Virtual Melanoma: When, Where and How Much to Cut Yang Kuang, Arizona State University

Transcription

1 Virtual Melanoma: When, Where and How Much to Cut Yang Kuang, Arizona State University Based on: Eikenberry S, Thalhauser C, Kuang Y. PLoS Comput Biol. 2009, 5:e

2 Mathematical Modeling of Melanoma Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma Eikenberry S, Thalhauser C, Kuang Y. PLoS Comput Biol. 2009, 5:e

3 Incidence Rates by Race Race/Ethnicity Male Female All Races 25.0 per 100,000 men 15.8 per 100,000 women White 28.9 per 100,000 men 18.7 per 100,000 women Black 1.1 per 100,000 men 1.0 per 100,000 women Asian/Pacific Islander 1.6 per 100,000 men 1.3 per 100,000 women American Indian/Alaska Native 3.9 per 100,000 men 2.8 per 100,000 women Hispanic 4.6 per 100,000 men 4.7 per 100,000 women

4 CDC - Skin Cancer Rates by State Light blue 7.3 to 16.6 Medium blue 16.7 to 19.1 Royal blue 19.2 to 21.3 Dark blue 21.4 to 30.9 Light Grey Did not meet USCS data quality criteria Alaska, Arizona, Arkansas, the District of Columbia, Illinois, Indiana, Louisiana, Mississippi, New York, North Dakota, South Dakota, and Texas Alabama, Florida, Iowa, Michigan, Missouri, Montana, Nebraska, Nevada, New Mexico, Oklahoma, Pennsylvania, and West Virginia California, Colorado, Georgia, Hawaii, Kansas, Minnesota, New Jersey, North Carolina, Ohio, Tennessee, Virginia, and Wyoming Connecticut, Delaware, Idaho, Kentucky, Maine, Massachusetts, New Hampshire, Oregon, Rhode Island, South Carolina, Utah, Vermont, and Washington Maryland and Wisconsin

5 Melanoma Cancer Clinical Progression Melanoma is a cancer of the melanocytes specialized melanin producing cells found in the skin scattered along the epidermis-dermis border. Melanoma invasion proceeds through two clinical stages: (1) Radial Growth Melanoma (RGM) (2) Vertical Growth Melanoma (VGM)

6 The Invasion Cascade Switch to Vertical Growth Separating the epithelium and underlying stroma is the basement membrane (BM), a tough sheet of extracellular matrix (ECM). Penetration the basement membrane and angiogenesis are two steps that occur in parallel, and are hallmarks of malignant invasion. This requires the active cooperation of stromal cells that produce: - matrix metalloproteinases (MMPs) MMPs degrade the ECM. - Angiogenic factors Vascular endothelial growth factor (VEGF) and basic fibroblast growth factor (bfgf).

7 Angiogenesis In general, cells must be within.2 mm of a capillary to receive sufficient nourishment through diffusion. Therefore all tumors must induce angiogenesis to grow beyond a very small size. Pro-angiogenic factors such as VEGF, bfgf, and TGF-β factors attract endothelial cells (ECs) and encourage proliferation. Endothelial cells (primarily) then construct a leaky, irregular tumor vasculature.

8 Geometry of the Tumor Microenvironment The geometry of a malignant tumor can be characterized as a collection of several compartments (1) A necrotic core. (2) The primary tumor mass composed of melanoma cells, infiltrating stroma cells (mainly fibroblasts), infiltrating immune cells, and an intratumoral vascular network. (3) A periturmoral region dense in mast cells, fibroblasts, macrophages, and endothelial cells. (4) Unaffected surrounding tissue.

9 Immune Response The immune system is capable of mounting an effective response against most tumors. Clinical tumors are likely the exception rather than the rule most cancers are destroyed before they cause symptoms or are detected. The tumor microenvironment is highly immunosuppresive. Immune cells can actually aid tumor invasion through the release of growth factors.

10 Modeling at the Tumor-Host Level Dynamics Models that accurately describe cancer cell growth, spread, and the interactions between cancer cells and the host tissue must be derived. Treatment Modeling of various treatment regimes surgery, radiation therapy, chemotherapy. Predicting optimum treatment regimes. Recurrence Models can be used to predict cancer recurrence in response to different treatments.

11 Melanoma Treatment Primary melanoma tumors are surgically removed along with a safety margin of healthy tissue intended to the risk of local recurrence. In the past a 5 cm margin was used regardless of tumor size. Today, margins of 1-2 cm are used, depending on tumor thickness. In thick melanoma tumors, margins of 3 cm were more effective at preventing cancer recurrence than 1 cm margins. How low safety margins can be reduced remains an open question.

12 Figures 5, 6 Reproduced from [5]. Treatment and Metastasis Recurrence: Case Study Initial state of melanoma lesion. Patient reported that it had been growing slowly for three years. Widespread cancer recurrence in satellite metastases occurred 30 days after excision.

13 Model Motivation We will attempt to address several questions using a mathematical model: 1. Can a mathematical model give plausible dynamics of tumor invasion? 2. Macrophages (immune cells) can either aid or benefit tumor progression. What is the mechanism that causes this qualitative change in behavior? 3. Can the phenomenon of metastasis recurrence following primary tumor excision be explained?

14 Model Formulation Geometry and Variables Seven variables are considered in the basic model:

15 Skin Geometry

16 Basic Assumptions 1. Cell motility is achieved according to Fick s Law, which yields a diffusion term. 2. Cancer cell motility additionally depends upon contact with other cancer cells and oxygen concentration. 3. Oxygen is the limiting nutrient and diffuses into the skin. 4. Proliferation of all cells is mediated by the amount of space available and the concentration of oxygen. 5. Cancer cell death is mediated by oxygen concentration 6. Cancer cells that die in response to hypoxic stress become necrotic debris.

17 Basic Assumptions 7. Cancer cells produce angiogenic factors both constitutively and in response to hypoxia. 8. Endothelial cells migrate into the system in response to angiogenic factors and form the tumor vasculature. 9. A basement membrane separates the epidermis from the dermis and restricts cell migration.

18 Full Model Equations

19 Cancer Cells Hill Terms Density dependent diffusion that increases (saturates) with O 2 concentration Lots of space, O 2 high birth rate Low O 2 high death rate

20 Healthy Cells Simple (linear) diffusion Lots of space, O 2 high birth rate: Low O 2 high death rate: Normal turnover (death):

21 Tumor Angiogenic Factor Linear diffusion Constitutive production: Hypoxic production: Normal degradation: Uptake by endothelial cells:

22 Blood Vessel Endothelial Cells When cell density is low, vessels new or unstable. Implies proliferation can occur if sufficient TAF present: When TAF is below the threshold value V T, endothelial cells undergo apoptosis: Baseline death rate:

23 Necrotic Debris Tumor cells that die because of hypoxia become necrotic debris Debris diffuses by linear diffusion and undergoes first-order degradation.

24 Oxygen Oxygen enters from tumor vascular at rate proportional to vascular surface area (A b b) and a permeability coefficient (P), and the difference between oxygen concentration in the capillary and tissue (r C = Capillary oxygen partial pressure) Consumption due to baseline demand (u 0 ) and additional use due to enhanced cancer metabolism.

25 Basement Membrane Basement membrane (BM) is degraded at maximum per capita rate β l by endothelial cells. The switch from radial to vertical growth requires both the degradation of the BM and angiogenesis, and the two processes are tightly coupled. The EC response to TAF can be thought to encapsulate all mechanisms by which the BM is degraded.

26 Basic Model Boundary Conditions Recall the cylindrical geometry of the model: No-flux boundary conditions are used for most variables, except To represent TAF induced endothelial cell influx from the vascular bed a Neumann B.C. is used at the base:

27 Basic Model Boundary Conditions Dirichlet B.C.s are used to hold oxygen concentration constant at the skin surface and at the vascular bed:

28

29 Basic Pattern of Invasion

30 Basic Pattern of Invasion

31 Basic Pattern of Invasion

32 Basic Pattern of Invasion

33 Model Extension - Cellular Immune Response We extend the model by including a class of cells that are directly cytotoxic to tumor cells. We assume that this cell has the properties of macrophages and natural killer (NK) cells and call it (surprisingly) an immune cell (IC). Two new variables are introduced:

34 Cellular Immune Response - Assumptions 1. IAF is produced at a constant rate by tumor cells. 2. Immune cells migrate up an IAF gradient. 3. Immune cells can devote energy to either cleaning necrotic debris or destroying tumor cells. 4. Contact with tumor cells and necrotic debris activates immune cells 5. Activated immune cells produce TAF. 6. Tumor cells are toxic to immune cells approximation of immunosuppressive tumor microenvironment.

35 Immune Response Γ [0,1] is the level of activation

36 Immune Response Model Extensions Γ ² [0, 1] is the level of activation.

37 Cellular Immune Response ICs are imagined to expend energy at some per capita rate $\tau$ which can be devoted either to debris cleanup or killing cancer cells. The amount of energy devoted to either task is proportional to the presence of debris and tumor cells. Energy devotion to debris cleanup is measured as:

38 Cellular Immune Response Suppose that τ can be scaled in such a way as to measure maximum rate of tumor cell destruction - τ c, or dead cell destruction - τ d.

39 Parametrization of Immune Extension

40 Dynamics with Immune Response In some cases, the immune response first inhibits, then helps tumor invasion. - Due to production of angiogenic factors by the immune cells. - Two-phase pattern of invasion

41 Dynamics with Immune Response

42 Another Model Extension: Metastastic Spread Model seeding of metastases into nearby skin tissue as a Poisson process. That is, the time to the next seeding event is an exponential random variable with rate parameter λ seed. Also assume distance from tumor edge described by an exponential RV with rate parameter λ distance. Thus, process can be completely described by two exponential rate parameters: λ seed and λ distance.

43 Immune Response in a Metastasizing Tumor Find that immune response directed against primary tumor can suppress nearby metastases in passing. If a metastasis does succeed in growing, the new immune response suppresses other nearby mets. Moderately metastatic tumors activate a strong regional immune response that suppresses all metastases.

44 Regional Immune Activation

45 Simulation of Surgical Treatment (1) The simulation is begun and the tumor is allowed to grow for a specified amount of time. (2) Then a surgical excision is simulated by setting all the value of all variables within a certain radius from the origin to zero. (3) The simulation is continued for several more years to determine if cancer will reoccur.

46 Surgical Treatment Primary Tumor with Immune Response The immune response typically causes a sharp border between the tumor and the surrounding healthy cells. Therefore fairly narrow margins will generally prevent recurrence, although recurrence will occur if margins are excessively narrow.

47 Surgical Treatment in a Metastasizing Tumor The immune response directed against primary tumor can suppress nearby metastases in passing. Surgical removal of primary tumor can lead either to aggressive metastatic recurrence or long-term metastatic suppression

48 Metastatic Recurrence After Surgery

49 Metastatic Persistence After Surgery

50 Conclusions and Predictions Tumor-immune interaction plays a central role in metastatic disease The model predicts that the immune response to the primary tumor can destroy or hold in check the vast majority of metastases that are produced. Somewhat nonintuitively, moderately metastatic tumors optimally activate a regional immune response.

51 Conclusions and Predictions Clinical Implications In the clinical case by De Giorgi et al., the tumor was known to be very slow growing, suggesting a delicate tumor-immune balance slightly in favor of the tumor. In simulations, only strongly immunogenic tumors were able to control metastatic disease after primary tumor resection. The tumor was also very large, and therefore great deal of associated immune tissue (tumors can be 70% infiltrating macrophages). Thus, perturbation of the immune response is a possible explanation for recurrence.

52 Conclusions and Predictions Clinical Implications Moderately metastatic tumors optimally activated the immune response. Therefore, we may predict that stage II/III tumors may respond best to immunostimulatory therapy.