I find the subject of cancer therapies fascinating. My particular interest is in the possibility of obtaining new insights by modelling and what role mathematics can play in this endeavour. I have heard many talks related to these subjects, both live and online. I was stimulated to write this post by a video of Martin McMahon, then at UCSF. It made me want to systematize some of the knowledge I have obtained from that video (which is already a few years old) and from other sources. First I should fix my terminology. I use the term ‘modern cancer therapies’ to distinguish a certain group of treatments from what I will call ‘classical cancer therapies’. The latter are still of central importance today and the characteristic feature of those I am calling ‘modern’ here is that they have only been developed in the last few years. I start by reviewing the ‘classical therapies’, surgery, radiotherapy and chemotherapy. Surgery can be very successful when it works. The aim is to remove all the cancerous cells. There is a tension between removing too little (so that a few malignant cells could remain and restart the tumour) and too much (which could mean too much damage to healthy tissues). A particularly difficult case is that of the glioma where it is impossible to determine the extent of the tumour by imaging techniques alone. An alternative to this is provided by the work of Kristin Swanson, which I mentioned in a previous post. She has developed techniques of using a mathematical model of the tumour (with reaction-diffusion equations) to predict the extent of the tumour. The results of a simulation, specific to a particular patient, is given to the surgeon to guide his work. In the case of radiotherapy radiation is used to kill cancer cells while trying to avoid killing too many healthy cells. A problematic aspect is that the cells are killed by damaging their DNA and this kind of damage may lead to the development of new cancers. In chemotherapy a chemical substance (poison) is used with the same basic aim as in radiotherapy. The substance is chosen to have the greatest effect on cells which divide frequently. This is the case with cancer cells but unfortunately they are not the only ones. A problem with radiotherapy and chemotherapy is their poor specificity.
Now I come to the ‘modern’ therapies. One class of substances used is that of kinase inhibitors. The underlying idea is as follows. Whether cells divide or not is controlled by a signal transduction network, a complicated set of chemical reactions in the cell. In the course of time mutations can accumulate in a cell and when enough relevant mutations are present the transduction network is disrupted. The cell is instructed to divide under circumstances under which it would normally not do so. The cells dividing in an uncontrolled way constitute cancer. The signals in this type of network are often passed on by phosphorylation, the attachment of phosphate groups to certain proteins. The enzymes which catalyse the process of phosphorylation are called kinases. A typical problem then is that due to a mutation a kinase is active all the time and not just when it should be. A switch which activates the signalling network is stuck in the ‘on’ position. This can in principal be changed by blocking the kinase so that it can no longer send its signals. An early and successful example of this is the kinase inhibitor imatinib which was developed as therapy for chronic myelogenous leukemia (CML). It seems that this drug can even cure CML in many cases, in the sense that after a time (two years) no mutated cells can be detected and the disease does not come back if the treatment is stopped. McMahon talks about this while being understandibly cautious about using the word cure in the context of any type of cancer. One general point about the ‘modern’ therapies is that they do not work for a wide range of cancers or even for the majority of patients with a given type of cancer. It is rather the case that cancer can be divided into more and more subtypes by analysing it with molecular methods and the therapy only works in a very specific class of patients, having a specific mutation. I have said something about another treatment using a kinase, Vemurafenib in a previous post. An unfortunate aspect of the therapies using kinase inhibitors is that while they provide spectacular short-term successes their effects often do not last more than a few months due to the development of resistance. A second mutation can rewire the network and overcome the blockade. (Might mathematical models be used to understand better which types of rewiring are relevant?) The picture of this I had, which now appears to me to be wrong, was that after a while on the drug a new mutation appears which gives the resistance. The picture I got from McMahon’s video was a different one. It seems that the mutations which might lead to resistance are often there before treatment begins. They were in Darwinian competition with other cells without the second mutation which were fitter. The treatment causes the fitness of the cells without the second mutation to decrease sharply. This removes the competition and allows the population of resistant cells to increase.
Another drug mentioned by McMahon is herceptin. This is used to treat breast cancer patients with a mutation in a particular receptor. The drug is an antibody and binds to the receptor. As far as I can see it is not known why the binding of the antibody has a therapeutic effect but there is one idea on this which I find attractive. This is that the antibodies attract immune cells which kill the cell carrying the mutation. This gives me a perfect transition to a discussion of a class of therapies which started to become successful and popular very recently and go under the name of cancer immunotherapy, since they are based on the idea of persuading immune cells to attack cancer cells. I have already discussed one way of doing this, using antibodies to increase the activities of T cells, in a previous post. Rather than saying more about that I want to go on to the topic of genetically modified T cells, which was also mentioned briefly here.
I do not know enough to be able to give a broad review of cellular immunotherapy for cancer treatment and so I will concentrate on making some comments based on a video on this subject by Stephan Grupp. He is talking about the therapy of acute lymphocytic leukemia (ALL). In particular he is concerned with B cell leukemia. The idea is to make artificial T cells which recognise the surface molecule CD19 characteristic of B cells. T cells are taken from the patient and modified to express a chimeric T cell receptor (CAR). The CAR is made of an external part coming from an antibody fused to an internal part including a CD3 -chain and a costimulatory molecule such as CD28. (Grupp prefers a different costimulatory molecule.) The cells are activated and caused to proliferate in vitro and then injected back into the patient. In many cases they are successful in killing the B cells of the patient and producing a lasting remission. It should be noted that most of the patients are small children and that most cases can be treated very effectively with classical chemotherapy. The children being treated with immunotherapy are the ‘worst cases’. The first patient treated by Grupp with this method was a seven year old girl and the treatment was finally very successful. Nevertheless it did at first almost kill her and this is not the only case. The problem was a cytokine release syndrome with extremely high levels of IL-6. Fortunately this was discovered just in time and she was treated with an antibody to IL-6 which not only existed but was approved for the treatment of children (with other diseases). It very quickly solved the problem. One issue which remains to be mentioned is that when the treatment is successful the T cells are so effective that the patient is left without B cells. Hence as long as the treatment continues immunoglobulin replacement therapy is necessary. Thus the issue arises whether this can be a final treatment or whether it should be seen a preparation for a bone marrow transplant. As a side issue from this story I wonder if modelling could bring some more insight for the IL-6 problem. Grupp uses some network language in talking about it, saying that the problem is a ‘simple feedback loop’. After I had written this I discovered a preprint on BioRxiv doing mathematical modelling of CAR T cell therapy of B-ALL and promising to do more in the future. It is an ODE model where there is no explicit inclusion of IL-6 but rather a generic inflammation variable.