Conceived and designed the experiments: SE CT YK. Performed the experiments: SE. Analyzed the data: SE CT YK. Wrote the paper: SE.
The authors have declared that no competing interests exist.
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.
Melanoma is a deadly skin cancer that invades into the dermis and metastasizes into the surrounding tissue. In clinical cases, surgical excision of the primary tumor has led to widespread and accelerated growth in metastases. We develop a mathematical model describing the basic process of melanoma invasion, metastatic spread, and the anti-tumor immune response. This model is formulated using partial differential equations that describe the spatial and temporal evolution of a number of different cellular populations, and it uses a realistic skin geometry. Using simulations, we examine the importance of the immune response when a primary tumor is spawning satellite metastases. We find that local metastases can be suppressed by the immune response directed against the primary tumor, but grow aggressively following surgical treatment. We also find that moderately metastatic tumors optimally activate the local immune response against disseminated disease, and in this case tumor excision may have profound effects on metastatic growth. We conclude that surgical perturbation of the immune response controlling local metastases is one mechanism by which cancer can recur. This could have implications as to the appropriate clinical management of melanomas and other solid tumors.
Melanoma, the most dangerous form of skin cancer, arises in the melanocytes and progresses through two well-defined clinical stages. Following a period of radial growth in the epidermis, melanomas may switch to malignant, vertical growth melanoma (VGM)
Angiogenesis is induced primarily by the release of angiogenic factors by melanoma cells and associated stromal cells and through the restructuring of the extracellular matrix (ECM) that occurs in concert with invasion
Although some angiogenesis may occur before penetration of the basement membrane, intense angiogenesis requires ECM remodeling, which in turn requires the cooperation of stromal cells. Even if the tumor is releasing large amounts of VEGF, most of it is sequestered in the ECM
Endothelial cells are the predominant cell type in the formation of new vasculature. Endothelial cell migration into the tumoral region is essential for angiogenesis and is facilitated by MMP mediated matrix remodeling
The immune system is able to effectively mobilize against tumor invasion. This occurs mainly through direct tumoricidal action by natural killer (NK) cells and phagocytes, such as macrophages, and through the T cell response
Melanoma has a strong tendency to metastasize, with most metastases occurring in the skin, lymph nodes, and lungs
In this paper we develop a spatially explicit model using partial differential equations (PDEs) to capture the dynamics of melanoma invasion in the skin. We first present a basic model that does not consider an immune response and examine tumor invasion in a cylindrical section of skin. Then we extend this model to include a cellular immune response and carry out a number of numerical experiments using this extended model. In these, we examine the possible dynamics of tumor invasion under different levels of immune response. We develop a method to realistically model the stochastic process of local metastatic spread, and surgical treatment is simulated in both locally metastasizing and non-metastasizing melanomas. These simulations are motivated by a clinical case where, following resection of a primary melanoma, widespread recurrence was seen in local satellite metastases
Based on our simulations we make several observations and predictions. First we observe that angiogenesis is strongly tumorigenic, which is very well known from previous experimental and theoretical work. In accordance with the biological observations in
We show that the immune response can either aid or inhibit tumor progression; the outcome depends on the balance between angiogenic factors released by immune cells and the growth-inhibitory and cytotoxic effects of the same immune cells. We also make several predictions concerning treatment and metastasis. We observe that a relatively small safety margin is necessary to remove all primary tumor material and prevent local recurrence due to “local persistence” (as defined in
We propose a spatially explicit model of melanoma invasion in the skin, formulated in terms of cell densities. We first present the full system with a detailed derivation of the governing equation for each variable. We then describe the model geometry and boundary conditions.
The basic model considers seven variables:
The basic assumptions we use to derive the model follow
Cell motility is achieved according to Fick's Law, which yields a diffusion term. Cancer cell motility additionally depends upon contact with other cancer cells and oxygen concentration.
Oxygen is the limiting nutrient and diffuses into the skin from the skin surface and a subcutaneous vascular bed
Proliferation of both healthy and cancerous cells is mediated by the amount of space available and the concentration of oxygen
Cancer cell death is mediated by oxygen concentration, and cancer cells that die in response to hypoxic stress become necrotic debris
Cancer cells produce angiogenic factors both constitutively and in response to hypoxia
Endothelial cells migrate into the system in response to angiogenic factors and form the tumor vasculature, which supplies additional oxygen
A basement membrane separates the epidermis from the dermis and restricts cell migration
The mathematical model is formulated as follows:
Where
H(x) is the Heaviside step function:
Cancer cell motility is achieved through density-dependent diffusion proportional to oxygen concentration. This is based on the assumption that cancer cell motility requires contact with other malignant cells as well as oxygen sufficient to provide the energy required for migration.
The density dependent term is similar to that used by Tohya
The Heaviside term,
We assume that cancer cell growth is dependent upon both the local cell density and oxygen concentration. Dependence on density is approximated by a logistic growth term with maximum per capita growth rate
We assume that cancer cells also undergo apoptosis in response to extreme hypoxia at maximum rate
Healthy cells are a generic type representing both the epidermal keratinocytes and dermal fibroblasts that make up most of the skin. We assume all healthy cell motility is due to simple diffusion and that cells cannot cross regions of high basement membrane density. Healthy cells grow logistically with carrying capacity
TAF can be viewed as an aggregation of those angiogenic factors released by melanomas. The most important of these is VEGF, and wherever possible (e.g. in parametrization) we treat TAF as though it were VEGF. TAF diffuses by simple diffusion and degrades at rate
Endothelial cells (ECs) are the primary cell type involved in capillary construction. We do not explicitly model the tumor vascular network, but instead assume that the density of microvessels is directly proportional to endothelial cell density. Therefore, we only consider the averaged microvessel density and omit the details of capillary construction, which is spatially complex and marked by local irregularities. Since our model considers tumor invasion at the macro-scale, this simplification has a minimal effect on the dynamics.
In this model we assume EC motility is achieved only through simple diffusion. Most models of angiogenesis also consider chemotaxis in response to a TAF gradient
EC proliferation in response to TAF appears to occur mainly when EC density is low. This occurs at the leading edge of a migratory EC front and in existing microvasculature that has been destabilized by TAF to the point that the ECs are pulled away from each other
ECs have been observed to undergo apoptosis when VEGF levels are below a certain critical value
Those cancer cells that die become necrotic debris, hence the addition of the cancer cell death term to the right side of the equation. We assume that normal homeostatic mechanisms in the skin prevent healthy cells from contributing significantly to necrotic material. This debris is assumed to disperse at a low rate through simple diffusion and degrade at some small rate
Oxygen diffuses through the skin where it is consumed by both melanomas and normal skin cells, and it is supplied by the tumor vasculature represented by endothelial cell density. Oxygen diffuses by simple diffusion with the diffusion coefficient
The source term represents the supply of oxygen from the tumor vasculature and is modeled according to the principle of solute transport as given in
Oxygen is consumed by melanomas and normal skin cells. Here, we include the cancer cell birth term as
Finally, if no is oxygen present, then clearly none can be consumed. Therefore,
The basement membrane (BM) is the most important geometric feature of the skin and has been included. It is a static “wall” across which cancer (and healthy) cells cannot cross. We assume the presence of endothelial cells degrades the basement membrane at rate
This assumption can be justified biologically as the switch from radial to vertical growth appears to require both the degradation of the BM and angiogenesis, and the two processes are tightly coupled
As our model is intended to capture the process of primary melanoma invasion in the skin we must build a reasonable approximation to this geometry. Therefore, we generally consider a three-dimensional domain using cylindrical coordinates –
Considering a cylindrical section of skin, let
Finally, the basement membrane separates the epidermis and dermis, serving as a barrier to migration for most cell types. The basement membrane is modeled using initial conditions in simulations - a thin strip of membrane is thought to exist at a depth of approximately 0.15 mm. A simple schematic of the modeled geometry is shown in
The epidermis is separated from the dermis by a sheet of basement membrane. At the base of the dermis is a vascular bed from which all endothelial cells migrate. Oxygen diffuses into the domain from the vascular bed and at the skin surface. The melanoma tumor originates in the epidermis. All other tissue is initially healthy cells.
All boundary conditions are no-flux with several exceptions for oxygen and endothelial cells. The Dirichlet boundary conditions for oxygen are 150 mmHG for the skin surface
We also provide for an influx of ECs from the vascular bed and circulation through a Neumann boundary condition at
The maximum EC influx is
While this model contains a large number of parameters, we can make at least a reasonable order of magnitude estimate for all of them from empirical biological data.
A wide range of population doubling times has been observed for different melanoma lines, with more advanced tumors typically having higher growth rates. Doubling times range between approximately 1 and 4 days
The linear diffusion coefficient for human keratinocytes was measured between .002 and .07 mm2 day−1 in
In
In a tissue scaffold model the diffusion coefficient for VEGF has been measured as
Using data from
The baseline oxygen consumption rate,
All parameters and values with references (if applicable) are given in
Param. | Meaning | Value Range | Ref. |
Melanoma density dependent diffusion coeff. | 7.0×10−7–4.0×10−9 | ||
Maximum TC growth rate | .17–.69 day−1 | ||
Measures sensitivity of TC growth to |
5–10 mmHg | ||
Measures sensitivity of TC death to |
1–5 mmHg | ||
Total cell density above which TCs don't prolif. | 1.0×105–5.0×105 TC mm−3 | ||
Healthy cell diffusion coefficient | .002–.07 mm2 day−1 | ||
Maximum healthy cell growth rate | .1 day−1 | ||
Normal healthy cell turnover rate | .03 day−1 | ||
Total cell density above which HCs don't prolif. | 1.0×105–5.0×105 HC mm−3 | ||
TAF diffusion coefficient | 0.497–6.048 mm2 day−1 | ||
Maximum hypoxic TC expression of TAF | |||
Constitutive TC expression of TAF | |||
Maximum per capita uptake of TAF by ECs | |||
TAF density of half maximal EC response | |||
TAF degradation rate | 19.96 day−1 | ||
EC diffusion coefficient | .000864–.070848 mm2 day−1 | ||
Critical density above which ECs don't prolif. | 1000 EC mm−3 | ||
Maximum EC growth/death rate | .30–.90 day−1 | ||
TAF level below which ECs undergo apoptosis | 5.0×10−13 g mm−3 | ||
Baseline EC death | .003–.005 day−1 | ||
Maximum EC influx | |||
Debris diffusion coefficient | 1×10−6 mm2 day−1 | ||
Necrotic debris degradation rate | 0.0–.01 day−1 | ||
8/85/200 mm2 day−1 | |||
Capillary surface area per EC | 1.492×10−4–2.293×10−4 mm2 EC−1 | ||
O2 capillary permeability coefficient | 8.64–6.74×104 mm day−1 |
||
Capillary |
30–40 mmHg | ||
Baseline skin |
1.2×106 mmHg g−1 day−1 | ||
Melanoma maintenance |
0 | ||
Maximum melanoma |
≤6.68 mmHg TC−1 day−1 | ||
Maximum rate of EC induced BM degradation | 1.0×10−4 g EC−1 day−1 | ||
Thresh. BM density above which migration inhibited | .025 g mm−3 | ||
BM density of half-maximal degradation | .01 g mm−3 |
We extend the model to include a class of immune cells that are directly cytotoxic to cancer cells. The most important of these cells are macrophages, dendritic cells (DCs), natural killer cells (NKs), and cytotoxic T cells (
Melanomas secrete inflammatory cytokines that attract circulating monocytes to the site of invasion. These monocytes can differentiate into macrophages or dendritic cells
Natural killers do not need any education to recognize and destroy neoplastic cells. On encountering aberrant cells NKs can initiate a large scale immune response through the release of cytokines that recruit other effector immune cells, the most important being tumor necrosis factor
A strong response by effector immune cells is probably more harmful to the tumor than helpful. The presence of tumor-infiltrating lymphocytes (TILs) has been associated with a good clinical prognosis in a number of cancers. Patient survival is 1.5 to 3 times longer in melanoma patients with high numbers of TILs compared to patients with few TILs
Similar to the TAF formulation,
We assume that ICs are attracted by IAF and are activated by contact with either tumor cells or necrotic debris. Tumor cells express IAF at a constant constitutive level. ICs express TAF and IAF and have a cytotoxic effect on tumor cells. Contact with tumor and dead cells is assumed to activate ICs. This activation causes immune cells to express both TAF and IAF and increases anti-tumor and debris cytotoxicity. Given that the tumor microenvironment is often immunosuppressive and even directly toxic to immune cells
(1) Cancer cells produce factors that attract immune cells, primarily macrophages and natural killers, to the tumor site. (2) In response to direct contact with either tumor or necrotic cells, immune cells are activated. (3) These activated immune cells kill cancer cells and clear necrotic debris at an increased rate; they also produce both TAF and IAF, enhancing the immune response while potentially inducing angiogenesis. (4) The tumor microenvironment is strongly immunosuppressive, and this is modeled by assuming that direct contact with tumor cells is cytotoxic to immune cells.
ICs are imagined to expend energy at some per capita rate
We assume activated ICs produce TAF at a rate proportional to the level activation, with a maximum of
IAF diffuses by simple diffusion, is produced by tumor cells at the constant rate
Boundary conditions are no-flux except for IAF. When IAF concentration is high enough we expect an influx of ICs into the domain, representing macrophage, NKs, and
The IAF density at which half-maximal IC migration occurs is given by
We have derived a baseline set of values for most parameters from empirical data. However, a wide range of values is allowed in simulations for those parameters representing cytotoxicity, as these are assumed to be quite variable among strains of melanoma.
We have been unable to find any estimates for the diffusion coefficient for macrophages. However, given that measurements for keratinocytes and endothelial cells fall within the same rough range, .002–.07
The molecular weight of MCP-1 was measured to be about 4 kDa in
Data on MCP-1 production by melanoma cells is given in
The turnover rate for human natural killer cells has been measured to be about 2 weeks
To determine the rate at which immune cells kill tumor cells we use data on phagocytosis in macrophages given in
Solving numerically allows the slope of the kill tracking line (
The parameters
A tumor cell strain is considered immunoevasive when parameter values are used that give low anti-tumor cytotoxicity and/or a high cell density for activation. A cell strain is considered immunosuppressive if it is highly cytotoxic to immune cells. All immune extension parameters with values and references are given in
Parameter | Meaning | Value Range | Reference |
Immune cell diffusion coefficient | .000864–.071 mm2 day−1 | ||
IC chemotaxis coefficient | 1.0×109 mm5 g−1 day−1 | ||
Maximum IC death rate due to tumor cells | .001–.1 IC TC−1 mm−3 | ||
IC density of half-maximal TC induced death | |||
Baseline IC turnover | 0.0–.05 day−1 | ||
IAF diffusion coefficient | |||
Tumor cell production of IAF | 1.667×10−15 | ||
−1.333×10−13 g TC−1 day−1 | |||
Maximum immune cell production of IAF | 1.667×10−15 | ||
−1.333×10−13 g TC−1 day−1 | |||
IAF degradation rate | 2.773–16.636 day−1 | ||
Maximum immune cell production of TAF | 0.0–1.5×10−14 g IC−1 day−1 | ||
Inactivated IC tumor cell cytotoxicity | ≤0.218 TC IC−1 day−1 | ||
Fully activated IC tumor cell cytotoxicity | ≤2.027 TC IC−1 day−1 | ||
Inherent IC bias towards debris cleanup | 1.0–10.0 | ||
TC density of half-maximal IC cytotoxicity | 102–103 TC mm−3 | ||
Inactivated IC debris cleanup | ≤0.218 TC IC−1 day−1 | ||
Fully activated IC debris cleanup | ≤2.027 TC IC−1 day−1 | ||
debris density of half-maximal IC cleanup | 102–103 TC mm−3 | ||
Weighs relative ability of debris to activate ICs | 1.0–10.0 | ||
Weighs relative ability of TCs to activate ICs | 1.0–10.0 | ||
Weighted TC density of half-maximal IC activation | 102–104 TC mm−3 | ||
IAF density of half maximal IC response | |||
Maximum IC influx |
All simulations have been run using a finite difference method on the symmetric cylindrical domain described previously. We run several simulations using the base model without the immune response to characterize the basic dynamics. For biologically realistic parameter values, the model produces realistic patterns of invasion. Before the onset of angiogenesis, growth is restricted to the epidermis. TAF expression by melanoma cells causes an influx of endothelial cells into the domain, which leads to penetration of the basement membrane and vascularization of the tumor within several months. Following this angiogenic switch, density increases and invasion spreads throughout the domain. Live cancer cell density is highest at the skin surface and the vascular bed. Between these boundaries the effects of oxygen consumption by proliferating cancer cells can be seen. Hypoxia is most pronounced near the invasive edge, where oxygen demand is greatest. Necrotic debris is initially concentrated in a roughly spherical core. As the tumor continues to invade, this core expands as an annulus following the invasive edge. Thus, the model predicts that in the absence of an immune response a solid invasive tumor with a necrotic core will form. Angiogenesis is predicted to be strongly tumorigenic.
To thoroughly demonstrate the model results, a 3-D isosurface of the evolution of three key variables, cancer cells, basement membrane, and endothelial cells, over several months of invasion is shown in
Time steps are evenly spaced over roughly 4.5 months of invasion. (A) An small initial tumor seed in the epidermis. (B) The tumor has spread radially within the epidermis, and endothelial cells have begun migrating toward the tumor. (C) The basement membrane has been penetrated by the endothelial cells, and the cancer cells have begun to invade vertically into the tumor vasculature. (D) A growing vasculature leads the advancing edge of radial invasion. The tumor mass has penetrated to the base of the domain and the vascular bed.
(A) The normal skin is largely intact, but a small epidermal tumor has displaced some of the surrounding healthy cells. (B) The tumor mass has been penetrated by the endothelial cells and, because of the associated degradation of the basement membrane, is invading vertically. The tumor is somewhat hypoxic, a core of necrotic debris has begun to form, and healthy cells continue to be displaced by the tumor. (C) A large tumor has invaded to the base of the domain and continues to expand radially. Hypoxia is most severe at the edge of invasion; this is reflected by the annular expansion of the necrotic core.
Time steps are evenly spaced over roughly 4.5 months of invasion under the basic model. (A) The seed of a tumor has been planted at the base of the epidermis. (B) The tumor has expanded vertically into the upper epidermis, and has increased in density due to cancer cells out-competing the healthy cells. (C) Here, the basement membrane has been eliminated, and the tumor invades vertically into the existing tumor-associated vasculature. (D) A significant necrotic core has formed roughly in the center of the vertical domain, very few healthy cells remain, and cancer cells continue to grow into the tumor vasculature.
A second set of simulations is performed in which the cancer cell line does not produce TAF constitutively. In these simulations, TAF production in response to hypoxia is not sufficient to induce angiogenesis, and tumor growth is restricted to the epidermis. Even when TAF production in response to hypoxia is set to the highest value reported in
In simulations of primary tumor invasion with an immune response, tumor growth is generally slowed significantly. There appear to be three possible eventual outcomes:
The tumor is completely destroyed.
A pseudo steady state is reached where the tumor ceases growing, and immune cell infiltration levels remain constant.
The tumor succeeds in completely invading the domain.
All three outcomes occur in biologically reasonable parameter space. The pattern of immune cell infiltration differs between tumors that can be characterized as immunoevasive versus immunosuppressive. In immunoevasive tumors, immune cell levels are high in the core, while suppressive tumors result in a high peritumoral concentration with little core presence.
We have also examined the effect of TAF in expression by activated immune cells. This can cause a two-phase pattern of growth, where the immune response initially holds the tumor to a steady state within the epidermis. After a period of apparent quiescence, immune-induced angiogenesis leads to a second phase of more aggressive vertical growth. Surprisingly, the tumor can completely invade the domain in this second phase even if the immune response was sufficient to hold it to a steady state in the first phase. The results of such a simulation are shown in
This steady state persists for about 300 days, but the expression of angiogenic factors by activated immune cells induces angiogenesis that leads to aggressive, unchecked tumor invasion.
To simulate treatment, the tumor is allowed to grow for some specified amount of time, after which surgical excision is performed. To simulate excision the value of all variables is set to zero within a prescribed region. Then the simulation is allowed to continue for several years.
The results of a surgical excision using the basic model without an immune response are simply characterized. Immediately following excision the wound is quickly filled by healthy cells, and in the following absence of TAF what remains of the tumoral vasculature quickly dies off. However, within a year any surviving tumor cells begin invading again, and soon the tumor recovers to pre-surgery mass.
In simulations without an immune response the tumor border is typically sharply defined with little spread beyond the visible border. A margin of several millimeters beyond the visible edge of the tumor is generally sufficient to ensure no tumor cells survive.
To simulate a primary tumor seeding metastases in the nearby tissue a normal simulation is first run for some initial amount of time (typically 6 months). Then a very small metastasis is introduced some distance from the primary tumor at the level of the vasculature. Note that this metastasis is necessarily a “metastasis ring,” due to the symmetry of the domain. This is not an unreasonable approximation, as metastasis spread is be expected to be roughly symmetric, and a complete ring can be thought to represent a worst-case scenario.
We assume that metastasis seeding can be modeled as a Poisson process, i.e. the probability of a metastasis being created within a given time period is an exponential random variable. We furthermore assume that the distance from the primary tumor at which the metastasis is seeded is exponentially distributed, with the greatest probability next to the tumor edge. Thus, a biologically reasonable approximation of metastatic spread into the surrounding skin tissue can be described by two exponential rate parameters,
To explicitly differentiate between primary tumor and metastasis populations an additional cancer cell variable is introduced. A number of minor modifications must be made to the model; they are straightforward and we do not present the details.
The immune response is included in these simulations. If no treatment is performed the primary tumor invades normally while the metastases, if sufficiently close to the primary tumor, are generally destroyed or held to an extremely low density. We have found that if
The primary tumor is held to a steady state and metastases are present at undetectable levels prior to resection. Following resection, aggressive metastatic recurrence occurs. The asymptotic behavior of the metastases is the same as the primary tumor (i.e. steady state), but the overall growth rate is faster, and the final cancer load is much greater.
Primary tumor (blue), metastases (cyan), and the basement membrane (gray) are shown. (A) The primary tumor at a steady state along with suppressed metastases. The exact picture shown is representative of the fluctuating metastatic load, which varies somewhat, although the overall behavior is one of a quasi-steady state. (B) Metastatic recurrence 6 months after resection of the primary tumor. The resection was successful in removing all material from the primary tumor, but the total mass of the metastases already exceeds that of the primary tumor.
Resection can lead to aggressive metastatic recurrence, but in some cases where the immune response is very strong, resection can lead to a state where metastases persist but remain held to a small size. This state of persistence can last indefinitely, and an example is shown in
In this case, the immune response holds the metastatic load to a total mass much less than that of the primary tumor, and unlike in the cases shown in
Numerical investigation has yielded a somewhat nonintuitive result concerning the rate at which metastases are seeded. In a sensitivity analysis of
The primary tumor is held to a steady state by the immune response, and metastases are seeded beginning 6 months into the simulation. For the lowest value of
These results suggest that moderately aggressive primary tumors that seed many metastases can induce a widespread local immune response that is sufficient to keep these metastases in check. Furthermore, those metastases that do manage to grow to significant sizes are quickly eliminated by the immune response, even though this response cannot eliminate the primary tumor.
We have examined the macroscopic dynamics of melanoma tumor growth using a reaction-diffusion framework. This diffusion framework models cell populations as a continuous density field, and has the effect of washing out local inhomogeneities. However, tumors have an irregular and heterogeneous architecture, and angiogenesis in particular is a spatially complex phenomenon, with the tumor vasculature marked by irregular construction and heterogeneity in blood flow
Of more concern is the variability in nutrient supply, principally oxygen, caused by a regionally heterogeneous vasculature, and its potential effect on overall tumor growth. However, our model suggests that melanoma is rather unique among solid tumors, with significant hypoxia only occurring at the leading edge of invasion. This is due in large part to the significant amounts of oxygen diffusing into the tumor core from the skin surface. Thus, the unique geometry in which melanoma invades likely dampens the importance of vascular irregularity. Therefore, we argue that the diffusion approximation can reasonably be employed in examining melanoma tumor invasion at the macroscopic level.
Using this model as a framework for early investigation, we have observed a wide range of interesting and biologically reasonable patterns of tumor invasion. Angiogenesis is strongly tumorigenic in this model. In simulations using the basic model, following the onset of angiogenesis the tumor spreads throughout the dermis and a significant necrotic core forms. Hypoxia is always most severe at the leading edge of radial invasion, and the necrotic core expands as an annulus in sustained invasion. The constitutive production of TAF (particularly VEGF) is more important than production in response to hypoxia in inducing angiogenesis in melanomas. In simulations, even the most aggressive cancer cell strains are unable to induce angiogenesis without at least a low level of constitutive TAF production. Therefore, TAF must be produced constitutively by melanomas or by cooperating stromal cells.
When an immune response is considered, it usually inhibits tumor growth, often destroying invasive tumors or holding them at a steady state for many years. These outcomes are observed in biologically reasonable parameter space, implying the immune response often plays a clinically meaningful role in the control of cancer growth. However, immune cells expressing TAF can also aid melanoma invasion by inducing tumorigenic angiogenesis. This can lead to a qualitative change in tumor behavior as non-invasive melanoma tumors restricted to the epidermis become aggressively invasive following immune induced angiogenesis.
We have investigated a primary tumor seeding micro-metastases into the local skin tissue. This line investigation is motivated by the case study reported by De Giorgi
We propose that local metastatic spread can be reasonably modeled using two exponential rate parameters. The first,
This framework also suggests an explanation for the phenomenon of aggressive metastasis growth following surgical excision of a primary tumor. The immune response directed against a primary tumor can suppress nearby metastases. Following surgical excision most of the immune cells attacking the primary tumor are removed, as is the major source of cytokines attracting other immune cells to the sight. In the absence of this immune response, previously checked metastases can begin growing aggressively. The total mass and growth rate of these metastases can exceed that of the primary lesion, making this recurrence potentially more clinically dangerous. This phenomenon was also studied in a mathematical model by Boushaba
There is clearly a threshold distance beyond which local immune activation is insufficient to suppress metastases. This distance may be highest in moderately metastatic tumors. It is possible that immune activation plays the dominant role in suppression near the primary tumor, where most metastases are expected to extravasate. Further away, growth inhibition due to soluble factors may become dominant. However, our model has not taken into account circulating lymphocytes or antibodies that may play an important role. Despite its limitations, the overall implication of this work is that therapy targeting a primary tumor can perturb the host immune response in a way that allows increased growth in disseminated disease without altering any of the underlying parameters describing the system.
The full model framework presented here can be translated into more focused systems aimed at addressing specific questions concerning melanoma invasion and treatment. With our model we have demonstrated that disruption of the immune response caused by surgical excision of a primary tumor is a possible mechanism for increased metastasis growth. Following surgery the wound healing response and associated inflammatory response probably also plays a role in cancer recurrence. A more detailed examination of tumor excision with a wound healing cascade could give insight into the importance of immune disruption. The effects of additional treatments such as chemotherapy or radiation therapy on the immune response and metastasis growth should also be investigated. Mathematical modeling may be particularly suited to examining the effect of different treatment schedules. At the least, predictions could be made concerning the efficacy of pre- versus post-operative adjuvant therapy.
Finally, this framework provides an opportunity to investigate the nature and power of the natural selective forces at work driving the evolution of aggressive melanoma tumors. By incorporating multiple cell strains with differential parameter values, we can study the spatial and temporal requirements for successful mutant strain invasion of a pre-existing tumor and how allopathic intervention alters the balance of selective forces in and around the primary tumor.
We would like to thank the anonymous reviewers for their careful reading and valuable comments.