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Cancer as a Dynamic Network

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Group Org

Next Meeting

Monday, Week 3 (location <tba>, time <tba>)

Agenda
Feedback on pseudo-code generation

People

(please add if interested, there is work to do!)

Consulting to us!

Concept

Tumor growth has been well studied and is a very complicated process. However, there are various dynamics of the growth process that are not adequately captured by theoretical models. The main approach in the literature is to use differential equations to model the diffusion of (say) nutrients (glucose, oxygen) to the growing cells, to promote the cell growth observed.

This project aims to build a network based model of cancer growth. Phases that could be modelled are:

  • Initial growth (from stem cell to a group of cells, diffusion mediated);
  • Angiogenesis (appropriation of blood-vessel supply from the host);
  • Metastatis (the splitting and movement of stem cells to other sites in the host).

In the first place, purely the stem-cell to many cell phase will be modelled. The concept is to begin with a central node (the stem cell) and then allow this node to add (dynamically) other nodes to the growing cancer network. A spring-algorithm will be used for visualisation of the growing tumor.

Nice features of such an approach are the following:

  • Non differential equation model (if time-scales of change in the tumor are similar then differential equations may not capture dynamics);
  • Nutrient diffusion can be modelled by a novel 'leaf-node to core nodes' process where the leaf (outer) nodes are assumed to be at the systemic nutrient level, all other cells will gain nutrients only via a minimum-path to the leaf node;
  • Nodes can be made to build edges between them to solidify a matrix of cells, thus enabling a stable structure despite the formation of the necrotic core due to central cell death;
  • Angiogenisis can be captured by using an edge path into the tumor as a proxy for a growing vessel, feeding into the leaf-to-core model of diffusion above, re-generating the core region

Model

(from discussion on large post-it note)

Basic Rules

Suppose that a stem cell begins to generate new cells in an umediated tumor growth phase under the presence of diffused vascular CHO and O2 nutrients. Call this cell x, and apply the following rules: 

  i. a       ==> b--a                       stem division

 ii. b--a    ==> d--b--a--c                 branching

iii. b--a--c ==> d--b--a--d--c--e           'push'-branching
  1. Stem division: A parent stem cell divides and produces two daughter stem cells, one which replaces itself (a), the other which is new (b);
  2. Branching: A stem cell (a), which is connected with cell (b), produces two daughter cells, one which replaces itself (a) and one which is new (c), either differentiated or stem; at the same time cell (b) replaces itself and 'produce' cell (d);
  3. Push-branching: A stem-cell (a) acts in the same way as rule ii, but 'pushes' the new cell away from the core; stem cells (b) and (c) apply rule ii.

Additionally, to connect the growing mass of cells, older cells in the periphery of the structure become inter-connected by a process of edge creation (this has no physical meaning, but is used to form spatial structure in the growing tumor slice: 

iv. (x,y)  = (if r(x,y) < r*) => x--y            edge creation

Two nodes (x) and (y) form an edge between them if they are 'close' in proximity.

Nutrient supply

Suppose that nutrient supply is mediated to some cell x_i based on the following rule: 

N(x) = f(h_min,g_0)

where h_min is the minimum network 'hops' (length of path) from a leaf node (where nutrient levels are assumed to be at the maximum value 'g_0') to the position of the cell 'x_i'. Nutrients to x_i are increasing in g_0 but decreasing in h. The hop value is then being used as a proxy for diffussion distance (and inter-distance consumption by other cells) to the cell x_i. This process deals well with the two phases of growth and angiogenesis.

Growth: nutrients will be largely influenced by h. Angiogenesis: nutrients will be influenced by the intrusion of vessels into the tumor, causing inner-cells to become 'leaf-nodes', hence dramatically reduces the value of h_min. More than one path-way might be close to cell x_i in which case, the nutrients it is able to use will approach a limiting value (say) m_0, the basal metabolic rate for a cell.

Work Schedule

Current

  • develop pseudo-code for modelling approach (due Wk2-Friday)

Future

  • build basic NetLogo model, check:
    • growth (!)
    • network structure realistic ?
    • necrotic core ?

Done

  • discuss initial concept
  • formulate modelling approach

Past Group Meetings

Week 1(Tues-Fri)
various meetings between Monika and Simon
Week 1(Sun)
Meeting at 8:30pm in cafe to organise model


Refs

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