# Mathematical modelling of contact dermatitis from nickel and chromium

## Abstract

Dermal exposure to metal allergens can lead to irritant and allergic contact dermatitis (ACD). In this paper we present a mathematical model of the absorption of metal ions, hexavalent chromium and nickel, into the viable epidermis and compare the localised irritant and T-lymphocyte (T-cell) mediated immune responses. The model accounts for the spatial-temporal variation of skin health, extra and intracellular allergen concentrations, innate immune cells, T-cells, cytokine signalling and lymph node activity up to about 6 days after contact with these metals; repair processes associated with withdrawal of exposure to both metals is not considered in the current model, being assumed secondary during the initial phases of exposure. Simulations of the resulting system of PDEs are studied in one-dimension, i.e. across skin depth, and three-dimensional scenarios with the aim of comparing the responses to the two ions in the cases of first contact (no T-cells initially present) and second contact (T-cells initially present). The results show that on continuous contact, chromium ions elicit stronger skin inflammation, but for nickel, subsequent re-exposure stimulates stronger responses due to an accumulation of cytotoxic T-cell mediated responses which characterise ACD. Furthermore, the surface area of contact to these metals has little effect on the speed of response, whilst sensitivity is predicted to increase with the thickness of skin. The modelling approach is generic and should be applicable to describe contact dermatitis from a wide range of allergens.

## Keywords

Contact dermatitis Metal ions Immune response Mathematical model Numerical solution## Mathematics Subject Classification

92C50## 1 Introduction

Contact dermatitis is a common condition in the workplace. Eurostat data estimated the incidence of contact dermatitis as 5.5 cases per 1000 employees (De Craeker 2008). In the UK, annual incidence rate of 12.9 per 100,000 workers has been reported (Cherry et al. 2000). It commonly arises in occupations using soaps and cleaners, wet work, rubber chemicals and the use of Personal Protective Equipment (PPE) (in particular, disposable gloves which have an occlusive effect on the skin when worn for long periods) (HSE 2017). Exposure of the skin to metal allergens, via metal ion carriers (MIC) occurs in the construction, jewellery and paint manufacturing industries with nickel (occurring in its most active state nickel (\(\hbox {II}^{+}\))) being the most common causative agent. Other metals such as Hexavalent Chromium (CrVI) are also regarded as a risk for dermatitis although recent statistics suggest a decreased incidence of dermatitis associated with electroplating work and the reduced use of chromates in cement (HSE 2017). EPIDERM data during the period 1996–2017, shows that approximately 37% of cases of contact dermatitis were induced by an allergic response and 44% by an irritant one (the remainder was mixed or unspecified; HSE 2017).

Hexavalent chromium (CrVI) exposure results in an immediate localised irritant response to the exposed skin, caused by the reduction of Cr(VI) to Cr(III), whilst nickel causes an allergic immune response. Both metals result in a localised inflammation of the healthy skin tissue. Nickel sulphate is known to cause allergic contact dermatitis (ACD) by directly interacting with specific histidine residues in the human Toll-like receptor 4 (TLR4), which normally act as an innate immune receptor for bacterial lipopolysaccharide (LPS). In doing so it mimics pathogen-associated molecular patterns (PAMPs) activating intracellular pro-inflammatory signalling pathways. There is other evidence that nickel may bind to the major histocompatibility molecules (MHC) and as a consequence, cross linking surface receptors in T-cell receptors activate their cytotoxic responses. The development of ACD, a delayed hypersensitivity response therefore appears to require activation of the innate immune system (Schmidt and Goebeler 2015), i.e. the nonspecific defence mechanisms that are activated immediately or within hours of an antigen’s appearance in the body. It has been estimated that approximately 10–15% of the human population suffers from contact hypersensitivity to metals (Budinger and Hertl 2000), but this allergic response is considerably more common in women (10%) than men (2%) (Peltonen 1979). The adsorption and excretion of metals is controlled by genetic factors in humans and single changes in DNA nucleotide sequence of specific genes can affect uptake metabolic pathways for handling metals, accounting for individual variability to metal toxicity (Ng et al. 2015).

Exposure in the workplace is usually repetitive and hence, difficult to prevent (Kanerva et al. 2000). Whilst treatment for the condition can be as simple as removing the source, in many occupations this is not always possible and hence further understanding of the biological mechanisms, both local to the contact area and associated with the immune system, is required in order to provide better treatment protocols. In the mathematical modelling to follow we aim to get a better understanding of how nickel and chromium elicits contrasting immunological responses and, for example, how skin structure and area of exposure effects the outcome.

Human skin consists of four main regions: the stratum corneum, the viable epidermis, the dermis and the subcutaneous layer. The stratum corneum is the outermost layer and plays a key role as a barrier to the penetration of molecules. Keratinocytes occupy approximately 95% of the epidermal layer and are produced in the stratum basale, the region dividing the epidermal and dermal regions. The dermis consists of a matrix of connective tissue housing the vasculature, lymphatic system, hair follicles and sweat glands. The subcutaneous layer consists of fatty (adipose) tissue; although this layer consists of living cells, there seems to be little evidence to suggest that it experiences a significant inflammatory response during metal ion exposure, unlike the epidermis and dermis.

Whilst both the epidermal and dermal layers are exposed to the effects of the irritant metal, different host responses result from exposure to chromium and nickel. Chromium exposure leads to a localised irritant response where the immune response is relatively rapid (within minutes) and involves the action of a host of immune cells, including macrophages and neutrophils, and numerous cytokines produced both by affected keratinocytes and by immune cells (Williams and Kupper 1996). The cytokines perform a number of functions, acting as promoters and inhibitors of immunological activity and as chemoattractants recruiting more immune cells into the compromised area. For the work to be undertaken here we are primarily interested in the role of the pro-inflammatory cytokine, interleukin-1\(\alpha \) (IL-1\(\alpha \)) (released from damaged keratinocytes), based on experimental measurements of its activity in response to both chromium and nickel exposure (Curtis et al. 2007; Franks et al. 2008). The immune cells employ a range of cytotoxic processes to clear damaged keratinocytes in the irritated zones, both specific (applied directly to affected cells) and non-specific (killing all cells in the vicinity).

Nickel causes a similar localised response to chromium VI, but in addition, initiates a T-lymphocyte mediated delayed type-IV hypersensitivity reaction which occurs after previous exposure and sensitisation to the metal. High molecular-weight protein antigens (e.g. from an invading pathogen) typically stimulate humoral (i.e. that which is mediated by macromolecules found in the extracellular fluid) and innate immunity (i.e. the nonspecific defence mechanisms that are activated immediately or within hours of an antigen’s appearance in the body) when specialised antigen presenting cells such as Langerhans cells encounter these foreign antigens in the skin. They then migrate to local lymph nodes presenting the antigen to other immune cells e.g. T-cells, activating cytotoxic responses against pathogens. However, nickel, by activating these cytotoxic responses may stimulate ACD independently of foreign antigens. For approximately 2 days, the activated T-cells remain in the lymph node undergoing two to three cell divisions. The cells then disperse into the blood stream and reach the affected area of skin.

Following removal of these two metals, the skin may undergo a process of regrowth and repair, the latter mediated by macrophages and fibroblasts. Furthermore, activated T-cells remain in the body (at a low level), and will rapidly reactivate and divide on subsequent exposures to the MIC. We note that the recovery processes will not be taken into account in the modelling below.

There is an extensive literature regarding the transport of agents through skin (see Jepps et al. 2013, Naegel et al. 2013 and references therein), skin disease (see Mollee and Bracken 2007, Tanaka and Ono 2013 and references therein), skin disease (Mollee and Bracken 2007; Tanaka and Ono 2013; Zhang et al. 2015; the latter specifically focusing on dermatitis) and immunological response on a molecular level (Dominguez-Hüttinger et al. 2013; Le et al. 2015; Lorenzi et al. 2015; Palsson et al. 2013). To date there has been limited mathematical modelling of contact dermatitis in the literature. The most closely related to that of the work presented here is that of Dominguez-Hüttinger et al. (2017) who developed a dynamic multiscale model of atopic dermatitis to understand the interaction between known genetic and environmental factors affecting the disease. Using a system of three nonlinear ODES to describe the interactions between the skin barrier, immune response and the effect of environmental factors, the authors demonstrate their model exhibits bistable behaviour in predicting either eradiction or further development of the condition. In other work, Döpfer et al. (2012) used an ODE infectious disease model to demonstrate the importance of detecting and treating dermatitis lesions quickly in reducing the spread of digital dermatitis in dairy cattle. Maxwell et al. (2014) studied mathematical models of the CD8+ T-cell response in the presence of a sensitising chemical, a key pathway in the contact dermatitis response; they discuss the importance of using such mathematical models to replace animal models in future work.

Our work is the first we are aware of which takes a spatiotemporal approach to understanding the key features of acute and chronic contact dermatitis, namely sensitisation and the immune response on the whole body as well as local skin level. In Sect. 2 we present the mathematical model which provides a description of the localised skin response in the dermal and epidermal regions to insult by a toxic agent, either chromium or nickel, coupled with the effects of a time-mediated immune-system response, over the first few days of contact with the MIC. The spatio-temporal model developed allows the localised skin reaction to be compared between irritant and allergic cases and the effects of a T-lymphocyte immune-system response to be elucidated. A simplified form of the model (excluding spatial and immune response effects) has been shown to be in good agreement with in vitro experimental data (Curtis et al. 2007; Franks et al. 2008) regarding the exposure of keratinocytes to nickel and chromium. Results from that work are used to parameterise our model, allowing us to test the effect of each mechanism, local and non-local, on the localised skin response to metal exposure.

## 2 Model description

### 2.1 Model formulation

*I*), T-cell activity as the adaptive (humoral) response (density

*T*) and the cytokine/chemokine as just cytokines (density

*c*). The focus will be on the initial responses to the MIC, i.e. up to about 6–8 days; as a first approximation we will not take into account the healing processes that will be occurring, on the assumption that these processes are secondary in the initial phases of the allergic response. The model will be developed for a general spatial geometry, based on the setup depicted in Fig. 1. The ions are sourced from the MIC on the skin surface and diffuse through the dead cell layers of the stratum corneum into the living regions that are of modelling interest, namely the dermis and epidermis (the shaded grey region in Fig. 1). Although the dermis and epidermis are distinct zones, we will assume for simplicity that the modelling domain is homogeneous and that it is bounded between \(z=0\) (the location of the vasculature) and \(z=Z\) (the live skin and stratum corneum interface). We will ignore the possible contributory factor of MIC abrasion on the inflammation process. The model will be studied in Sect. 3 mainly in 1-D (depth based) and in 3-D with radially symmetry assumed (\(r\in [0,\infty ])\). The sequence of events, for which the model is intended to describe, is illustrated in Fig. 2. The scenario depicted is for the first-contact case where there are no activated T-cells present initially. Contact with the ion leads to a cascade of immune activity, firstly innate and then adaptive, whereby eventual removal of the ion carrier leads to a situation of low level adaptive immune activity in readiness for second contact. There will be a number of parameters in the model and in the interest of being systematic with their naming we define \(\beta _{{\mathrm{ab}}}\) and \(\delta _{{\mathrm{ab}}}\) to be the rate constants for the birth and death, respectively, of “

*a*” due to “

*b*”; \(\delta _{{\mathrm{a}}}\) is the natural decay rate constant of species

*a*. Details on dimensional parameter values sourced and estimated from the literature are given in Table 2. Table 1 lists the dependent variables in the model.

*n*), breakdown product of dead cells (

*p*) and extracellular fluid (fixed volume fraction \(w_0\)); the immune cells, although present, are assumed to occupy a very small fraction of space. We note that

*p*is a product of cytolysis (i.e. the dissolution of cells, especially by an external agent), whether by apoptosis or necrosis, which will be treated as distinct from the live cell and extracellular fluid. Hence, the volume fractions satisfy

*g*being the concentration of granules). The function

*H*(

*x*) is the Heaviside function, such that \(H(x) = 1\) for \(x \ge 0\) and \(H(x)=0\) for \(x < 0\); this function is a simple representation of the nonlinear activation response by immune cells to a range of cytokines (Callard et al. 1999). This term appears in several equations, ensuring that the model has a steady-state representing a healthy state and that there is no auto-immune activity in the absence of the irritant. T-cell activation is assumed to occur when \(F(A_e,c) > 1\), where

*i*and cytokines. On death, it is assumed that the entire volume of live cells become dead cell debris so it follows that

List of model variables and their interpretation

Variable | Description | Variable | Description |
---|---|---|---|

| Live cell volume fraction | | Dead cell volume fraction |

\(A_e\) | Extracellular ion conc. | \(A_N\) | Live cell ion conc. |

\(A_p\) | Dead cell ion conc. | | Cytokine concentration |

| Innate immune cell density | | Granule (cytotoxic product) concentration |

| Programmed T-cell density | | Lymph node activity |

*I*is thus

*n*. Furthermore, we assume that only activated immune cells respond to chemotactic cues. The flux \(\varvec{J_I}\) is therefore given by

*L*(boundary condition (18)). As with innate immune cells, the T-cell flux \(\varvec{J_T}\) is governed by random motion and, when activated, chemotaxis up cytokine gradients, hence

*L*(

*t*) governs inward flux of T-cells into the skin region. Prior to ion contact \(L=0\) so that \(T=0\). The value of

*L*(

*t*) will become non-zero following ion contact and will reach a peak when the lymph nodes are most actively releasing newly programmed T-cells. Following removal of the irritant, the value of

*L*(

*t*) will drop to a basal level \(\varepsilon \), so that there will be a continued presence of programmed T-cells in the skin, albeit at a much lower level. The activity in the lymph nodes are assumed to be governed by a Michaelis–Menten function of the total amount of extracellular ions \(A_e\) (assumed to be proportional to the amount picked up by dendritic cells) over the previous few days. The passage of dendritic cells to the lymph nodes and the recruitment of naive T-cells typically takes around 2 days (Stoitzner et al. 2003), though, we will assume the timescale for this process can be distributional, given by function \(k(\tau )\). To describe the winding down process of \(L\rightarrow \varepsilon \), whilst permitting \(L=0\) to be a steady-state as well, we use, for simplicity, a logistic term. The evolution of

*L*(

*t*) is assumed to follow

*V*is the skin domain. In the biologically relevant case of \(\varepsilon \ll 1 \), the maximum value of

*L*(

*t*) will be a little over unity. The kernel \(k(\tau )\) in the integrals of (14) is assumed to have the following property

### 2.2 Boundary and initial conditions

*i*and the \(Q_*\)s are mass transfer coefficients. Here, the influx rate of T-cells from the blood is assumed proportional to lymph node activity

*L*and \(r \in [0, R]\) is the region of metal ion infiltration at \(z=Z\); due to the large aspect ratio of likely MIC diameter and skin thickness,

*R*will be approximately the width of the MIC (see Fig. 1). At “large” \(r > R\), we impose “far-field” distributions for cytokine \(c_I(z)\), innate immune cells \(I_I(z)\) and T-cells, \(T_\infty (z,t)\), that reflect those for healthy tissue (being too far from the MIC source to be affected); these distributions thus satisfy

The full governing system of equations is summarised and non-dimensionalised in “Appendix A”. Non-dimensionalising was undertaken to aid in parameter estimation as well as numerical solutions of the system.

### 2.3 Parameter values

We have parameterised our model with values from the literature where they have been available as shown in Tables 2 and 3. Where values have not been available, we have made informed estimates to the non-dimensional parameters by comparing the magnitude and importance of each of the relevant mechanisms in the model. The non-dimensional parameter values are shown in Table 4.

List of non-ion specific model parameters

Parameter | Description | Value | Units | Sources |
---|---|---|---|---|

| Skin thickness | 0.2 | cm | Laurent et al. (2007) |

\(w_0\) | Extracellular water volume fraction | 0.05 | Dimensionless | Moore (1987) |

\({\bar{\tau }}\) | Mean lymph node response time | 2.0 | Day | Mempel et al. (2004) |

\(\beta _{cn}\) | SC cytokine background prod. rate | \(1.5 \times 10^{-3}\) | \(\upmu \hbox {M day}^{-1}\) | |

\(\beta _{cI}\) | Activated IIC cytokine prod. rate | – | \(\upmu \hbox {M day}^{-1}\) | |

\(\beta _{ceT}\) | Activated PTC cytokine prod. rate | – | \(\upmu \hbox {M day}^{-1}\) | |

\(\beta _{gc}\) | Cytokine mediated granule prod. rate | – | \(\hbox {day}^{-1}\) | |

\(\beta _{gi}\) | Ion induced granule prod. rate | – | \(\hbox {day}^{-1}\) | |

\(\beta _{Ic}\) | Activated IC growth rate | 1.6 | \(\hbox {day}^{-1}\) | Ganusov et al. (2005) |

\(\beta _{T}\) | Activated TC growth rate | 1.6 | \(\hbox {day}^{-1}\) | \(=\beta _{Ic}\) |

\(\delta _{c}\) | Cytokine decay rate constant | 3.2 | \(\hbox {day}^{-1}\) | Franks et al. (2008) |

\(\delta _{ng}\) | Granule induced SC death rate | – | \(\upmu \hbox {M}^{-1}\hbox { day}^{-1}\) | |

\(\delta _{nTi}\) | PTC induced SC death rate | – | \(\hbox {day}^{-1}\,\upmu \hbox {M}^{-1}\) | |

\(\delta _{eD}\) | Background ion removal rate const. | – | \(\hbox {day}^{-1}\) | |

\(\delta _I\) | IIC death rate const. | 1.2 | \(\hbox {day}^{-1}\) | Brach et al. (1992) |

\(\delta _T\) | PTC death rate const. | – | \(\hbox {day}^{-1}\) | |

\(\delta _g\) | Granule decay rate const. | 8.3 | \(\hbox {day}^{-1}\) | Enrique et al. (1999) |

\(\lambda \) | Lymph node activation rate const. | – | \(\hbox {Day}^{-1}\) | |

\(\varepsilon \) | Long term lymph node activity | – | Dimensionless | |

\(c_I\) | Cytokine conc. for IIC activation | – | \(\upmu \hbox {M}\) | |

\(c_T\) | Cytokine conc. for PTC activation | – | \(\upmu \hbox {M}\) | |

\(M_g\) | SC death per granule | – | Dimensionless | |

\(M_c\) | IIC removed per granule produced | – | Dimensionless | |

\(D_c\) | Cytokine diffusion rate | 0.10 | \(\hbox {cm}^2\hbox { day}^{-1}\) | Nugent and Jain (1984) |

\(D_I\) | IC diffusion coefficient | – | \(\hbox {cm}^2\hbox { day}^{-1}\) | |

\(\chi _I\) | IIC chemotaxis coefficient | – | \(\hbox {cm}^2\,\upmu \hbox {M}^{-2}\hbox { day}^{-1}\) | |

\(D_T\) | PTC diffusion coefficient | – | \(\hbox {cm}^2\hbox { day}^{-1}\) | |

\(\chi _{Ti}\) | PTC chemotaxis coefficient | – | \(\hbox {cm}^2\,\upmu \hbox {M}^{-2}\hbox { day}^{-1}\) | |

\(Q_c\) | Cytokine mass transfer const. | – | \(\upmu \hbox {M cm day}^{-1}\) | |

\(Q_I\) | IIC mass transfer const. (m.t.c.) | – | \(\upmu \hbox {M cm day}^{-1}\) | |

\(Q_T\) | PTC mass transfer const. (m.t.c.) | – | \(\upmu \hbox {M cm day}{-1}\) |

Dimensional model parameter values that are different for nickel and chromium

Parameter | Description | | | Units | Sources |
---|---|---|---|---|---|

\(\delta _{ni}\) | Cell death rate from ion | \(6.5 \times 10^{-4}\) | \(6.3 \times 10^{-5}\) | \(\upmu \hbox {M}^{-1}\hbox { day}^{-1}\) | Franks et al. (2008) |

\(\beta _{ci}\) | Live cell cytokine prod. | \(7.8 \times 10^{-6}\) | 0 | \(\hbox {Day}{-1}\) | Franks et al. (2008) |

\(k_n\) | Live cell-ion binding rate | 1.4 | 320 | \(\hbox {Day}^{-1}\) | Franks et al. (2008) |

\(k_p\) | Dead cell-ion binding rate | 1.4 | 320 | \(\hbox {Day}^{-1}\) | \(k_p=k_n{}^\mathrm{a}\) |

\(\mu _n\) | Live cell ion partition coeff. | 8.3 | 2.2 | Dimensionless | Franks et al. (2008) |

\(\mu _p\) | Dead cell ion partition coeff. | 8.3 | 2.2 | Dimensionless | \(\mu _p=\mu _n{}^\mathrm{a}\) |

\(\delta _{nTi}\) | TC induced cell death rate | – | – | \(\upmu \hbox {M}^{-2}\hbox { day}^{-1}\) | |

\(\beta _{gi}\) | Ion induced granule prod. | – | – | \(\upmu \hbox {M}^{-1}\hbox { day}^{-1}\) | |

\(K_{L_i}\) | Lymph node activation const. | – | – | mol | |

\(A_{e_i}\) | TC activation const. | – | – | mol | |

\(A_{c_i}\) | TC cytokine prod. const. | – | – | mol | |

\(D_e\) | Ion diffusion coefficient | 1.3 | 2 | \(\hbox {cm}^2\hbox { day}^{-1}\) |

Dimensionless parameters and their values used in the “standard simulation”

Parameter | Value | Parameter | Value | Parameter | | |
---|---|---|---|---|---|---|

\(A_{\text{ surface }}{}^{\mathrm{b}}\) | 0.3 | \(\beta _{cET}\) | 150 | \(\delta _{ni}\,^\mathrm{a}\) | 1.3 | 0.13 |

\(\beta _{cI}\) | 0 | \(\beta _{gc}\) | 1 | \(\beta _{ci}\,^\mathrm{a}\) | 35 | 0 |

\(\beta _{I}\) | 1.2 | \(\beta _{T}\) | 1.2 | \(k_n\,^\mathrm{a}\) | 2.8 | 640 |

\(c_I\) | 1.5 | \(c_T\) | \(\infty ^{\mathrm{c}}\) | \(k_p\,^\mathrm{a}\) | 2.8 | 640 |

\(\chi _{Iz}\) | 0.3 | \(\chi _{Ir}\) | 0.012 | \(\mu _n\,^\mathrm{a}\) | 8.3 | 2.2 |

\(D_{cz}\,^\mathrm{a}\) | 5 | \(D_{cr}\,^\mathrm{a}\) | 0.2 | \(\mu _p\,^\mathrm{a}\) | 8.3 | 2.2 |

\(D_{Iz}\) | 0.1 | \(D_{Ir}\) | 0.004 | \(D_{er}\,^\mathrm{a}\) | 2.6 | 4.0 |

\(D_{Tz}\) | 0.1 | \(D_{Tr}\) | 0.004 | \(D_{ez}\,^\mathrm{a}\) | 65.0 | 100.0 |

\(\delta _c\) | 6.4 | \(\delta _{eD}\) | 0.1 | \(\chi _{Ti z}\) | 0.1 | 0.1 |

\(\delta _g\,^\mathrm{a}\) | 17 | \(\delta _I\,^\mathrm{a}\) | 2.4 | \(\chi _{Ti x}\) | 0.004 | 0.004 |

\(\delta _{Ic}\) | 0.9 | \(\delta _{ng}\) | 10 | \(\delta _{nTi}\) | 2.5 | 2.5 |

\(\delta _T\) | 0.006 | \(\varepsilon \) | 0.001 | \(A_{e_{i}}\) | 0.02 | 0.02 |

\(\lambda \) | 1.4 | \(M_c\) | 100,000 | \(A_{c_i}\) | 0.1 | 0.1 |

\(M_g\) | 0.001 | \(N_{remove}\) | 0.8 | \(\beta _{gi}\) | 300 | 300 |

\(Q_c\) | 1 | \(Q_I\) | \(0.503^{\mathrm{d}}\) | \(K_{L_i}^{\mathrm{f}}\) | 1.6 | 1.6 |

\(Q_T\) | \(6.186^{\mathrm{d}}\) | \(\hbox {SA}_{sim}\) | 3.141\(^{\mathrm{e}}\) | |||

\(Z_D\) | 1 |

## 3 Model analysis and results

Numerical solutions to the non-dimensional model (Appendix A) were undertaken using the method of lines, where central difference approximations were used for the spatial derivatives. The resulting system of ordinary differential equations approximated using the Numerical Algorithms Group (NAG) routine D02NJF, a routine that uses an implicit method designed for stiff problems that yields a sparse Jacobian matrix. For 1-D model simulations, the system was efficiently solved using a uniform grid, however, for the 3-D simulations (effectively 2-D by assuming radial symmetry), in order to deal with rapid variations in the *x* direction near the line \(x = X\), a non-uniform grid was used with most of the points concentrated about this line. As will be demonstrated in Sects. 3.2 and 3.3, there are only small differences between the results from equivalent 1-D and 3-D simulations, so the majority of results discussed here are for the 1-D case.

There are a large number of parameters in this system with only a few that can be reliably estimated. The choice of parameters in Table 4 gave solutions that possess notable differences in responses between chromium and nickel and between first and subsequent contacts. To minimise any contrivances in our results, the only difference between the parameters regarding chromium and nickel are those measured by Franks et al. (2008) and the diffusion coefficients. In all simulations, the parameters used are those listed in Table 4, with any differences noted in the figure captions and main text. With the lack of detailed, relevant spatio-temporal data of events occurring in the skin during the first few days of contact, we cannot be certain how close the simulated results presented are to reality. The parameters are chosen so that the resulting simulated results show clearly a contrast in events between chromium and nickel ions and between first and second contacts, using as a basis the data measured by Franks et al. (2008). The model can produce results to describe the spatial-temporal evolution to a much higher level of detail than that which is currently achievable using experimental techniques and clinical observation. The aim of the simulations is to gain insights into this system with the aim of informing potential areas of experimental study.

### 3.1 The standard simulation in 1-D

*N*(

*r*,

*t*) of cells, defined as

*r*variable is for the 3-D simulations discussed in Sect. 3.2. As expected for both metals, Fig. 3 shows that the period of time for MIC removal is longer in the first contact case, and considerably longer in the case of nickel. At the surface concentration of \(A_e= 0.3\) (equivalent to 0.3 \(\upmu \)M), Cr(VI) is much more toxic than Ni(II), and in these simulations the toxicity of Cr(VI) is sufficient to cause MIC removal before adaptive immune response comes into play. As such, MIC removal was deemed to occur when 20\(\%\) of the localised skin tissue became damaged. However, the pre-existence of specialist T-cells activates a rapid response and for both Cr(VI) and Ni(II) the MIC is removed by \(t=0.3\) (about 15 h). For the first contact Ni(II) case, the concentration of ion is barely toxic and it is only when the adaptive response is activated and T-cells have successfully migrated from the blood stream to where Ni(II) concentration is highest does notable damage of the skin occur, consequently the MIC remains in contact until about \(t = 1.75\) (equivalent to about 3.5 days).

*L*is at a maximum level in Fig. 3).

Figures 4 and 5 show “heat maps” of the simulated evolution of the spatial distribution of the main variables for chromium and nickel, respectively. The horizontal axis shows the depth variable *z* and vertical axis is time *t*. For each variable, the colour scale is the same for each metal ion so that the corresponding distributions can be compared. In each case, the ion distribution in the skin equilibrates fairly rapidly during initial contact and on removal of the MIC, characterised by the rapid drop in \(A_e\) concentration. The equilibration of nickel ions between extracellular space and skin cells is much more rapid than chromium ions (see \(A_n\) figures), with the mass transfer coefficient ratio being \(k_n(\text{ Ni })/k_n(\text{ Cr }) \approx 230\); consequently nickel builds up and drains out quickly, whilst on removal of the MIC, chromium ions in cells acts an ion reservoir and hence they linger in the system. As can be seen from Fig. 4, immune cell activity in the chromium ion case is significantly enhanced on second contact (see plots of *I* and *T*), leading to greater damage of skin. On first contact, the relatively low immune cell activity suggest that skin cell death is largely due to toxicity of the chromium ions. On second contact, the rather delayed emergence of the innate immune cells (peaking around \(t\approx 2\)) suggests that death is due to ion toxicity and T-cell action; the notable peak in granules will have little effect as the skin cells have already died off there. Furthermore, T-cells are able to penetrate throughout the skin, unlike the innate cells, due to their initial presence in the skin and their relative longevity. In contrast, the immune cell activity in the nickel case becomes much more intense in the first contact than on second contact (see Fig. 5), particularly in T-cell activity. Here, nickel has less intrinsic toxicity than chromium, and skin cell death is associated with peaks in cytokines and hence caused by immune cell activity. However, on second contact, the area of skin cell death is beyond where innate immune cells have penetrated, so their role is secondary in this simulation.

### 3.2 The standard simulation in 3-D with radial symmetry

*N*as defined by (27)), external ion concentration \(Ae_T\) defined as

### 3.3 Effect of contact surface area

*S*is the surface area of contact, noting that \(w_0 D_{ey} \partial A_e(1,t) / \partial y\) is the ion influx rate at time

*t*.

### 3.4 Effect of ion concentration

The rate at which ions can traverse the stratum corneum into the living tissue will depend on, for example, contact location (e.g. corneum thickness), contact material, sweat levels and breaks in skin. So the range of suitable values for the surface ion concentration is likely to be very extensive. Figure 9 shows the effects of surface ion concentration (\(A_e(1,t)\)) on removal time (when \(N=0.8\)) and the steady-state levels of overall ion influx and survival fraction (\(N_\infty \)) following a period of continuous exposure of chromium (left) and nickel (right) on first contact (dashed) and second contact (solid).

We note that the model is only intended to describe events after the first few days of initial contact, so the simulations corresponding to where the removal time \(T_r > 4\) (indicated by the dashed line in the top two graphs) are not likely to be biological relevant; it is expected that skin growth will compensate for much of damage in this case, so contact may be tolerated. As expected, the higher concentrations lead to shorter removal time, though the overall influx of ions following contact increases in the case of chromium and to a rather more complicated relationship in the nickel case. In these simulations, T-cells are activated when \(A_e > 0.02\), consequently when \(A_e(1,t) < 0.02\) then the response is due to the inherent toxicity of the ion and very little immune activity is occurring. In the case of nickel, at surface concentrations in the region of \(0.02 \le A_e(1,t) \le 1\), much of the cell death is due to immune cell activity, whilst for \(A_e(1,t) > 1\) nickel is sufficiently toxic to displace immune cell activity as the main source of death and consequently “accelerates” removal time; these transitions are evident from the various bumps along the curves for the nickel case. We note that typically more damage occurs on second contact, though there is not a significant difference in the case of chromium.

### 3.5 Role of skin thickness

### 3.6 Occupationally relevant exposures

### 3.7 Effect of distributional time delay

## 4 Summary and discussion

The influx of metal ions through skin contact with metal, or products containing metals, leads to a complex response both directly, via toxic activity on cells, and indirectly, via immunological activity. Furthermore, different ionic species, Cr(VI) and Ni(II) being relevant to this paper, can produce significantly different skin reactions as can the history of contact from the same species. Much of what we know regarding cellular response to these ions is from in vitro studies, but how these responses interact in situ to result in dermatitis is much less well understood. The aim of this paper was to formulate a mathematical model which incorporates many of the important known factors that lead to dermatitis development in a spatio-temporal setting, that can offer insights as to why there are differences in skin responses between two ionic species, namely Cr(VI) and Ni(II), and between first and second contact. In the interest of keeping the model manageable, we reduced the complexity by focusing on metal ion toxicity (using data from Franks et al. 2008), the innate and adaptive immune response and regulation via a single, generic cytokine. Despite the number of simplifications in the modelling assumptions, the model possesses a large number of parameters. In the simulations we used values that are available from the literature. Uninformed parameters were estimated and were tuned to produce results showing the desired differences in skin response to differing circumstances.

- 1.
Cr(VI) seems to linger in the skin longer than that of Ni(II), leading to prolonged damage and lymph-node/T-cell mediated activity.

- 2.
To generate the large response difference between first and second contact cases for Ni(II), the T-cell infiltration rate has to be relatively slow (governed by parameter \(Q_T\)) and T-cells have to die off slowly (i.e. small death rate \(\delta _T\)); thus ion-specific T-cells are predicted to occupy the contact site for a long time (i.e. weeks) after the contact has been removed.

- 3.
The typically small aspect ratio of skin depth to contact surface area means that there is little difference in the overall outcomes predicted by the 1-D and the radially symmetric 3-D simulations. However, latter simulations do hint at the possibility of extra skin damage occurring in a local region near the vicinity of the contact edge; this being due to the chemotactic infiltration of immune cells from the contact region periphery.

- 1.
The contact surface area has little effect on removal time. This is partly due to our criteria for metal ion carrier (MIC) removal being when there is 20% localised damage and that over the contact surface distributions of ions and immune cells are fairly uniform (see 3-D simulations). However, the larger surface area means that more ions enter the body, which may lead to other detrimental effects.

- 2.
Removal time only decreases by a small amount as skin thickness decreases and that thicker skin will experience greater damage in the longer term.

- 3.
Intermittent contact appears not to make a significant difference in skin response qualitatively, though, due to the lingering of ions in the skin, the overall contact time of irritant will be less than it would be for continuous contact.

- 4.
Assuming a single point or a simple distributed time delay in lymph-node activity does not effect results too much, particularly in the continuous contact case (results not shown). However, with intermittent contact case, it is possible that a spreading of response can lead to earlier removal.

The parameter gaps in the current model are mainly centred around the various immune cells, such as diffusion and chemotaxis rate coefficients in tissues, typical concentrations of cytotoxic granules and production rates and cytokine or ion concentration levels for immune cell activation. Estimates of these parameters from, if possible, in vitro studies would be invaluable for model verification and improvement, which will help assess the adequacy of the current assumptions and/or the need to include other factors to formulate a more dependable model. In our interest of making a detailed model as simple as possible, we made a number of simplifications to reality. These include the blending of epidermis and upper dermis into homogeneous living tissue, absence of skin regrowth and recovery, assuming a single pro-inflammetory cytokine etc.; including these mechanisms will doubtless improve the model, but will come at a cost of more, as yet undetermined, parameters. Nevertheless, the current model has made a number of interesting predictions, which will hopefully motivate further and more directed investigations into contact dermatitis.

## Notes

### Acknowledgements

MJT acknowledges support from the University of Oxford and a Wellcome Trust Value in People Award whilst undertaking this research. SJF acknowledges support from the Health and Safety Executive and the Health and Safety Laboratories Internal Research Programme whilst undertaking this work and more recently the Daphne Jackson Trust.

## References

- Brach M, deVos S, Gruss H, Herrmann F (1992) Prolongation of survival of human polymorphonuclear neutrophils by granulocyte-macrophage colony stimulating factor is caused by inhibition of programmed cell death. Blood 80(11):2920–2924Google Scholar
- Budinger L, Hertl M (2000) Immunologic mechanisms in hypersensitivity reactions to metal ions: an overview. Allergy 55:108–115CrossRefGoogle Scholar
- Callard R, George A, Stark J (1999) Cytokines, chaos, and complexity. Immunity 11:507–513CrossRefGoogle Scholar
- Cherry N, Meyer J, Adisesh A, Brooke R, Owen-Smith V, Swales C, Beck M (2000) Surveillance of occupational skin disease: EPIDERM and OPRA. Brit J Derm 142(6):1128–1134CrossRefGoogle Scholar
- Curtis A, Morton J, Balafa C, MacNeil S, Gawkrodger D, Warrem N, Evans G (2007) The effects of nickel and chromium on human keratinocytes: differences in viability, cell associated metal and IL-1alpha release. Toxicol In Vitro 21(5):809–819CrossRefGoogle Scholar
- De Craeker W, Roskams N, de Beeck R (2008) Occupational skin diseases and dermal exposure in the european union (eu-25): policy and practice overview. European Agency for Safety and Health at WorkGoogle Scholar
- Dominguez-Hüttinger E, Christodoulides P, Miyauchi K, Irvine A, Okada-Hatakeyama M, Kubo M, Tanaka R (2017) Mathematical modeling of atopic dermatitis reveals “double switch” mechanisms underlying four common disease phenotypes. J Allergy Clin Immunol (in press)Google Scholar
- Dominguez-Hüttinger E, Ono M, Barahona M, Tanaka R (2013) Risk factor-dependent dynamics of atopic dermatitis: modelling multi-scale regulation of epithelium homeostasis. Interface Focus 3(2):20120090CrossRefGoogle Scholar
- Döpfer D, Holzhauer M, van Boven M (2012) The dynamics of digital dermatitis in populations of dairy cattle: model-based estimates of transition rates and implications for control. Vet J 193:648–653CrossRefGoogle Scholar
- Enrique E, Garcia-Ortega P, Sotorra O, Graig P, Richart C (1999) Usefulness of UniCAP-Tryptase fluoroimmunoassay in the diagnosis of anaphylaxis. Allergy 54(6):602–606CrossRefGoogle Scholar
- Franks S, Ward J, Tindall M, King J, Curtis A, Evans G (2008) A mathematical model to predict differences in keratinocyte responses to chromium and nickel. Toxicol In Vitro 22(4):1088–1093CrossRefGoogle Scholar
- Ganusov V, Pilyugin S, de Boer R, Murali-Krishna K, Ahmed R, Antia R (2005) Quantifying cell turnover using CFSE data. J Immun Meth 298(1–2):183–200CrossRefGoogle Scholar
- Hausinger R (1993) Biochemistry of nickel. Plenum Press, New YorkCrossRefGoogle Scholar
- Health and Safety Executive (2017) Work-related skin disease in great Britain. HSE report www.hse.gov.uk/statistics/causdis/dermatitis/skin.pdf
- Jepps O, Dancik Y, Anissimov Y, Roberts M (2013) Modeling the human skin barrier—towards a better understanding of dermal absorption. Adv Drug Deliv Rev 65(2):152–168CrossRefGoogle Scholar
- Kanerva L, Jolanki R, Estlander T, Alanko K, Savlea A (2000) Incidence rates of occupational allergic contact dermatitis caused by metals. Am J Contact Dermat 11(3):155–160CrossRefGoogle Scholar
- Laurent A, Mistretta F, Bottigioli D, Dahel K, Goujon C, Nicolas J, Hennino A, Laurent P (2007) Echographic measurement of skin thickness in adults by high frequency ultrasound to assess the appropriate microneedle length for intradermal delivery of vaccines. Vaccine 25(34):6423–6430CrossRefGoogle Scholar
- Le D, Miller J, Ganusov V (2015) Mathematical modeling provides kinetic details of the human immune response to vaccination. Front Cell Infect Microbiol 4:177CrossRefGoogle Scholar
- Lin F, Nguyen C, Wang S, Saadi W, Gross S, Jeon N (2004) Effective neutrophil chemotaxis is strongly influenced by mean IL-8 concentration. Biochem Biophys Res Comm 319(2):576–581CrossRefGoogle Scholar
- Lorenzi T, Chisholm R, Melensi M, Lorz A, Delitala M (2015) Mathematical model reveals how regulating the three phases of T-cell response could counteract immune evasion. Immunology 146(2):271–280CrossRefGoogle Scholar
- Luster A, Alon R, von Andrian U (2005) Immune cell migration in inflammation: present and future therapeutic targets. Nat Immunol 6:1182–1190CrossRefGoogle Scholar
- Maxwell G, MacKay C, Cubberley R, Davies M, Gellatly N, Glavin S, Gouin T, Jacquoilleot S, Moore C, Pendlington R, Saib O, Sheffield D, Stark R, Summerfield V (2014) Applying the skin sensitisation adverse outcome pathway (aop) to quantitative risk assessment. Toxicol in Vitro 28:8–12CrossRefGoogle Scholar
- Mempel T, Henrickson S, von Andrian U (2004) T-cell priming by dendriticcells in lymph nodes occurs in three distinct phases. Nature 427(6670):154–159CrossRefGoogle Scholar
- Mollee T, Bracken A (2007) A model of solute transport through stratum corneum using solute capture and release. Bull Math Biol 69(6):1887–1907MathSciNetCrossRefzbMATHGoogle Scholar
- Moore J (1987) Death of cells and necrosis in tumour. In: Potten CS (ed) Perspectives in mammalian cell death. Oxford University Press, OxfordGoogle Scholar
- Naegel A, Heisig M, Wittum G (2013) Detailed modelling of skin penetration—an overview. Adv Drug Deliv Rev 65(2):191–207CrossRefGoogle Scholar
- Ng E, Lind P, Lindgren C, Ingelsson E, Mahajan A, Morris A, Lind L (2015) Genome-wide association study of toxic metals and trace elements reveals novel associations. Hum Mol Genet 24:4739–4745CrossRefGoogle Scholar
- Nugent L, Jain R (1984) Extravascular diffusion in normal and neoplastic tissues. Cancer Res 44(1):238–244Google Scholar
- Palsson S, Hickling T, Bradshaw-Pierce E, Zager M, Jooss K, O’Brien P, Spilker M, Palsson B, Vicini P (2013) The development of a fully-integrated immune response model (firm) simulator of the immune response through integration of multiple subset models. BMC Syst Biol 7:95CrossRefGoogle Scholar
- Pan L, Yuan P, Lin L (2002) 3D electroplated microstructures fabricated by a novel height control method. Microsyst Technol 8(6):391–394CrossRefGoogle Scholar
- Peltonen L (1979) Nickel sensitivity in the general population. Contact Dermatol 5:27–32CrossRefGoogle Scholar
- Schmidt M, Goebeler M (2015) Immunology of metal allergies. J Ger Soc Dermatol 13(7):653–659Google Scholar
- Shrivastava R, Upreti R, Seth P, Chaturvedi U (2002) Effects of chromium on the immune system. FEMS Immunol Med Microbiol 34:1–7CrossRefGoogle Scholar
- Stoitzner P, Holzmann S, McLellan A, Ivarsson L, Stossel H, Kapp M, Kammerer U, Douillard P, Kampgen E, Koch F, Saeland S, Romani N (2003) Visualization and characterization of migratory langerhans cells in murine skin and lymph nodes by antibodies against langerin/cd207. J Invest Dermatol 120:266–274CrossRefGoogle Scholar
- Tanaka R, Ono R (2013) Skin disease modeling from a mathematical perspective. J Invest Dermatol 133(6):1472–1478CrossRefGoogle Scholar
- Tantemsapya N, Meegoda J (2004) Estimation of diffusion coefficient of chromium in colloidal silica using digital photography. Environ Sci Technol 38(14):3950–3957CrossRefGoogle Scholar
- Williams I, Kupper T (1996) Immunity at the surface: homeostatic mechanisms of the skin immune system. Life Sci 58(18):1485–1507CrossRefGoogle Scholar
- Zhang H, Hou W, Henrot L, Schnebert S, Dumas M, Heusèle C, Yang J (2015) Modelling epidermis homoeostasis and psoriasis pathogenesis. J R Soc Interface 12(103):UNSP 20141071CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.