Cancer for dummies: A mathematical introduction to cancer
Explaining cancer growth and treatment from a mathematical point of view
Have you ever been interested in a topic that strands so far away from your previous studies that you feel too overwhelmed about all that you would need to learn before you can even begin to understand it? I had precisely this problem with the topic of cancer and that is why I decided to write about it.
I am no health professional or biologist, I am a computer scientist, so my understanding (and consequently explanation) of cancer is super limited and there are many things about the disease that I haven’t even begin to understand. So even though I will explain it in the best of my capacities, be aware that I might provide some small inaccuracies, mainly in the naming of some phenomenon. While my previous degrees are in computer science, this year I began a PhD in Math and Stats and I am doing my doctoral research at the Computational Biology program at the Peter MacCallum Cancer Centre, doing cancer treatment research. It has been quite an ordeal because I will have to produce breakthrough science in a field of study where I do not even understand what most of the words mean, or how concepts relate to each other, whenever I read a very technical academic paper. However, I began reading papers explaining cancer from a mathematical point of view, and that has allowed me to understand it in a very detailed, yet specific, manner thanks to my engineering background without requiring a thorough understanding of medicine or biology.
So what causes cancer and how can we prevent it?
This is a topic I will not talk about because I frankly have no idea. I merely created this section to specify that I will only focus on cancer growth and treatment, starting from the point where there is exactly one cancerous cell already existing.
Tumor formation and progression
The Bozic Model
Cancer is a very complex disease in terms of biological pathways, cells, enzymes, etc. However, it can be modelled mathematically in a rather simple manner if we ignore the biological reasoning behind it.
First, it starts with a normal cell that has a small chance to mutate on every life cycle. After a certain threshold number of mutations are acquired, the cell becomes malignant. This is known as tumorigenesis and it is where cancer begins. Once there is a cancer cell, it continues following this process. On every generation the cell can either split into new cells, acquire a new mutation, or die. When an already cancerous cell acquires a new mutation, it starts a new branch, known as subclone, of cells with that particular mutation. Then that branch continues proliferating at an even faster rate than its parent.
A basic model of cancer progression, proposed by Ivana Bozic, depends on 4 main factors:
- Probability of cell death (d)
- Probability of cell birth (b)
- Probability of cell mutation (m)
- Selective advantage (s), which is the difference between b and d (b-d) specifically on the first cancer cell.
Based on this model, known as the branching model of cancer, the disease can behave very differently depending mainly in the initial values of m and s.
Using the Bozic model and considering possible values for the 4 parameters it is based on, one can understand cancer progression since the moment there is a single cell until it grows into a big tumor. And even though the selective advantage already dictates that cells have a higher chance of duplicating than of dying, leading to growth, as new mutations are acquired the difference between d and b continues increasing, following a couple particular mathematical patterns. This leads to exponential growth where child subclones eventually outgrow their parent clones in size due to a faster proliferation rate.
The following table summarizes cancers with different combinations of parameters and explains how they behave in terms of reaching a size big enough to be detected.
- Fast-growing cancers (those with s=0.1 and big values of m) have a higher probability of progressing into a big and dangerous tumor. And when they do, they do it quickly. Their times to detection in years lie within the single digits.
- Slow cancers are much less likely to become a problem, and when they do, it takes decades for them to even reach that size.
Based on these facts, which cancer do you think poses a greater threat: a fast cancer, or a slow cancer?
The fact is that a fast-growing cancer is very scary and dangerous because it can grow too much, out of control, before it is detected. And because of this, it is very likely to metastasize, expanding to all the body and quickly turning deadly. However, while less imposing, a slow-growing cancer comes with its own threats which can make it just as dangerous, just in a different way.
The previous graphs show how fast and slow cancers grow. They show the number of cells (the log of this value) vs time (in generations) for each subclone (line), and the color indicates the number of mutations that particular subclone has. As it can be seen, a fast-growing cancer gets to a dangerous size quite swiftly, but thanks to that, they do not have much time to mutate, so at the time of detection the composition consists mainly of cells with 1 or 2 mutations.
Slow cancers take way longer to grow big. But thanks to that, they have plenty of time to acquire mutations and become a much more complex mass by the time they reach detection. The pie graphs on the left show the proportional composition of the tumor at the time of detection. The issue with this is that as cells acquire more mutations, they also acquire more resistance to therapy. From the previous graphs a red cancer would be the easiest to deal with, followed by the blue, green, etc.
Knowing this and looking at the previous graphs you might start understanding why slow growing cancers can be dangerous.
The previous images showed that a fast cancer can indeed kill you very quickly, but a slow cancer has a higher chance of becoming unstoppable. This table compares the proportion of curable and uncurable cells in tumors that took a couple of years to grow vs tumors that took decades. As it can be seen, almost half of a fast-growing tumor is sensitive to therapy at the time of detection, while less than 10% of the tumor is curable when it took longer to develop. Due to this, comparing cancers that do get detected on time, one is much more likely to be able to successfully suppress and even eliminate a fast-growing cancer than a slower one. Sadly, in many scenarios there is nothing that can be done to stop a slow growing tumor once it has reached a certain size and resistance, and the most probable outcome in those scenarios is quite negative.
Now that we have a slight understanding of cancer formation, growth and how it can kill us, let’s get into how we can fight it.
Just as we managed to understand cancer mathematically without really understanding the biological explanation behind it, we can also understand therapy through math. While there are many different types of therapies available, all with their different reasoning and implications, many share similarities in their mathematical implementation. So for the sake of simplicity, I will simply discuss therapy in a general way rather than particularly.
The math behind therapy is more complex than that of growth because it is intertwined with the math of resistance, so I will not go into much detail. The most important thing one has to understand is that therapy increases the death probability (d) of sensitive cells, and resistance lowers it for resistant cells. And once d is greater than b the cancer is effectively shrinking rather than growing.
However, as the simulations of tumor growth have shown, a fraction of cells is usually already resistant to treatment at the time of detection. And since growth is exponential, it is usually a matter of time before that fraction grows and becomes the bulk of the tumor while other cells are killed, even while on active treatment. Because of this, there has been a recent spike on research regarding therapy strategies in order to effectively kill the tumor, or at least to maximize the time it takes before it becomes immune.
Based on what we think we know so far, what do you think is more effective. Constant maximum dosage, or intermittent drug treatment?
If you answered constant maximum dosage, you are in line with the prevalent medical belief until recently. It is quite intuitive that maximizing dosage of cancer killing drugs would lead to the maximum effectiveness, but recent studies have demonstrated otherwise.
In the last decade, doctors have been exploring adaptative therapies where medication is turned on and off instead of keeping the patient on the maximum tolerated dosage constantly, and it has actually led to better clinical outcomes. It might seem counterintuitive, but it actually makes sense and it can be explained using Darwin’s theory of evolution and competition where there is some survival of the fittest. This is because drug resistance for cells, at times where no drug threat is present, comes at a cost on the cell’s overall fitness.
In an attempt to explain the adaptative therapy interaction in intuitive terms, let’s try to visualize cancer as a race where:
- Each competitor is a cancer cell.
- Each competitor can be better or less prepared for good performance depending on its own characteristics and on the race’s challenges.
- Competitors can slow down or quit the race, representing cell death.
- Competitors ahead in the race can invite friends in, mimicking cell birth.
- Changing what discipline the race is on can represent different treatments.
If the race starts off as a marathon on an average day (no treatment), participants with a better overall fitness and the lightest marathoning gear will be the best performing. So, all of those will get ahead in the race, while random participants that for some reason decided to wear various layers of winter clothing and boots will start lagging behind or simply quitting the race. But let’s imagine the race suddenly turns into an alpine adventure up the Everest (start drug treatment). In that very instant, many of the participants running on shorts and t-shirts will either start drastically slowing down or plainly dropping out of the race, and the number of participants still competing will decrease by a huge amount.
If this harsh environment only lasts for a bit before going back to being a marathon, the few marathoners that struggled but withstood during this time will regain their pace, get back to the top positions and once again start inviting more marathoners into the race while those with mountain gear stay at the back of the pack and in small numbers. However, if this harsh environment is maintained for the entire remainder of the race, most if not all the marathoners will indeed quit the race, but the random people that were wearing winter clothing and snow boots (resistance to that particular treatment) will start taking over. Eventually the majority or entirety of the race will be experienced mountaineers extremely prepared for the harsh environment, and since they are even enjoying it, they will start inviting their alpine friends and it will grow steadily. At this point, once there are no more marathon runners (no more cells sensitive to treatment) the race/tumor will once again resume its exponential growth speed where all the participants/cells involved are already adapted to the environment. Even if the race returned back to a marathon setting, mountaineers will continue ahead at whatever pace they want since they do not need to compete against the fit marathoners anymore.
This is the logic behind the recently proposed adaptative therapy.
- A) If no treatment is applied, sensitive cancer cells will thrive.
- B) If treatment is constantly applied, tumor will be temporarily slowed down, but resistant cells will eventually (quickly) thrive.
- D) If a strategy where treatment is applied until it kills a specific proportion of the population, and stopped until it reaches certain size again, tumor is controlled for way longer before it becomes immune to therapy.
And clinical trials exploring this method proved it significantly increases the time to immunity. A recent study  where 546 patients were put in intermittent therapy has had outstanding results. In the control group (constant dosage, b) 14 of the 16 patients had tumor progression before 27 months, with a median of 16 months.
While only 1 of the 11 (out of the 546, those that have had intermittent therapy for at least a year) patients has reached progression before the 33-month mark. Which suggests a drastically higher median time, at least 3.38 times greater.
This is a great finding, and it is very promising for several reasons. First because it extends the patient’s life, but also because it reduces the amount of drugs being administered to the patient. This reduces both costs and adverse effects of therapy.
Fortunately, as previously mentioned, there are different possibilities for therapy, so even if cancer gains resistance to one therapy, there is still hope by relying on alternative therapies. However, there is also interconnection between resistance acquisition when multiple therapies are in play. Without going into much detail, because it keeps getting more complex as we go on, research has found that simultaneous therapy is much more effective than sequential therapy.
Due to the way resistances are acquired, it is better to introduce multiple challenges or treatments at the same time rather than applying one and then another one once that one fails.
Following the race analogy, finding an alpine hiker and ice climber (dual-resistant cell) within the initial population of marathoners is much less likely to happen than finding one from a population that already consists of only mountaineers (mono-resistant cells). From a population of mountaineers, it is more likely that some also climb, so that if after hiking you now introduced climbing into the race, some would be able to go on and eventually invite more ice climbers until eventually all participants are ice climber mountaineers (dual-resistant).
This was further demonstrated by Ivana Bozic when exploring therapies on patients with and without dual resistors.
Genetic vs Epi-Genetic Mutations
In my previous examples, it has been assumed and exemplified that acquiring a resistant-related mutation implies acquiring resistance. This has been the consensus and the generalized belief of the scientific and medical community so far, but recent research seems to prove otherwise. Up to earlier this year, cancer resistance has always been believed to follow a 1:1 genotype to phenotype ratio, which means that once the cell gets the genetic alteration (change in genotype) there will be a deterministic change in phenotype (how the genotype is expressed in the cell, given the environment). However, recent studies by Sui Huang show that cancer does not follow this 1:1 ratio. In fact, for every genotype there are many possible phenotypes, and every phenotype can be due to many possible genotypes.
This is revolutionary and it will change how cancer is understood and studied from now on, because it implies a slightly different method of resistance acquisition than what I described before.
Traditionally, the prevalent belief was that cells have a fixed fitness for any given genotype. Position in x (genotype) can be mapped to specific position in y (phenotype).
However, it has just recently been shown that this is not the case. Actually, cells with a given genotype (or series of mutations, talking about cancer) seem to be mapped to an entire 3-D space known as the gene regulatory network which shows an entire epigenetic landscape of possible phenotypes.
What this means for cancer is that if there was a mutation believed to confer resistance, in fact it only increases the possibility of acquiring it.
The full scope of the implications this entails are still not entirely clear because the results have been out for less than a month. However, two of the biggest implications that we can already foresee are that:
- Resistance-related mutation does not necessarily mean resistance.
- Resistance could be reversed.
Going back to the example of the race. This would mean that instead of having the participants be either wearing or not specific gear, each participant could be carrying a backpack with different gear which they decide when to wear and use as the environment requires it. So participants with identical backpacks (genotypes) could follow different strategies (phenotypes) due to environmental issues. And participants following identical strategies could have different things in their backpacks, as long as both include the possibility of that specific chosen strategy. And participants with any given backpack would be able to change their strategy back and forward as the environment requires and allows for, and this could lead to very different outcomes to the race than those previously envisioned.
This last concept, the changing of the strategy without the changing of the being is known in cancer as an epi-genetic mutation, and the possibility of having epi-genetic mutations is known as cancer plasticity (referencing plastic’s malleability).
Now that I have shared pretty much everything I know about cancer so far, I also want to share why it is useful. Understanding how any given phenomenon behaves, in this case cancer, allows us to create computer models that emulate that behavior. And leveraging the fact that we have real data to compare those models to, one can fine-tune a model to match either the average behavior of the disease, or that of specific patients.
In the case of a patient-by-patient model matching, once a model accurately emulates its behavior, one can use it to run tests and predictive analysis. And this is exactly what my thesis will focus on. Using these mathematical principles, and the data available through the cancer center where I am doing my research, we can create mathematical models to match the current clinical state of a patient and run many tests on the model in order to find the most effective treatment. Through models one can explore a virtually infinite number of therapeutic strategies, investigate different modes of resistance acquisition, look for ways to leverage plasticity to reverse immunity, among many other possibilities. And this way one can generate insights and come to conclusions that create new clinically applicable science, without having any formal academic background in the field.
I hope someone finds this useful. It took me some work getting up to speed with cancer research given my background, so I wanted to create this to make it easier for anyone else wanting to enter the medical or biological field coming from an engineering background. Or even if it is not for people pretending to give it that use, I also wanted to share another perspective on cancer that a common folk like myself can understand. And finally, to shed some light on the applicability of computer science and engineering in pretty much any field anyone could want to embark on. With a logical, mathematical, or even algorithmical understanding of a phenomenon, someone with the thought process and skillset of an engineer can tackle it and bring valuable contributions in any field he is interested in.
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