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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.

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 [

The impacts of sound waves on humans are complicated and they depend on the frequency, sound pressure level, and duration. In [

They also validated the dose-response curve against historical human data from rifle noise tests [

In this paper, we carry out a mathematical analysis based on the dose-response relation and the dose combination rule developed in [

We organize the rest of the paper as follows. In Section 2 we review the relevant results in [

In [

P ( SELA comb ) = 1 1 + exp ( − α ( SELA comb − ID 50 ) ) (1)

In the dose-response relation (1):

• SELA comb 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);

• ID 50 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 [

depicts a hypothetical dose-response relation with α = 0.2 / ( dBA ) , twice the value from experimental data [

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 ID 50 = 170 dBA unless specified otherwise.

In the dose-response relation (1), the dose is quantified by SELA comb . For a single shot (one impulse), the sound exposure (SE) is defined as the time integral of squared A-weighted sound pressure:

E = ∫ ( A-weighted p ( t ) ) 2 d t (2)

Conventionally, the sound exposure in the air is measured relative to the reference sound exposure E 0 ≡ 400 ( μ Pa ) 2 ⋅ s . The sound exposure level (SELA) expresses the sound exposure in the unit of dBA as

SELA = 10 log 10 ( E E 0 ) dBA = 10 log 10 ( ∫ ( A-weighted p ( t ) ) 2 d t 400 ( μ Pa ) 2 ⋅ s ) dBA (3)

For multiple shots of equal intensity, the effective combined SELA for N shots is calculated using the dose combination rule described in [

SELA comb = SELA + λ log 10 N (4)

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.

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

E = E 0 10 SELA 10 = E 0 e x p ( l n ( 10 ) SELA 10 ) (5)

The sound exposure energy corresponding to the effective combined SELA value (4) can be written out in the same fashion as above

E comb = E 0 e x p ( l n ( 10 ) SELA comb 10 ) = E 0 e x p ( l n ( 10 ) SELA + λ l o g 10 N 10 ) = E ⋅ N λ 10 (6)

where E is the sound exposure energy of each impulse given in (5). For λ = 10 , E comb is simply the sum of individual energies.

E comb = N ⋅ E (7)

For λ < 10 , E comb is less than the sum of individual energies:

E comb < N ⋅ E (8)

In [

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 [

We introduce some short notations to facilitate the discussion.

• S : the effective combined SELA of an event where the event may be a single impulse or a composite event consisting of a sequence of heterogeneous events.

• S j : the effective combined SELA of event j.

• G { S 1 , S 2 , ⋯ , S N } : the effective combined SELA for a sequence of N events, respectively with individual SELA values { S 1 , S 2 , ⋯ , S N } . Here each event in the sequence may itself be a composite event consisting of sub-events. Mathematically, function G { S 1 , S 2 , ⋯ , S N } describes the general dose combination rule, which is yet to be specified.

In [

G { S , S , ⋯ , S } ︸ N impulses = S + λ l o g 10 ( N ) (9)

One of the goals in our study is to extend the dose combination rule (9) to the general case where the SELA values { S 1 , S 2 , ⋯ , S N } are not all equal. In the dose-response relation developed in [

The self-consistency can be seen by considering a sequence of 2N impulses, each with SELA value S. Let us call it view #1. The sequence can also be viewed as two big events, each big event consisting of N impulses and each big event having the effective combined SELA value S + λ l o g 10 ( N ) . Let us call this view #2. The effective combined SELA value calculated using (9) is the same for both view #1 and view #2:

View # 1 : G { S , S , ⋯ , S } ︸ 2 N impulses = S + λ l o g 10 (2N)

View # 2 : G { G { S , S , ⋯ , S } ︸ N impulses , G { S , S , ⋯ , S } ︸ N impulses } = G { S , S , ⋯ , S } ︸ N impulses + λ l o g 10 ( 2 ) = ( S + λ l o g 10 ( N ) ) + λ l o g 10 ( 2 ) = S + λ l o g 10 (2N)

For the purpose of extending the formulation to the general case of hete- rogeneous 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 E 1 ; and group the rest n impulses into a composite event E 2 . Thus, the sequence of ( m + n ) homo- geneous impulses can be viewed as two composite events E 1 and E 2 with different effective combined SELA values.

S , S , ⋯ , S ︸ m impulses ︷ Event E 1 , S , S , ⋯ , S ︸ n impulses ︷ Event E 2

The effective combined SELA value of event E 1 is given by (9)

S E 1 = S + λ log 10 ( m ) (10)

The effective combined SELA value of event E 2 is

S E 2 = S + λ log 10 ( n ) (11)

The effective combined SELA value of events E 1 and E 2 is

S E 1 + E 2 = S + λ l o g 10 ( m + n ) (12)

With the effective combined SELA values, the injury risks in these composite events are given by the dose-response relation:

P E 1 = 1 1 + e x p ( − α ( S E 1 − ID 50 ) ) (13)

P E 2 = 1 1 + e x p ( − α ( S E 2 − ID 50 ) ) (14)

P E 1 + E 2 = 1 1 + e x p ( − α ( S E 1 + E 2 − ID 50 ) ) (15)

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 E 1 and event ( E 1 + E 2 ) are indeed stand-alone, event E 2 is not. Event E 2 occurs after event E 1 in the sequence of events E 1 and E 2 . If events E 1 and E 2 are independent of each other, then the three injury pro- babilities should satisfy the relation

( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) = 0 (16)

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, ( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) > 0 , 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 E 1 increases one’s conditional probability of escaping injury in the sound exposure of event E 2 , and thus, increases the overall no-injury probability of the composite state E 1 + E 2 .

Conversely, if the probability of no injury in the composite event is less than the product of no-injury probabilities in individual element events, ( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) < 0 , 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 E 1 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 E 2 , and thus decreases the overall no-injury probability of the composite state E 1 + E 2 .

We examine the sign of ( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) to assess the indepen- dence, immunity, or fatigue. First we express P E 1 as

P E 1 = 1 1 + e x p ( − α ( S E 1 − ID 50 ) ) = 1 1 + ( 10 S E 1 − ID 50 λ ) − α λ l n 10 ≡ 1 1 + W E 1 − η (17)

Parameter η and intermediate variable W E 1 are defined as

η ≡ α λ ln 10 , (18)

W E 1 = W ( S E 1 ) ≡ 10 S E 1 − ID 50 λ = m ⋅ 10 S − ID 50 λ (19)

where we have used the expression of S E 1 given in (10). In a similar fashion, we can write P E 2 and P E 1 + E 2 as

P E 2 = 1 1 + W E 2 − η , W E 2 = n ⋅ 10 S − ID 50 λ (20)

P E 1 + E 2 = 1 1 + W E 1 + E 2 − η , W E 1 + E 2 = ( m + n ) ⋅ 10 S − ID 50 λ (21)

Notice directly from (19), (20) and (21) that quantities W E 1 , W E 2 and W E 1 + E 2 are related by a simple additive relation

W E 1 + E 2 = W E 1 + W E 2 (22)

We replace W E 1 + E 2 by W E 1 + W E 2 and write ( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) as

( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) = ( 1 − 1 1 + ( W E 1 + W E 2 ) − η ) − ( 1 − 1 1 + W E 1 − η ) ( 1 − 1 1 + W E 2 − η ) = W E 1 η + W E 2 η − ( W E 1 + W E 2 ) η + ( W E 1 W E 2 ) η ( 1 + ( W E 1 + W E 2 ) η ) ( 1 + W E 1 η ) ( 1 + W E 2 η ) (23)

Since the denominator is always positive, it follows that the numerator determines the sign of ( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) . We introduce a theorem.

Theorem 1: When η < 1 , we always have

W E 1 η + W E 2 η − ( W E 1 + W E 2 ) η + ( W E 1 W E 2 ) η > 0 (24)

for all W E 1 > 0 and W E 2 > 0 .

The proof of the theorem is presented in Appendix. From Theorem 1, we see that when the parameter η ≡ α λ l n 10 is less than 1, the quantity

( 1 − P E 1 + E 2 ) − ( 1 − P E 1 ) ( 1 − P E 2 ) is always positive, which clearly rules out the independence and indicates negative synergy (immunity) in the injury mechanism between preceding and subsequent events. In [

or 10. For these parameter values, the quantity η ≡ α λ l n 10 is 0.15 or 0.34, well

within the range of η < 1 . Therefore, in the framework of the logistic dose- response relation and the dose combination rule [

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):

1 − P ( S ) = 1 e α ( S − ID 50 ) + 1

In the case of independent events, the probability of no injury for the whole sequence is

( 1 − P ( S ) ) N = ( 1 e α ( S − ID 50 ) + 1 ) N (25)

When we use the dose combination rule (4) to combine the N impulse shots into one effective SELA value S comb = S + λ l o g 10 ( N ) , the probability of no injury has the expression

1 − P ( S comb ) = 1 e α ( S comb − ID 50 ) + 1 = 1 N η ⋅ e α ( S − ID 50 ) + 1 (26)

where η is given in (18).

to the number of shots N. In the logarithmic scale, the exponential decay yields a downward constant slope. In sharp contrast, the no-injury probability corres- ponding to the dose combination rule in [

Building on the insight gained in the analysis above, we extend the dose combination rule for a homogeneous sequence of impulse shots [

In Equations (17), (20), and (21) above, we wrote the dose-response relation in terms of an intermediate variable W ( S ) . When we divide a long sequence of impulses of the same intensity into two segments and treat each segment as a composite event, the two segments and the whole sequence are each described by its effective combined SELA value ( S E 1 , S E 2 , S E 1 + E 2 ) . In particular, the injury probability of each segment is described by the dose-response relation with the effective combined SELA value (when it is treated as a stand-alone composite event, excluding the effects of sound exposure in preceding events) . While it is not directly obvious how we can combine S E 1 and S E 2 to obtain S E 1 + E 2 , in the analysis above a key observation is that the intermediate variable W ( S ) is additive:

W ( S E 1 + E 2 ) = W ( S E 1 ) + W ( S E 2 ) (27)

Thus, we propose to extend the dose combination rule by summing the intermediate variable W ( S ) over all events in the sequence. More specifically, consider a sequence of N sound exposure events where each event in the sequence may be a composite event consisting of sub-events. Each sound exposure event in the sequence is described by its effective combined SELA value. The whole sequence is specified by SELA values of the N events: { S j , j = 1 , 2 , ⋯ , N } . When we view the sequence as an overall composite event, the effective combined SELA value for the whole sequence is denoted by G { S 1 , S 2 , ⋯ , S N } . Mathematically, function G { S 1 , S 2 , ⋯ , S N } is specified via the corresponding

intermediate variable W ( G { S 1 , S 2 , ⋯ , S N } ) ≡ 10 G { S 1 , S 2 , ⋯ , S N } − ID 50 λ , which is given as the sum of W ( S j ) over individual events in the sequence.

W ( G { S 1 , S 2 , ⋯ , S N } ) = ∑ j = 1 N W ( S j ) (28)

This relation on the intermediate variable W leads to the generalized dose combination rule:

G { S 1 , S 2 , ⋯ , S N } = λ l o g 10 ( ∑ j = 1 N 10 S j − ID 50 λ ) + ID 50 (29)

Note that when all S j ’s are equal to S, (29) is reduced to

G { S , S , ⋯ , S } = S + λ l o g 10 N , (30)

Therefore, the generalized dose combination rule (29) is consistent with the special case dose combination rule described in [

The generalized dose combination rule, as given in (29), appears to be affected by the median injury dose ID 50 . Actually it is not affected by ID 50 . We can get rid of ID 50 from (29). Let S max ≡ m a x k S k . We re-write the generalized dose combination rule as

G { S 1 , S 2 , ⋯ , S N } = S max + λ log 10 ( ∑ j = 1 N 10 S j − S max λ ) , (31)

where S max ≡ m a x k S k .

Let us look at a few examples of combining two sound exposure events.

G { S , S } = S + λ l o g 10 ( 2 ) = S + 0.3 λ (dB)

G { S , ( S + 3 ) } = ( S + 3 ) + λ l o g 10 ( 1 + 10 − 3 λ ) = { ( S + 3 ) + 0.19 , λ = 3.44 ( S + 3 ) + 1.76 , λ = 10

The effective combined SELA of two events with S and (S + 3) dB is 0.19 dB and 1.76 dB above the larger one, respectively for λ = 3.44 and λ = 10 . In terms of SELA values, 3 dB difference means a factor of 2 in sound exposure energy.

G { S , ( S + 10 ) } = ( S + 10 ) + λ log 10 ( 1 + 10 − 10 λ ) = { ( S + 10 ) + 0.002 , λ = 3.44 ( S + 10 ) + 0.41 , λ = 10

The effective combined SELA of two events with S and (S + 10) dB is 0.002 dB and 0.41 dB above the larger one, respectively for λ = 3.44 and λ = 10 . In terms of SELA values, 10 dB difference means a factor of 10 in sound exposure energy. In general, for Δ S ≥ λ , the effective combined SELA of two events with S and S + Δ S is approximately

G { S , ( S + Δ S ) } = ( S + Δ S ) + λ log 10 ( 1 + 10 − Δ S λ ) ≈ ( S + Δ S ) + λ ln ( 10 ) 10 − Δ S λ

For moderately large Δ S / λ , the combined SELA is just the larger one of the two event.

The generalized dose combination rule (31) is an extension of the rule from [

We study the hypothetical situation where all sound exposure events are independent of each other. We want to write out and examine the dose combination rule corresponding to this situation. Consider a sequence of N sound exposure events S E Q = { E j , j = 1 , 2 , ⋯ , N } with individual SELA values { S j , j = 1 , 2 , ⋯ , N } . The probability of no injury in each individual event is calculated from the dose-response relation (1):

1 − P E j = 1 − 1 1 + e x p ( − α ( S − ID 50 ) ) = 1 e x p ( α ( S − ID 50 ) ) + 1

When all events are independent of each other, the probability of no injury for the whole sequence is the product of no-injury probabilities of individual events

1 − P S E Q = ∏ j = 1 N ( 1 − P E j ) = 1 ∏ j = 1 N ( e x p ( α ( S j − ID 50 ) ) + 1 ) (32)

On the other hand, the probability of no injury for the whole sequence is governed by the dose-response relation with the effective combined dose denoted by G Indp { S 1 , S 2 , ⋯ , S N } , which is what we want to find. Here we use the subscript “Indp” to empasize that the function refers to the case of independent events.

1 − P S E Q = 1 e x p ( α ( G Indp { S 1 , S 2 , ⋯ , S N } − ID 50 ) ) + 1 (33)

Mathematically, function G Indp { S 1 , S 2 , ⋯ , S N } is defined by equating (32) and (33).

e x p ( α ( G Indp { S 1 , S 2 , ⋯ , S N } − ID 50 ) ) = ∏ j = 1 N ( e x p ( α ( S j − ID 50 ) ) + 1 ) − 1 (34)

Solving for G Indp { S 1 , S 2 , ⋯ , S N } , we arrive at:

G Indp { S 1 , S 2 , ⋯ , S N } = 1 α l n ( ∏ j = 1 N ( e x p ( α ( S j − ID 50 ) ) + 1 ) − 1 ) + ID 50 (35)

(35) is the dose-combination rule for the case of independent events. We compare (35) to the empirical rule (31). We first look at the dependence on N for a homogeneous sequence of N events, each with SELA value S. The empirical rule is a linear function of log 10 ( N ) .

G Empr { S , S , ⋯ , S } = S + λ log 10 (N)

For large N, the combined dose of independent events asymptotically behaves like

G Indp { S , S , ⋯ , S } ≈ N ⋅ 1 α l n ( e α ( S − ID 50 ) + 1 ) + ID 50

Unlike the empirical rule (31), the combined dose of independent events (35) does depend on the median injury dose ID 50 . To see the dependence on ID 50 , we examine the combined dose of two independent events ( N = 2 ), each with SELA value S.

G Indp { S , S } = S + 1 α l n ( e α ( S − ID 50 ) + 2 ) ≈ { S + 1 α l n ( 2 ) , α ( S − ID 50 ) moderately large, negative S + 1 α l n ( 3 ) , S = ID 50 2 ( S − ID 50 ) + ID 50 , α ( S − ID 50 ) moderately large, positive (36)

For S moderately below the median injury dose ID 50 , the combined dose of independent events behaves the same way as the empirical rule with the

increment for empirical rule λ l o g 10 ( 2 ) replaced by 1 α l n ( 2 ) . Recall that parameters α and λ are related by parameter η defined in (18): η = α λ ln ( 10 ) . As a result, the two increments are related by

1 α ln ( 2 ) = 1 η × λ log 10 (2)

Thus, for η significantly below 1, the increment for independent events is significantly higher than that in the empirical rule. For example, for η = 0.15 , the two increments differ by a factor of 6.67.

For S moderately above the median injury dose ID 50 , however, the combined dose for independent events behaves very differently from the empirical rule. Instead of adding a fixed increment of λ log 10 ( 2 ) , the combined dose for independent events doubles the amount of S above the median injury dose ID 50 .

To illustrate the behaviors of G Indp { S , S } described in (36), we plot ( G Indp { S , S } − S ) , the resulting increment in combining two independent events of equal SELA values (S).

higher than S and the increment ( G Indp { S , S } − S ) is approximately ( S − ID 50 ) . For S below ID 50 , the increment ( G Indp { S , S } − S ) is approximately constant with respect to S but is much higher than that for the empirical rule.

To demonstrate the dependence on the median injury dose ID 50 , in

We start this section by summarizing the extended empirical dose-response formulation for multiple sound exposure events with heterogeneous SELA values { S 1 , S 2 , ⋯ , S N } . The formulation contains two components: 1) the rule for combining the heterogeneous doses { S 1 , S 2 , ⋯ , S N } into one effective dose S comb and 2) the dose-response relation mapping the effective dose S comb to the occurrence probability of injury.

• The extended empirical dose combination rule:

S comb = S max + λ l o g 10 ( ∑ j = 1 N 10 S j − S max λ ) , S max ≡ m a x k S k (37)

• The dose-response relation:

P ( S comb ) = 1 1 + e x p ( − α ( S comb − ID 50 ) ) (38)

In this section, we study the immunity in the dose-response formulation (37) and (38). We consider the case of two composite sound exposure events E 1 and E 2 , respectively with effective combined SELA values S E 1 and S E 2 . In the previous section, we wrote the dose-response relation (38) in terms of interme-

diate variable W ( S ) ≡ 10 S − ID 50 λ and parameter η ≡ α λ l n ( 10 ) .

P ( S ) = 1 1 + W ( S ) − η , W ( S ) ≡ 10 S − ID 50 λ , η ≡ α λ l n ( 10 ) (39)

We examine several probabilities via intermediate variable W ( S ) .

Pr ( no injury in E 1 ) = 1 − P ( S E 1 ) = 1 1 + W ( S E 1 ) η = 1 1 + W 1 η , W 1 ≡ W ( S E 1 )

Pr ( no injury in E 1 and no injury in E 2 ) = 1 − P ( S E 1 + E 2 ) = 1 1 + ( W 1 + W 2 ) η , W 2 ≡ W ( S E 2 )

Pr ( no injury in E 2 | no injury in E 1 ) = Pr ( no injury in E 1 and no injury in E 2 ) Pr ( no injury in E 1 ) = 1 − P ( S E 1 + E 2 ) 1 − P ( S E 1 ) = 1 + W 1 η 1 + ( W 1 + W 2 ) η (40)

When we treat E 2 as a stand-alone event (i.e. excluding any effect left over from the sound exposure in the preceding event E 1 ), the probability of no injury in event E 2 is

Pr ( no injury in E 2 , excluding the effect of E 1 ) = 1 − P ( S E 2 ) = 1 1 + W 2 η (41)

From Theorem 1 in Section 3, we have

W 1 η + W 2 η + ( W 1 W 2 ) η > ( W 1 + W 2 ) η (42)

Adding 1 to each side of the inequality and then factoring the left-hand side, we obtain

( 1 + W 1 η ) ( 1 + W 2 η ) > 1 + ( W 1 + W 2 ) η (43)

which yields

1 + W 1 η 1 + ( W 1 + W 2 ) η > 1 1 + W 2 η (44)

Comparing (40) and (41) gives us

Pr ( no injury in E 2 | no injury in E 1 ) > Pr ( no injury in E 2 , excluding the effect of E 1 ) (45)

Therefore, we conclude that the effect of sound exposure in E 1 provides an immunity in the subsequent event E 2 . We assess the effect of immunity quantitatively as a percentage of decrease in probability of injury in event E 2 due to the preceding event E 1 , and call it the immunity factor of event E 1 on event E 2 . Mathematically, the immunity factor is defined as

ϕ ( S E 1 , S E 2 ) ≡ 1 − Pr ( injury in E 2 | no injury in E 1 ) Pr ( injury in E 2 , excluding the effect of E 1 ) (46)

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

Pr ( injury in E 2 | no injury in E 1 ) = ( 1 − ϕ ) × Pr ( injury in E 2 , excluding the effect of E 1 ) (47)

Below we study the behavior of immunity factor in four regimes of doses.

We write probabilities in terms of intermediate variable W ( S ) , and express the immunity factor as a function of W 1 and W 2 :

ϕ ( S E 1 , S E 2 ) = ( 1 − 1 1 + W 2 η ) − ( 1 − 1 + W 1 η 1 + ( W 1 + W 2 ) η ) 1 − 1 1 + W 2 η = W 1 η + W 2 η − ( W 1 + W 2 ) η + ( W 1 W 2 ) η W 2 η [ 1 + ( W 1 + W 2 ) η ] (48)

We consider regimes described in terms of intermediate variable W ( S ) ≡ 10 S − ID 50 λ .

• W = O ( 1 ) :

The corresponding SELA value has the range S = ID 50 + λ l o g 10 ( O ( 1 ) ) which is a small neighborhood around S = ID 50

• W = O ( ϵ ) :

The corresponding SELA value has the range S = ID 50 − λ log 10 ( O ( 1 / ϵ ) ) which is moderately below the median injury dose ID 50 .

• W = O ( 1 / ϵ ) :

The corresponding SELA value has the range S = ID 50 − λ log 10 ( O ( 1 / ϵ ) ) which is moderately above the median injury dose ID 50

In the analysis below, we assume η ≡ α λ ln ( 10 ) < 1 . In particular, we will look at the case of η = 0.15 , corresponding to α = 0.1 and λ = 3.44 , the parameter values from [

Regime 1: The second dose ( S E 2 ) is moderately above the first dose ( S E 1 ).

S E 2 − S E 1 = λ log 10 ( O ( 1 / ϵ ) )

which in turn yields

W 1 W 2 = 10 S E 1 − S E 2 λ = O ( ϵ ) = small

We use the small parameter W 1 W 2 = O ( ϵ ) to derive an asymptotic expression for the immunity factor ϕ . In the expression of ϕ given in (48), dividing both the numerator and the denominator by W 2 η , we write ϕ as a function of W 1 W 2 and W 2 as follows:

ϕ ( S E 1 , S E 2 ) = ( W 1 W 2 ) η ( 1 + W 2 η ) + 1 − ( 1 + W 1 W 2 ) η 1 + W 2 η ( 1 + W 1 W 2 ) η (49)

Using the Taylor expansion ( 1 + W 1 W 2 ) η = 1 + η W 1 W 2 + ⋯ , and noticing that for η < 1 , we have W 1 W 2 = O ( ϵ ) ≪ O ( ϵ η ) = ( W 1 W 2 ) η , we obtain

ϕ ( S E 1 , S E 2 ) = ( W 1 W 2 ) η ( 1 + W 2 η ) + 1 − ( 1 + η W 1 W 2 + ⋯ ) 1 + W 2 η ( 1 + η W 1 W 2 + ⋯ ) = ( W 1 W 2 ) η ( 1 + W 2 η ) + ⋯ 1 + W 2 η + ⋯ = ( W 1 W 2 ) η + ⋯ = 10 S E 1 − S E 2 λ η + ⋯ = O ( ϵ η ) = small

Conclusion for Regime 1: The immunity caused by the preceding dose on the subsequent dose of moderately higher SELA value, is small.

Regime 2: The second dose ( S E 2 ) is moderately below the first dose ( S E 1 ).

S E 2 − S E 1 = − λ log 10 ( O ( 1 / ϵ ) )

which leads to

W 2 W 1 = 10 S E 2 − S E 1 λ = O ( ϵ ) = small

We use the small parameter W 2 W 1 = O ( ϵ ) to derive an asymptotic expression for the immunity factor ϕ . Dividing both the numerator and the denominator by W 1 η in the formula of ϕ in Equation (48), we express ϕ in terms of W 2 W 1 and W 1 :

ϕ ( S E 1 , S E 2 ) = ( W 2 W 1 ) η ( 1 + W 1 η ) + 1 − ( 1 + W 2 W 1 ) η ( W 2 W 1 ) η [ 1 + W 1 η ( 1 + W 2 W 1 ) η ] (50)

Substituting in the Taylor expansions ( 1 + W 2 W 1 ) η = 1 + η W 2 W 1 + O ( ϵ 2 ) , and dividing the numerator and the denominator by ( W 2 W 1 ) η ( 1 + W 1 η ) , we get

ϕ ( S E 1 , S E 2 ) = ( W 2 W 1 ) η ( 1 + W 1 η ) − η W 2 W 1 + O ( ϵ 2 ) ( W 2 W 1 ) η [ ( 1 + W 1 η ) + W 1 η ( η W 2 W 1 + O ( ϵ 2 ) ) ] = 1 − η 1 + W 1 η ( W 2 W 1 ) 1 − η + O ( ϵ 2 − η ) 1 + W 1 η 1 + W 1 η 1 + W 1 η ( η W 2 W 1 + O ( ϵ 2 ) ) (51)

Applying the expansion ( 1 1 + z ) = ( 1 − z + ⋯ ) to the denominator, we arrive at

ϕ ( S E 1 , S E 2 ) = 1 − η 1 + W 1 η ( W 2 W 1 ) 1 − η − η W 1 η 1 + W 1 η W 2 W 1 + ⋯ = 1 − η ( W 2 W 1 ) 1 − η ( 1 + W 2 η 1 + W 1 η ) + ⋯ (52)

When S E 1 and S E 2 move together from low to high with the difference ( S E 2 − S E 1 ) unchanged, the pair ( W 1 , W 2 ) increases in magnitude from small

to large with the ratio W 2 W 1 = 10 S E 2 − S E 1 λ fixed. As a result, the factor ( 1 + W 2 η 1 + W 1 η ) varies from 1 at low ( S E 1 , S E 2 ) to ( W 2 W 1 ) η = O ( ϵ η ) at high ( S E 1 , S E 2 ) .

1 + W 2 η 1 + W 1 η ≈ { 1, ( S E 1 , S E 2 ) moderately below ID 50 ( W 2 W 1 ) η , ( S E 1 , S E 2 ) moderately above ID 50 (53)

With the result of (53), we write the immunity factor as

ϕ ( S E 1 , S E 2 ) ≈ { 1 − η 10 S E 2 − S E 1 λ ( 1 − η ) , ( S E 1 , S E 2 ) moderately below ID 50 1 − η 10 S E 2 − S E 1 λ , ( S E 1 , S E 2 ) moderately above ID 50

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

From

Regime 3: The two doses are close to each other.

S E 2 − S E 1 = l o g ( O (1))

which is equivalent to W 1 W 2 = O ( 1 ) .

We start with the expression of the immunity factor ϕ given in (49) as a function of W 1 W 2 and W 2 . In Regimes 1 and 2, we see that when the ratio W 1 W 2 is varied from small (Regime 1) to large (Regime 2), the immunity factor changes dramatically from ϕ ≈ 0 to ϕ ≈ 100 % . When W 1 W 2 is fixed (Regime 3), however, the immunity factor ϕ varies much slower with respect to W 2 . Now we prove that when the ratio W 1 W 2 is fixed, the immunity factor ϕ increases monotonically with respect to W 2 .

To facilitate the mathematical analysis, let us introduce variables

r ≡ W 1 W 2 , u ≡ W 2 η

The immunity factor as a function of ( r , u ) has the expression

ϕ = r η ( 1 + u ) + 1 − ( 1 + r ) η 1 + u ( 1 + r ) η (54)

Taking the derivative of ϕ with respect to u, yields

d ϕ d u = [ ( 1 + r ) η − r η ] [ ( 1 + r ) η − 1 ] [ 1 + u ( 1 + r ) η ] 2 > 0

This tells us immediately that the immunity factor ϕ is an increasing function of W 2 . The range of ϕ with respect to W 2 is readily obtained from expression (54).

ϕ | W 2 = 0 = r η + 1 − ( 1 + r ) η

ϕ | W 2 = ∞ = r η ( 1 + r ) η

In the case of η = 0.15 and r ≡ W 1 W 2 = 1 , the range ( ϕ | W 2 = 0 , ϕ | W 2 = ∞ ) is fairly tight:

ϕ | W 2 = 0 = 2 − 2 η = 0.8904 (55)

ϕ | W 2 = ∞ = 2 − η = 0.9013 (56)

Conclusion for Regime 3: The immunity caused by the preceding dose on the subsequent dose of similar magnitude (small difference measured in dBA), is significant and fairly large. The immunity factor remains almost unchanged as the two SELA values ( S E 1 and S E 2 ) are varied from being significantly below the median injury dose ID 50 to being significantly above ID 50 as long as the two doses are varied together with the difference S E 1 − S E 2 fixed.

With the expression of ϕ in terms of r ≡ W 1 W 2 and u ≡ W 2 η given in (54),

we continue to explore the immunity factor ϕ as a function of r while u is fixed.

Regime 4: Instead of focusing on a small parameter region that gives a particular asymptotic value of the immunity factor, here we consider the trend behavior of the immunity factor when the first dose ( S E 1 ) is varied from low to high while the second dose ( S E 2 ) is fixed.

Differentiating (54) with respect to r, we get

d ϕ d r = η ( 1 + u ) r η − 1 − ( 1 + r ) η − 1 + u r η − 1 ( 1 + r ) η − 1 [ 1 + u ( 1 + r ) η ] 2 (57)

Since parameter η is less than 1, we have η − 1 < 0 , and it follows that r η − 1 − ( 1 + r ) η − 1 > 0 holds for all positive r. Applying this result to the numerator of (57) gives us

d ϕ d r > 0 , for r > 0 and u > 0 (58)

The range of ϕ with respect to r = W 1 W 2 is calculated from (54).

ϕ | r = 0 = 0

ϕ | r = ∞ = 1

Therefore, we conclude that when W 2 is fixed, the immunity factor ϕ increases monotonically with respect to W 1 . 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

We have theoretically examined the results in [

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.

The authors would like to thank Joint Non-Lethal Weapons Directorate of US Department of Defense for supporting this work. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the US Government.

Wang, H., Burgei, W.A. and Zhou, H. (2017) Interpreting Dose-Response Relation for Exposure to Multiple Sound Impulses in the Framework of Immunity. Health, 9, 1817-1842. https://doi.org/10.4236/health.2017.913132

Here we give a proof of Theorem 1 appeared in Section 3.

For η ≡ α λ ln ( 10 ) < 1 , we have ( η − 1 ) < 0 . Consider function

g ( w ) = W E 1 η + w η − ( W E 1 + w ) η

It satisfies

g ( 0 ) = 0

g ′ ( w ) = η [ w η − 1 − ( W E 1 + w ) η − 1 ] > 0 for all w > 0

It follows that g ( w ) > 0 for all w > 0 . Or, equivalently,

W E 1 η + W E 2 η − ( W E 1 + W E 2 ) η > 0 for all W E 2 > 0 ( and W E 1 > 0 )

From this, we see that

W E 1 η + W E 2 η − ( W E 1 + W E 2 ) η + W E 1 η W E 2 η > 0 for all W E 1 > 0 and W E 2 > 0

which completes the derivation.