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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.

The rapid development of applications such as wireless communications, security scanning, tissue diagnosis, and non-lethal weapons for crowd control or perimeter security has considerably increased human exposure to high-frequency millimeter waves (MMW) ranging from 30 to 300 gigahertz (GHz). For the purpose of biological risk assessment, it is vital to understand the effects of this irradiation on humans.

Many experiments have shown that exposure to MMW at sufficiently high intensities primarily produces fast heating near the surface of the skin [

Nociceptors are sensory nerve cells that respond to painful stimuli by sending out signals to the spinal cord via a chain of nerve fibers. When the collective signal becomes strong enough, the withdrawal reflex is triggered and the subject moves away from the exposure [

In this section, we study the mathematical formulation when the nociceptor density is a function of depth in the skin. We start by introducing proper mathematical notations:

· y: the depth coordinate (the skin surface is y = 0 ).

· r : 2-D coordinates on the skin surface; ( r , y ) is the 3-D coordinates.

· T ( r , y ) : 3-D spatial temperature profile of the skin.

· T act : activation temperature of nociceptors; given T ( r , y ) , the activation status of a nociceptor at ( r , y ) is governed by the indicator function

I ( T ≥ T act ) ( r , y ) ≡ { 1, if T ( r , y ) ≥ T act 0, if T ( r , y ) < T act

· ρ ( y ) : nociceptor density at depth y (number per volume).

· X act : total number of activated nociceptors in the skin,

X act = ∫ ρ ( y ) I ( T ≥ T act ) ( r , y ) d r d y

· ρ 0 : characteristic reference nociceptor density.

When the nociceptor density is uniform, ρ ( y ) = ρ 0 , the activated volume is proportional to the number of activated nociceptors. In this case, we adopted the activated volume as the single metric predictor variable (the input dose) for predicting withdrawal reflex [

Input dose in the case of uniform density:

z ≡ ∫ I ( T ≥ T act ) ( r , y ) d r d y (1)

The advantages of this input dose are 1) it makes the dose quantity independent of the nociceptor density ρ 0 , and 2) it shifts the effect of ρ 0 into the dose threshold z c . In the dose response model, withdrawal reflex occurs when z ≥ z c where the effects of ρ 0 and all other factors are reflected the single metric quantity z c .

In the case of non-uniform nociceptor density, the activated volume is no longer proportional to the number of activated nociceptors, and thus is no longer a valid candidate for the input dose. We like to define the dose such that it contains (1) as a special case. For that purpose, we define the equivalent activated volume z equ based on reference density ρ 0 , and use z equ as the input dose.

Input dose in the case of non-uniform density:

z equ ≡ 1 ρ 0 X act = ∫ ρ ( y ) ρ 0 I ( T ≥ T act ) ( r , y ) d r d y (2)

The dose response relation has the same form as in the case of uniform density.

Outcome ( z equ ) = { 1 ( withdrawal reflex ) , if z equ ≥ z c 0 ( no withdrawal reflex ) , if z equ < z c

In this study, we consider the hypothetical situation where the time of withdrawal reflex and the spatial temperature profile at reflex are measurable in experiments. With these two measurable entities, we explore the behaviors of several parameterized nociceptor density types. The objectives of the study are 1) to distinguish these candidate density types from each other based on the measurable entities, and 2) to infer the parameter values.

The calculation of z equ defined in (2) requires only the relative density ρ ( y ) / ρ 0 . The effect of ρ 0 is contained in the dose threshold z c . When ρ ( y ) / ρ 0 is given, the dose response model has only two unknown parameters: T act and z c . In a test, the measured temperature profile T ( r , y ) at reflex provides a constraint on ( T act , z c ) . Mathematically, we construct constraint function z c ( T act ) as follows. For any value of T act , by definition, the corresponding value of z c is the value of z equ calculated based on T act and T ( r , y ) .

z c ( T act ) = ∫ ρ ( y ) ρ 0 I ( T ≥ T act ) ( r , y ) d r d y (3)

when ρ ( y ) / ρ 0 is given, function z c ( T act ) is completely determined by the measured T ( r , y ) . Constraint functions z c ( T act ) based on T ( r , y ) measured at different test conditions are potentially distinct from each other. All these constraint functions have one common intersection, which gives the true values of ( T act , z c ) .

When the true relative density ρ ( y ) / ρ 0 is unknown, we work with a trial relative density r ( try ) ( y ) . We use r ( try ) ( y ) to replace ρ ( y ) / ρ 0 in (3) and construct trial constraint function z c ( T act ) from the test data. Note that the true values of ( T act , z c ) satisfy only the true constraint function calculated using the true ρ ( y ) / ρ 0 . When r ( try ) ( y ) deviates from the true ρ ( y ) / ρ 0 , the true values of ( T act , z c ) are not on the trial constraint curve calculated using r ( try ) ( y ) . Consequently, for a pair of trial constraint functions calculated using r ( try ) ( y ) (based on measured T ( r , y ) of two distinct test conditions), their intersection is not at the true values of ( T act , z c ) , and the intersection varies with the test conditions of the pair. The test-condition-dependence of the intersection serves as an indication that the trial density r ( try ) ( y ) is incorrect. The specific behavior of test-condition-dependence of the intersection provides a venue for us to tune r ( try ) ( y ) toward the true ρ ( y ) / ρ 0 .

We examine several types of parameterized density. We study the test-condition dependence of 1) the reflex time and 2) the intersection of a pair of trial constraint functions. The goal is to identify system behaviors that a) help us distinguish these density types from each other and b) guide us to tune the trial parameter toward its true value.

We carry out the analysis in the idealized situation where

· the electromagnetic heating is uniform over the beam cross-section (with area A) and it decays exponentially with depth y;

· the initial temperature is uniform everywhere;

· the heat conduction is included only in the depth direction.

This is the same as case B in our previous study [

{ ρ m C p ∂ T ( y , t ) ∂ t = K ∂ 2 T ( y , t ) ∂ y 2 + P dep μ e − μ y ∂ T ( y , t ) ∂ y | y = 0 = 0 , T ( y ,0 ) = T 0

where

· ρ m is the mass density of skin,

· C p is the specific heat capacity of the skin,

· K is the thermal conductivity of the skin,

· μ is the absorption coefficient of the skin,

· P dep is the beam power density deposited on (absorbed into) the skin, and

· T 0 is the initial temperature of the skin.

In this idealized situation, the region of activated nociceptors ( T ( y ) ≥ T act ) is a cylinder with depth y act governed by T ( y act ) = T act . By definition, T act , y act | ( reflex ) and the reflex time t ref are constrained by the temperature distribution, which we assume is measurable.

T act = T ( y act | ( reflex ) , t ref ) (4)

We consider a general 1-parameter form for the relative density of nociceptors

ρ ( y ) ρ 0 = f ( β y )

where f ( s ) can be any positive function. The activated depth y act and the dose z equ both vary with the activation temperature T act , and are related by Equation (2) as

z equ = A ∫ 0 y act f ( β y ) d y = A β F ( β y act ) (5)

y act = 1 β F − 1 ( β z equ A ) (6)

where A is the beam spot area, F ( s ) ≡ ∫ 0 s f ( z ) d z , and F − 1 ( u ) is the inverse function of F ( s ) . Since f ( s ) is positive, function F ( s ) is monotonically increasing and the inverse function F − 1 ( u ) is well-defined over the range of F ( s ) . By definition, functions F ( s ) and F − 1 ( u ) satisfy F ( 0 ) = 0 and F − 1 ( 0 ) = 0 . Recall that the dose threshold is defined as z c ≡ z equ | ( reflex ) . Equation (6) gives

y act | ( reflex ) = 1 β F − 1 ( β z c A ) (7)

Equation (4) in combination with (7) provides a constraint on T act , z c , β and t ref , which can be used for different purposes, depending on which parameters are known. When t ref and T ( y , t ) are measured, (4) gives us a constraint on z c , T act and β . On the other hand, when z c , T act and β are given, (4) can be viewed as a governing equation for t ref . This is useful, for example, for examining the behavior of t ref vs. A.

In the analysis of subsequent sections, we need the expansions of f ( s ) and F ( s ) , F − 1 ( u ) and their derivatives. We now derive these expansions. We first write out the Taylor expansion of f ( s ) around s = 0 .

f ( s ) = a 0 + a 1 s + a 2 s 2 2 ! + ⋯ where a k = f ( k ) ( 0 )

F ( s ) ≡ ∫ 0 s f ( z ) d z = a 0 s + a 1 s 2 2 ! + a 2 s 3 3 ! + ⋯

The expansion of F − 1 ( u ) is derived from that of F ( s ) using an iterative method. It depends on which term in the expansion of F ( s ) is the leading non-zero term. Let u ≡ F ( s ) . The inverse function F − 1 ( u ) maps u back to s. We discuss two cases.

· Case 1: f ( 0 ) ≠ 0 . Function u = F ( s ) has the expansion

u = a 0 s + a 1 s 2 2 ! + ⋯ where a k = f ( k ) ( 0 )

Based on that, we built an iteration formula:

s j + 1 = 1 a 0 ( u − a 1 1 2 ! s j 2 ) , s 0 = 0

The iteration gives us expansions of F − 1 ( u ) and ( F − 1 ) ′ ( u )

F − 1 ( u ) = 1 a 0 u − a 1 2 a 0 3 u 2 + ⋯ for f ( 0 ) ≠ 0 (8)

( F − 1 ) ′ ( u ) = 1 a 0 − a 1 a 0 3 u + ⋯ for f ( 0 ) ≠ 0 (9)

· Case 2: f ( 0 ) = 0 but f ′ ( 0 ) ≠ 0 . Function u = F ( s ) has the expansion

u = a 1 s 2 2 ! + a 2 s 3 3 ! + ⋯ where a k = f ( k ) ( 0 )

Based on that, we construct an iteration formula for s 2 :

s j + 1 2 = 2 a 1 ( u − a 2 1 3 ! ( s j 2 ) 3 / 2 ) , s 0 2 = 0

which yields the expansion of s 2 in terms of u

s 2 = 2 a 1 u ( 1 − a 2 6 ⋅ 2 a 1 ( 2 a 1 u ) 1 / 2 + ⋯ )

Taking square roots of both sides, we obtain

F − 1 ( u ) = 2 a 1 u 1 / 2 − a 2 3 a 1 2 u + ⋯ for f ( 0 ) = 0 but f ′ ( 0 ) ≠ 0 (10)

( F − 1 ) ′ ( u ) = 1 2 a 1 u − 1 / 2 − a 2 3 a 1 2 + ⋯ for f ( 0 ) = 0 but f ′ ( 0 ) ≠ 0 (11)

with the mathematical results above, we study the behavior of t ref vs A.

When T act , z c and β are given, (4) with (7) governs t ref vs A. In our previous study [

( T ( y , t ) − T 0 ) K μ P dep = H ( y nd , t nd ) (12)

where the non-dimensional depth y nd and the non-dimensional time t nd are defined as

y nd ≡ μ y , t nd ≡ t K μ 2 ρ m C p

The normalized temperature H ( y nd , t nd ) has the expression

H ( y , t ) ≡ ∫ 0 t G ( y , s ) d s

G ( y , t ) ≡ 1 2 erfc ( 2 t − y 4 t ) e t − y + 1 2 erfc ( 2 t + y 4 t ) e t + y (13)

It is important to notice that the normalized temperature H ( y , t ) is parameter-free. The non-dimensional version of (4) with (7) has the form

( T act − T 0 ) K μ P dep = H ( y act , nd , t ref , nd ) (14)

y act , nd ≡ μ β F − 1 ( β μ A nd ) , A nd ≡ A μ z c , t ref , nd ≡ t ref K μ 2 ρ m C p

In the above, A nd ≡ A μ z c is the non-dimensional beam spot area. In the limit of A → ∞ , we have y act , nd → 0 and the equation for t ref , nd becomes

( T act − T 0 ) K μ P dep = h ( t ref , nd | A → ∞ )

where h ( t nd ) ≡ H ( 0, t nd ) . Let t 0 ≡ h − 1 ( ( T act − T 0 ) K μ P dep ) . We have t ref , nd | A → ∞ = t 0 . We examine the asymptotic behavior of this convergence. We seek an expansion of the form

t ref , nd ( A nd ) = t 0 [ 1 + c A ( 1 A nd ) α + ⋯ ] (15)

Expanding H ( y nd , t nd ) around ( 0, t 0 ) and substituting (15), we get

0 = ∂ H ( 0, t 0 ) ∂ t t 0 c A ( 1 A nd ) α + 1 2 ∂ 2 H ( 0, t 0 ) ∂ y 2 ( μ β F − 1 ( β μ A nd ) ) 2 (16)

Exponent α and coefficient c A are determined using the leading term expansion of F − 1 ( u ) . The result depends on whether the nociceptor density at the skin surface is zero.

· Case 1: f ( 0 ) ≠ 0 .

The expansion of F − 1 ( u ) given in (8). Substituting it into (16), we have

α = 2, c A = 1 2 t 0 ( f ( 0 ) ) 2 ⋅ 1 − erfc ( t 0 ) e t 0 erfc ( t 0 ) e t 0 (17)

Here we have used ∂ H ( 0, t 0 ) ∂ t = erfc ( t 0 ) e t 0 and ∂ 2 H ( 0, t 0 ) ∂ y 2 = erfc ( t 0 ) e t 0 − 1 derived in our previous study.

· Case 2: f ( 0 ) = 0 but f ′ ( 0 ) ≠ 0

The expansion of F − 1 ( u ) given in (10). Substituting it into (16), we arrive at

α = 1, c A = μ t 0 f ′ ( 0 ) β ⋅ 1 − erfc ( t 0 ) e t 0 erfc ( t 0 ) e t 0 (18)

Returning to the physical quantities before the non-dimensionalization, the reflex time vs beam spot area is given by

t ref ( A ) = t 0 ρ m C p K μ 2 [ 1 + c A ( μ z c A ) α + ⋯ ] , t 0 = h − 1 ( ( T act − T 0 ) K μ P dep ) (19)

When t ref and T ( y , t ref ) are measured, (4) with (7) serves as a constraint on T act , z c and β with beam spot area A as a parameter describing the test condition. We denote the constraint function as T act ( z c ; β , A ) and use (4) to write it as

T act ( z c ; β , A ) = Φ ( 1 β F − 1 ( β z c A ) ; 1 A ) (20)

where Φ ( y ; v ) T ( y , t ref ( 1 / v ) ) is the temperature profile at reflex with beam spot area A = 1 / v . We represent the effect of A via variable v = 1 / A since t ref is a smooth function of v = 1 / A as A → ∞ . To facilitate the discussion, we introduce two sets of mathematical notations. These two sets of notations are used to distinguish a quantity’s true value from its role as a variable in a function.

· β * : true value of coefficient β .

· β : a trial value of coefficient β .

· ( T act * , z c * ) : true values of model parameters T act and z c .

· T act ( z c ; β , A ) : constraint function (20) calculated using trial value β ; in T act ( z c ; β , A ) , z c denotes the independent variable, and T act the dependent variable.

Note that the data is generated with the true value β * . In the calculation constraint function (20), we use the measured data and a trial value β since β * is unknown. When β = β * , the constraint function shares one common intersection for all values of A:

T act ( z c * ; β * , A ) = T act * for all A (21)

when β ≠ β * , in general ( T act * , z c * ) is not on constraint curve T act ( z c ; β , A ) , and the intersection of a pair of constraint functions varies with the test conditions (values of A). We study the behavior of the intersection vs A. We make use of (21) and expand T act ( z c ; β , A ) around ( z c * , β * ) . For conciseness, we introduce ε ≡ β z c A and write (20) as:

T act ( z c ; β , A ) = Φ ( ξ ; 1 A ) , ξ ( β , ε ) ≡ 1 β F − 1 ( ε ) , ε ( β , z c ) ≡ β z c A

We apply the chain rule to calculate partial derivatives.

∂ T act ∂ z c = ∂ ∂ y Φ ( ξ ; 1 A ) ( F − 1 ) ′ ( ε ) 1 A (22)

∂ T act ∂ β = ∂ ∂ y Φ ( ξ ; 1 A ) ( F − 1 ) ′ ( ε ) 1 A ⋅ z c β [ − F − 1 ( ε ) ε ( F − 1 ) ′ ( ε ) + 1 ] ≡ ∂ T act ∂ z c ⋅ η where η ≡ z c β [ − F − 1 ( ε ) ε ( F − 1 ) ′ ( ε ) + 1 ] (23)

Using these derivatives, we write out the expansion of T act ( z c ; β , A ) .

T act ( z c ; β , A ) = T act * + ∂ T act ∂ z c | ( z c * , β * ) ⋅ [ ( z c − z c * ) + η | ( z c * , β * ) ( β − β * ) ] (24)

The slope of T act ( z c ; β , A ) vs. z c is ∂ T act ∂ z c | ( z c * , β * ) , which is given in (22). In Appendix A, we show that the slope converges to zero as A → ∞ . It follows that

T act ( z c ; β , A = ∞ ) = T act * for all β (25)

(25) shows that in the limit of A → ∞ , the constraint curve is a horizontal line at T act * , independent of z c and β . For finite A, ∂ T act ∂ z c ≠ 0 and T act in (24) varies with z c and β . We consider the intersection of T act ( z c ; β , A ) and T act ( z c ; β , ∞ ) ≡ T act * . The T act -coordinate of the intersection is T act * . Let z c ( I ) ( β , A ) denote the z c -coordinate of the intersection. Solving for z c ( I ) ( β , A ) from (24) and (25) leads to

z c ( I ) ( β , A ) = z c * − η | ( z c * , β * ) ( β − β * ) (26)

The dependence of z c ( I ) ( β , A ) on A is contained in η | ( z c * , β * ) given in (23). Using the expansion of F − 1 ( ε ) ε ( F − 1 ) ′ ( ε ) derived in (73) in Appendix A, we investigate the behavior of η at large A.

· Case 1: f ( 0 ) ≠ 0 .

η | ( z c * , β * ) = z c β ( − a 1 2 a 0 2 ε ) | ( z c * , β * ) = − ( z c * ) 2 f ′ ( 0 ) 2 ( f ( 0 ) ) 2 ⋅ 1 A for large A

In case 1, as A → ∞ , the intersection z c ( I ) ( β , A ) given by (26) converges to the true value z c * regardless of trial value β .

l i m A → ∞ z c ( I ) ( β , A ) = z c * for all β

The residual in convergence is proportional to ( β − β * ) / A and has the same sign as f ′ ( 0 ) ( β − β * ) . More specifically, we have

z c ( I ) ( β , A ) = z c * + ( z c * ) 2 f ′ ( 0 ) 2 ( f ( 0 ) ) 2 ⋅ β − β * A for large A (27)

· Case 2: f ( 0 ) = 0 but f ′ ( 0 ) ≠ 0

η | ( z c * , β * ) = z c β ( − 1 − 2 a 2 3 a 1 3 / 2 ε 1 / 2 ) | ( z c * , β * ) = z c * β * ( − 1 − 2 β * z c * f ″ ( 0 ) 3 ( f ′ ( 0 ) ) 3 / 2 ⋅ 1 A ) → − z c * β * for large A (28)

In case 2, as A → ∞ , the intersection z c ( I ) ( β , A ) given by (26) converges to

l i m A → ∞ z c ( I ) ( β , A ) = z c * − ( − z c * β * ) ( β − β * ) = z c * β β *

which is proportional to trial value β . The residual in convergence is proportional to ( β − β * ) / A and has the same sign as f ″ ( 0 ) ( β − β * ) . Asymptotically, z c ( I ) is

z c ( I ) ( β , A ) = z c * β β * + ( z c * ) 3 / 2 ( β * ) 1 / 2 ⋅ 2 f ″ ( 0 ) 3 ( f ′ ( 0 ) ) 3 / 2 ⋅ β − β * A for large A (29)

with the analytical preparations above, we examine four types of parametric form for nociceptor density vs depth, depicted in

For type 1 nociceptor density, the relative density takes the parametric form

ρ ( y ) ρ 0 = f ( β y ) , where f ( s ) = e − s (30)

The graph of type 1 f ( s ) is plotted in

f ( 0 ) = 1 ≠ 0 , f ′ ( 0 ) = − 1 , max 0 ≤ s < ∞ f ( s ) = 1

The reflex time vs beam spot area is given by (19) and (17), namely,

t ref ( A ) = t 0 ρ m C p K μ 2 [ 1 + c A ( μ z c A ) 2 + ⋯ ] , c A = 1 − erfc ( t 0 ) e t 0 2 t 0 erfc ( t 0 ) e t 0 (31)

As A increases, t ref converges to its limit with the residual proportional to 1/A^{2}:

t ref ( A ) − t ref ( ∞ ) ~ 1 A 2

It is interesting to notice that expansion (31) is independent of β * . Consequently, the measurements of t ref ( A ) vs. A do not contain any information for estimating β * .

We consider the intersection of the pair T act ( z c ; β , A ) and T act ( z c ; β , ∞ ) ≡ T act * . The z c -coordinate of the intersection, z c ( I ) ( β , A ) , is generally described by (26). For type 1 density, we have f ( s ) = e − s , f ( 0 ) = 1 ≠ 0 and f ′ ( 0 ) = − 1 , and the specific expression of z c ( I ) ( β , A ) is given by (27).

z c ( I ) ( β , A ) = z c * − ( z c * ) 2 2 ( β − β * ) ⋅ 1 A (32)

At β = β * , the intersection z c ( I ) ( β * , A ) = z c * is independent of A. When β ≠ β * , the trend of z c ( I ) ( β , A ) vs. A tells us whether β > β * or β < β * .

· For β > β * , z c ( I ) ( β , A ) ascends toward z c * from below as A increases.

· For β < β * , z c ( I ) ( β , A ) descends toward z c * from above as A increases.

When it is known that the nociceptor density has the parametric form of type 1 given in (30), we can tune the trial value β down or up toward the true value β * depending on whether the calculated z c ( I ) ( β , A ) increases or decreases with A.

We now consider the situation where the type of parametric form of ρ ( y ) / ρ 0

is unknown. With no information on the type of density form, we use the uniform density as the trial density in calculating the constraint function. Let T act , uni ( z c ; A ) denote the constraint function based on the test data (which is generated using type 1 density (30) with true value β * ) and calculated using framework (20) with the uniform trial density ρ ( y ) / ρ 0 ≡ 1 . Notice that the uniform density is a member of type 1 family (30) with β = 0 . Thus, the two constraint functions T act , uni ( z c ; A ) and T act ( z c ; β , A ) are related by T act , uni ( z c ; A ) = T act ( z c ; β = 0, A ) . Let z c , uni ( I ) ( A ) denote the z c -coordinate of the intersection of the pair T act , uni ( z c ; A ) and T act,uni ( z c ; A = ∞ ) ≡ T act * . Setting β = 0 in (32), we obtain

z c , uni ( I ) ( A ) = z c * + ( z c * ) 2 2 β * ⋅ 1 A (33)

Since the uniform density is parameter-free, the calculation of constraint function T act,uni ( z c ; A ) and intersection z c , uni ( I ) ( A ) is based solely on the test data. It does not require any input parameter or knowledge of the function form of the true density. Once the test data is available, T act,uni ( z c ; A ) and z c , uni ( I ) ( A ) can be calculated. When the true underlying density is type 1 given in (30), result (33) predicts that z c , uni ( I ) ( A ) converges to z c * as A → ∞ with the difference proportional to 1/A.

z c * = c 0 , β * = 2 c 1 c 0 2

Before we end this section, we clarify that result (33) predicts the behavior of constraint function T act,uni ( z c ; A ) calculated using the uniform trial density when the true underlying density affecting the test data is type 1 given in (30). When the true underlying density is of a different type, the behavior will be different. One objective of examining T act,uni ( z c ; A ) and z c , uni ( I ) ( A ) is to identity the type of nociceptor density from the observed behavior of z c , uni ( I ) ( A ) vs A, based on the theoretically predicted behaviors for a list of density types. In the subsequent sections, we will study more density types.

For type 2 nociceptor density, the relative density has the parametric form

ρ ( y ) ρ 0 = f ( β y ) , where f ( s ) = s e 1 − s (34)

The graph of type 2 f ( s ) is shown in

f ( 0 ) = 0 , f ′ ( 0 ) = e ≠ 0 , f ″ ( 0 ) = − 2 e , max 0 ≤ s < ∞ f ( s ) = 1

The reflex time vs beam spot area is described by (19) and (18).

t ref ( A ) = t 0 ρ m C p K μ 2 [ 1 + c A ⋅ μ z c A + ⋯ ] , c A = μ ( 1 − erfc ( t 0 ) e t 0 ) t 0 e β * erfc ( t 0 ) e t 0 (35)

As A increases, t ref converges to its limit with the difference proportional to 1/A.

t ref ( A ) − t ref ( ∞ ) ~ 1 A

^{2} for type 1 nociceptor density (30). To distinguish between these two density forms, we introduce an auxiliary quantity Q

Q ≡ t ref ( A ) − t ref ( 2 A ) t ref ( 2 A ) − t ref ( 4 A ) (36)

The theoretical prediction above tells us

Q = { 4 if ρ ( y ) = ρ 0 e − β y ( type 1 ) 2 if ρ ( y ) = ρ 0 ⋅ ( β y ) e 1 − β y ( type 2 ) (37)

In (35), coefficient c A does contain β * . However, in (35) β * is tangled with

other parameters. It is not possible to extract the value of β * unless all other parameters are known.

We consider the intersection of the pair T act ( z c ; β , A ) and T act ( z c ; β , ∞ ) ≡ T act * . The z c -coordinate of the intersection, z c ( I ) ( β , A ) , is generally described by (26). For type 2 density, we have f ( s ) = s e 1 − s , f ( 0 ) = 0 , f ′ ( 0 ) = e ≠ 0 and f ″ ( 0 ) = − 2 e , and the specific expression of z c ( I ) ( β , A ) is given by (29).

z c ( I ) ( β , A ) = z c * β β * − ( 2 z c * ) 3 / 2 3 e β * ( β − β * ) ⋅ 1 A

At β = β * , the intersection z c ( I ) ( β * , A ) = z c * is independent of beam spot area A. When β ≠ β * , the trend of z c ( I ) ( β , A ) vs A tells us whether β > β * or β < β * .

· For β > β * , z c ( I ) ( β , A ) ascends toward z c * β β * as A increases.

· For β < β * , z c ( I ) ( β , A ) descends toward z c * β β * as A increases.

The increase/decrease trend of z c ( I ) ( β , A ) vs. A is qualitatively the same as that for type 1 density given in (30). There are two differences. For type 2 density (34), we have 1) l i m A → ∞ z c ( I ) ( β , A ) = z c * β β * , which varies with trial value β , and 2) the residual in convergence is proportional to 1 / A , instead of 1/A.

When it is known that the nociceptor density has the parametric form of type 2 given in (34), we can tune the trial value β down or up toward the true value β * depending on whether the calculated z c ( I ) ( β , A ) increases or decreases with A.

When the type of parametric form of ρ ( y ) / ρ 0 is unknown, we like to design a method to identify the true underlying density type among a set of candidate density types based on test data measured in experiments. In the analysis of reflex time above, we used quantity Q defined in (36), with behaviors described in (37), to distinguish between type 1 and type 2 densities. Now we look into using constraint functions as a tool for that purpose. With no information on the density type, we use the uniform density as the trial density in calculating the constraint function.

Let T act,uni ( z c ; A ) denote the constraint function based on the test data (which is generated using type 2 density (34) with true value β * ) and calculated using framework (20) with the uniform trial density ρ ( y ) / ρ 0 ≡ 1 . We select the uniform density to make the calculation of constraint function parameter-free, based solely on the test data.

Note that the uniform density is not a member of type 2 family (34). Consequently, constraint function T act,uni ( z c ; A ) is not a special case of T act ( z c ; β , A ) . To study the behavior of T act,uni ( z c ; A ) , we try to connect it to T act ( z c ; β * , A ) , the true constraint function. Both T act,uni ( z c ; A ) and T act ( z c ; β , A ) are constructed from the same test data generated with the true underlying density (34). Each of them is calculated in framework (20) using a different type of trial density, and has different features:

· T act ( z c ; β * , A ) is the constraint function calculated using the correct density type (34) with the true parameter value β * . Operationally, the calculation of T act ( z c ; β * , A ) is not realistic unless we know β * . The theoretical advantage of T act ( z c ; β * , A ) is that it shares a common intersection ( z c * , T act * ) for all values of A.

· T act,uni ( z c , uni ; A ) is the constraint function calculated using the uniform trial density. Operationally, we can always calculate T act,uni ( z c , uni ; A ) from test data. However, the point ( z c * , T act * ) in general is not on the curve T act,uni ( z c , uni ; A ) .

The two constraint functions above are unified in Φ ( y ; 1 / A ) via y act , as described in (4).

T act = Φ ( y act ; 1 A ) (38)

The mapping between y act and z c , however, depends on the trial density. As a result, variable z c is related to y act differently in the two constraint functions. For clarity of the discussion, we use z c ,uni to denote the variable in T act,uni ( z c , uni ; A ) , and use z c for the variable in T act ( z c ; β * , A ) . The difference lies in the trial density used in the formulation. For the true density, z c is related to y act in (5). For the uniform density, z c ,uni = A ⋅ y act . Expressing y act in terms of z c or z c ,uni , we obtain

y act = { 1 β * F − 1 ( β * z c A ) for ρ ( y ) / ρ 0 = ( β * y ) e 1 − β * y z c , uni A for ρ ( y ) / ρ 0 = 1 (39)

Combining (39) and (38), we write out T act,uni ( z c , uni ; A ) and T act ( z c ; β * , A )

T act ( z c ; β * , A ) = Φ ( 1 β * F − 1 ( β * z c A ) ; 1 A )

T act,uni ( z c , uni ; A ) = Φ ( z c , uni A ; 1 A ) (40)

Since T act ( z c ; β * , A ) is calculated using the true density, it satisfies

T act * = Φ ( 1 β * F − 1 ( β * z c * A ) ; 1 A ) for all A

Taking the limit as A → ∞ yields

T act * = l i m A → ∞ Φ ( 1 β * F − 1 ( β * z c * A ) ; 1 A ) = Φ ( 0 ; 0 )

In (40), letting A → ∞ and using Φ ( 0 ; 0 ) = T act * , we obtain

T act,uni ( z c ,uni ; ∞ ) = T act *

Let z c ,uni ( I ) ( A ) denote the intersection of the pair T act,uni ( z c ,uni ; A ) and T act,uni ( z c ,uni ; ∞ ) ≡ T act * . Both z c ,uni ( I ) ( A ) and z c ( I ) ( β * , A ) ≡ z c * are calculated by first mapping T act * to y act * and then mapping to z c ( I ) . Given the measured temperature profile at reflex, Equation (38) maps T act * to a unique y act * , independent of the trial density. It follows that in (39), line 1 with z c ( I ) ( β * , A ) ≡ z c * and line 2 with z c ,uni ( I ) ( A ) are both equal to y act * .

1 β * F − 1 ( β * A z c * ) = y act * = z c ,uni ( I ) ( A ) A

Therefore, z c , uni ( I ) ( A ) and z c * are related by

z c , uni ( I ) ( A ) = A β * F − 1 ( β * A z c * ) (41)

For type 2 density (34), the expansion of F − 1 ( u ) is given in (10) with a 1 = f ′ ( 0 ) = e and a 2 = f ″ ( 0 ) = − 2 e . Substituting the expansion of F − 1 ( u ) into (41) yields

z c , uni ( I ) ( A ) = 2 A e β * z c * + 2 3 e z c * (42)

z c * = 3 e 2 c 0 , β * = 3 c 0 c 1 2

z c , uni ( I ) ( A ) is the intersection of two constraint functions calculated from test data using the uniform trial density. Given the test data, the process of calculating z c , uni ( I ) ( A ) is parameter-free. However, z c , uni ( I ) ( A ) depends on the test data, which is affected by the underlying true density. When the true density has the type 2 form given in (34), result (42) predicts that z c , uni ( I ) ( A ) increases linearly with A , unbounded as A → ∞ . Result (42) for type 2 density is in sharp contrast

with the situation for type 1 density, described in (33) of Section 5, where z c , uni ( I ) ( A ) converges to the true value z c * as A → ∞ . This difference in the behavior of z c , uni ( I ) ( A ) vs. A provides another mechanism of distinguishing between type 1 and type 2 densities, in addition to the method of examining quantity Q, described in (37).

We write type 3 density in the same parametric form as that of types 1 and 2.

ρ ( y ) ρ 0 = f ( β y ) , f ( s ) = θ ( s − 1 ) , β ≡ 1 y 0 (43)

where θ ( s ) is the Heaviside step function.

θ ( s ) = { 0 , s < 0 1 , s ≥ 0 (44)

The graph of type 3 f ( s ) is shown in

F ( s ) = ∫ 0 s θ ( s − 1 ) d s = { 0 , s < 1 s − 1 , s ≥ 1

F − 1 ( u ) = u + 1 for u > 0

The mapping between y act and z c is described by (7) and has the specific expression

y act | ( reflex ) = 1 β F − 1 ( β z c A ) = z c A + y 0 , β ≡ 1 y 0 (45)

With the expression of y act in (45), Equation (14) for t ref,nd becomes

( T act − T 0 ) K μ P dep = H ( y act,nd , t ref,nd )

y act,nd ≡ μ y act = 1 A nd + μ y 0 , A nd ≡ μ z c A

In the limit of A → ∞ , we have y act,nd → μ y 0 and the equation for t ref,nd converges to

( T act − T 0 ) K μ P dep = H ( μ y 0 , t ref,nd | A → ∞ )

Consider h ( t | q ) ≡ H ( q , t ) as a function of t with q as a parameter. Let h − 1 ( u | q ) be the inverse of h ( t | q ) and τ 0 ( q ) ≡ h − 1 ( ( T act − T 0 ) K μ P dep | q ) . As A → ∞ , we have

t ref,nd | A → ∞ = τ 0 ( μ y 0 )

By definition, τ 0 ( q ) satisfies H ( q , τ 0 ( q ) ) = ( T act − T 0 ) K μ P dep . We expand function H ( y nd , t nd ) around ( μ y 0 , τ 0 ( μ y 0 ) ) and write the equation for t ref,nd as

0 = ∂ H ( μ y 0 , τ 0 ( μ y 0 ) ) ∂ t ( t ref,nd − τ 0 ( μ y 0 ) ) + ∂ H ( μ y 0 , τ 0 ( μ y 0 ) ) ∂ y ( 1 A nd ) (46)

Substituting t ref,nd ( A nd ) = τ 0 ( μ y 0 ) [ 1 + c A ( 1 A nd ) α + ⋯ ] into (46) to calculate α and c A , and then mapping back to the physical quantities, we obtain

t ref ( A ) = τ 0 ( μ y 0 ) ρ m C p K μ 2 [ 1 + c A ( μ z c A ) + ⋯ ] , c A = − ∂ H ( μ y 0 , τ 0 ( μ y 0 ) ) ∂ y τ 0 ( μ y 0 ) ∂ H ( μ y 0 , τ 0 ( μ y 0 ) ) ∂ t

As A increases, t ref ( A ) converges to its limit with the residual proportional to 1/A.

t ref ( A ) − t ref ( ∞ ) ~ 1 A

This behavior is similar to that of type 2 density. For both type 2 and type 3, the nociceptor density is zero at the skin surface.

For type 3 density, it is more sensible to use y 0 as the parameter since it has the clear physical meaning of the depth at which the nociceptor density jumps from 0 to ρ 0 . In the unified parametric form in (43), the generic parameter is β ≡ 1 / y 0 . Mathematically, working with β ≡ 1 / y 0 allows us to use general results of the unified parametric form obtained in previous sections. In the analysis below, we will go back and forth between y 0 and β .

We adopt the general convention of using y 0 * to denote the true value and y 0 to represent the variable. The general form of constraint function with trial value β is given in (20). Using β = 1 / y 0 and F − 1 ( u ) = u + 1 , we write it as

T act ( z c ; y 0 , A ) = Φ ( z c A + y 0 ; 1 A ) (47)

In the limit of A → ∞ , T act ( z c ; y 0 , A ) converges to a horizontal line

T act ( z c ; y 0 , A = ∞ ) = Φ ( y 0 ; 0 )

At y 0 * , the constraint function shares the common intersection ( z c * , T act * ) for all A.

T act ( z c * ; y 0 * , A ) = Φ ( z c * A + y 0 * ; 1 A ) = T act * for all A

In particular, we have Φ ( y 0 * ; 0 ) = T act * . We expand Φ ( z c A + y 0 ; 1 A ) around ( z c * A + y 0 * ; 1 A ) and expand Φ ( y 0 ; 0 ) around ( y 0 * ,0 ) to write them respectively as

Φ ( z c A + y 0 ; 1 A ) = T act * + ∂ ∂ y Φ ( z c * A + y 0 * ; 1 A ) ( z c − z c * A + y 0 − y 0 * ) (48)

Φ ( y 0 ; 0 ) = T act * + ∂ ∂ y Φ ( y 0 * ; 0 ) ( y 0 − y 0 * ) (49)

Let z c ( I ) ( y 0 , A ) denote the z c -coordinate of the intersection of T act ( z c ; y 0 , A ) and T act ( z c ; y 0 , ∞ ) . z c ( I ) ( y 0 , A ) is governed by equating the RHSs of (48) and (49).

∂ ∂ y Φ ( z c * A + y 0 * ; 1 A ) ( z c − z c * A + y 0 − y 0 * ) = ∂ ∂ y Φ ( y 0 * ; 0 ) ( y 0 − y 0 * )

Solving for z c ( I ) ( y 0 , A ) yields

z c ( I ) ( y 0 , A ) = z c * + A ( ∂ ∂ y Φ ( y 0 * ; 0 ) ∂ ∂ y Φ ( z c * A + y 0 * ; 1 A ) − 1 ) ( y 0 − y 0 * ) (50)

We expand ∂ ∂ y Φ ( z c * A + y 0 * ; 1 A ) with respect to 1 A into the form

∂ ∂ y Φ ( z c * A + y 0 * ; 1 A ) = ∂ ∂ y Φ ( y 0 * ; 0 ) ( 1 + b 1 A + b 2 A 2 + ⋯ ) (51)

Substituting the expansion into (50), we obtain

z c ( I ) ( y 0 , A ) = z c * + ( − b 1 + ( b 1 2 − b 2 ) 1 A ) ( y 0 − y 0 * ) (52)

In Appendix B, we show that the coefficients satisfy

b 1 > 0 and ( b 1 2 − b 2 ) > 0 (53)

At y 0 = y 0 * , the intersection z c ( I ) ( y 0 * , A ) = z c * is independent of beam spot area A. When y 0 ≠ y 0 * , the trend of z c ( I ) ( y 0 , A ) vs A implies whether y 0 < y 0 * or y 0 > y 0 * .

· For y 0 < y 0 * , z c ( I ) ( y 0 , A ) ascends toward z c * − b 1 ( y 0 − y 0 * ) > z c * as A increases.

· For y 0 > y 0 * , z c ( I ) ( y 0 , A ) descends toward z c * − b 1 ( y 0 − y 0 * ) < z c * as A increases.

In terms of β ≡ 1 / y 0 , the increase/decrease trend of z c ( I ) vs A for type 3 density (43) resembles that for type 2 density (34). Both types of densities share the common feature that the nociceptor density is zero at skin surface: ρ ( 0 ) = 0 .

When it is known that the nociceptor density has the parametric form of type 3 given in (43), we can tune the trial value y 0 up or down toward the true value

y 0 * depending on whether the calculated z c ( I ) ( y 0 , A ) increases or decreases with A.

When the type of parametric form of ρ ( y ) / ρ 0 is unknown, we apply the uniform trial density in calculating the constraint function and use it as a tool for probing the density type. This approach has the advantage of being operationally practical. Once the test data is available, the calculation of constraint function does not require any input parameters.

Let T act,uni ( z c ; A ) denote the constraint function based on the test data (which is generated using type 3 density (43) with true value y 0 * ) and calculated using framework (47) with the uniform trial density ρ ( y ) / ρ 0 ≡ 1 . For type 3 parametric family (43), the uniform density is a special case at y 0 = 0 . However, expansions (48) and (49) are only for the case of y 0 near y 0 * , away from the skin surface. At y 0 = 0 , the expansions will be different because of the insulated boundary condition ∂ ∂ y Φ ( 0 , t ) = 0 . At y 0 = 0 , we have

T act,uni ( z c ; A ) = Φ ( z c A ; 1 A )

T act,uni ( z c ; ∞ ) = Φ ( 0 ; 0 )

We expand Φ ( z c A ; 1 A ) around ( 0 ; 0 ) . It follows that

Φ ( z c A ; 1 A ) = Φ ( 0 ; 0 ) + ∂ ∂ v Φ ( 0 ; 0 ) 1 A + 1 2 ∂ 2 ∂ y 2 Φ ( 0 ; 0 ) z c 2 A 2 + 1 2 ∂ 2 ∂ v 2 Φ ( 0 ; 0 ) 1 A 2 + ∂ 2 ∂ y ∂ v Φ ( 0 ; 0 ) z c A 2 + ⋯

Let z c ,uni ( I ) ( A ) denote the intersection of T act,uni ( z c ; A ) and T act,uni ( z c ; ∞ ) . It satisfies

Φ y y ( z c , uni ( I ) ) ( A ) 2 + 2 Φ y v z c , uni ( I ) ( A ) + 2 Φ v A + Φ v v + ⋯ = 0

All derivatives are evaluated at ( 0 ; 0 ) . Solving for z c , uni ( I ) ( A ) , we obtain

z c ,uni ( I ) ( A ) = Φ v − Φ y y ⋅ A + 2 Φ y v − Φ y y + ⋯ (54)

Result (54) predicts that when the true underlying density is type 3 given in (43), z c , uni ( I ) ( A ) increases linearly with A , unbounded as A → ∞ . This behavior is similar to that for type 2 density. Both type 3 and type 2 share the common feature of ρ ( 0 ) = 0 .

We represent type 4 density in the same parametric form as that of types 1 - 3.

ρ ( y ) ρ 0 = f ( β y ) , f ( s ) = 1 − 1 2 θ ( s − 1 ) , β ≡ 1 y 0 (55)

where θ ( s ) is the Heaviside step function defined in (44). The graph of type 4 f ( s ) is shown in

Because of the discontinuity in density profile (8), the Taylor expansions of F ( s ) and F − 1 ( u ) derived in Section 3 are no longer valid. We write out F ( s ) and F − 1 ( u ) directly.

F ( s ) = ∫ 0 s ( 1 − 1 2 θ ( s − 1 ) ) d s = { s , s < 1 1 2 ( s + 1 ) , s ≥ 1

F − 1 ( u ) = { u , u < 1 2 u − 1 , u ≥ 1 (56)

The activated depth at reflex given in (7) has the expression

y act | ( reflex ) ( A ) = y 0 F − 1 ( z c y 0 A ) = { z c A , z c A < y 0 2 z c A − y 0 , z c A ≥ y 0 (57)

The non-dimensional reflex time t ref,nd is related to A via y act ( A ) in Equation (14).

( T act − T 0 ) K μ P dep = H ( μ y act ( A ) , t ref,nd ) (58)

Conventionally, we view t ref,nd as a function of A since in experiments the beam spot area is prescribed in test design and the corresponding reflex time is observed. To facilitate the mathematically analysis, we view A as a function of t ref,nd . We study function A ( t ref,nd ) for type 4 density and connect it to its counterpart for the uniform density. Let

· A ( tp4 ) ( t ref,nd ) : function A vs. t ref,nd for type 4 density

· A ( uni ) ( t ref,nd ) : function A vs. t ref,nd for the uniform density

· t ref,nd ( tp4 ) ( 1 / A ) : function t ref,nd vs. 1/A for type 4 density

· t ref,nd ( uni ) ( 1 / A ) : function t ref,nd vs. 1/A for the uniform density.

For any given value of t ref,nd , the corresponding y act ( t ref,nd ) is completely determined by Equation (58), independent of the nociceptor density. We use y act ( t ref,nd ) to connect A ( tp4 ) ( t ref,nd ) and A ( uni ) ( t ref,nd ) . For type 4 density, y act ( A ( tp4 ) ) is given in (57). Its inverse function is

A ( tp4 ) ( y act ) = z c y 0 ⋅ 1 F ( y act y 0 ) = { z c y act , y act < y 0 2 z c y act + y 0 , y act ≥ y 0 (59)

For the uniform density, f ( s ) = 1 , F ( s ) = s , and A ( uni ) ( y act ) is given by

A ( uni ) ( y act ) = z c y 0 ⋅ 1 F ( y act y 0 ) = z c y act (60)

Combining (59) and (60), we can express 1 / A ( uni ) in terms of 1 / A ( tp4 ) .

1 A ( uni ) = { 1 A ( tp4 ) , 1 A ( tp4 ) < y 0 z c 2 A ( tp4 ) − y 0 z c , 1 A ( tp4 ) ≥ y 0 z c (61)

(61) described the relation between 1 / A ( tp4 ) ( t ref,nd ) and 1 / A ( uni ) ( t ref,nd ) . Now we treat t ref,nd as the dependent variable and view it as a function of 1/A. (61) leads to

t ref,nd ( tp4 ) ( 1 / A ) = { t ref,nd ( uni ) ( 1 / A ) , 1 A < y 0 z c t ref,nd ( uni ) ( 2 A − y 0 z c ) , 1 A ≥ y 0 z c (62)

Equation (62) reveals that t ref,nd ( tp4 ) ( 1 / A ) is obtained from t ref,nd ( uni ) ( 1 / A ) by a piecewise linear scaling on the independent variable 1/A. Function t ref,nd ( uni ) ( 1 / A ) is smooth. After the piecewise linear scaling, t ref,nd ( tp4 ) ( 1 / A ) has a discontinuity in derivative.

For type 4 density, it is more sensible to choose y 0 as the parameter since it has the clear physical meaning of the depth at which the nociceptor density drops sharply from ρ 0 to ρ 0 / 2 . In the unified parametric form in (55), the generic parameter is β ≡ 1 / y 0 . We adopt the general convention of using y 0 * to denote the true value and y 0 to represent the variable. The general form of constraint function with trial value β is given in (20). Using β = 1 / y 0 and F − 1 ( u ) given in (56), we write it as

T act ( z c ; y 0 , A ) = { Φ ( z c A ; 1 A ) , z c A < y 0 Φ ( 2 z c A − y 0 ; 1 A ) , z c A ≥ y 0 (63)

when beam spot area A is sufficiently large to make z c * A < y 0 * , (57) gives y act < y 0 * and the measured data for type 4 density is the same as that for the uniform density. Constraint function (63) is based on the data for true value y 0 * and is calculated with trial value y 0 . For any positive trial value y 0 > 0 and a finite interval of z c near z c * in consideration, when A is sufficiently large, we have Interval of z c A < m i n ( y 0 * , y 0 ) and

T act ( z c ; y 0 , A ) = Φ ( z c A ; 1 A ) for large A

Here Φ ( z c A ; 1 A ) is the constraint function based on the data for the uniform density and is calculated using the uniform trial density. Φ ( z c A ; 1 A ) is independent of trial value y 0 , and passes through ( z c * , T act * ) for all values of A.

Φ ( z c * A ; 1 A ) = T act * for all A

In the limit of 1 / A → 0 , we have Φ ( 0 ; 0 ) = T act * . Let z c ( I ) ( y 0 , A ) denote the z c -coordinate of the intersection of T act ( z c ; y 0 , A ) and T act ( z c ; y 0 , ∞ ) . Our analysis above shows that

· For any y 0 , T act ( z c ; y 0 , A ) = Φ ( z c A ; 1 A ) when A is sufficiently large.

· For any y 0 , T act ( z c ; y 0 , A = ∞ ) = Φ ( 0 ; 0 ) = T act * .

· For any y 0 , z c ( I ) ( y 0 , A ) = z c * when A is sufficiently large.

More specifically, to have z c ( I ) ( y 0 , A ) = z c * , we only need T act ( z c ; y 0 , A ) = Φ ( z c A ; 1 A ) for z c near z c * . The condition Interval of z c A < m i n ( y 0 * , y 0 ) becomes A > z c * m i n ( y 0 * , y 0 ) . We conclude that

z c ( I ) ( y 0 , A ) = z c * for A > z c * m i n ( y 0 * , y 0 ) (64)

Thus, for large A, z c ( I ) ( y 0 , A ) is independent of y 0 . To probe the position of trial value y 0 relative to true value y 0 * , we need to examine the behavior of z c ( I ) ( y 0 , A ) for A < z c * m a x ( y 0 * , y 0 ) . In this range of A, constraint function (63) takes the form

T act ( z c ; y 0 , A ) = Φ ( 2 z c A − y 0 ; 1 A )

At true value y 0 * , T act ( z c ; y 0 * , A ) shares the common intersection ( z c * , T act * ) for all A.

Φ ( 2 z c * A − y 0 * ; 1 A ) = T act *

We expand Φ ( 2 z c A − y 0 ; 1 A ) around ( 2 z c * A − y 0 * , 1 A ) .

Φ ( 2 z c A − y 0 ; 1 A ) = T act * + ∂ ∂ y Φ ( 2 z c * A − y 0 * ; 1 A ) ( 2 z c − z c * A − ( y 0 − y 0 * ) )

Setting the LHS to T act * and solving for z c , we obtain z c ( I ) ( y 0 , A )

z c ( I ) ( y 0 , A ) = z c * + A 2 ( y 0 − y 0 * ) for A < z c * m a x ( y 0 * , y 0 ) (65)

For y 0 < y 0 * and z c * y 0 * < A < z c * y 0 , the activated depth at reflex is y act * = z c * A > y 0 . Substituting the expression of y act * into (57) and solving for z c , we have

z c ( I ) ( y 0 , A ) = z c * − 1 2 ( z c * − A y 0 ) for z c * y 0 * < A < z c * y 0 (66)

For y 0 > y 0 * and z c * y 0 < A < z c * y 0 * , the activated depth at reflex is y act * = 2 z c * A − y 0 * . This y act * may or may not be beyond depth y 0 . The case of y act * > y 0 corresponds to A < 2 z c * y 0 + y 0 * . Substituting the expression of y act * into (57) and solving for z c yields

z c ( I ) ( y 0 , A ) = z c * + A 2 ( y 0 − y 0 * ) for z c * y 0 < A < 2 z c * y 0 + y 0 * (67)

Similarly, the case of y act * < y 0 corresponds to A > 2 z c * y 0 + y 0 * . We get

z c ( I ) ( y 0 , A ) = z c * + ( z c * − A y 0 * ) for 2 z c * y 0 + y 0 * < A < z c * y 0 * (68)

Summarizing results for various cases, we write out a complete description of z c ( I ) ( y 0 , A ) .

The case of y 0 < y 0 * ,

z c ( I ) ( y 0 , A ) = { z c * + A 2 ( y 0 − y 0 * ) for A < z c * y 0 * z c * − 1 2 ( z c * − A y 0 ) for z c * y 0 * < A < z c * y 0 z c * for A > z c * y 0

The case of y 0 > y 0 * ,

z c ( I ) ( y 0 , A ) = { z c * + A 2 ( y 0 − y 0 * ) for A < 2 z c * y 0 * + y 0 z c * + ( z c * − A y 0 * ) for 2 z c * y 0 * + y 0 < A < z c * y 0 * z c * for A > z c * y 0 * (69)

At y 0 = y 0 * , we have z c ( I ) ( y 0 * , A ) = z c * for all A. When y 0 ≠ y 0 * , z c ( I ) ( y 0 , A ) has non-trivial dependence on A. The trend of z c ( I ) ( y 0 , A ) vs A tells us whether y 0 < y 0 * or y 0 > y 0 * .

· When y 0 < y 0 * , from small A to large A, z c ( I ) ( y 0 , A ) starts below z c * , decreases further below z c * , and then reverts rapidly back to z c * and stays there.

· When y 0 > y 0 * , from small A to large A, z c ( I ) ( y 0 , A ) starts above z c * , increases further above z c * , and then reverts rapidly back to z c * and remains there.

When it is known that the nociceptor density has the parametric form of type 4 given in (55), we can tune the trial value y 0 up or down toward the true value y 0 * depending on the behavior of the calculated z c ( I ) ( y 0 , A ) vs. A.

When the type of parametric form of ρ ( y ) / ρ 0 is unknown, we use the uniform

trial density in calculating the constraint function. The goal is to use the resulting constraint function as a tool to probe the underlying unknown density type.

Let T act,uni ( z c ; A ) denote the constraint function based on the test data (which is generated using type 4 density (55) with true value y 0 * ) and calculated using framework (63) with the uniform trial density ρ ( y ) / ρ 0 ≡ 1 . For type 4 parametric family (55), the uniform density is a special case with large y 0 .

It follows that the behavior of T act,uni ( z c ; A ) is the same as that of T act ( z c ; y 0 , A ) for large y 0 , which we analyzed in the previous subsection.

Let z c , uni ( I ) ( A ) denote the intersection of the pair T act,uni ( z c ; A ) and T act,uni ( z c ; ∞ ) ≡ T act * . Intersection z c , uni ( I ) ( A ) has the same behavior as z c ( I ) ( y 0 , A ) for large y 0 . Based on result (69) for z c ( I ) ( y 0 , A ) in the previous subsection, we conclude for z c , uni ( I ) ( A ) that

z c , uni ( I ) ( A ) = { z c * + ( z c * − A y 0 * ) for A < z c * y 0 * z c * for A > z c * y 0 * (70)

Result (70) predicts that when the true underlying density is type 4 given in (55), z c , uni ( I ) ( A ) starts at slightly below 2 z c * for small A; decreases linearly toward z c * as A increases; arrives at z c * at A = z c * y 0 * and stays there for A > z c * y 0 * . The qualitative behavior of z c , uni ( I ) ( A ) converging to z c * for large A is similar for type 1 and type 4 densities.

In this study, we investigate the nociceptor density of the form: ρ ( y ) = ρ 0 f ( β y ) , and its effect on heat-induced withdrawal reflex. We examine 4 types of f ( s ) illustrated in

We consider the situation where the reflex time and the temperature profile at the reflex are measurable in tests. We build the mathematical formulation for extracting 3 key parameters from test data:

· the activation temperature T act for heat-sensitive nociceptors,

· the critical threshold z c on the equivalent activated volume, and

· the parameter β in the relative density.

Our general strategy is to identify distinct behaviors for different densitytypes and distinct behaviors for different regions of parameter values. We compare these theoretical patterns with the observed patterns from test data to pinpoint the underlying unknown density type. We inspect the behavior calculated using a trial parameter value in the parametric form to determine whether the trial value is below or above the true value. Then we use that information to tune the trial value up or down accordingly toward the true value. The process is repeated with the new trial value until convergence. To best illustrate each key module individually in its own setting, we divide the task of finding the density type and the parameter value into stages and consider several problems. Mathematically, we proceed from the simplest problem to the realistic one in which both the density type and the parameter value are unknown. The solution of a simpler problem provides the building blocks for solving a more complicated problem.

Problem 1:

When both the density type f ( s ) and the true scaling parameter β * are known for the nociceptor density, the test data allow us to construct true constraint functions on ( z c , T act ) . Multiple constraint functions, obtained from tests at different values of beam spot area A, share a common point at the true value of ( z c , T act ) . Parameters z c and T act are determined by finding the intersection of these distinct constraint functions.

Problem 2:

When the density type f ( s ) is given but the true scaling parameter β * is unknown, we construct trial constraint functions using trial values of β . Let T act ( z c ; β , A ) denote the trial constraint function that is based on the measured data (which is generated with true value β * ) and that is calculated using the given parametric form with trial value β . When the trial value β is different from the true value β * , in general, the true value ( z c * , T act * ) is not on trial constraint functions, and trial constraint functions at different values of beam spot area A do not share a common intersection. The behaviors of trial constraint functions T act ( z c ; β , A ) are demonstrated for the 4 density types, respectively in

Problem 3:

When both the density type f ( s ) and the true scaling parameter β * are unknown, we construct trial constraint functions using the uniform trial density ρ ( y ) / ρ 0 ≡ 1 , which is parameter-free. The trial constraint function using the uniform density is denoted by T act,uni ( z c ; A ) . When the true density is not the uniform density, the true value ( z c * , T act * ) is not on trial constraint function T act,uni ( z c ; A ) , and for different values of A trial constraint functions T act,uni ( z c ; A ) do not share a common point. Similar to what we did on trial constraint functions T act ( z c ; β , A ) in Problem 2 (where the density type is given), here we look at the intersection of a pair of trial constraint functions: T act,uni ( z c ; A ) and T act,uni ( z c ; ∞ ) . Let z c , uni ( I ) ( A ) denote the intersection of the pair, which is affected only by the beam spot area A used in tests. Given a collection of measured data sets at several values of A, we examine the trend of z c , uni ( I ) ( A ) vs. A. The trend behavior of z c , uni ( I ) ( A ) vs. A is displayed for the 4 density types, respectively in

The goal of our study is to develop a methodology that utilizes test data to 1) identify the density type for the underlying nociceptor density, 2) find by trial and error the true scaling parameter in the parametric form, and 3) determine the activation temperature of nociceptors and the critical threshold on the equivalent activated volume. The results of 1) and 2) basically specify the nociceptor density profile vs depth. We assume the test data available include measurements of the reflex time and of the spatial temperature profile at reflex. Combining the procedures outlined in Problems 1, 2 and 3 above, we obtain such a methodology exactly for this purpose.

The authors thank Dr. John Biddle and Dr. Stacy Teng of Institute for Defense Analyses for introducing the ADT CHEETEH-E model and for many helpful discussions. The authors acknowledge the Joint Intermediate Force Capabilities Office of U.S. Department of Defense and the Naval Postgraduate School 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 U.S. Government.

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, H.Y., Burgei, W.A. and Zhou, H. (2020) Effect of Depth-Dependent Nociceptor Density on the Heat-Induced Withdrawal Reflex. Applied Mathematics, 11, 788-824. https://doi.org/10.4236/am.2020.118053

Derivation of l i m A → ∞ T act ( z c ; β , A ) = T act * in (25)

We show the convergence in three steps.

Step 1: We first look at the factor ( F − 1 ) ′ ( ε ) 1 A in the expression of ∂ T act ∂ z c | ( z c * , β * ) in (22). As A → ∞ we have ε → 0 . We expand ( F − 1 ) ′ ( ε ) as ε → 0 . In both cases 1 and 2, we have l i m ε → 0 ( F − 1 ) ′ ( ε ) ⋅ ε = 0 . It follows that

l i m A → ∞ ( F − 1 ) ′ ( ε ) 1 A = l i m ε → 0 ( F − 1 ) ′ ( ε ) ⋅ ε 1 β z c = 0 (71)

Step 2: Next we look at the factor ∂ ∂ y Φ ( ξ ; 1 / A ) in the expression of ∂ T act ∂ z c | ( z c * , β * ) in (22). With l i m A → ∞ ξ = 0 and the insulated boundary condition ∂ T ( 0, t ) ∂ y = 0 , we have

l i m A → ∞ ∂ ∂ y Φ ( ξ ; 1 A ) = l i m ε → 0 ∂ ∂ y T ( ξ , t ref ( A ) ) = 0 (72)

Taking the limit as A → ∞ in (22), and using results (71) and (72), we conclude

l i m A → ∞ ∂ T act ∂ z c = l i m A → ∞ ∂ ∂ y Φ ( ξ ; 1 A ) ( F − 1 ) ′ ( ε ) 1 A = 0

Step 3: We show that the term η | ( z c * , β * ) in (24) is bounded as A → ∞ . In the expression of η in (23), we look at F − 1 ( ε ) ε ( F − 1 ) ′ ( ε ) , the component that varies with ε . Using expansions (8) and (9) in case 1 or expansions (10) and (11) in case 2, we get

F − 1 ( ε ) ε ( F − 1 ) ′ ( ε ) = { 1 + a 1 2 a 0 2 ε + ⋯ for case 1 2 + 2 a 2 3 a 1 3 / 2 ε 1 / 2 + ⋯ for case 2 (73)

In both cases, we have l i m ε → 0 η = finite .

Combining the results of three steps above, we conclude that for any trial value β , as A → ∞ , the constraint function converges to T act * , as described in (25) in the main text.

Derivation of property (53)

Property (53) describes coefficients in the expansion of ∂ ∂ y Φ ( z c * A + y 0 * ; 1 A ) with respect to 1 / A . Recall that Φ ( y ; v ) ≡ T ( y , t ref ( 1 / v ) ) where v ≡ 1 / A . Function T ( y , t ) as given in (12) and (13), has the properties below

1) ∂ ∂ y T ( y , t ) is always negative.

2) At any given depth y, the absolute value of ∂ ∂ y T ( y , t ) increases with time t.

3) At any given time t, the absolute value of ∂ ∂ y T ( y , t ) start with 0 value at y = 0 , increases with y until the inflection point at y = κ ( t ) , and then decreases to zeros beyond the inflection point. The depth of inflection point y = κ ( t ) increases with t.

We consider the situation of y 0 * < κ ( t 0 ) . That is, the nociceptor density jump occurs before the inflection point of temperature profile at reflex. Combining properties 1 and 3 of T ( y , t ) with y 0 * < κ ( t 0 ) , we have

− ∂ ∂ y T ( z c * A + y 0 * , t ref ( A ) ) ︸ positive > − ∂ ∂ y T ( y 0 * , t ref ( A ) ) ︸ positive for all A (74)

As beam spot area A increases t ref ( A ) decreases and is bounded from below by t ref ( A ) ≥ t ref ( A = ∞ ) = τ 0 ( μ y 0 * ) . In turn, τ 0 ( μ y 0 * ) is bounded by τ 0 ( μ y 0 * ) ≥ τ 0 ( 0 ) = t 0 > 0 , reflecting that the presence of a skin layer with no nociceptor delays the occurrence of withdrawal reflex. Thus, we have

t ref ( A ) ≥ t ref ( A = ∞ ) ≥ t 0 > 0 independent of A and y 0 *

Applying property 2 of T ( y , t ) to (74) with t ref ( A ) ≥ t ref ( A = ∞ ) , we obtain

− ∂ ∂ y Φ ( z c * A + y 0 * ; 1 A ) > − ∂ ∂ y Φ ( y 0 * ; 0 ) > 0 (75)

The first part of (53), b 1 > 0 , follows from inequality (75) and definition of b 1 in (51). The second part of (53), ( b 1 2 − b 2 ) > 0 , is verified in simulations.