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.


Introduction
Hearing loss is the third most common health problem in the US and more than 28 million Americans have lost some hearing. There are three basic types of hearing loss: conductive hearing loss, sensorineural hearing loss and mixed hearing loss [1]. This classification is based on which part of the auditory system is damaged. Conductive hearing loss occurs when there is damage to the eardrum and the tiny bones of the middle ear. It may be caused by ear infection or impacted earwax and it results in a reduction in the ability to hear faint sounds. Luckily, conductive hearing loss can often be medically or surgically reversed. Sensorineural hearing loss stems from damage to the inner ear or auditory nerve. For people with sensorineural hearing loss, sound appears unclear or muffled. Unfortunately, sensorineural hearing loss is permanent. It cannot be corrected but patients with sensorineural hearing loss can be helped through the use of hearing aids. When a conductive hearing loss occurs together with a sensorineural hearing loss, the hearing loss is called a mixed hearing loss.
Symptoms of mixed hearing loss include sounds turning softer in volume and becoming more difficult to understand. One possible cause of sensorineural hearing loss is exposure to loud noise or blast, which is very common in a battlefield. In fact, hearing loss is one of the most prevalent military medical problems. According to the US Department of Veterans Affairs [2] [3], in 2014 more than 933,000 Veterans have hearing loss disability, and nearly 1.3 million experiencing tinnitus which refers to the perception of a ringing, buzzing, or other kind of noise in the ears [4]. With an increasing trend in hearing loss among veterans, it is important to be able to assess the risk of significant hearing loss injury when exposed to blast explosions [5].
The impacts of sound waves on humans are complicated and they depend on the frequency, sound pressure level, and duration. In [6] Dr. Chan and co-workers assembled experimental data of chinchillas exposed to multiple impulse noise shots in the laboratory [7]. They chose chinchillas to model human hearing response because the size, structure, and function of the chinchilla's ear are very 1819 Health similar to those of humans. They used the A-weighted sound exposure level (SELA) as an effective single metric (the dose) for predicting the injury risk [8].
Based on the test data they constructed a logistic dose-response relation for unprotected human ears. They adopted an empirical dose combination rule to combine multiple identical sound impulses, uniformly distributed in time, into one effective combined dose [6] [9]. The injury risk over the combined event (i.e., over the sequence of multiple noise shots) is governed by the dose-response relation with the effective combined SELA as the dose.
They also validated the dose-response curve against historical human data from rifle noise tests [10] and proposed a temporary threshold shift (TTS) recovery model in the form of a log-linear function.
In this paper, we carry out a mathematical analysis based on the dose-response relation and the dose combination rule developed in [6]. The motivations of our analysis are: 1) to understand the risk of hearing loss injury caused by multiple noise impulses from the point of view of individual sound exposure events; 2) to extend the dose combination rule developed in [6] to accommodate the general case where we need to combine multiple heterogeneous impulses of non-uniform SELA values into one effective combined SELA value. In the framework of the dose-response relation and the dose combination rule, we examine the synergy for causing injury (fatigue) or negative synergy (immunity) or independence among sound exposure events. The sign and magnitude of synergy will shed light on the role of acoustic reflex (which tends to decrease the injury risk for subsequent sound exposures) and on the role of fatigue/partial damage (which tends to increase the injury risk for subsequent sound exposures). Our primary goal is to establish a unified dose-response formulation for assessing the injury risk of unprotected ears caused by a heterogeneous sequence of sound exposures with non-uniform SELA values. This extended dose-response formulation will provide the foundation for assessing the injury risk in realistic situations.
We organize the rest of the paper as follows. In Section 2 we review the relevant results in [6]. Section 3 contains an analysis of unconditional injury probabilities and conditional probabilities as governed by the dose-response relation and by the dose combination rule, with the goal of determining whether the interactive effect among sequential doses is immunity or fatigue. In Section 4, we extend the dose combination rule to the general case of a heterogeneous sequence of sound exposure events with non-uniform doses. We then construct the dose combination rules for the case of independent events and we compare the two rules. In Section 5, we investigate quantitatively the effect of immunity measured by the immunity factor defined as the percentage of decrease in the injury probability attributed to the sound exposure in the preceding event. We carefully work through the detailed behaviors of the immunity factor in four regimes of parameter values. Finally, we summarize the extended dose-response formulation and the main conclusions in Section 6, and provide a mathematical proof of a theorem in the Appendix.

Review of Relevant Results in [6]
In [6], experimental data of permanent threshold shift (PTS) after exposure to multiple impulse noise shots was studied. A dose-response relation was constructed for PTS of various cut-off levels. The empirical dose-response relation is expressed in the form of a logistic model: In the dose-response relation (1): is the combined A-weighted sound exposure level, a combined single metric quantifying the overall effect of multiple impulse shots;  P is the probability of PTS of a given cut-off level (for example, PTS > 40 dB);  50 ID denotes median injury dose at which the injury risk P is 50%; and  α is the coefficient describing the steepness of the curve around the median injury dose.
In this study, we define injury as PTS above 40 dB, which has a median injury dose of 170 dBA. In the subsequent analysis, we shall use 50 ID 170 dBA = unless specified otherwise.
In the dose-response relation (1), the dose is quantified by comb SELA . For a single shot (one impulse), the sound exposure (SE) is defined as the time integral of squared A-weighted sound pressure: Conventionally, the sound exposure in the air is measured relative to the reference sound exposure For multiple shots of equal intensity, the effective combined SELA for N shots is calculated using the dose combination rule described in [6]: where N is the number of impulses, and SELA is sound exposure level of each individual impulse. Here "effective combined SELA" or "dose combination" means "combining multiple sound exposure events into one composite event with an effective combined SELA value as a single metric quantifying the overall effect of multiple sound exposure events". We call this "dose combination" instead of "dose accumulation" to distinguish it from the situation where the dose from a preceding sound exposure event has some positive or negative influence on the injury risk in a subsequent event. In summary, "combined dose" means the effective dose for the overall composite event, counting the injury in all element events. In contrast, "cumulative dose" refers to the effective dose for one event, counting the injury in only one event, including both the effect of the current event's dose and the left-over effects from preceding events' doses.
To further delineate the difference between "combined dose" and "cumulative dose", let us consider the simple case where all sound exposure events are independent of each other. In this case, the "cumulative dose" is just the dose of the current event since preceding events have no influence on the injury risk of the current event due to independence. On the other hand, the "combined dose" for two identical and independent events is larger than the dose of each event since the total injury risk in the two events is certainly larger than that in each individual event. Health Before we end this section, we examine the connection between the sound exposure energy and the dose combination rule (4). Recall from the definition of SELA (3) that the sound exposure energy of each impulse is ( ) The sound exposure energy corresponding to the effective combined SELA value (4) can be written out in the same fashion as above SELA log exp ln 10 10 where E is the sound exposure energy of each impulse given in (5). For For 10 λ < , comb E is less than the sum of individual energies:

Independence or Synergy or Negative Synergy among a Sequence of Doses for Causing Injury
We study analytically the effects of multiple impulse shots based on the mathematical framework of the logistic dose-response relation and the dose combination rule proposed in [6]. The process of analysis will guide us in finding a self-consistent and reasonable way of extending the dose combination rule from the special case of multiple shots of equal intensity ("homogeneous doses") [6] to the general case of multiple shots of non-uniform intensities ("heterogeneous doses"). We introduce some short notations to facilitate the discussion. , , , N G S S S describes the general dose combination rule, which is yet to be specified. In [6], the dose combination rule is given for the special case where all events in the sequence are single impulses of the same intensity ("homogeneous doses"). One of the goals in our study is to extend the dose combination rule (9) For the purpose of extending the formulation to the general case of heterogeneous events, we need to bring events with different doses into a problem that can be solved in the special case of homogeneous events. We consider a sequence of ( ) m n + impulses of equal intensity, each with SELA value S. We group the first m impulses into a composite event 1 E ; and group the rest n impulses into a composite event 2 E . Thus, the sequence of ( )

S S S S S S
The effective combined SELA value of event 1 E is given by (9) ( ) The effective combined SELA value of event 2 E is ( ) The effective combined SELA value of events 1 E and 2 E is ( ) Notice that each injury probability above is directly from the dose-response relation using the SELA value of the event, not including effects from any other events. This is for the case where the event under consideration is treated as a stand-alone event, i.e., not preceded by any other event(s).
While event 1 E and event ( ) E and 2 E are independent of each other, then the three injury probabilities should satisfy the relation The independence implies that the probability of no injury in the composite event is the product of no-injury probabilities in individual element events.
If the probability of no injury in the composite event is more than the product of no-injury probabilities in individual element events,

then it rules out the independence and indica-
tes some kind of negative synergy in the injury mechanism among individual element events (i.e., immunity passed onto subsequent events). Here immunity means that having experienced the sound exposure but not injured in event 1 E increases one's conditional probability of escaping injury in the sound exposure of event 2 E , and thus, increases the overall no-injury probability of the composite Conversely, if the probability of no injury in the composite event is less than the product of no-injury probabilities in individual element events, then it also rules out the independence. In this case, it indicates some kind of positive synergy in the injury mechanism among individual element events (i.e., fatigue damage passed onto subsequent events).
Here fatigue damage means that even if the sound exposure (dose) in event 1 E does not directly cause injury, it nevertheless weakens the subject or otherwise makes the subject more vulnerable so as to increase the subject's conditional injury probability in the sound exposure of event 2 E , and thus decreases the overall no-injury probability of the composite state 1 We examine the sign of ( ) ( )( ) to assess the independence, immunity, or fatigue. First we express Notice directly from (19), (20) and (21) We replace Since the denominator is always positive, it follows that the numerator determines the sign of ( ) ( )( ) We introduce a theorem. within the range of 1 η < . Therefore, in the framework of the logistic doseresponse relation and the dose combination rule [6], we draw two conclusions: 1) a subsequent event is not independent of the presence of preceding event(s), and 2) the effect of a preceding event on the subsequent events is manifested in the form of immunity instead of fatigue; that is, a sequence of sound exposure events demonstrates negative synergy in causing injury.
To further distinguish the dose combination rule (4) from the case of independent events, we consider a sequence of N impulse shots, each with SELA value S. We look at the difference between these two cases in the probability of no injury as a function of N. For each individual shot, when viewed as a stand-alone event, the probability of no injury is given by the dose-response relation (1) where η is given in (18). Figure 2 displays the plots of no-injury probability vs N for these two cases, which clearly demonstrates the difference. In the case of independent events, the no-injury probability decays exponentially with respect

Dose Combination Rules
Building on the insight gained in the analysis above, we extend the dose combination rule for a homogeneous sequence of impulse shots [6], to the general case of N sound exposure events of non-uniform SELA values where each event in the sequence may be a composite event consisting of sub-events.
Then we construct the dose combination rule for the case of independent events and compare the two rules.

Extension of the Dose Combination Rule to the General Case of Multiple Heterogeneous Doses
In Equations (17) When we view the sequence as an overall composite event, the effective combined SELA value for the whole sequence is denoted by { } Therefore, the generalized dose combination rule (29) is consistent with the special case dose combination rule described in [6].
The generalized dose combination rule, as given in (29)  , , , N G S S S when necessary. This will set up the proper notation framework allowing us to discuss the dose combination rule for independent events and compare it with the empirical rule, which is the subject of the next subsection.

Dose Combination Rule for the Case of Independent Events and Comparison with the Empirical Rule
We study the hypothetical situation where all sound exposure events are independent of each other. We want to write out and examine the dose     In this section, we study the immunity in the dose-response formulation (37) and ( Pr no injury in and no injury in

Behaviors of Immunity in the Extended Empirical Dose-Response Formulation
Therefore, we conclude that the effect of sound exposure in 1 E provides an immunity in the subsequent event 2 E . We assess the effect of immunity quantitatively as a percentage of decrease in probability of injury in event 2 E due to the preceding event 1 E , and call it the immunity factor of event 1 E on event 2 E . Mathematically, the immunity factor is defined as In terms of immunity factor φ , the conditional probability of injury given the prior sound exposure and the unconditional probability of injury are related by ( ) ( ) ( ) We consider regimes described in terms of intermediate variable Using the Taylor expansion , and noticing that for Applying the expansion ( ) With the result of (53), we write the immunity factor as Conclusion for Regime 2: The immunity caused by the preceding dose on the subsequent dose of moderately lower SELA value, is large and approaches 100%.
We numerically demonstrate the asymptotic behaviors of regime 1 and regime 2. In Figure 6, we plot the immunity factor ( ) Health We start with the expression of the immunity factor φ given in (49)   Therefore, we conclude that when 2 W is fixed, the immunity factor φ increases monotonically with respect to 1 W . Below, we recast this statement in terms of the SELA values of the two events.
Conclusion for Regime 4: The immunity caused by the preceding dose on the subsequent dose increases monotonically with respect to the first dose while the second dose is fixed. It goes from near 0 when the first dose is significntly below the second dose, to near 100% when the first dose is raised to above the second dose.
We numerically visualize the trend behavior predicted in Regime 4. In Figure   8, we plot the immunity factor ( ) value for the case of independent events is asymptotically a linear function of N for large N. In comparison, the empirical dose combination rule is a linear function of ( ) 10 log N , much smaller than that for independent events. We also studied the effect of immunity quantitatively via the immunity factor defined as the percentage of decrease in the injury probability attributed to the sound exposure in the preceding event. Our main results on immunity are: 1) the immunity factor is primarily a function of the difference in SELA value between the two sound exposure events; it is virtually independent of the magnitude of the two SELA values as long as the difference between the two is fixed; 2) the immunity factor increases monotonically with respect to the difference in SELA values; 3) when the first dose (SELA) is significantly below the second dose, the immunity factor is close to 0; 4) when the two doses are comparable, the immunity factor is fairly large, approaching 100%; 5) when the first dose is moderately above the second dose, the immunity factor is close to 100%.
Our extended empirical dose-response formulation provides the theoretical foundation for assessing the injury risk in realistic situations where the sound exposure consists of multiple heterogeneous noise impulses with non-uniform SELA values. Future sound exposure experiments are needed for testing, validating and refining the extended empirical dose-response formulation.