to become less well-defined. Numeric calculations showed that the addition of side holes or slits resulted in a 9-30% reduction of the velocity of contrast material exiting the end hole of the angiocatheter. With the catheter tip directed obliquely to the wall, the maximum wall shear stress was always highest for the unmodified catheter and always lowest for the 4 side slit catheter. Modified angiocatheters may have the potential to reduce extravasation events in patients by reducing vessel wall shear stress.

The next project discussed involves studying the effect of leading-edge tubercles on cavitation characteristics for marine rudders. Three different rudders were constructed and tested in a water tunnel: baseline, 3-tubercle leading edge, and 5-tubercle leading edge. In the linear (non-stall) regime, tubercled rudders performed equally to the smooth rudder. Hydrodynamic stall occurred at smaller angles of attack for the tubercled rudders than for the smooth rudder. When stall did occur, it was more gradual for the tubercled rudders, whereas the smooth rudder demonstrated a more dramatic loss of lift. At lower Re, the tubercled rudders also maintained a higher value of lift post-stall than the smooth rudder. Cavitation onset for the tubercled rudders occurred at lower angles of attack and higher values of cavitation number than for the smooth rudder, but cavities on the tubercled rudders were localized in the slots as opposed to the smooth rudder where the cavity spread across the entire leading edge.

In the final project discussed, modeling of mass transfer in amphibian cone outer segments, a detailed derivation of a simplified (continuum, one-dimensional) mathematical model for the radio-labeled opsin density profile in the amphibian cone outer segment is presented. This model relies on only one free parameter, which was the mass transfer coefficient between the plasmalemma and disc region. The descriptive equations were nondimensionalized, and scale analysis showed that advective effects could be neglected as a first approximation for early times so that a simplified system could be obtained. Through numeric computation the solution behavior was found to have three distinct stages. The first stage was marked by diffusion in the plasmalemma and no mass transfer in the disc region. The second stage first involved the plasmalemma reaching a metastable state whereas the disc region density increased, then involved both the plasmalemma and disc regions increasing in density with their distributions being qualitatively the same. The final stage involved a slow relaxation to the steady-state solution.

Item Open Access The Impact of Weighting Marginal DCS Events as Non-Events, Pharmacokinetic Gas Content Models, and Optimal Decompression Schedule Calculation(2017) Murphy, Francis GregoryDecompression sickness (DCS) in man is a condition associated with reduction in ambient pressure. These reductions may result from a return to normobaric pressure from hyperbaric pressure, or an ascent to hypobaric pressure. Previous approaches to mitigate the risk of DCS have been based on both deterministic and probabilistic decompression algorithms. Deterministic decompression algorithms generate ascent schedules with binary outcomes. Following the prescribed ascent schedule should prevent the onset of DCS. Failure to comply with the prescribed schedule should make the onset of DCS an inevitable outcome. However, in practice DCS may occur in individuals that follow deterministic decompression schedules precisely and may not occur in individuals that fail to comply with these schedules. Probabilistic algorithms do not provide a decompression schedule with a binary outcome, instead they generate a schedule associated with a target probability of DCS occurring.

In this work, several aspects of probabilistic algorithms are investigated using the techniques of survival analysis and numerical optimization. All work was completed using computer software coded in the C# programming language. Model optimization and evaluation techniques described below were completed using U.S. Navy standard dive data sets. These dive data sets consist of time series recordings of pressure, inspired gas, DCS outcome, last known time at which test subjects were healthy, and time at which DCS symptoms were definitely present if applicable. This data did not require institutional review board approval for use and has been previously described in the literature.

Models investigated were evaluated with their optimized parameters using statistical tests to determine how well they described both the training data and data not included in the training set. Statistical techniques used to evaluate the models included the Akaike information criterion (AIC), Pearson Х2 test, and occurrence density functions. AIC had to be used as the models examined during this work were often not nested within each other. When models were nested, the log likelihood difference test was used to compare the candidate models. In addition to these rigorous statistical tests, graphs were frequently used to present qualitative information about the underlying models and/or data.

There is no diagnostic test for DCS, and as such, the outcome of an exposure is not always clear. Test subjects may experience transient symptoms of DCS which spontaneously resolve prior to recompression treatment. These mild events which spontaneously resolve are termed marginal DCS events. During optimization, marginal DCS events are typically assigned a fractional weight of either 0.5 or 0.1 with a full DCS event being weighed as 1.0 and a non-event as 0.0. Previous work has shown that the overall quality of model fit to the data can be improved by assigning a weight of 0.0 to marginal DCS events during optimization. In this work the U.S. Navy LE1 and LEM models were re-optimized against the BIG292 and NMRI98 data sets with marginal DCS events weighted as 0.0. Features were added incrementally to the EE1 model until LE1 or LEM were formed to evaluate if the features were still statistically justified. All features were found to be statistically relevant; these features were a linear gas washout and a threshold term in the case of LE1 and linear gas washout, a threshold term, and the inclusion of oxygen as a participating gas in the case of LEM. Further, the addition of these features enabled both models to more accurately predict the observed incidences and times of occurrence of DCS for the profiles in their training sets. Prior to re-optimization, LE1 had incorrectly ascribed the risk of marginal DCS events entirely to bounce dives. LEM gave undue weight to saturation dives in the training set due to the bulk of marginal events occurring during saturation dive exposures, despite saturation dives only comprising 14.4% of the data. It is concluded that marginal DCS events should not be assigned a fractional weight, but should be accommodated by another mechanism in the model optimization process.

Pharmacokinetic gas content models have been shown to well describe gas uptake and washout in the skeletal muscle and cerebral tissue of sheep. These models differ from the EE1, LE1, and LEM models in that they feature multi-exponential kinetics. The multi-exponential kinetics arise from using a series of compartments coupled by either perfusion or diffusion in lieu of a collection of parallel independent perfusion limited compartments. Eleven models incorporating coupled compartments were investigated in this work along with one model that consisted of a single perfusion-limited compartment. Six of the models had been previously investigated using sheep and five were novel.

No pharmacokinetic gas content model described the overall NMRI98 data better than the LEM model by weighted AIC index. However, several data subsets including single air, repetitive and multilevel air, and oxygen decompression dives were better described by pharmacokinetic gas content models than either LEM or the single perfusion limited compartment. A single perfusion limited compartment outperformed both LEM and all of the pharmacokinetic gas content models for saturation exposures. No one model being the best descriptor of all dive data types indicates that multiple compartment structures are needed to best describe the data.

The four best performing pharmacokinetic gas content models; Central Serial Two Tissue (CS2T/CS2T_3) and Perfusion Diffusion Base (PDB/PDB_10); were augmented with the incorporation of oxygen as a participating gas. Oxygen was incorporated by the addition of a sink term to the differential equations describing oxygen uptake and washout. A sink term was used to allow for oxygen to be scaled in future work with the addition of exercise information. Unlike a model based on a collection of parallel uncoupled compartments (LEM), none of the pharmacokinetic gas content models were improved by the addition of oxygen as a participating gas. This suggests that the benefit from including oxygen as a participating gas is a function of the underlying model structure and not the data.

Combining multiple pharmacokinetic gas content models was also explored in this work. The combinations of CS2T3_PLB, PDB_PLB, CS2T3_PDB, and PDBX2 were all tested. PLB stands for perfusion limited base model (the single perfusion limited compartment used above) and PDBX2 is two copies of the PDB model in parallel. Notionally, using a collection of pharmacokinetic gas content models in parallel should allow for one compartment (PLB) to describe the saturation dive data and another collection of compartments such as CS2T_3 to describe the rest of the data. In practice this did not happen, each of the models combined optimized to each bear a fraction of the risk for all dives. Pharmacokinetic gas content models consisting of a collection of different parallel compartment structures did not better describe the data than LEM.

The final problem considered in this work is the problem of calculating an ascent path to end a hyperbaric exposure with the shortest possible ascent time and without exceeding a specified target risk (probability of DCS occurring). Current methods for solving this problem are based upon searching through hundreds to thousands of possible schedules until an acceptable solution is found. However, it is possible to directly calculate the shortest ascent path which does not exceed the target risk. The necessary calculations to determine the shortest path within a target risk are described. However, this path is not necessarily unique and further work will be needed prior to this work becoming a complete solution to the ascent search problem.

This work provides significant advances to probabilistic algorithms for mitigating the risk of DCS during diving. The LE1 and LEM models were re-optimized with marginal DCS events weighted as non-events. Re-optimization resulted in more accurate prediction of risk by the models when compared to their respective training data. Pharmacokinetic gas content models were investigated in depth and they were found to be good predictors for air diving and oxygen decompression diving. The need for oxygen as a participating gas was ruled out for pharmacokinetic gas content models. Combinations of multiple pharmacokinetic gas content models were found to not optimize well when naively combined, suggesting a need for a more complex likelihood function. Finally, an outline for directly calculating the optimal ascent path for a given hyperbaric exposure as part of a probabilistic decompression algorithm was provided. Together this work provides a significant step toward probabilistic decompression algorithms being a viable replacement for deterministic decompression algorithms in all underwater operations.

Item Open Access Trinomial probabilistic modeling of full, marginal, and no decompression sickness(2018) Andriano, Nicholas RyanDecompression sickness (DCS) is a condition associated with reductions in ambient pressure during underwater diving. Determining the risk of DCS from a dive exposure remains an active area of research, to develop safe decompression schedules to mitigate the occurrences of DCS. This thesis develops a probabilistic model for the trinomial outcome of full, marginal, and no DCS. These models determine probabilities of the various outcomes for a given dive schedule. Six variants of exponential-exponential (EE) and linear-exponential (LE) decompression models were used for optimization of model parameters to best fit dive outcomes from empirical data of 3,322 exposures. Using the log likelihood difference test, the LE1 model was determined to provide the best fit for the data when considering full events along with marginal DCS events as separate, hierarchical events with a weighting of 0.1. The LE1 trinomial marginal model can be used to better understand decompression schedules, expanding upon binomial models which treat full DCS as an event and marginal DCS as a non-event. Future work could investigate whether the use of marginal DCS cases improves probabilistic DCS models. Model improvement could include the addition of a fourth outcome, where full DCS is split and categorized as serious or mild DCS, creating a tetranomial model with serious, mild, marginal, and no DCS outcomes for comparison with the presently developed model.