How to mathematically model cancer?

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Emman.78
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I am considering doing an extended project for school that uses mathematical modelling, specifically ordinary differential equations, to either model or predict the effects of a therapy on tumour growth. How would I go about this? Please give any advice regarding any of the following:
how to construct mathematical models (what the typical process would be in this case)
How to model a tumour - I understand that I could use either an exponential, logistic or gompertz function to model the growth under normal circumstances, but any extra advice on this would be appreciated

How would I model the therapy (lets take chemotherapy in this example, although if it is not too difficult I will try and model more recent ones). What type of function would I use for this and why? (this is probably one of the main things I am struggling with). I will need to show the effect of this on the tumour of course.

Where can i find experimental data for the growth of tumours with a treatment (it can really be any). I'm struggling to find this so any sites would be greatly appreciated.

When modelling, especially to observe/predict the effect of a drug, do you usually find experimental data, and begin to build your model around that or would you construct a model then find some data that matches that.

Apologies if some of the questions are worded incorrectly, I'm still very new to this and really want to challenge myself and do a project like this for my sixth form project (it is not an EPQ as it will not be assessed, but may serve a similar role on my university application)
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Muttley79
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(Original post by Emman.78)
I am considering doing an extended project for school that uses mathematical modelling, specifically ordinary differential equations, to either model or predict the effects of a therapy on tumour growth. How would I go about this? Please give any advice regarding any of the following:
how to construct mathematical models (what the typical process would be in this case)
How to model a tumour - I understand that I could use either an exponential, logistic or gompertz function to model the growth under normal circumstances, but any extra advice on this would be appreciated

How would I model the therapy (lets take chemotherapy in this example, although if it is not too difficult I will try and model more recent ones). What type of function would I use for this and why? (this is probably one of the main things I am struggling with). I will need to show the effect of this on the tumour of course.

Where can i find experimental data for the growth of tumours with a treatment (it can really be any). I'm struggling to find this so any sites would be greatly appreciated.

When modelling, especially to observe/predict the effect of a drug, do you usually find experimental data, and begin to build your model around that or would you construct a model then find some data that matches that.

Apologies if some of the questions are worded incorrectly, I'm still very new to this and really want to challenge myself and do a project like this for my sixth form project (it is not an EPQ as it will not be assessed, but may serve a similar role on my university application)
Surely this is what you need to find out - we can't do your project for you.
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RDKGames
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(Original post by Emman.78)
I am considering doing an extended project for school that uses mathematical modelling, specifically ordinary differential equations, to either model or predict the effects of a therapy on tumour growth. How would I go about this? Please give any advice regarding any of the following:
how to construct mathematical models (what the typical process would be in this case)
How to model a tumour - I understand that I could use either an exponential, logistic or gompertz function to model the growth under normal circumstances, but any extra advice on this would be appreciated

How would I model the therapy (lets take chemotherapy in this example, although if it is not too difficult I will try and model more recent ones). What type of function would I use for this and why? (this is probably one of the main things I am struggling with). I will need to show the effect of this on the tumour of course.

Where can i find experimental data for the growth of tumours with a treatment (it can really be any). I'm struggling to find this so any sites would be greatly appreciated.

When modelling, especially to observe/predict the effect of a drug, do you usually find experimental data, and begin to build your model around that or would you construct a model then find some data that matches that.

Apologies if some of the questions are worded incorrectly, I'm still very new to this and really want to challenge myself and do a project like this for my sixth form project (it is not an EPQ as it will not be assessed, but may serve a similar role on my university application)
This is the type of question professional mathematical modellers ask themselves, so you're not doing yourself any favours when you decide to jump into answering such a big question when you say that you're very new to this.

Unless you are prepared to work in a collobaration with experts on cancer, or you manage to dig up some reliable data on cancer treatment, I am afraid that this is too much for a 6th former to handle. You would be required to know the maths *and* the biology of cancer treatment and tumor growth.

Alas, I will give you tips on mathematical modelling in general which you would find useful.

Always begin by understanding the full process and identifying the variables and parameters that play a role in whatever process it is that you are modelling. How are you measuring tumor growth, for instance? Is the number of cancerous cells present in the body over time? If so, you need to also look at modelling the rate at which a normal cell becomes a cancerous cell. Also, the age/weight/medical history/family history of a patient play a role in such a model. How do we fit these into the model?

Once the variables and parameters are identified, begin by constructing a very simple equation which can model the general behaviour. Accuracy isn't too important at this stage, it is the principle of getting onto the right tracks. Now, this is where your knowledge of the subject (or your expert friend) comes in; because in order to develop an equation, you need to be familiar with the process. Generally, you drag this very basic model through a computer like MATLAB in order to help you analyse it. Are there any particular values for the parameters where something goes a bit bananas? Why? Can you also explain analytically from the equation any behaviour that you observe? And perhaps the most important aspect of all; is your equation dimensionally consistent? This is extremely important because if it isn't, your model straight up falls flat on the ground. There is a process called non-dimensionalisation which is very useful in Applied Mathematics and modelling, and the purpose of it is to scale your variables so that you can eliminate some of the parameters in your model (this choice is often important!) hence end up with less parameters to vary when testing your model.

From here, you begin developing your model. This process is you would call extending the model. The point is to more accurately capture the behaviour of whatever process you are modelling. Are some variables/parameters more important than others? Does the extended model capture everything the simpler model does, and more? If not, go back and sort it out. How do the predictions differ? Are they agreeing more with any experimental data?


Anyway, I merely give you the snapshot of a mathematical modelling process. If you want my advice, I'd suggest you model something simpler. Look at perhaps modelling a healing of a wound. Skin cells are trying to close up a wound, while bacteria cells are attacking the skin cells. This would lead you to look at competing population growth models such as Lotka-Volterra.
Last edited by RDKGames; 8 months ago
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Emman.78
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(Original post by RDKGames)
This is the type of question professional mathematical modellers ask themselves, so you're not doing yourself any favours when you decide to jump into answering such a big question when you say that you're very new to this.

Unless you are prepared to work in a collobaration with experts on cancer, or you manage to dig up some reliable data on cancer treatment, I am afraid that this is too much for a 6th former to handle. You would be required to know the maths *and* the biology of cancer treatment and tumor growth.

Alas, I will give you tips on mathematical modelling in general which you would find useful.

Always begin by understanding the full process and identifying the variables and parameters that play a role in whatever process it is that you are modelling. How are you measuring tumor growth, for instance? Is the number of cancerous cells present in the body over time? If so, you need to also look at modelling the rate at which a normal cell becomes a cancerous cell. Also, the age/weight/medical history/family history of a patient play a role in such a model. How do we fit these into the model?

Once the variables and parameters are identified, begin by constructing a very simple equation which can model the general behaviour. Accuracy isn't too important at this stage, it is the principle of getting onto the right tracks. Now, this is where your knowledge of the subject (or your expert friend) comes in; because in order to develop an equation, you need to be familiar with the process. Generally, you drag this very basic model through a computer like MATLAB in order to help you analyse it. Are there any particular values for the parameters where something goes a bit bananas? Why? Can you also explain analytically from the equation any behaviour that you observe? And perhaps the most important aspect of all; is your equation dimensionally consistent? This is extremely important because if it isn't, your model straight up falls flat on the ground. There is a process called non-dimensionalisation which is very useful in Applied Mathematics and modelling, and the purpose of it is to scale your variables so that you can eliminate some of the parameters in your model (this choice is often important!) hence end up with less parameters to vary when testing your model.

From here, you begin developing your model. This process is you would call extending the model. The point is to more accurately capture the behaviour of whatever process you are modelling. Are some variables/parameters more important than others? Does the extended model capture everything the simpler model does, and more? If not, go back and sort it out. How do the predictions differ? Are they agreeing more with any experimental data?


Anyway, I merely give you the snapshot of a mathematical modelling process. If you want my advice, I'd suggest you model something simpler. Look at perhaps modelling a healing of a wound. Skin cells are trying to close up a wound, while bacteria cells are attacking the skin cells. This would lead you to look at competing population growth models such as Lotka-Volterra.
Thank you! This was extremely insightful. The biology of cancer treatment is not too difficult to understand, it’s just the maths that is becoming increasingly difficult . I will research this topic further, but I can always pick another topic (such as the one you suggested if the need arises).
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Emman.78
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(Original post by Muttley79)
Surely this is what you need to find out - we can't do your project for you.
I was more asking for tips on how to model since all the articles I look at jump straight into it without providing much of an explanation of why they use them. I also found that a lot of articles are modelling vastly different system, so I thought it would be helpful to get some advice about where to start. It certainly wouldn’t have been “doing my project for me”.
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Gregorius
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(Original post by Emman.78)
I am considering doing an extended project for school that uses mathematical modelling, specifically ordinary differential equations, to either model or predict the effects of a therapy on tumour growth. How would I go about this?
I applaud your ambition; this is quite a difficult project to do at sixth form level, but if you are comfortable with the ordinary differential equations that you’ve studied at this level, you should be able to do something interesting.

Please give any advice regarding any of the following:

how to construct mathematical models (what the typical process would be in this case)
How to model a tumour - I understand that I could use either an exponential, logistic or gompertz function to model the growth under normal circumstances, but any extra advice on this would be appreciated.
The very basic models for tumour growth are based on ideas nicked from population biology: the tumour cells are considered to be one “species” competing against the non-tumour cells. So, exponential models are derived from little or no competition, and the logistic model is a classic example of a model of species competition. Similarly, the Gompertz model, and the Bertalanffy model. I would think that at your level, a project comparing just these basic models would be quite enough.

Are you able to get hold of a copy of Britton’s “Essential Mathematical Biology”? It’s got a good chapter introducing these models. Also, perhaps, take a quick look at this paper:

https://bmccancer.biomedcentral.com/...885-019-5911-y

It provides a not-too intimidating introduction to these sort of models, has some tumour growth data and looks at model fit. It’s probably also right at the limit of what’s accessible to you.

How would I model the therapy (lets take chemotherapy in this example, although if it is not too difficult I will try and model more recent ones). What type of function would I use for this and why? (this is probably one of the main things I am struggling with). I will need to show the effect of this on the tumour of course.
Don’t! This is way too hard. Modern chemotherapies can be as complex as targeting individual cell-signalling pathways in tumour cell – you’re looking at the prospect of stochastic partial differential equations here. Scary!

Where can i find experimental data for the growth of tumours with a treatment (it can really be any). I'm struggling to find this so any sites would be greatly appreciated.
Best place is probably to extract it from the supplementary materials of open-access journal papers online. Google search “tumor growth data” turns up quite a lot.

When modelling, especially to observe/predict the effect of a drug, do you usually find experimental data, and begin to build your model around that or would you construct a model then find some data that matches that.
A lot of the early models in this field were phenomenological models – taking observed data and trying to fit simple differential equations to that data. But pretty soon, theory had to get involved, as the early models were more than a bit approximate. So the logistic model already involves the idea of tumour cells and non-tumour cells as competing forces. Almost all models these days are based on theory: adding on what we know of the processes of cellular biology, immunology, tumour vascularization etc. etc etc. It all gets very complex very quickly!

Apologies if some of the questions are worded incorrectly, I'm still very new to this and really want to challenge myself and do a project like this for my sixth form project (it is not an EPQ as it will not be assessed, but may serve a similar role on my university application)
A key thing, I think, is for you to not take on too much – and as it’s not an assessed project, make sure you enjoy it! So start with the basics, and try and do that well.
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drvramachandran
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(Original post by Emman.78)
I am considering doing an extended project for school that uses mathematical modelling, specifically ordinary differential equations, to either model or predict the effects of a therapy on tumour growth. How would I go about this? Please give any advice regarding any of the following:
how to construct mathematical models (what the typical process would be in this case)
How to model a tumour - I understand that I could use either an exponential, logistic or gompertz function to model the growth under normal circumstances, but any extra advice on this would be appreciated

How would I model the therapy (lets take chemotherapy in this example, although if it is not too difficult I will try and model more recent ones). What type of function would I use for this and why? (this is probably one of the main things I am struggling with). I will need to show the effect of this on the tumour of course.

Where can i find experimental data for the growth of tumours with a treatment (it can really be any). I'm struggling to find this so any sites would be greatly appreciated.

When modelling, especially to observe/predict the effect of a drug, do you usually find experimental data, and begin to build your model around that or would you construct a model then find some data that matches that.

Apologies if some of the questions are worded incorrectly, I'm still very new to this and really want to challenge myself and do a project like this for my sixth form project (it is not an EPQ as it will not be assessed, but may serve a similar role on my university application)
The commonest mathematical tumour model is the Gompertizian curve.
ther are many limitations to this curve, but is adequate for describing the tumour growth.
The effects of chemotherapy is based on Log Cell Kill model. Two other hypothesis of effective delivery of chemotherapy and resistant mutations are the Goldie-Coleman and Simon Norton hypothesis.
for Radiotherapy the linear-quadratic model of cell kill with Alfa/beta Ratio.
There is also effects of hypoxia, repopulation, angiogenesis etc.
there is a graph of therapeutic efficacy/ toxicity curves, which gives the therapeutic ratio.
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Emman.78
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(Original post by Gregorius)
I applaud your ambition; this is quite a difficult project to do at sixth form level, but if you are comfortable with the ordinary differential equations that you’ve studied at this level, you should be able to do something interesting.



The very basic models for tumour growth are based on ideas nicked from population biology: the tumour cells are considered to be one “species” competing against the non-tumour cells. So, exponential models are derived from little or no competition, and the logistic model is a classic example of a model of species competition. Similarly, the Gompertz model, and the Bertalanffy model. I would think that at your level, a project comparing just these basic models would be quite enough.

Are you able to get hold of a copy of Britton’s “Essential Mathematical Biology”? It’s got a good chapter introducing these models. Also, perhaps, take a quick look at this paper:

https://bmccancer.biomedcentral.com/...885-019-5911-y

It provides a not-too intimidating introduction to these sort of models, has some tumour growth data and looks at model fit. It’s probably also right at the limit of what’s accessible to you.



Don’t! This is way too hard. Modern chemotherapies can be as complex as targeting individual cell-signalling pathways in tumour cell – you’re looking at the prospect of stochastic partial differential equations here. Scary!



Best place is probably to extract it from the supplementary materials of open-access journal papers online. Google search “tumor growth data” turns up quite a lot.



A lot of the early models in this field were phenomenological models – taking observed data and trying to fit simple differential equations to that data. But pretty soon, theory had to get involved, as the early models were more than a bit approximate. So the logistic model already involves the idea of tumour cells and non-tumour cells as competing forces. Almost all models these days are based on theory: adding on what we know of the processes of cellular biology, immunology, tumour vascularization etc. etc etc. It all gets very complex very quickly!



A key thing, I think, is for you to not take on too much – and as it’s not an assessed project, make sure you enjoy it! So start with the basics, and try and do that well.
This is brilliant! I'm going to try and see if I can get a copy of that book you mentioned, but I will also read the paper you've provided now!
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Emman.78
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(Original post by drvramachandran)
The commonest mathematical tumour model is the Gompertizian curve.
ther are many limitations to this curve, but is adequate for describing the tumour growth.
The effects of chemotherapy is based on Log Cell Kill model. Two other hypothesis of effective delivery of chemotherapy and resistant mutations are the Goldie-Coleman and Simon Norton hypothesis.
for Radiotherapy the linear-quadratic model of cell kill with Alfa/beta Ratio.
There is also effects of hypoxia, repopulation, angiogenesis etc.
there is a graph of therapeutic efficacy/ toxicity curves, which gives the therapeutic ratio.
Thanks, I'm going to try and research those and see which one would be best for me to model.
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