A modular modeling approach for investigating wool critical buckling from biologically variable along-fiber microstructure

Abstract

Mammalian hair fibers are internally sophisticated. We introduce a modeling approach aimed at use in research that derives value from understanding how microstructural organization generates effects at the macroscopic level in the context of natural biological variation. Critical buckling load is solved using a numerical approach applied to a modular fiber microstructure model where fibers of arbitrary length are made up of snippets composed of serial cross-sections at 25 micrometer intervals. As an example, the model is applied to investigate how much effect changes to single microstructural properties (fiber ellipticity, cortical cell type distribution and cell type proportion) have on critical buckling load in the context of prickle. Potential uses and key weak areas in our knowledge of wool fiber morphology and biophysics are discussed.

Keywords

hair biomechanics mechanomics numerical model orthocortex paracortex cuticle critical buckling load textile prickle

In biological materials such as wool and hair, macro-scale mechanical properties are often an emergent result of contributions from many components, each of which has inherent biological variation. This study investigates how we might use naturally variable biological data from keratin fibers to better understand the various contributions of different structural features to a macroscopic functional property. One current challenge in the wool industry is to find new high-value uses for wool that take advantage of its inherent properties. Keratin fibers, hair and wool, have a high degree of internal molecular and structural complexity that span from the mesoscale to the macro scale.^1–3 Single fiber properties can include length, diameter, curvature, bending stiffness and along-fiber changes. These are important for defining the mechanical properties of individual fibers, which in turn affect the performance of multi-fiber constructs, whether they be within a mammalian coat or within a textile.⁴ Ultimately the challenge is to understand how patterning of proteins within fibers and their organization across multiple fibers scales up to whole-material properties. Such an approach, which allows a connection between genomic control and material function, is sometimes called mechanomics.⁵ There are, however, many gaps in our knowledge of keratin fiber development⁶ and also in the relationships among fiber composition, structure and mechanical properties.^7,8 In this context, the paper addresses one such knowledge gap, namely the contribution of along-fiber microstructural variation to function.

Fiber buckling is chosen as a functional mechanical property. Such stiffness-related properties are relevant to a wide range of real-world scenarios within animal coat function, human hair and textiles, most notably as prickle. Prickle is the discomfort felt by a wearer when a short fiber end, constrained by neighboring fibers, pokes out of the material and into the wearer's skin. Prickle has been well described^9,10 and is generally controlled in the industry by excluding high-diameter fibers from next-to-skin products, but has also received significant attention in the past in the context of fiber modeling. Here we investigate how along-fiber variation in key properties of wool fibers affects resistance to buckling, and we evaluate the potential for using fiber models constructed in a modular fashion from data from actual wool fibers.

The two different mechanical discomfort sensations arising from garments, prickle and itch, are caused by protruding fibers activating nerve receptors in the skin. These are associated with two independent neural coding mechanisms, namely Aδ nociceptors and C-fiber nociceptors, respectively.^11,12 A fiber that generates this form of discomfort causes a local indentation in the skin, with the resulting stress being propagated through underlying tissues to the transduction sites of nerve receptors.¹² In addition, the fiber must be stiff enough to resist buckling at the threshold force for nerve activation; in the average human, it may be as low as 75 mg force, or 0.74 mN.¹¹ Garnsworthy et al.¹¹ suggested that the minimum length required for a fiber to cause prickle is about 200 µm because otherwise the indentation of the skin would lead to increased contact area of fabric and skin, reducing the pressure at the fiber end. Recent data, however, suggest that peak nociceptor location may occur at a depth of 130 µm in glabrous skin and 50 µm in hairy skin.¹³ Thus, although studies have used nominal lengths of 2000 µm in studies of prickle, very short fibers may cause discomfort to some people, and it is clear that variation occurs in the skin.

Wool fibers also vary in properties that affect their tendency to buckle, but there has been little evaluation of the effects of properties other than changes in average diameter. The minimum axial force required to buckle a fiber, its critical buckling load (CBL), is a function of its length, and its longitudinal profile and composition. A fiber subjected to the CBL can exist in either its original state (but unstable under an infinitesimal lateral force) or a buckled (deformed) state. Experimental studies investigating prickle have often used cylindrical, synthetic fibers that are of uniform diameter and composition along their length.^14,15

The CBL of wool fibers is governed by the Euler-Bernoulli beam theory.¹⁶ Analytical solutions are readily obtained for the case where fibers are uniform, and cylindrical, and have been applied with these assumptions to animal and plant fibers.^10,14 However, explicit solutions are not generally available for irregular shapes, necessitating numerical approaches. For example, the effect on CBL of smooth sinusoidal variations in diameter along the length of a fiber has been investigated using finite element modeling (FEM).^17,18

Recently we presented a numerical approach¹⁹ that improved upon the accuracy of the calculations of an earlier FEM,¹⁷ which we found sometimes to give inaccurate predictions.¹⁹ Indeed, Burkhart²⁰ cautioned against automatic acceptance of FEM predictions. Our method additionally promised the ability to calculate CBLs for fibers of arbitrary variation in composition, diameter and ellipticity along the length of the fiber. We regard this as essential for studying CBL of any complex mammalian hair fiber because they are highly heterogeneous in composition and irregular in shape, and natural variation along and between fibers is significant. Approaches to understanding the impact of this biological variation may require testing a population of model fibers, which is an important consideration for the ease of automation of model design and also the calculations to solve CBL. In this paper, we describe and apply our approach using publicly available measurements of composition and diameter from segments of real wool fibers.

Methods

Equations for CBL

The CBL of a thin column or rod can be described by the Euler-Bernoulli beam theory,¹⁶ which is applicable to textile fibers. When a fiber is not subject to axial restraint it can be represented for all combinations of boundary conditions by the following fourth-ordinary differential equation (ODE)²¹ which is given in the following in non-dimensional form:

\frac{d^{2}}{d x^{2}} (ζ (x) \frac{d^{2} w}{d x^{2}}) + β \frac{d^{2} w}{d x^{2}} = 0

(1)

x where

0 \leq x \leq 1

is the normalized distance from one end of the fiber (of length L),

w \equiv w (x)

is the displacement of the fiber at each point

x,

β is the normalized CBL, and ζ is the normalized flexural rigidity:

ζ (x) \equiv E (x) I (x) / ζ_{0} ζ_{0} = E_{| x = 0} I_{| x = 0}

(2)

where E and I are functions that give respectively the elastic modulus and moment of inertia along the fiber.

The actual CBL, P (in N) can be obtained from β by:

P \equiv β ζ_{0} / L^{2}

(3)

For the purposes of modeling we assume that the end of the fiber that is anchored in the garment is clamped at the surface of the garment (at $x = 0)$ , while the other end is pinned or hinged (at $x = 1$ ); i.e. fixed into the skin and able to rotate but not slide. These assumptions are represented by the following boundary conditions:²²

Clamped end (x = 0) : w = 0 and \frac{dw}{dx} = 0

(4)

Pinned end (x = 1) : w = 0 and \frac{d^{2} w}{d x^{2}} = 0

(5)

Along-fiber variation in morphology and composition from fiber data

Along-fiber morphology data for our experiments were mined from a confocal microscopy study of short wool snippets, providing cross-section data of ellipticity, and cross-sectional areas for orthocortex and paracortex across a variety of fiber diameters ranging from 13 to 37 µm.²³ For each fiber, cross-sections were taken every 25 µm along a length of either 100 or 150 µm of fiber, capturing local along-fiber variation (Figure 1). Measurement data are available in the online repository Figshare (https://doi.org/10.6084/m9.figshare.5500873.v1).

Figure 1.

Data on along-fiber variation from confocal microscopy of wool snippets. (a) Snippet mounted for three-dimensional imaging, box indicates area of measurement in aqueous medium. (b) A longitudinal optical section about halfway through the fiber indicating the location (dashed lines) of transverse optical sections. (c) Differential staining of orthocortex and paracortex cell types allowed measurements (d) of their cross-sectional area to be made. For details of methods for confocal microscopy and staining see Harland et al.²³

The confocal analysis did not include the amount of fiber cross-section taken up by the fiber cuticle (Figure 2(a)). Cuticle in wool is a single layer which overlaps adjacent to the paracortex cell patch. The cuticle is thought to have a uniform cell thickness independent of fiber diameter. To calculate the area of the cuticle in fiber cross-sections (used to provide cuticle volume fraction, $V_{c}$ ), we used data of cuticle thicknesses adjacent to the orthocortex and paracortex cell patches measured from transmission electron microscope cross-sections of 420 fibers (nominally classed as 21 micron) from merino sheep.^24–26 The mean thickness of cuticle adjacent to orthocortex was 602 nm (standard error 24.4 nm) and that adjacent to paracortex was 753 nm (standard error 30.6 nm) (Figure 2(b)).

Figure 2.

Summary of model parameters. (a) Transmission electron micrograph (TEM) of a merino wool fiber cross-section, showing paracortex (P); orthocortex (O); and cuticle (Cu) (see Thomas et al.²⁷). Cuticle data were derived from TEM. (b) Summary of combined measurements from which model parameters were derived: area of paracortex (A_Cx,p); area of orthocortex (A_Cx,o); major cortex axis length (a); minor cortex axis length (b); mean thickness of cuticle adjacent to paracortex and orthocortex respectively. (c) Model fibers composed of snippets (5 or 7 cross-sections) joined to other snippets, e.g. 300 µm long fiber. At interfaces, data from adjoining cross-sections were averaged.

The final model fibers were created by joining snippets together end to end (Figure 2(c)). We chose to join snippets by averaging the data from the two end sections to create a join section. This prevented the problems associated with large differences in either diameter and/or cell type proportions and/or ellipticity, between snippets.

One approach for dealing with multi-component wool fiber models was presented by Liu and Bryson²⁸ who calculated the elastic modulus of the fiber as the weighted average of cuticle, orthocortex and meso/paracortex elastic modulus, while assuming the fiber to be homogeneous and of uniform diameter along its length. However, we wished to investigate how both the composition and structure of the fiber affected CBL. Liu and Bryson²⁹ presented a three-component model (orthocortex, paracortex and cuticle) in which fiber was treated as an elliptical rod containing another elliptical rod representing the cortex. In the cross-section view, the cortex ellipse was concentric with and fully contained within the ellipse that represented the outer edge of the cuticle, with the major axes of the two ellipses not necessarily collinear. The paracortex was represented by a segment of the cortical ellipse, with the defining chord being parallel to an ellipse axis. Liu and Bryson²⁹ assumed the composition and structure to be constant along the fiber.

For each measurement point along the fiber, we used a model for the cross-section that was similar to that of Liu and Bryson, with a key difference being the introduction of a second cuticle layer over the paracortex side. Data shows a segment representation is appropriate for some fibers, but for other fibers a more symmetrical representation would be more accurate. Thus, we considered two alternative cross-section models. In the first, the paracortex was represented by the segment defined by $y = d_{1}$ while in the other, the paracortex was represented by both the chord $y = d_{2}$ and its reflection in that chord (Figure 3). To ensure the same area, d₂ > d₁. In addition (for one cross-section model) we investigated the effect of a variation in paracortex to orthocortex ratio (a 50% reduction in orthocortex area versus increasing ellipticity, while maintaining cortical area at each measurement point).

Figure 3.

Two alternative models for paracortex structure in the cross-section of a fiber with bilateral cortical cell type distribution: (a) symmetric-distributed paracortex, (b) chord-distributed paracortex.

Letting $s = 1$ represent the segment paracortes model, and $s = 2$ represent the symmetric paracortex model, the area of the paracortex is given by:

A_{P} = sab (\frac{π}{2} - \frac{d}{b} \sqrt{1 - \frac{d^{2}}{b^{2}}} - \arcsin (\frac{d}{b}))

(6)

where we define a and b as the radii of the minor and major axes respectively of the cortex (obtained from

A_{P} + A_{O} = π ab

) and

d = d_{1}, d_{2}

for

s = 1, 2

defines the paracortex chord. The value of d can be obtained numerically from Equation (6). The second cuticle layer lies on the primary cuticle layer over the region defined (in polar coordinates) by

θ = [φ, π - φ]

where φ is defined by:

tan φ = \frac{bd}{a \sqrt{b^{2} - d^{2}}}

(7)

In both cross-section models, ellipses representing the outer edges of the cortex and primary cuticle layer were assumed to be concentric, with major axes collinear along the y-axis. The paracortex was assumed to be symmetrical about the y-axis with its centroid displaced from the x-axis. For both cross-section models this configuration ensured a larger second moment of area about the x-axis than the y-axis, and thus the neutral plane would run through the y-axis (major axis).

The second moment of inertia about the y-axis for the second cuticle layer can be calculated as:

I_{y, Q 2} = I_{y, φ, a + c + h, b + c + h} - I_{y, φ, a + c, b + c}

(8)

where

I_{y, φ, α, β}

is the second moment of area about the y-axis of the ellipse sector defined by

θ = [φ, π - φ]

, with minor and major axes of a and b:

I_{y, φ, α, β} = \frac{α^{3} β}{4} (\frac{π}{2} - \frac{α β tan φ}{β^{2} + α^{2} {tan}^{2} φ} - \arctan (\frac{α}{β} tan φ))

(9)

The second moment of area of the first cuticle layer is obtained by differencing the moments of area for concentric ellipses:

I_{y, Q 2} = ((a + c) 3 (b + c) - a^{3} b) \frac{π}{4}

(10)

The second moment of area for the paracortex can be calculated as:

I_{y, P} = \frac{a^{3} bs}{4} (\frac{π}{2} - \arcsin \frac{d}{b} - \frac{2 d}{3 b} (\frac{5}{2} - \frac{d^{2}}{b^{2}}) \sqrt{1 - \frac{d^{2}}{b^{2}}})

(11)

where

s = 1

and

d = d_{1}

for the chord paracortex model, and

s = 2

d = d_{2}

for the symmetric model.

The second moment of area of the orthocortex is obtained by subtracting $I_{y, P}$ from $I_{y}$ for the cortical ellipse:

I_{y, O} = \frac{a^{3} b π}{4} - I_{y, P}

(12)

The flexural rigidity for the cross-section is obtained by taking the weighted sums of the second moments of area of the components, where the weight for each component is its elastic modulus.

Mechanical properties of fiber components

Cortical regions are composed of cell remnants which are packed with bi-composite bundles (macrofibrils) of keratin intermediate filaments (KIFs) embedded in a matrix of relatively formless keratin associated proteins (KAPs). Orthocortex and paracortex differ in macrofibril organization and protein composition,³ and most notably in disulfide-bond rich KAPs.³⁰ Orthocortical cells have fewer species of high-disulfide KAPs and their macrofibrils have a helically twisted architecture, producing discrete columns which have circular profiles as visualized using transmission electron microscopy (Figure 2(a)). Paracortical cells contain a greater number of high-sulfur KAP species, the KIFs tend to be further apart, and helical twisting is of low intensity or absent. Paracortical macrofibrils are often fused together at their margins (Figure 2(a)). These combinations lead to the expectation that orthocortex and paracortex have different mechanical properties. Mesocortex, an additional cell type, is sometimes defined by macrofibrils that have highly ordered hexagonal packing of KIFs.³¹ Recent research suggests that mesocortex is not a specific cell type,^27,32 and, like others,^28,33 we assume here that paracortex and mesocortex have the same mechanical properties. Liu and Bryson²⁸ used values for cuticle and cortex elasticity derived from direct measurements from fiber cross-sections using force-volume atomic force microscopy.^34,35

The elastic modulus values used here for the different fiber components were obtained from Liu et al.,³³ probably derived from Caldwell and Bryson³⁵ and unpublished data from Caldwell, who gave mean values of 3.0, 3.3 and 3.7 GPa for cuticle, orthocortex and paracortex, respectively.

The data sources for our model are summarized in Table 1.

Table 1.

Data sources for the model

Property	Direct (D) or calculated (C)	Origin	Reference
Cross-section areas	D	Confocal microscopy	Harland et al.²³
Cross-section ellipticity	D	Confocal microscopy	Harland et al.²³
Cuticle cross-section area	C	Transmission electron microscopy thickness data	Harland et al.²⁵
Cortex elastic modulus	C	Atomic force microscopy	Liu et al.³⁶
Cuticle elastic modulus	D	Atomic force microscopy	Liu et al.³⁶

Solution of CBL problem

Continuous case

We presented a numerical integration and gradient search method approach that expressed Equation (1) as a system of four first-order differential equations, and used the boundary conditions of clamped/pinned fiber to determine the values of w and its first, second and third derivatives at $x = 0$ , and hence the initial conditions for the four-dimensional system.¹⁹

\begin{matrix} w (0) = 0, w^{'} (0) = 0, w^{″} (0) = - ζ (0), \\ w^{″'} (0) = ζ (0) + ζ^{'} (0) \end{matrix}

(13)

This ODE system was solved numerically and β, the only unknown, was found using a Levenberg-Marquard gradient search.¹⁹

Discrete data case

While in an ideal case we would have data available for a practically continuous function of x, for this study we mined data based on cross-sectional measurements made at 25 µm intervals, and we approximated the wool fibers in linear piecewise fashion. This presented an obstacle to using the fourth-order formalism of Equation (1), since it involves the first and second derivatives of the flexural rigidity, ζ. For our discrete representation, however, ζ is not differentiable. Thus, we took the alternative approach of integrating Equation (1) twice and using the boundary conditions to obtain

\begin{matrix} ζ (x) \frac{d^{2} w}{d x^{2}} + β w + a (1 - x) = 0, a = ζ (0) (\frac{d^{2} w}{d x^{2}})_{| x = 0} \end{matrix}

(14)

This was converted into a two-dimensional ODE system by making the definition $u \equiv \frac{dw}{dx}$ and substituting into Equation (14) to give

\begin{matrix} \frac{dw}{dx} = u, \frac{du}{dx} = \frac{a (1 - x) - β w}{ζ} \end{matrix}

(15)

and solved using numerical integration and a gradient search for β, along the lines described previously. We chose to carry this out in the numerical computation package MATLAB® and used the numerical integration routine ode113 with stringent error tolerances, and solved for β using fminsearch which implements the Levenberg-Marquard gradient search algorithm. Alternative software such as Octave would be equally effective.

Results

Model fibers can be varied significantly from the data. As an illustration of the power of the approach we consider one model fiber that is 500 µm long and is made of up five 100 µm snippets, each snippet composed of five cross-sections from the confocal data set. The resulting fiber (Figure 4(a)) presents an example of normal variation observed in wool fibers and allows us then to explore how variation in combinations of structural parameters can influence single-fiber buckling. The assemblies of the fiber cross-sections for the two paracortex distribution models, and the model with the ratio of orthocortex to paracortex (ortho:para) changed are shown in Figures 4(b) and (c), respectively. In a fourth example model, we took the symmetric paracortex model and increased the ellipticity of all segments by 20%.

Figure 4.

Assembly of example alternative fiber models and varied orthocortex to paracortex (ortho:para) ratio: (a) symmetric paracortex, (b) segment paracortex, (c) symmetric, increased paracortex.

In this set, the diameter of the individual segments ranged from 14 to 19 microns, while the ortho:para ratio ranged between 68:32 and 45:55, with a mean ortho:para ratio of 56.5:43.5. This overall ortho:para ratio was the one used in the CBL calculations for the following three model fibers: symmetric paracortex, segmented paracortex and symmetric paracortex with increased ellipticity. It was reversed to an ortho:para ratio of 43.5:56.5 in the CBL calculations for the symmetric increased paracortex model fiber.

Visualization of the fibers by plotting their cross-sectional areas in interpolated form along the fiber was used for experimental discussions between wool scientists and modelers to ensure clear communication of how models and experimental variation matched with wool fiber knowledge (Figure 5).

Figure 5.

Composition of fiber along its length: (a) symmetric paracortex with increased ellipticity, (b) symmetric, increased paracortex.

Figure 6.

Flexural rigidity and flexural rigidity multiplied by ellipticity (ellipticity-adjusted) plotted against fiber cross-sectional area (CSA). Flexural rigidity showed a 1.66 power relationship with CSA, instead of the expected power of 2. The ellipticity-adjusted flexural rigidity showed a 2.04 power relationship with CSA, indicating that the reduced apparent power is due to largely due to the effect of ellipticity, and that variation in fiber composition along its length had negligible contribution.

The data on fiber ellipticity, and cross-sectional area (CSA) for orthocortex and para-/mesocortex that were used to construct the simulated fibers are shown in Table 2 for sequential positions along the fiber from 0 to 500 µm in 25 µm steps, along with the CSA for the cuticle 1 and 2 layers, total fiber CSA, and the calculated flexural rigidity for the symmetric paracortex fiber.

Table 2.

Measured data for fiber ellipticity, and cross-sectional area (CSA) for orthocortex and para-/mesocortex shown for sequential positions along the fiber from 0 to 500 µm in 25 µm steps, along with the model calculations for CSA for the cuticle 1 and 2 layers, total fiber CSA, and the calculated flexural rigidity for the symmetric paracortex fiber

Measured data				Model constructed properties
Position	Aspect	Ortho-	Para-meso	Cuticle 1	Cuticle 2	Fiber	Flexural rigidity
(µm)	(ratio)	CSA (µm²)	CSA (µm²)	CSA (µm²)	CSA (µm²)	CSA (µm²)	(% of max)
0	1.05	80.9	73.1	27.6	2.6	184.3	46
25	1.17	82.9	70.0	27.6	2.5	183.0	41
50	1.07	77.5	48.8	25.1	2.2	153.6	31
75	1.07	82.0	80.3	28.3	2.7	193.4	49
100	1.12	94.9	59.6	27.7	2.4	184.6	43
125	1.03	97.2	57.0	27.6	2.4	184.2	46
150	1.07	104.4	48.2	27.5	2.2	182.3	43
175	1.09	96.5	52.6	27.2	2.3	178.6	41
200	1.22	90.0	68.8	28.2	2.5	189.4	42
225	1.43	86.5	79.2	29.1	2.6	197.3	39
250	1.47	93.7	69.0	28.9	2.4	194.0	37
275	1.47	100.0	66.7	29.2	2.4	198.3	38
300	1.30	94.7	81.6	29.7	2.6	208.7	48
325	1.29	118.8	81.8	31.6	2.7	234.8	61
350	1.35	111.4	67.7	30.0	2.4	211.5	47
375	1.47	116.9	75.6	31.3	2.5	226.2	50
400	1.30	133.2	90.5	33.3	2.8	259.8	74
425	1.39	165.6	105.3	36.7	3.0	310.6	100
450	1.41	154.7	115.1	36.7	3.1	309.5	98
475	1.31	124.8	117.2	34.6	3.1	279.7	86
500	1.38	125.4	122.1	35.1	3.2	285.8	86

Note: Flexural rigidity has been normalized to the maximum value.

In our previous paper¹⁹ we calculated the following dimensionless inconsistency score, γ, to use as an indicator of how well the solutions satisfied both Equation (14) and the boundary conditions

\begin{matrix} γ = \sqrt{\frac{\int_{0}^{1} (ζ (x) w^{″} + β w - a (1 - x)) 2 d x}{β^{2} \int_{0}^{1} w^{2} d x}} + \sqrt{\frac{w (1) 2 + w^{″} (1) 2}{w^{″} (0) 2 + w^{″'} (0) 2}}, a = ζ (0) w''_{| x = 0} \end{matrix}

(16)

Consistent predictions for β and w should yield a value near zero for γ.

The consistency score to check the validity of the results was low for all cases (Table 3), indicating accuracy of the solution. The critical buckling profiles for the symmetric paracortex fiber are shown in Figure 7. The boundary conditions ensure that the displacement of the fiber is zero at both ends. The magnitude of the bending moment is equal to the flexural rigidity multiplied by the second derivative of the displacement.³⁷ Thus, the second boundary condition for the pinned end constrains the bending moment to be zero at $x = 500 μ m$ . Equally, the second boundary condition for the clamped end (that the first derivative of displacement is zero) results in a non-zero bending moment at $x = 0 μ m$ : since the fiber increases its rate of displacement as x increases from 0, $dw / dx$ must also be non-zero as x increases from 0, and hence $d^{2} w / d x^{2}$ must also be non-zero since it is the rate of change of $dw / dx$ .

Figure 7.

Predicted critical buckling profiles for the fiber with symmetric paracortex configuration. Graphs for the other configurations were qualitatively similar and are not shown.

Table 3.

Critical buckling load (CBL) predicted for the four fibers analyzed

	Normalized CBL	Actual CBL	Consistency score
Fiber	(dimensionless)	(mN)	(dimensionless)
Symmetric paracortex	9.34	0.69	1e-7
Segment paracortex	9.38	0.71	2e-8
Symmetric increased paracortex	9.32	0.72	1e-7
Symmetric increased ellipticity	9.36	0.59	5e-9

The significant increase in flexural rigidity in the x = 400 $μ m$ to x = 500 $μ m$ region reflects the increase in CSA of fiber cross-sections, as can be seen from Table 2 and Figures 4 and 5. For a perfectly cylindrical fiber, the flexural rigidity would increase with the square of CSA (fourth power of radius), all other factors being constant. However, from our working shown previously (and in particular see Equation (12)), the second moment of area (and thus flexural rigidity) will decrease inversely with the ratio of ellipticity expressed as the ratio of the major to minor axes. In addition, variation in composition along the fiber will contribute to changes in flexural rigidity. Comparing trends indicated that the power relationship between increase in flexural rigidity and CSA was 1.66 instead of 2, while the power relationship between flexural rigidity multiplied by ellipticity was 2.04 (Figure 6). For this data set, ellipticity increased as CSA increased along the length of the fiber, countering and masking the contribution of increased CSA to a moderate extent. Multiplying flexural rigidity by ellipticity removed this offset, and reveals that variation in fiber composition along the length of the fiber was much less significant than CSA or flexural rigidity.

The CBL for the symmetric model was 0.69 mN. Despite the symmetric paracortex model having slightly more secondary cuticle than the chord model (with both having the same amount of paracortex, orthocortex and primary cuticle layer), the chord model had a slightly (3%) greater CBL, at 0.71 mN. The difference between the models is probably due to the greater elastic modulus of the paracortex relative to the orthocortex, combined with the chord model being on average distributed further from the y-axis than the symmetric model. Decreasing the ratio of ortho:para in the symmetric model had a slightly (4%) greater effect, increasing the CBL to 0.72 mN (Table 3). Increasing ellipticity in the symmetric model by 20% decreased the CBL by 15% to 0.59 mN.

Discussion

We used our fiber model approach based on fiber cross-sections and data from historical studies of fiber structure and mechanics (Table 1) to calculate how CBL is affected by along-fiber morphological variation. This modular approach provided considerable power in our ability to create alternative fiber profiles in a controlled way from a basis in natural morphological variation, which was amenable to an experimental statistical approach.

What the approach can be used to do

The model fiber used in this calculation was assembled from data from a set of short lengths of real merino fibers that both varied in diameter and proportion of orthocortex and paracortex. Our approach allows exploration of one or more morphological parameters in fiber models containing natural variation. This could be a universal change to all parts of a fiber, as we illustrate here with our example fiber for cell type distribution, cell type ratio and fiber ellipticity. However, specific localized phenomena such as a narrow point/defect, or a single part of a fiber that differs, could easily be investigated within the context of many fibers that otherwise contain natural variation. It is also possible to ask questions that go beyond natural, such as, “How much does a property, e.g. cell type proportion, have to change before it is statistically effective over biological variation?”

We can use the approach to predict which aspects of single fiber morphology can be simplified/assumed in further modeling efforts. A key example is the distribution pattern of orthocortical and paracortical cells types across the fiber. Fibers from merino sheep, as for other low-diameter wool fibers, are generally described as bilateral,³¹ but it is unknown how small a change in distribution pattern is required before mechanical changes become evident. We can see examples of symmetric and chord versions of bilateral distribution in real fibers (Figure 8). This is important when considering future work on cell type distribution patterns found in other wools and mammalian hairs, including multiple patches of paracortex.³² We may ask, “Is it worth modeling details of cell type patch shape, or do all fibers with bilateral distribution act the same?” With the example fiber presented we find a 3% difference in bending stiffness introduced by changing the shape of the bilateral distribution. Our intention here is not to investigate in depth the results from a biological perspective, but we hope this illustrates how the modeling approach can be used as a predictive tool for decision making in future research. How much of a difference is important will depend on the purpose of a model. For example, in the context of prickle, a difference in CBL amounting to only a few percentage points could be significant from a physical perspective if the CBL of a fiber is near the sensitivity threshold (which could vary between individuals and for receptor type).

Figure 8.

Distributions of paracortical cells in wool fibers and illustrating variation in shape profile and diameter. (a) Symmetric distribution of paracortex in a fiber with an elliptical profile. (b) Chord distribution of paracortex in a circular profile fiber. Figures modified from work originally presented in Plowman et al.³²

Significant gaps in knowledge revealed

Building a model tests our understanding of the biology that underpins a system or process because we must include enough detail of all sub-components and variation within them (often where parts have been scientifically investigated in isolation). Solutions from models rely on interactions between sub-components. When developing mathematical models of biological systems, those sub-components where data is missing (relying on an assumption) or is pulled from studies where biological variation has not been investigated become particularly salient; let us call these data paucity warnings. Here we identified three data paucity warnings:

Cell type elastic modulus is a key parameter for which few studies have been carried out. The studies have assumed that paracortex has a higher elastic modulus than orthocortex based on the work of Caldwell and Bryson.^33,38 On this basis, switching from a high orthocortex fiber to one high in paracortex appears to have little effect on the CBL (Table 3). However, this parameter will affect all others, including the effects of cell distribution and numbers of layers of cuticle.

Cell type distribution across and along fibers is not well described and is largely unknown other than in coarse terms (e.g., bilaterial vs annular^31,32) between breeds. Our results suggest that small changes in the shape of the distribution can have noticeable effects (e.g., our symmetric and segment models would both be counted as bilateral).

Our results on along-fiber variation came from a single data set³⁹ from fibers from a single breed of sheep, which had originally been selected to understand fiber curvature.²³ Beyond this one data set, we do not have a clear understanding of what biological variation exists within fibers, and between fibers and breeds.

We conclude that publicly available data sets on normal biological variation in along-fiber morphology using confocal microscopy, serial sectioning or other similar methods will also enhance the value of the modeling approach described here to advance our understanding of how microstructure scales up to generate macro-scale mechanical function.

Footnotes

Acknowledgements

We gratefully acknowledge discussions with Surinder Tandon, Andy Cooper and Anita Grosvenor during this work. We acknowledge important data collected as parts of projects funded by Australian wool growers and the Australian government through Australian Wool Innovation Limited (AWI) and and also by Kao Corporation, Japan.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received the following financial support for the research, authorship, and/or publication of this article: AgResearch Smart and Sustainable Biomaterials Programme (New Zealand Government Ministry of Business Innovation & Employment, Strategic Science Investment Fund).

ORCID iD

Jeffrey E Plowman

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