Thermal Effect of a Revolving Gaussian Beam on Activating Heat-Sensitive Nociceptors in Skin

We consider the problem of inducing withdrawal reflex on a test subject by exposing the subject’s skin to an electromagnetic beam. Heat-sensitive nociceptors in the skin are activated wherever the temperature is above the activation temperature. Withdrawal reflex occurs when the activated volume reaches a threshold. Previously we studied static beams with 3 types of power density distribution: Gaussian, super-Gaussian, and flat-top. We found that the flaptop is the best and the Gaussian is the worst in their performance with regard to 1) minimizing the time to withdrawal reflex, 2) minimizing the energy consumption and 3) minimizing the maximum temperature increase. The less-than-desirable performance of Gaussian beams is attributed to the uneven distribution of power density resulting in low energy efficiency: near the beam center the high power density does not contribute proportionally to increasing the activated volume;outside the beam effective radius the low power density fails to activate nociceptors. To overcome the drawbacks of Gaussian beams, in this study, we revolve a Gaussian beam around a fixed point to make the power density more uniformly distributed. We optimize the performance over two parameters: the spot size of static beam and the radius of beam revolution. We find that in comparison with a static Gaussian beam, a revolving Gaussian beam can reduce the energy consumption, and at the same time lower the maximum temperature.

Non-Dimensional Analysis of Thermal Effect on Skin Exposure to an Electromagnetic Beam

We consider the problem of inducing withdrawal reflex on a test subject by exposing the subject’s skin to an electromagnetic beam. Heat-sensitive nociceptors in the skin are activated wherever the temperature is above the activation temperature. Withdrawal reflex occurs when the activated volume reaches a threshold. We non-dimensionalize the problem to write the temperature as the product of a parameter-free function of non-dimensional variables and a function of beam parameters. This formulation allows studying beam parameters without knowing skin material parameters. We examine the effects of spot size, total power and distribution type of the electromagnetic beam on 3 quantities at reflex: 1) the time to reflex, 2) the maximum temperature increase, and 3) the total energy consumption. We find that the flat-top beam is the best, with the lowest energy consumption and the smallest maximum temperature increase. The Super-Gaussian beam is only slightly inferior to the flat-top. The Gaussian beam has by far the worst performance among these three.

Inference of Median Subjective Threshold in Psychophysical Experiments

We consider the response of a test subject upon a skin area being heated with an electromagnetic wave or a contact surface. When the specifications of the electromagnetic beam are fixed, the stimulus is solely described by the heating duration. The binary response of a subject, escape or no escape, is determined by the stimulus and a subjective threshold that varies among test realizations. We study four methods for inferring the median subjective threshold in psychophysical experiments: 1) sample median, 2) maximum likelihood estimation (MLE) with 2 variables, 3) MLE with 1 variable, and 4) adaptive Bayesian method. While methods 1 - 3 require samples of time to escape measured in the method of limits, method 4 utilizes binary outcomes observed in the method of constant stimuli. We find that a) the adaptive Bayesian method converges and is as efficient as the sample median even when the assumed model distribution is incorrect;b) this robust convergence is lost if we infer the mean instead of the median;c) for the optimal performance in an uncertain situation, it is best to use a wide model distribution;d) the predicted error from the posterior standard deviation is unreliable, dominated by the assumed model distribution.

Effect of Depth-Dependent Nociceptor Density on the Heat-Induced Withdrawal Reflex

Previously we introduced a concise dose-response model for the heat-induced withdrawal reflex caused by millimeter wave radiation. The model predicts the occurrence of withdrawal reflex from the given spatial temperature profile. It was formulated on the assumption that the density of nociceptors in skin is uniform, independent of the depth. The model has only two parameters: the activation temperature of heat-sensitive nociceptors and the critical threshold on the activated volume for triggering withdrawal reflex. In this study, we consider the case of depth-dependent nociceptor density in skin. We use a general parametric form with a scaling parameter in the depth direction to represent the nociceptor density. We analyze system behaviors for four density types of this form. Based on the theoretical results, we develop a methodology for 1) identifying from test data the density form of nociceptors distribution, 2) finding from test data the scaling parameter in the density form, and 3) determining from test data the activation temperature of nociceptors.

Internal Temperature of Skin when Surface Temperature Is Controlled with an Electromagnetic Beam

We study the thermal effect on skin exposed to an electromagnetic beam of time-dependent power. We consider two types of beam power time schedules. In the controlled temperature exposure, the skin surface temperature is increased quickly to a prescribed level using a high beam power;then the surface temperature is maintained at the prescribed level by adjusting the beam power adaptively. In the constant power exposure, the applied beam power is relatively low and stays unchanged over the time. We start both types of exposures at the same time and compare their internal temperatures of skin when they have the same surface temperature. In a non-dimensionalized formulation, we show that at the moment when both exposure types reach the same prescribed surface temperature level, the controlled temperature exposure has a higher internal temperature at all depths. This conclusion is mathematically rigorous and is independent of skin material properties.

Extracting Human Reaction Time from Observations in the Method of Constant Stimuli

We consider the psychophysical experiments in which the test subject’s binary reaction is determined by the prescribed exposure duration to a stimulus and a random variable subjective threshold. For example, when a subject is exposed to a millimeter wave beam for a prescribed duration, the occurrence of flight action is binary (yes or no). In experiments, in addition to the binary outcome, the actuation time of flight action is also recorded if it occurs;the delay from the initiation time to the actuation time of flight action is the human reaction time, which is not measurable. In this study, we model the random subjective threshold as a Weibull distribution and formulate an inference method for estimating the human reaction time, from data of prescribed exposure durations, binary outcomes and actuation times of flight action collected in a sequence of tests. Numerical simulations demonstrate that the inference of human reaction time based on the Weibull distribution converges to the correct value even when the underlying true model deviates from the inference model. This robustness of the inference method makes it applicable to real experimental data where the underlying true model is unknown.

Minimum Energy Requirement for Inducing Withdrawal Reflex in Millimeter Wave Exposures

We consider the problem of inducing withdrawal reflex on a test subject via exposure to a millimeter wave beam. In our physical model, there are 10 physical parameters affecting the occurrence of withdrawal reflex. Our goal is to pinpoint the roles of these physical parameters in inducing withdrawal reflex. We first carry out non-dimensionalization to reduce the model to a non-dimensional system of only 3 composite parameters: non-dimensional beam power density, non-dimensional beam radius, and non-dimensional exposure time. If the beam power is kept on and steady, withdrawal reflex occurs eventually;the shortest exposure time for inducing withdrawal reflex corresponds to the smallest energy consumption at the given power density and beam radius. In the 2D space of power density and beam radius, the overall minimum energy occurs at the corner of very large power density and very small beam radius, which also produces a very large value of maximum skin temperature and a long time to withdrawal reflex. To reduce the burn injury risk, we introduce a cap on the maximum skin temperature. At each given total beam power, we carry out optimizations with respect to the beam radius, constrained by the prescribed temperature cap. The energy consumption varies negatively with the prescribed temperature cap: a lower temperature cap can be accommodated only with a higher energy consumption via the venue of a larger beam radius. The energy consumption is relatively flat with respect to the total beam power and attains a minimum at a moderately large total beam power. The time to withdrawal reflex is approximately inversely proportional to the total beam power. Our analysis demonstrates that a moderately large total beam power is a good compromise to achieve both low energy consumption and short time to withdrawal reflex.

Interpreting Dose-Response Relation for Exposure to Multiple Sound Impulses in the Framework of Immunity

Hearing loss is a common military health problem and it is closely related to exposures to impulse noises from blast explosions and weapon firings. In a study based on test data of chinchillas and scaled to humans (Military Medicine, 181: 59-69), an empirical injury model was constructed for exposure to multiple sound impulses of equal intensity. Building upon the empirical injury model, we conduct a mathematical study of the hearing loss injury caused by multiple impulses of non-uniform intensities. We adopt the theoretical framework of viewing individual sound exposures as separate injury causing events, and in that framework, we examine synergy for causing injury (fatigue) or negative synergy (immunity) or independence among a sequence of doses. Starting with the empirical logistic dose-response relation and the empirical dose combination rule, we show that for causing injury, a sequence of sound exposure events are not independent of each other. The phenomenological effect of a preceding event on the subsequent event is always immunity. We extend the empirical dose combination rule, which is applicable only in the case of homogeneous impulses of equal intensity, to accommodate the general case of multiple heterogeneous sound exposures with non-uniform intensities. In addition to studying and extending the empirical dose combination rule, we also explore the dose combination rule for the hypothetical case of independent events, and compare it with the empirical one. We measure the effect of immunity quantitatively using the immunity factor defined as the percentage of decrease in injury probability attributed to the sound exposure in the preceding event. Our main findings on the immunity factor are: 1) the immunity factor is primarily a function of the difference in SELA (A- weighted sound exposure level) between the two sound exposure events;it is virtually independent of the magnitude of the two SELA values as long as the difference is fixed;2) the immunity factor increases monotonically from 0 to 100% as the first dose is varied from being significantly below the second dose, to being moderately above the second dose. The extended dose-response formulation developed in this study provides a theoretical framework for assessing the injury risk in realistic situations.

Risk of Hearing Loss Caused by Multiple Acoustic Impulses in the Framework of Biovariability

We consider the hearing loss injury among subjects in a crowd with a wide spectrum of individual intrinsic injury probabilities due to biovariability. For multiple acoustic impulses, the observed injury risk of a crowd vs the effective combined dose follows the logistic dose-response relation. The injury risk of a crowd is the average fraction of injured. The injury risk was measured in experiments as follows: each subject is individually exposed to a sequence of acoustic impulses of a given intensity and the injury is recorded;results of multiple individual subjects were assembled into data sets to mimic the response of a crowd. The effective combined dose was adjusted by varying the number of impulses in the sequence. The most prominent feature observed in experiments is that the injury risk of the crowd caused by multiple impulses is significantly less than the value predicted based on assumption that all impulses act independently in causing injury and all subjects in the crowd are statistically identical. Previously, in the case where all subjects are statistically identical (i.e., no biovariability), we interpreted the observed injury risk caused by multiple impulses in terms of the immunity effects of preceding impulses on subsequent impulses. In this study, we focus on the case where all sound exposure events act independently in causing injury regardless of whether one is preceded by another (i.e., no immunity effect). Instead, we explore the possibility of interpreting the observed logistic dose-response relation in the framework of biovariability of the crowd. Here biovariability means that subjects in the crowd have their own individual injury probabilities. That is, some subjects are biologically less or more susceptible to hearing loss injury than others. We derive analytically the distribution of individual injury probability that produces the observed logistic dose-response relation. For several parameter values, we prove that the derived distribution is mathematically a proper density function. We further study the asymptotic approximations for the density function and discuss their significance in practical numerical computation with finite precision arithmetic. Our mathematical analysis implies that the observed logistic dose-response relation can be theoretically explained in the framework of biovariability in the absence of immunity effect.

A Concise Model and Analysis for Heat-Induced Withdrawal Reflex Caused by Millimeter Wave Radiation

In this study, we consider the heat-induced withdrawal reflex caused by exposure to an electromagnetic beam. We propose a concise dose-response relation for predicting the occurrence of withdrawal reflex from a given spatial temperature profile. Our model is distilled from sub-step components in the ADT CHEETEH-E model developed at the Institute for Defense Analyses. Our model has only two parameters: the activation temperature of nociceptors and the critical threshold on the activated volume. When the spatial temperature profile is measurable, the two parameters can be determined from test data. We connect this dose-response relation to a temperature evolution model for electromagnetic heating. The resulting composite model governs the process from the electromagnetic beam deposited on the skin to the binary outcome of subject’s reflex response. We carry out non-dimensionalization in the time evolution model. The temperature solution of the non-dimensional system is the product of the applied power density and a parameter-free function. The effects of physical parameters are contained in non-dimensional time and depth. Scaling the physical temperature distribution into a parameter-free function greatly simplifies the analytical solution, and helps to pinpoint the effects of beam spot area and applied power density. With this formulation, we study the theoretical behaviors of the system, including the time of reflex, effect of heat conduction, biological latency in observed reflex, energy consumption by the time of reflex, and the strategy of selecting test conditions in experiments for the purpose of inferring model parameters from test data.

Asymptotics and Well-Posedness of the Derived Distribution Density in a Study of Biovariability

In our recent work (Wang, Burgei, and Zhou, 2018) we studied the hearing loss injury among subjects in a crowd with a wide spectrum of heterogeneous individual injury susceptibility due to biovariability. The injury risk of a crowd is defined as the average fraction of injured. We examined mathematically the injury risk of a crowd vs the number of acoustic impulses the crowd is exposed to, under the assumption that all impulses act independently in causing injury regardless of whether one is preceded by another. We concluded that the observed dose-response relation can be explained solely on the basis of biovariability in the form of heterogeneous susceptibility. We derived an analytical solution for the distribution density of injury susceptibility, as a power series expansion in terms of scaled log individual non-injury probability. While theoretically the power series converges for all argument values, in practical computations with IEEE double precision, at large argument values, the numerical accuracy of the power series summation is completely wiped out by the accumulation of round-off errors. In this study, we derive a general asymptotic approximation at large argument values, for the distribution density. The combination of the power series and the asymptotics provides a practical numerical tool for computing the distribution density. We then use this tool to verify numerically that the distribution obtained in our previous theoretical study is indeed a proper density. In addition, we will also develop a very efficient and accurate Pade approximation for the distribution density.

Dose-Injury Relation as a Model for Uncertainty Propagation from Input Dose to Target Dose

We study a general framework for assessing the injury probability corresponding to an input dose quantity. In many applications, the true value of input dose may not be directly measurable. Instead, the input dose is estimated from measurable/controllable quantities via numerical simulations using assumed representative parameter values. We aim at developing a simple modeling framework for accommodating all uncertainties, including the discrepancy between the estimated input dose and the true input dose. We first interpret the widely used logistic dose-injury model as the result of dose propagation uncertainty from input dose to target dose at the active site for injury where the binary outcome is completely determined by the target dose. We specify the symmetric logistic dose-injury function using two shape parameters: the median injury dose and the 10 - 90 percentile width. We relate the two shape parameters of injury function to the mean and standard deviation of the dose propagation uncertainty. We find 1) a larger total uncertainty will spread more the dose-response function, increasing the 10 - 90 percentile width and 2) a systematic over-estimate of the input dose will shift the injury probability toward the right along the estimated input dose. This framework provides a way of revising an established injury model for a particular test population to predict the injury model for a new population with different distributions of parameters that affect the dose propagation and dose estimation. In addition to modeling dose propagation uncertainty, we propose a new 3-parameter model to include the skewness of injury function. The proposed 3-parameter function form is based on shifted log-normal distribution of dose propagation uncertainty and is approximately invariant when other uncertainties are added. The proposed 3-parameter function form provides a framework for extending skewed injury model from a test population to a target population in application.

Integrating Test Data and ATBM Simulations into Dose Propagation Uncertainty Formulation for Bone Fracture Risk Assessment

We consider the problem of assessing bone fracture risk for a subject hit by a blunt impact projectile. We aim at constructing a framework for integrating test data and Advanced Total Body Model (ATBM) simulations into the risk assessment. The ATBM is a finite element model managed by the Joint Non-Lethal Weapons Directorate for the purpose of assessing the risk of injury caused by blunt impacts from non-lethal weapons. In ATBM simulations, the quantity that determines arm bone fracture is the calculated maximum strain in the bone. The main obstacle to accurate prediction is that the calculated strain is incompatible with the measured strain. The fracture strain measured in bending tests of real bones is affected by random inhomogeneity in bones and uncertainty in measurement gauge attachment location/orientation. In contrast, the strain calculated in ATBM simulations is based on the assumption that all bones are perfectly elastic with homogeneous material properties and no measurement uncertainty. To connect test data and ATBM simulations in a proper and meaningful setting, we introduce the concept of elasticity-homogenized strain. We interpret test data in terms of the homogenized strain, and build an empirical dose-injury model with the homogenized strain as the input dose for predicting injury. The maximum strain calculated by ATBM has randomness due to uncertainty in specifications of ATBM setup parameters. The dose propagation uncertainty formulation accommodates this uncertainty efficiently by simply updating the shape parameters in the dose-injury model, avoiding the high computational cost of sampling this uncertainty via multiple ATBM runs.

Risk of Hearing Loss Injury Caused by Multiple Flash Bangs on a Crowd

A flash bang is a non-lethal explosive device that delivers intensely loud bangs and bright lights to suppress potentially dangerous targets. It is usually used in crowd control, hostage rescue and numerous other missions. We construct a model for assessing quantitatively the risk of hearing loss injury caused by multiple flash bangs. The model provides a computational framework for incorporating the effects of the key factors defining the situation and for testing various sub-models for these factors. The proposed model includes 1) uncertainty in the burst point of flash bang mortar, 2) randomness in the dispersion of multiple submunitions after the flash bang mortar burst, 3) decay of acoustic impulse from a single submunition to an individual subject along the ground surface, 4) the effective combined sound exposure level on an individual subject caused by multiple submunitions at various distances from the subject, and 5) randomness in the spatial distribution of subjects in the crowd. With the mathematical model formulated, we seek to characterize the overall effect of flash bang mortar in the form of an effective injury area. We carry out simulations to study the effects of uncertainty and randomness on the risk of hearing loss injury of the crowd. The proposed framework serves as a starting point for a comprehensive assessment of hearing loss injury risk, taking into consideration all realistic and relevant features of flash bang mortar. It also provides a platform for testing and updating component models.

A Two-Parameter Description for the Injury Risk of Flash Bangs with Uncertainty

We study the random injury outcome caused by multiple flash bang submunitions on a crowd. We are particularly interested in the fluctuations in injury outcome among individual realizations. Previously, to simulate the distribution of the actual number of injured, we developed a comprehensive Monte Carlo model. While the full computational model is important for thorough theoretical investigations, in practical operations, it is desirable to characterize the phenomenological behavior of injury outcome using a concise model with only one or two parameters. Conventionally, the injury outcome is indicated by the average fraction of injured, which is called the risk of significant injury (RSI). The single metric RSI description fails to capture fluctuations in the injury outcome. The number of injured in the crowd is influenced by many random factors: the aiming error of flash bang mortar, the dispersion of submunitions after mortar burst, the amount of acoustic dose reaching individual subjects, and the biovariability of individual subjects’ reactions to a given acoustic dose. We aim to include these random factors properly and concisely. In this study, we represent the random injury outcome as a compound binomial model, in which the hidden injury probability is drawn from a two-parameter model distribution. We formulate and examine six model distributions for the injury probability. The best performer is a mixture of uniform and triangle distributions, parameterized by (RSI, dp) where dp is the standard deviation of the hidden injury probability. This mixture model predicts the behavior of injury outcome with uncertainty, based solely on the two parameters (RSI, dp) in the flash bang description. For example, we can predict the probability of the injury outcome not exceeding a prescribed tolerance. We advocate the adoption of this two-parameter characterization for flash bangs to replace the single-parameter RSI description. Whenever we need to give a high level coarse description of a flash bang situation, we state that the injury risk is represented by (RSI, dp).