Cancer as a Dynamic Network

---

Update

Simon

I have tried to run the file, but there is following error:

??? Undefined function or variable 'addlib'.

Error in ==> C:\Documents and Settings\mpiotrowska\Desktop\TUMOUR PROJECT\tumor.m

On line 3 ==> addlib({'lib-misc'});

Are you sure that you used the matlab function? Which matlab function (instead of addlib) can I use?

Sangus 19:10, 17 June 2007 (MDT) ... oooops..! Sorry, addlib is a function I use to load Matlab libraries that I have written. The only function required is 'choose.m' and this is included. So just comment out the addlib line and all should (!) be fine...?

no.. won't be fine.. now you will need this: Choose.zip

---

Monika -- here is the current work, in case you would like to do something on it today (Sat). I will be back Sunday.

• Download zip here
• As you will see, I have added the fact that the cells grow before differentiating -- this is to overcome 'explosive' events when a new cell is born, pushing others far away
• As you can also see, this visualization is in 2D, but it will be easy to convert to 3D now - I had to get this growth process working though
• There are very nice visualizations in Matlab for drawing spheres, or even elliptical objects, which I have it set up to do (but commented out), though this takes some time
• One possible thing to do, would be to go back to NetLogo and try to code the physics stuff in a similar way to here (now that we know what works) .. since NL has better (faster) visualizations for spheres in 3D ...?
• See you Sunday (or today?!)

---

Group Org

Next Meeting

Monday, Week 3 (location Student Centre Cafe, time 6:30pm)

Agenda

Discuss model as it is, what needs to be added (and in what priority); current literature -- who has done similar to this before???

People

• Monika Piotrowska (monika AT mimuw.edu.pl)
• Simon Angus (s.angus AT unsw.edu.au)
• ...

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

Consulting to us!

• Will Ludington

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.

Model Parameters

Recent research data shows that glucose concentrations below 2,5mM seemed to be extreme difficult to most of the cancer cell lines. For comparison, normal (physiological) blood glucose levels are around 5 mM (normally before a meal) up to 10mM or slightly higher within two hours after a meal.

The oxygen pressure varies in dependency where the tissue is located, in other words how far away is the next blood vessel. Oxygen pressure in arteries is 90mmHg and in veins it is around 40mmHg, so that physiological conditions are somewhere in between. Consumption of oxygen (by tumour cells) depends on the different mutations a tumour could have. Most surviving (in patients) tumours have developed the ability to induce angiogenesis, by secreting VEGF, (EGF) and FGF - which are known growth factors. Depending on the tumour ability to induce angiogenesis, tumours growth deprnds on oxygen levels between 40 and 90 mm Hg. But, there are reports about tumours growing totally without oxygen – the tumours contained the cells growing under anaerobe conditions forming lactate out of glucose.

Hence, we have to look at each single condition for each single tumour or, to set of parameters describing the tumour environment.

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

Week 2(Thu)

Meeting at 6:30pm in cafe

Week 2(Fri)

Meeting at 1:00pm in cafe

Refs

• Quick overview of Cancer from recent Nature article
• Cancer modelling database clearing house: http://calvino.polito.it/~mcrtn/index.html