Tumor-Immune Interaction, Surgical Treatment, and Cancer Recurrence in a Mathematical Model of Melanoma

Abstract

Malignant melanoma is a cancer of the skin arising in the melanocytes. We present a mathematical model of melanoma invasion into healthy tissue with an immune response. We use this model as a framework with which to investigate primary tumor invasion and treatment by surgical excision. We observe that the presence of immune cells can destroy tumors, hold them to minimal expansion, or, through the production of angiogenic factors, induce tumorigenic expansion. We also find that the tumor–immune system dynamic is critically important in determining the likelihood and extent of tumor regrowth following resection. We find that small metastatic lesions distal to the primary tumor mass can be held to a minimal size via the immune interaction with the larger primary tumor. Numerical experiments further suggest that metastatic disease is optimally suppressed by immune activation when the primary tumor is moderately, rather than minimally, metastatic. Furthermore, satellite lesions can become aggressively tumorigenic upon removal of the primary tumor and its associated immune tissue. This can lead to recurrence where total cancer mass increases more quickly than in primary tumor invasion, representing a clinically more dangerous disease state. These results are in line with clinical case studies involving resection of a primary melanoma followed by recurrence in local metastases.

Curvature-Dependent Elastic Properties of Liquid-Ordered Domains Result in Inverted Domain Sorting on Uniaxially Compressed Vesicles

“…Experimental values and comparisons with clinical observations are discussed thereafter in the case of melanoma where atmospheric oxygen is the main source of nutrients [19].…”

“…Taking into account experimental values ( [19] and references therein) for the model parameters (À $ 0:2-0:67 day À1 , n $ 1190-3030 day À1 , D n $ 8:64-86:4 mm 2 day À1 ) we estimate the stability properties of different growing melanoma. Taking oxygen as the limiting nutrient [19], our model predicts for rapidly growing melanoma (i.e., nodular melanoma, having V e $ 1:5 mm month À1 ) a control parameter T < 1, a stable symmetrical growth, while for slowly growing melanoma types (V e ¼ 0:12-0:4 mm month À1 , [16]) the control parameter is in the range T ¼ 0:95-16:2, predicting contour instabilities localized in an outer ring of size comparable to the nutrient penetration length, typically 0:07-0:35 mm.…”

“…Taking oxygen as the limiting nutrient [19], our model predicts for rapidly growing melanoma (i.e., nodular melanoma, having V e $ 1:5 mm month À1 ) a control parameter T < 1, a stable symmetrical growth, while for slowly growing melanoma types (V e ¼ 0:12-0:4 mm month À1 , [16]) the control parameter is in the range T ¼ 0:95-16:2, predicting contour instabilities localized in an outer ring of size comparable to the nutrient penetration length, typically 0:07-0:35 mm. The predicted threshold radius is less than 0.68 mm, to compare with the 3 mm threshold given in the ABCDE criteria [10], for a typical growth rate of 0.33 mm per month, suggesting that contour instabilities of melanoma are observable in the early growth of the tumor.…”

“…Considering the velocities involved in tumor growth [16,17], the nutrient advection is negligible, if compared to the diffusion, so that their transport is simply described by a diffusion consumption equation @n=@t ¼ D n Án À n c n. Cellular growth depends linearly on the local nutrient concentration [18], a limiting factor being oxygen supply [19] causing often hypoxia and necrosis of the tumor center [20]. Cell death rate is taken constant, giving the volume transfer rate from liquid to cellular phase Àð c ; nÞ ¼ c c n À c c .…”

Addressing current challenges in cancer immunotherapy with mathematical and computational modelling

“…While partial differential equation (PDE) models have been much more sparsely employed to model immunotherapy, they can be especially useful when modelling a large number of interacting cells in a tissue, in which case building a corresponding ABM would be too computationally costly. For example, Eikenberry et al [36] developed a PDE of melanoma with immune infiltrate, and showed that surgical removal of primary tumours with high levels of immune infiltrate could promote growth of satellite metastases, as was observed clinically, thereby providing a model-based hypothesis for tumour classification with respect to responsiveness to therapy (in this case, surgery).…”