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. 2011 Mar 6;8(56):345-68.
doi: 10.1098/rsif.2010.0285. Epub 2010 Jul 23.

The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis

P Ciarletta  1 , L Foret, M Ben Amar
Affiliations

The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis

P Ciarletta et al. J R Soc Interface. .

Abstract

Cutaneous melanoma is disproportionately lethal despite its relatively low incidence and its potential for cure in the early stages. The aim of this study is to foster understanding of the role of microstructure on the occurrence of morphological changes in diseased skin during melanoma evolution. The authors propose a biomechanical analysis of its radial growth phase, investigating the role of intercellular/stromal connections on the initial stages of epidermis invasion. The radial growth phase of a primary melanoma is modelled within the multi-phase theory of mixtures, reproducing the mechanical behaviour of the skin layers and of the epidermal-dermal junction. The theoretical analysis takes into account those cellular processes that have been experimentally observed to disrupt homeostasis in normal epidermis. Numerical simulations demonstrate that the loss of adhesiveness of the melanoma cells both to the basal laminae, caused by deregulation mechanisms of adherent junctions, and to adjacent keratynocytes, consequent to a downregulation of E-cadherin, are the fundamental biomechanical features for promoting tumour initiation. Finally, the authors provide the mathematical proof of a long wavelength instability of the tumour front during the early stages of melanoma invasion. These results open the perspective to correlate the early morphology of a growing melanoma with the biomechanical characteristics of its micro-environment.

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Figures

Figure 1.
Figure 1.
Schematic of the layered structure of human skin (a), and of the cellular organization in the lower epidermis (b). The epidermal structure includes keratinocytes (Kc), melanocytes (Mc) and malignant melanoma cells (mm), the latter shown both in the proliferating (dark grey) and in the necrotic state (grey). The reticular dermis is attached to the lower epidermis through the basal laminae, being characterized by a dispersion of fibroblasts (Fb) inside the ECM network of collagen type I fibres (Coll). The epidermal structure on the right side has been partially drawn using the schematics employed by [18].
Figure 2.
Figure 2.
Schematic of the anchoring complex between a basal keratinocyte and the ECM network of the reticular dermis. The ultrastructure of the basal laminae of human skin is depicted, with indication of the three structural layers, as visible in electronic microscopy, and of the main molecular constituents.
Figure 3.
Figure 3.
Evolution of the volume fraction of melanoma cells ϕM over the dimensionless time t̄ in a two-dimensional grid (260 × 260). The material parameters used for this numerical simulation are Ā = B̄ = 1, Ā1 = B̄1 = 1.11, L̄0 = 6, κ/c1 = 20 and σ̄yield = 0.
Figure 4.
Figure 4.
Spatial evolution over x̄ of the volume fraction ϕ̄M of the melanoma cells (a) and of the nutrient concentration n̄ (b), shown at different dimensionless times t̄. The depicted values of the two-dimensional simulation refer to an axial symmetric initial distribution of melanoma cells, and are shown in a mid-section over the x̄ axis. The initial growth phases are depicted at times t̄ = 0, 1, 2, 3 (dashed lines), while later evolution is shown at time intervals t̄ = 5, 10, 15 (solid lines) and t̄ = 20 (bold line). The material parameters used for this numerical simulation are Ā = B̄ = 1, Ā1 = B̄1 = 1.11, κ/c1 = 20 and σ̄yield = 0.
Figure 5.
Figure 5.
Evolution of the volume fraction ϕ̄m of the basal laminae for σ̄yield = 0 (a) and for σ̄yield = 0.06 (b), over the dimensionless x̄ axis. The volume fractions are depicted at times t̄ = 0,0.375, 3.75, 7.5, 11.25, 15. The material parameters used for this numerical simulation are Ā = B̄ = Ā1 = B̄1 = 1, and κ/c1 = 20.
Figure 6.
Figure 6.
Evolution of the ratio L̄/L̄0 for the tumour border over the dimensionless time t̄. The variability of such a ratio is shown in function of c1/χM ((a) ΠmM/ΠlM = 1, B̄ = B̄1 = 1), and of ΠmM/ΠlM ((b) c1/χM = 1, Ā = B̄ = 1). The material parameters used for this numerical simulation are σ̄yield = 0, and κ/c1 = 20 ((a) solid orange line, c1 = χM = 1; solid blue line, c1 = χM = 3; solid brown line, c1 = χM = 5; solid black line, c1 = χM = 7; solid pink line, c1 = χM = 9. (b) Solid orange line, ΠmM/ΠlM = 0.1; solid blue line, ΠmM/ΠlM = 0.3; solid brown line, ΠmM/ΠlM = 0.5; solid black line, ΠmM/ΠlM = 0.7, solid pink line, ΠmM/ΠlM = 0.9.
Figure 7.
Figure 7.
Evolution of the ratio L̄/L̄0 for the tumour border over the dimensionless time t̄. The variability of such a ratio is shown in function of σ̄yield ((a) L̄0 = 8: solid orange line, σ̄yield = 0; solid blue line, σ̄yield = 0.02; solid brown line, σ̄yield = 0.04; solid black line, σ̄yield = 0.06; solid pink line, σ̄yield = 0.08), and of L̄0 ((b) σ̄yield = 0: solid orange line, L̄0 = 8; solid blue line, σ̄yield = 10; solid brown line, σ̄yield = 12; solid black line, σ̄yield = 14). The material parameters used for this numerical simulation are Ā = Ā1 = B̄ = B̄1 = 1, and κ/c1 = 20.
Figure 8.
Figure 8.
Evolution of the volume fraction of melanoma cells ϕM over the dimensionless time t̄ in a two-dimensional grid (160 × 160). The material parameters used for this numerical simulation are c1/χM = c1/χK = 1, L̄0 = 6, κ/c1 = 20, and ΠKM = 100ΠMl = 100ΠKl = 50ΠMm = 25ΠKm.
Figure 9.
Figure 9.
Spatial evolution over x̄ of the volume fraction ϕ̄M of the melanoma cells (a) and of the keratinocytes ϕ̄K (b) shown at different times t̄. The depicted values of the two-dimensional simulation (grid 120 × 120) refer to an axial symmetric initial distribution of melanoma cells, and are shown in a section over the x̄ axis. The growth phases are depicted at times t̄ = 0 (bold, dark red), 4.6875 (dashed, red), 37.5 (bold, red), 56.25 (dashed, purple), 93.75 (bold, purple),131.25 (dashed, blue), 168.75 (dashed, bold, blue), 206.25 (bold, blue). The material parameters used for this numerical simulation are c1/χM = c1/χK = 1, κ/c1 = 20, and ΠKM = 20ΠMl = 20ΠKl = 10ΠMm = 5ΠKm.
Figure 10.
Figure 10.
Evolution of the ratio L̄/L̄0 for the tumour border over the dimensionless time t̄. The variability of such a ratio is shown in function of c1/χM ((a) ΠKM = 10ΠMm: solid orange line, C1/χM = 1; solid blue line, C1/χM = 10; solid brown line, C1/χM = 102; solid black line, C1/χM = 103; solid pink line, C1/χM = 5·103; solid yellow line, C1/χM = 104), and of ΠKM/ΠmM ((b) c1/χM = 1: solid orange line, ΠKM/ΠmM = 10; solid blue line, ΠKM/ΠmM = 15; solid brown line, ΠKM/ΠmM = 20; solid black line, ΠKM/ΠmM = 25). The material parameters used for this numerical simulation are 4ΠMl = 4ΠKl = 2ΠMm = ΠKm, σ̄yield = 1000, L̄0 = 9 and κ/c1 = 20.
Figure 11.
Figure 11.
Evolution of the ratio L̄/L̄0 for the tumour border over the dimensionless time t̄. The variability of such a ratio is shown in function of L̄0. The material parameters used for this numerical simulation are c1/χM = c1/χK = 1, κ/c1 = 20, and ΠKM = 20ΠMl = 20ΠKl = 10ΠMm = 5ΠKm (solid orange line, L̄0 = 6; solid blue line, L̄0 = 9; solid brown line, L̄0 = 13; solid black line, L̄0 = 16).
Figure 12.
Figure 12.
Different shapes of the elastic stress function Σ(Φ), shown for ϕext = 0.8 (a), and instability diagram in function of ϕext (b) for different values of the constitutive parameter ℵ (ℵ = 2, solid line: ℵ = 3, dashed line; ℵ = 5, dot-dashed line; ℵ = 10, x-marked line).
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