A Mechanochemical Model of Striae Distensae ⇑ Stephen J

Mathematical Biosciences 240 (2012) 141–147

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Mathematical Biosciences

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A mechanochemical model of striae distensae ⇑ Stephen J. Gilmore a, Benjamin L. Vaughan Jr. b, , Anotida Madzvamuse c, Philip K. Maini b,d a Dermatology Research Centre, University of Queensland, School of Medicine, Princess Alexandra Hospital, Brisbane, Australia b Centre for Mathematical Biology, Mathematical Institute, University of Oxford, UK c Department of Mathematics, University of Sussex, UK d Oxford Centre for Integrative Systems Biology, Department of Biochemistry, University of Oxford, UK article info abstract

Article history: Striae distensae, otherwise known as stretch marks, are common skin lesions found in a variety of clinical Received 6 October 2011 settings. They occur frequently during adolescence or pregnancy where there is rapid tissue expansion Received in revised form 28 June 2012 and in clinical situations associated with corticosteroid excess. Heralding their onset is the appearance Accepted 29 June 2012 of parallel inflammatory streaks aligned perpendicular to the direction of skin tension. Despite a consid- Available online 14 July 2012 erable amount of investigative research, the pathogenesis of striae remains obscure. The interpretation of histologic samples – the major investigative tool – demonstrates an association between dermal lympho- Keywords: cytic inflammation, elastolysis, and a scarring response. Yet the primary causal factor in their aetiology is Stretch marks mechanical; either skin stretching due to underlying tissue expansion or, less frequently, a compromised Mathematical modelling Numerical simulation dermis affected by normal loads. In this paper, we investigate the pathogenesis of striae by addressing the coupling between mechanical forces and dermal pathology. We develop a mathematical model that incorporates the mechanical properties of cutaneous fibroblasts and dermal extracellular matrix. By using linear stability analysis and numerical simulations of our governing nonlinear equations, we show that this quantitative approach may provide a realistic framework that may account for the initiating events. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction served in monozygotic twins [9]. Much emphasis has been placed on the effects of skin stretching in the pathogenesis of striae [10] Striae distensae, commonly known as stretch marks, are benign since the lesions are found to be aligned perpendicular to the direc- skin lesions associated with considerable cosmetic morbidity. Her- tion of skin tension. Some investigators have suggested that alding their onset is the appearance of parallel inflammatory mechanical rupture of dermal components is an important initiat- streaks aligned perpendicular to the direction of skin tension. ing event. However, there is some debate about the relative impor- The evolution of striae is characterised by at least two phases: an tance of skin stretching in the aetiology of striae: one group could initial inflammatory phase known as striae rubra and a later, not find any relationship between striae and the increase in chronic phase known as striae alba [1]. Striae may occur in a wide abdominal girth among pregnant females [11] and it has been variety of clinical settings but most commonly develop initially in noted that striae are rare over the extensor surfaces of joints (re- either adolescence [2] or pregnancy [3]. Striae may also occur in gions of skin over a joint that is stretched when the joint is flexed) conditions where the dermis is abnormal: Cushing’s syndrome where the skin is subject to physiologic stretching [12]. Less con- [4], prolonged application of topical steroids [5], or Marfan’s syn- troversy exists regarding the possible role of glucocorticoids in drome [4] are examples. Finally, striae may develop in association the pathogenesis of striae. This is largely due to the known associ- with changes to body habitus such as weight loss [6], cachexia [7], ations between alterations to hormonal status observed in preg- obesity [8], or body-building. nancy, weight changes, and adolescence on one hand and the Despite their ubiquity and considerable investigation into their more obvious effects of hormonal changes observed in Cushing’s origins, the pathogenesis of striae distensae remains unknown. syndrome and topical steroid application on the other. In addition, Genetic factors are likely to be important since striae have been ob- the catabolic effect of both adrenocorticotropic hormone (ACTH) and cortisol are well known. These hormones may modulate fibroblast activity directly leading to reduced mucopolysaccharide ⇑ Corresponding author. Present address: Department of Mathematical Sciences, secretion, possible changes to elastic fibres, and reduced collagen University of Cincinnati, Cincinnati, Ohio, USA. via either reduced production or increased collagenase secretion E-mail addresses: [email protected] (S.J. Gilmore), [email protected] (B.L. Vaughan Jr.), [email protected] (A. Madzvamuse), maini@maths. or both. Finally, increased levels of steroid hormones and their ox.ac.uk (P.K. Maini). metabolites have been found in patients exhibiting striae [13].

From the pathologic perspective, the earliest changes are sub- stability analysis and investigate mode selection. Section 4 de- clinical and are only detectable by electron microscopy. These scribes our numerical results of the full nonlinear model. We con- changes involve mast cell degranulation (the release histamine clude this paper in Section 5 with a discussion of our results and along with other molecules from granules in the mast cell’s cyto- the implications for pathogenesis. plasm into the extracellular space) and the presence of activated macrophages in association with mid-dermal elastolysis [14]. 2. Model description While mast cell activation has been reported in association with elastolysis in actinically affected skin (pigmentary changes and Our model is based on the simple assumption that in the pre- elastolysis secondary to chronic UV exposure), anetoderma (cir- clinical phase of striae development there exists spatial inhomoge- cumscribed areas of slack skin due to loss of dermal elastin) has neity in the density of one or more constituents of the dermis. We been reported in skin affected by mastocytosis. These findings sup- thus focus on its two most important components: fibroblasts and port the concept that the release of enzymes, possibly elastases the extracellular matrix (ECM). Fibroblasts are spindle-shaped cells (enzymes that are capable of degrading the elastin molecule within embedded within the ECM; they play an essential role in dermal the dermis), from mast cells plays a very early and important role homeostasis, wound healing, and recently have been shown to ex- in the pathogenesis of striae [14]. When lesions initially become press a Hox code that accounts for the regional specificity of the visible, collagen bundles begin to show structural alterations, epidermal phenotype [25]. Proteoglycans and mucopolysaccha- fibroblasts become prominent, and mast cells are absent [14]. rides constitute the ECM. While collagen gives the skin its tensile On light microscopy, the earliest changes in striae rubra involve strength, the proteoglycans and mucopolysaccharides are gel-like dermal oedema, perivascular lymphocyte cuffing (the appearance substances that trap water, thus facilitating molecular diffusion of lymphocytes in the surrounding small blood vessels within the and cell transport. dermis), and an associated increase in the glycosaminoglycan con- Since striae are frequently observed to align in a direction per- tent of the dermis [15]. Examination of early lesions shows fine pendicular to the direction of skin tension, they are found to devel- elastic fibres predominating throughout the dermis in association op as parallel inflammatory streaks in the skin. Hence, without loss with thick and tortuous fibres toward the periphery [16]. An inte- of generality, we can reduce a potential two-dimensional problem gral component of elastic fibres, fibrillin microfibrils, are found to to a one-dimensional problem since the patterning is translation- be reduced in striae rubra [17]. In contrast to early inflammatory ally invariant in the direction perpendicular to the direction of skin lesions, striae alba are characterised by epidermal atrophy, loss tension. We thus consider a one-dimensional model, definedon a of appendages, and a densely packed region of thin eosinophilic periodic domain, similar to the model developed by Oster et al. collagen bundles aligned horizontal to the surface. Later stage le- [26] and modified by Vaughan, Jr. et al. [27] where the skin is trea- sions are thus indistinguishable, from the perspective of light ted as a visco-elastic medium. The model derived below and the microscopy, from a dermal scar. model in Vaughan, Jr. et al. [27] generalise the model discussed A number of studies suggest that fibroblasts play a key role in in Oster et al. [26] and Murray [21] by keeping the inertial terms the pathogenesis of striae. Compared with normal fibroblasts, and writing the governing equations in the material frame of refer- expression of fibronectin and both type I and III procollagen were ence. This derivation keeps the nonlinear terms that arise from found to be significantly reduced in fibroblasts from striae, sug- transformation from spatial to material frames of reference in gesting that there exist fundamental aberrations of fibroblast the spatial derivatives, which increases the range of parameters metabolism in striae distensae [18]. From a bio-mechanical per- where the solution evolves to a bounded steady state [27]. spective, ex-vivo fibroblasts from patients with early striae disten- The model consists of conservation equations for the fibroblast sae were found to exhibit high levels of alpha-smooth muscle actin cell density, ^c, and the ECM density, q^, in a deformed frame of refer- and were able to generate higher contractile forces in comparison ence coupled through a force balance equation governing the with fibroblasts from later stage striae [19]. mechanical interaction of the fibroblasts with the ECM, which is de- Taken together, the foregoing discussion suggests at least two fined in the undeformed (reference) frame. In this formulation, we major factors play important roles in the aetiology of striae disten- will transform the equations governing the cell and ECM densities sae: mechanical stretching of the skin and pre-existing dermal from the deformed frame to the reference frame, which will simplify pathology. The relative effects of these factors are unknown. For the methods used to solve this model numerically. Note that vari- example, it is unknown to what extent steroid hormones in preg- ables with a hat denote a variable in the deformed frame of refer- nancy or adolescence may pre-condition the skin such that it may ence and variables without a hat are defined in the reference frame. be predisposed to developing striae when subjected to stretching. The stress tensor, r, satisfies the force balance equation in the In this paper we develop a mathematical model that attempts reference frame, to capture early changes in striae distensae development. We are 2 encouraged by the success of similar models used to describe @ u @ q0 ¼ ðÞþr þ sðÞ^c; q^ q0F; ð1Þ wound healing [20] and earlier contact guidance models of straie @t2 @x by Murray [21] and Hariharan [22] and we are motivated by the where u is the material displacement, sðÞ^c; q^ is the traction due to ability of mathematical models to incorporate the postulated rele- cell-ECM interactions and depends on the cell and ECM densities vant elements of early striae development including fibroblast con- in the deformed frame of reference, q0 is the ECM density in the ref- tractility, fibroblast motility and remodelling of the extra-cellular erence frame, and F is an external body force. We follow the Oster– matrix. Our model allows us to quantify the degree of mechanical Murray–Harris model [26] and treat the ECM as a linear, isotropic, stretching (given by a single parameter) and dermal stiffness (gi- visco-elastic material. Hence, the stress tensor in one dimension is ven by an independent parameter) so that we are able to explore @2u @u the relevant contributions of skin stretching and glucocorticoid-af- r ¼ A þ B : ð2Þ fected skin [23,24]. Finally, as a model of pattern formation in the @t@x @x skin, we are able to investigate how microscopic events may lead Here, A ¼ l1 þ l2, where l1 and l2 are the shear and bulk viscosi- to macroscopic patterns. ties of the ECM, respectively, and The remainder of this paper is organised as follows: Section 2 ðÞ1 m describes the derivation of our model and a non-dimensionalisa- B ¼ E ; tion of our governing equations. In Section 3, we perform a linear ðÞ1 þ m ðÞ1 2m S.J. Gilmore et al. / Mathematical Biosciences 240 (2012) 141–147 143 where E is the Young’s modulus and m is the Poisson’s ratio of @c @ 1 @ c ¼ D : ð10Þ the ECM. It is assumed that the ECM material is attached to @t @x 1 þ @u=@x @x 1 þ @u=@x the subcutaneous fascia by fibrous bands that resist the lateral displacement of the overlying dermis. We model this attach- Next, we non-dimensionalise Eqs. (10) and (7) using the ment as a linear spring and the body force in the force balance relations: equation, (1),is x t c u x ¼ ; t ¼ ; c ¼ ; u ¼ ; F ¼su; ð3Þ L T c0 L 1 A B where s is a positive spring constant. k ¼ kc2; a ¼ ; a ¼ ; b ¼ ; ð11Þ 0 sT2 L2Ts L2s We model the traction exerted by the cell-matrix interactions in q0 q0 DT s c b the deformed frame of reference as d ¼ ; s ¼ 0 0 ; b ¼ 0 ; ! L2 L2s L2 ^c @2q^ sðÞ¼^c; q^ s q^ þ b ; ð4Þ where L is the characteristic length scale and T is the characteristic 0 1 þ k^c2 0 @^x2 time scale. The non-dimensionalised equations, after dropping the stars, are: where s0 is the traction strength, k is a constant that accounts, in a phenomenological way, for contact inhibition, b0 is the strength of @c @ 1 @ c ¼ d ð12Þ the long-range traction that arises from the fibrous nature of the @t @x 1 þ @u=@x @x 1 þ @u=@x ECM, which can extend the range of the traction force exerted by the fibroblasts, and @=@^x is the spatial derivative in the deformed and frame of reference. @2u @3u @2u a ¼ a þb We assume that there is no production or degradation of the @t2 @x2@t @x2 "# ECM, so we can relate the ECM density in the deformed frame to @ c @ 1 @ 1 þs 1þb u: ð13Þ the ECM density in the reference frame using the relation @x ðÞ1þ@u=@x 2 þkc2 @x 1þ@u=@x @x 1þ@u=@x q0 ¼ Jq^, where J ¼ 1 þ @u=@x is the Jacobian of the deformation Note that we generalise the Oster–Murray–Harris model by gradient and q0 is a constant. Likewise, we transform the cell den- 2 2 sity from the deformed frame to the reference frame using the keeping the inertial term, a@ u=@t . Even though the a term is same relation, c ¼ J^c. Here, c is not assumed to be constant since small for physically relevant parameters, the acceleration of the 2 2 we will allow for the movement of cells by diffusion. medium, @ u=@t , can be large in the numerical simulations and We transform the derivatives in the long-range traction force it would not be appropriate to neglect this term for all time. into the reference frame by taking the derivative of the definition The above equations for the dimensionless fibroblast density, c, of the displacement, uxðÞ¼; t ^xxðÞ; t x, with respect to the spatial and the material displacement, u, are coupled with appropriate coordinate in the reference frame, to obtain boundary conditions. Since striae occur as parallel lines and are translationally invariant in the transverse direction, we assume @ 1 @ ¼ : ð5Þ in this paper that the solution is periodic in space and, hence, @^x 1 þ @u=@x @x we enforce periodic boundary conditions on the ends of the domain. Hence, the traction force in the reference frame is

cq0 3. Linear analysis sðÞ¼c;@u=@x s0 2 2 ðÞ1 þ @u=@x þkc @ 1 @ 1 The linearized versions of Eqs. (12) and (13) around the normal- 1 þ b0 : ð6Þ ized steady state c ¼ 1 and u ¼ 0 are @x 1 þ @u=@x @x 1 þ @u=@x ! @c @2c @3u See Appendix A for the details of the derivation of this term. ¼ d ð14Þ @t @x2 @x3 Inserting (2) and (6) into (1), the resulting force balance equation in one dimension is and "# ! @2u @ @2u @u @u @2u @3u s @2u q0 2 ¼ A þ B þ s c; sq0u ¼ 0: ð7Þ a ¼ a þ b 2 @t @x @x@t @x @x @t2 @x2@t ðÞ1 þ k 2 @x2 4 It is assumed that fibroblasts move randomly and are advected sðÞ1 k @c sb @ u þ u ¼ 0: ð15Þ with the medium. Thus the governing equation for the cell density, ðÞ1 þ k 2 @x 1 þ k @x4 ^c, in the deformed frame of reference is 2 By defining sk ¼ s=ðÞ1 þ k and bk ¼ bðÞ1 þ k and looking for solu- @^c @ @2^c þ ðÞ¼^cv D ; ð8Þ tions of the form @^t @^x @x^2 c ¼ zertþikx; ð16Þ where v ¼ @u=@t is the velocity of the medium, D is the diffusion u coefficient and ^t is time in the deformed frame. Relating the cell density in the deformed configuration, ^c, with the cell density in where z is the eigenvector, r is the linear growth rate, and k is the the initial configuration, c, using the relation c ¼ J1^c and the spatial wavenumber, we require jjA ¼ 0, for change in the spatial derivative, (5), and the change in the temporal 2 3 dk2 idk3 derivatives due to the change in frame, 6 r þ 7 6 7 A ¼ 4 5; ð17Þ @ @ @ 2 2 2 ¼ þ v ; ð9Þ iskðÞ1 k k ar þ ak r þ Bk @t @^t @^x we obtain where 144 S.J. Gilmore et al. / Mathematical Biosciences 240 (2012) 141–147 sffiffiffiffiffiffiffiffiffi 2 4 2 Bk ¼ skbkk þ ðÞb 2sk k þ 1: ð18Þ 2 1 kc ¼ : ð26Þ skbk The characteristic equation is Note that this case only arises if there is cell diffusion, d – 0. ar3 þ ðÞa þ ad k2r2 þ Bk2 þ adk4 r The third case will occur when 2 2 2 þ dk Bk þ sðÞ1 k k s 1 ad 2 k > and b 2 þ ðÞ1 k s b ad a k ¼ 0: ð19Þ 2 þ a ðÞ1 k

If we ignore inertial forces (a = 0) and cell diffusion (d = 0), we re- ¼ 4ðÞskbk þ daðÞþ ad : ð27Þ cover the characteristic equation for the basic Oster–Murray–Harris The critical wavemode is model. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we take the ratio sk=b as the bifurcation parameter, the real 2 kc ¼ skbk þ daðÞþ ad : ð28Þ part of r can become positive in three ways as sk=b. Applying the Routh–Horwitz conditions to (19), Note that when k P 1 þ 2a=ad, this case never produces a bifurca- Bk2 þ adk4 < 0; tion as sk is increased. If k – 1, diffusion causes the system to be able to reach a bifur- 2 2 d – 0 and Bk þ skðÞ1 k k < 0; cation for different values of critical sk, which depend on the ð20Þ parameters b; b; a; a, and d. When k ¼ 1, the bifurcation conditions or in the second case, (25), are reached first for all values of b; b; a; a, 2 2 2 2 2 4 adk Bk þ skðÞ1 k k P ðÞa þ ad k Bk þ adk and d and are identical to the bifurcation conditions for the original Oster–Murray–Harris model. are the conditions necessary for linear instability. The fixed, dimensional parameter values we use for this system The first case, Bk2 þ adk4 < 0, will occur when are [28–30,21,31,32] sk 1 2 2 > ; and ½b 2s ¼ 4ðÞs b þ ad : ð21Þ 5 9 cm 2 2 b 2 k k k A ¼ 10 poise; D0 ¼ 10 ; b ¼ 10 cm ; s ð29Þ 1 cell g The critical wavemode in this case is s ¼ 102 ; c ¼ 104 ; q ¼ 101 : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 0 cm3 0 cm3 2 1 k ¼ : ð22Þ Using the length scale L ¼ 1 cm and the time scale T ¼ 10 s, these c s b þ ad k k correspond to the non-dimensional parameters The presence of diffusion has a stabilizing effect in this case by a ¼ 104; a ¼ 103; d ¼ 108; b ¼ 102: ð30Þ increasing the critical value of sk required for a bifurcation from qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1 We set k = 1 and the two parameters, b and s, are chosen so that the s ¼ þ b þ bðÞ2b þ b ; ð23Þ k 2 2 uniform steady state is linearly unstable to small random perturbations. Fig. 1 shows the dispersion relation for the root of (19) with a in the original Oster–Murray–Harris model to positive real part for various values of s with b ¼ 1:012 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b 1 (B ¼ 10:12 dynes=cm ). A bifurcation occurs for s > 2:469 5 4 sk ¼ þ b þ bðÞþ2b þ b 4ad : ð24Þ (s > 2:469 10 dynes cm =cell mg) where only one root has a 2 2 0 positive real part for k2 – 0 and the other two have negative real The second case, d – 0 and Bk2 ðÞ1 k k2 < 0, will occur parts for k2 – 0. þ sk pffiffiffi when For k2 ¼ 0, the three roots are r ¼ 0; i= a. The r = 0 root corresponds to a uniform increase/decrease in the cell density. Since s 1 k > and ½b ðÞ1 þ k s 2 ¼ 4s b : ð25Þ we can scale any perturbations in the total cell density out, we b 1 þ k k k k pffiffiffi can neglect this root. The r ¼i= a roots correspond to oscilla- The critical wavemode is tory translation of the medium. These oscillations do not affect

Fig. 1. Dispersion relation as the bifurcation parameter s is increased. The other two roots of the characteristic Eq. (19), r2 and r3, are not shown and Rfgr2; r3 6 0. The roots are real except for a narrow region (shown in the inset), 0 6 k2 6 2 104 for s ¼ 2:5933, where the roots are complex with a negative real part. For k2 ¼ 0, the root is purely complex. S.J. Gilmore et al. / Mathematical Biosciences 240 (2012) 141–147 145 the cell/ECM densities and do not contribute to the nonlinear in the displacement and cell/ECM densities at 45 and 90 days. Here, dynamics of the system. the maximum cell density has increased by a total 2% after 90 days. For k2 – 0, the root with a positive real part has a zero imagi- Any appreciable effect due to this slow growth will occur over a nary part except for a thin region, 0 < k2 < 2 104. This narrow long time frame and since we are interested in the onset of pat- range of wavemodes are not admissible unless the domain is of terns that can become precursors of striae distensae, we will focus sufficient length and are not admissible for biologically relevant on the initial development of patterning. domain sizes in this paper. 5. Discussion 4. Numerical results We have described in detail a mathematical model proposed to We numerically solve Eqs. (12) and (13) using finite differences capture the earliest events in the pathogenesis of striae distensae. in space and the backward Euler method in time. We take the do- Motivation for the model is twofold: first, we are interested in main to be periodic with length 2pcm discretised using a mesh exploring the relative contributions of both skin stretching and with 300 grid points and a time step of Dt ¼ 101. The uniform corticosteroid effects in the pathogenesis of striae; and second, steady state solution for u and c is perturbed using a uniform ran- we are interested in the process of pattern formation in striae. dom function with mean zero and the amplitude of the perturba- Since stretch marks are frequently observed to align perpendicular tions are 102. The numerical solutions are independent of grid to the direction of skin tension, it is likely that the effects of tissue size for grids that are sufficiently refined to resolve the significant forces play an important role in the genesis of striae, and in deter- spatial frequencies in the problem and are independent of the ex- mining the patterns that form. These considerations naturally lead act random initial condition subject to a phase shift. to a mechanochemical type model. From a biological point of view, we suggest the precursors of In this paper, we have reformulated Murray’s mechanochemical clinically evident striae may appear as regions of ECM density that modelling approach [21] by stating the equations in a common are below the uniform steady state value (u ¼ 0; q ¼ c ¼ 1). Fig. 2 frame of reference. In the limit of small displacements this reduces shows the dilation and the ECM density after four days for to the original model, but crucially, the model is now applicable 2 b ¼ 1:012 (B ¼ 10:12 dynes=cm ) and s ¼ 2:593 (s0 ¼ 2:593 over a much wider range of parameter space and boundary condi- 105 dynes cm4=cell mg). In the ECM density, we can see the tions [27]. The work by Hariharan [22] (under the supervision of development after four days of regions where there is a significant one of the authors) lists some preliminary results on this problem, decrease in density (min q 74%q0). This corresponds to a but these are significantly extended. 2:6 102 g=cm3 decrease in ECM density at the minima. The crit- We have shown that an intuitive model incorporating the den- ical wavemode is kc ¼ 3 and this corresponds to an interlesional sity of fibroblasts, the density of extracellular matrix, and a force distance of approximately 2:09 cm. The above numerical solution balance equation that accounts for the contractile forces generated is not a steady state solution. There is a second phase of growth by fibroblasts is able to predict, for biologically realistic parame- that is due to the diffusion of fibroblasts. We can see slow growth ters, periodic solutions for ECM density. In this periodic spatial

Fig. 2. Numerical solution of Eqs. (12) and (13) after 4, 45, and 90 days for b^ ¼ 1:012 and s^ ¼ 2:593. The ECM and cell densities are in the deformed variables in the deformed frame of reference. The minima after four days represent a 26% decrease in ECM density and the interlesional distance is approximately 2:09 cm. 146 S.J. Gilmore et al. / Mathematical Biosciences 240 (2012) 141–147 pattern, the precursors of striae are postulated to appear in close the average distance found between striae that are aligned in a par- proximity to regions of ECM density at their maximal densities. allel arrangement. As discussed in Section 1, an important unresolved issue with Although we have demonstrated that increases in the contrac- regard to the pathogenesis of striae distensae is the relative contri- tile forces exhibited by fibroblasts or increases to skin extensibility butions of skin stretching on one hand, and endocrine factors on may lead to dermal inhomogeneity of ECM density, we have been the other. While some investigators have suggested that striae unable to provide a definite link between these changes and the may simply result from tissue rupture due to mechanical loads, mast cell degranulation that is known to be an early event in the endocrine changes are present in association with many of the pathogenesis of striae. However, it is known that increases in clinical situations in which striae are encountered [12]. In our GAG (glycosoaminoglycan) density is a very early finding in striae model, we are able to quantify the degree of stretch imposed on rubra [15], and it is unclear at present whether these changes pre- the skin by our parameter s0 (which is a measure of the contractile date, coincide with, or follow the mast cell associated elastolysis. force exerted by fibroblasts on the surrounding extracellular ma- Our model, in predicting periodicity in ECM density along the axis trix) since the fibroblasts respond to external mechanical force of skin tension, adds weight to the hypothesis that local increases by opposing that force. Furthermore, we are able to characterize in GAG density (since the GAG is part of the ECM) may precede the skin stiffness by adjusting B, a parameter proportional to the mast cell degranulation. Since fibroblasts in early striae are known Young’s modulus. Since it is recognised that corticosteroid expo- to exhibit aberrant gene expression profiles, one possible patho- sure may increase skin extensibility [23,24], we can model the ef- genic mechanism is apparent: secreted fibroblast products may ex- fects of corticosteroid excess by reducing B. Our results suggest ist locally in higher concentrations where the ECM density is that the uniformity of cutaneous extracellular matrix may be ren- higher and thus lead to spatially dependent mast cell recruitment dered unstable by either increasing s0 or decreasing B. We obtain and subsequent elastolysis. solutions that are qualitatively identical via two distinct mecha- In summary, we present a conceptually simple but mathemati- nisms, and these distinct mechanisms correspond to either stretch- cally complex model that attempts to account for the earliest ing of the skin or increasing the extensibility of skin. From the events in the pathogenesis of striae distensae. We suggest that clinical perspective, these results suggest that either factor alone the results are sufficiently robust enough to provide evidence for can induce striae. For example, in adolescence striae may develop the existence of an important symmetry breaking mechanism that simply as a result of skin stretching (increasing s0); abnormal is able to distinguish between two fundamentally different and endocrine factors and changes to skin extensibility are not needed. clinically relevant causes. Conversely, the application of potent topical steroids to the skin results in a reduction in the mucopolysaccaride content [33] – and Acknowledgements this may account for the increases in skin extensibility reported [24] – with the subsequent development of striae under normal The authors thank Dr. Cameron Hall for his helpful insights. skin tension. Indeed, the skin is not subject to increased stretch B.L.V. was partially supported by St. John’s College, Oxford. P.K.M. in a number of clinical situations where striae distensae arise: ca- was partially supported by a Royal Society Wolfson Research Merit chexia (a wasting syndrome associated with malignancy or chronic Award. infection) and other causes of severe weight loss are two examples where decreased dermal substrate and the resultant increased Appendix A. Transformation of the traction force into the extensibility of skin may be the sole aetiological factor. Interest- material frame of reference ingly, of seven patients treated with high dose intravenous corticosteroid for alopecia areata (an autoimmune disease associated with We begin with the traction force exerted by the fibroblasts on circular patches of hair loss in the scalp) [23] one patient devel- the ECM in the spatial frame of reference: oped striae distensae on the thighs within four days of the infusion. ! In this study skin extensibility was measured and was shown to in- ^c @2q^ crease within hours of the infusion, reaching a maximal extensibil- sðÞ¼^c; q^ s q^ þ b ; ðA:1Þ 0 1 þ k^c2 0 @^x2 ity at around four days. In ageing skin it has been reported that there is a decrease in extensibility [34] (corresponding to an in- where the hats designate the spatial frame of reference. We trans- crease in B in our model) and perhaps explaining why striae are form ^c and q^ from the spatial frame to the reference frame using only rarely observed to develop in mature adults. Finally, our mod- the Jacobian of the deformation gradient, J ¼ 1 þ @u=@x, to get el is not inconsistent with both increased stretch and dermal changes acting together in the genesis of striae. For example, in c q ^c ¼ and q^ ¼ 0 ; ðA:2Þ pregnancy lesions may arise due to the synergistic effect of both 1 þ ux 1 þ ux increased skin extensibility on one hand, and the increased mechanical forces acting on the skin secondary to a rapidly dis- where ux ¼ @u=@x. Substituting (A.2) into (A.1) and simplifying tending abdomen on the other. yields the equation Striae distensae in the skin often exhibit typical patterns. ! 2 Although the individual lesions are linear and usually 5–10 cm cq0 @ 1 s0 1 þ bðÞ1 þ ux : ðA:3Þ 2 2 ^2 long, multiple lesions are the norm and are always aligned perpen- ðÞ1 þ ux þ kc @x 1 þ ux dicular to the axis of skin tension. Striae develop on the back of adolescent males as parallel streaks perpendicular to the direction Next, we convert the spatial derivatives to the reference frame of vertical growth. Conversely, striae on the breasts in females of- using ten have a radial pattern indicating that the lines of tension in the @ 1 @ skin in an enlarging breast are circumferential. Although our model ¼ ; ðA:4Þ ^ is one-dimensional, we are able to predict the existence of periodic @x 1 þ ux @x solutions that arise parallel to the direction of tension, consistent to obtain the traction force in the reference frame: with the clinical observations noted above. We have shown the @u cq0 @ 1 @ 1 dimensional value for the wavelength of our solutions can be ad- s c; ¼ s0 1þb : ðA:5Þ @x 2 2 @x 1 u @x 1 u justed to approximate 1 cm. This result is in good agreement with ðÞ1þux þkc þ x þ x S.J. Gilmore et al. / Mathematical Biosciences 240 (2012) 141–147 147

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