Wednesday, 24 February 2021

Adaptive Cancer Therapy: tackling cancer drug resistance by capitalising on competition

How to deal with resistance? This is the headline question these days with regards to COVID vaccines. But it is an important question also in cancer therapy. Over the past century, oncology has come a long way, but all too often cancers still recur due to the emergence of drug-resistant tumour cells. How to tackle these cells is one of the key questions in cancer research. The main strategy so far has been the development of new drugs to which the resistant cells are still sensitive. While this approach has been successful, it often doesn’t take too long for further mutants to arise, requiring yet another set of new treatments.

About ten years ago, a group at the Moffitt Cancer Center in Florida proposed that, alternatively, it may be possible to fight resistance through a change in treatment scheduling. Tumours are complex ecosystems in which cancer cells compete with each other for limited space and resources. The researchers hypothesised that it may be possible to leverage this competition to suppress the growth of resistant cells.

Cancer drugs are typically given so as to maximise the tumour cell kill. However, this quickly releases any surviving resistant cells from intra-tumoral competition, allowing them to proliferate (Figure 1). Instead, the researchers proposed “adaptive therapy” in which treatment is reduced once the tumour burden has been brought down to a tolerable level. In this way a pool of sensitive cells is maintained which will competitively suppress the resistant cells (Figure 1). This idea was inspired by similar approaches which are used in the management of pesticide resistance and invasive species, and was supported by prior mathematical modelling studies carried out, for example, at the Wolfson Centre for Mathematical Biology (WCMB) in the Mathematical Institute in Oxford. Adaptive therapy has subsequently been developed further by the Integrated Mathematical Oncology (IMO) department at Moffitt (the first mathematics department in a Cancer centre), and has recently been successfully tested in a clinical trial in prostate cancer, where it achieved both a 10 month increase in the time to progression, as well as a 53% reduction in cumulative drug usage.

But what are the factors which determine whether or not adaptive therapy will be beneficial? Despite promising pre-clinical and clinical results, and a number of elegant modelling studies, these factors have remained unclear. Yet this knowledge is important, as there is growing interest in testing adaptive therapy also in other settings. One factor which has been a point of particular controversy was the so-called “cost of resistance”. This refers to the concept that as resistant cells have to divert energy towards mechanisms to deal with the drug (e.g. pumps that remove drug from the cell), they will proliferate more slowly in a drug-free environment, i.e. they will pay a fitness cost for their resistance trait. It is intuitive that such a cost will facilitate keeping resistant cells at bay, and this had been one of the key arguments for adaptive therapy in its original development. But is it required? For example, an experimental collaborator of ours at Moffitt found that he could detect such a cost, if he grew cells in low nutrient conditions, but not if he grew them in normal culture medium. Given this context dependence, can adaptive therapy still work even if resistant cells do not pay any cost?

To address this question, we, Maxi Strobl, a DPhil student in the Systems Approaches to Biomedical Science Doctoral Training Centre (SABS DTC) and Philip Maini (Oxford Mathematics) in collaboration with a team of theoreticians, clinicians and experimentalists at the IMO (Jeffrey West, Mehdi Damaghi, Mark Robertson-Tessi, Joel Brown, Robert Gatenby and Sandy Andersonand mathematician Yannick Viossat (Universite Paris-Dauphine) analysed a simple ODE model. We assumed cells were either sensitive to treatment, or fully resistant, and that the two populations competed with each other in a Lotka-Volterra fashion. Subsequently, we compared two treatment protocols: i) continuous therapy at a constant, high dose, and ii) adaptive therapy, in which treatment is withdrawn upon a 50% burden reduction, and only reinstated when the tumour returns to its initial size (the algorithm used in the prostate cancer clinical trial). Using phase plane analysis we were able to show that adaptive therapy was beneficial under a wide range of conditions in our model.

This result was not only comforting, but the phase planes allowed us to visualise our argument in a way that was more easily understandable to our biological and clinical collaborators. Essentially, we plotted the growth rate of the resistant cells as a function of the cell densities of the two populations, and superimposed the trajectories of the tumour during continuous or adaptive therapy. This shows quite nicely how, by maintaining more sensitive cells for longer, adaptive therapy stays in regions of smaller resistant cell growth than continuous therapy. In fact, an artistic collage in which we plotted these surfaces in 3-D (the third dimension being the resistant cell growth rate), was chosen for the cover image of the February 15th issue of Cancer Research in which our paper has appeared. Phase planes on the cover of one of the leading cancer research journals; what a great sign for the increasing role of mathematics in oncology (Figure 1)!

But what about resistance costs? To our surprise we found that costs seemed to have only a limited effect. They slowed disease progression, but did not change the relative benefit of adaptive therapy in our model. While this was a useful result, it was a bit puzzling. Costs might not always be necessary, but it seemed unlikely that they had such little impact. It turned out we had forgotten an important factor, which determines the actual “cost” of a resistance cost: the rate of cell death (also referred to as cell turnover). If this is small, then even a large cost is not a big problem, as the resistant cells can essentially just “hang out” until the tumour is (re)treated. However, if turnover is fast, then cells have much fewer opportunities to divide in their lifespan, so that every delay has an impact. In fact, this holds true not only for inhibition caused by resistance costs, but also for interference through competition.

We concluded that tumour cell turnover is an important, but so far overlooked, factor in adaptive therapy. Moreover, when we tested this by fitting our model to data from prostate cancer patients, we found a pattern which was indeed consistent with our hypothesis: patients whose tumours progressed on treatment were characterised by smaller estimates for their values of resistance cost compared to those patients who did not progress and who had a similar estimated rate of turnover (As a side note: interestingly, including an explicit death term can fix a number of problems of the logistic growth model and the Lotka-Volterra model more generally. If you’re interested to learn more, check out this blog post).

So what? Our work contributes to the theoretical understanding of adaptive therapy, and our insights on the role of turnover have two important implications. Firstly, they indicate that turnover is a factor we should think about when interpreting results from experimental models, as cell death rates vary between the flask, the mouse, and the human. Secondly, our insights provide a potential route for improving adaptive therapy by modulating cell death rates by using several drugs. Intriguingly, resistance costs are also a controversial topic in agricultural pest research, which inspired adaptive therapy. Yet despite the uncertainty about costs, pest management schemes have been effective by virtue of combining multiple interventions. How to choose and time combination treatment is a complex question but we believe that, again, integrating biology and mathematics will help us find the answer.

Figure 1: Using phase planes to visualise the mathematics of adaptive therapy. Left: Our collage of phase planes which made the cover of Cancer Research. Shown are an arrangement of different parameterisations as well as different viewing angles. Right: The underlying mathematics. Plotting the growth rate of the resistant cells we see that continuous high-dose treatment quickly removes all sensitive cells, resulting in rapid resistant cell expansion. In contrast, adaptative therapy maintains cell density high, which slows the growth of resistance by competitive suppression.

Further reading:

Sunday, 21 February 2021

#WhatsonYourMind Series 3: A Sam Howison Special

Take a mathematician with an endless curiosity about the world around him & the capacity of his subject to interpret it, & you have Series 3 of our #WhatsonYourMind films: a Sam Howison Special featuring geometry, flying spiders, tennis, rain, Pascal's mystic hexagram &, of course, Professor Pointyhead.

Editor's note: #WhatsonYourMind is the opportunity for Oxford Mathematicians to let it all out in 58 seconds (2 seconds for credits).

You can also watch the first two series on the Oxford Mathematics YouTube Channel.





Friday, 19 February 2021

Do you want a taste of Graduate Life in Oxford Mathematics?

Applications are now open for the University’s 2021 graduate access programmes: UNIQ+, & UNIQ+Digital.

Our graduate access programmes, open to all students in the UK, are designed to encourage and support talented undergraduates who would find continuing into postgraduate study a challenge for reasons other than their academic ability. 

UNIQ+ remote internships offer paid summer research experience over six weeks, from 5 July 2021. Participants will work on research projects with regular support and supervision from Oxford students and staff, as well as training in key research skills. There are six projects in Maths ranging from 'Randomized algorithms in machine learning' to 'Investigating the size of Riemann zeta function.' The full list and more information about UNIQ+ can be found here.

In addition, from July to October 2021, UNIQ+ Digital will also offer students from under-represented groups a flexible, free and fully online programme of mentoring, events and digital content to support them all the way through from considering graduate study to submitting an application.

The deadline for all programmes is 12 midday on Friday 19 March 2021. For full information, including eligibility criteria, visit the Graduate Access webpages.

Friday, 12 February 2021

Can accounting for cross-immunity between related virus strains help us better forecast the spread of epidemics?

As we have seen over the past year, the ability to accurately predict the course of an epidemic is imperative when making decisions about how to control the spread of a virus. In this work, which focuses on influenza, Oxford Mathematicians Rahil Sachak-Patwa, Helen Byrne and Robin Thompson (now at the University of Warwick) investigated whether accounting for cross-immunity between related strains could improve forecast accuracy.

Influenza viruses mutate over time, and previous exposure to influenza confers partial protection against future infections with related strains. To understand whether including cross-immunity could improve forecasts of future epidemics, the team considered two ordinary differential equation models (Fig. 1).

In both models, the population is partitioned into distinct classes which represent susceptible, exposed (i.e. infected but not yet infectious), infected, and recovered individuals. In the ‘1-group model’, all individuals are assumed to be identical and cross-immunity is neglected. In the ‘2-group model’, the population is split into two groups; cross-immune individuals who have previously been infected by a related strain, and immunologically naïve individuals who have no immunoprotection. It is assumed that the cross-immune individuals are less likely to experience severe disease and therefore recover more quickly than immunologically naive individuals, and hence only cases of naïve individuals get recorded.

Figure 1 – Schematic of the two models.

The team fitted both models to estimated case notification data from the 2009 H1N1 influenza pandemic in Japan, and then generated synthetic data for a future outbreak by assuming that the 2-group model more accurately represents the influenza infection dynamics. They then used both models to generate real-time forecasts for future outbreaks as they occur. Model parameter values estimated from the 2009 epidemic were used as informative priors, motivated by the fact that without using prior information from 2009, the forecasts are highly uncertain.

In the scenario that was considered for an epidemic occurring 25 years into the future, the 1-group model only produced accurate outbreak forecasts once the peak of the epidemic had passed, while the 2-group model made accurate forecasts early on (Fig. 2). Based on these results, the team conclude that accounting for cross-immunity, driven by exposure to previous epidemics, improves the accuracy of forecasts during influenza outbreaks. These findings could also be applied to the ongoing coronavirus pandemic, where incorporating information about cross-immunity between strains into models could result in better predictions about the spread of the virus.

Figure 2 – Forecasts of a future epidemic occurring 25 years into the future using the 1-group and 2-group models.

Thursday, 4 February 2021

Ulrike Tillmann appointed Director of the Isaac Newton Institute for Mathematical Sciences

Oxford Mathematician Ulrike Tillmann FRS has been appointed Director of the Isaac Newton Institute for Mathematical Sciences and N.M. Rothschild & Sons Professor of Mathematical Sciences at the University of Cambridge. The Issac Newton Institute for Mathematical Sciences is the UK's national research institute for mathematics. She will take up the post on 1 October 2021 while continuing to work part-time in the Mathematical Institute in Oxford to continue her research collaborations.

Ulrike's research interests lie in Algebraic Topology and its applications. Her work on the moduli spaces of Riemann surfaces and manifolds of higher dimensions has been inspired by problems in quantum physics and string theory. More recently her work has broadened into areas of data science and she co-leads the Oxford Centre for Topology and Data Science. She is a Fellow of Merton College.

Ulrike is well-known for her many contributions to the broader mathematical community, serving on a range of scientific boards including membership of the Council of the Royal Society. She will become President of the London Mathematical Society (LMS) in November 2021.


Tuesday, 2 February 2021

Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time

Take a piece of rope and knot it as you wish. When you are done, glue the two extremities together and you will obtain a physical realisation of what mathematicians also call a knot: a simple closed curve in 3-dimensional space. Now, put the knotted rope on a table and take a picture of it from above. It is now a planar projection of your knot. The mathematical equivalent of it is a knot diagram with multiple crossings as shown in the figure. The mathematical challenge is to decide whether a given knot diagram is actually the ‘unknot’, which means that by moving the knot around, you could deform it to a round circle. This question was formulated by Max Dehn in 1910, and was highlighted by Alan Turing in his final published paper:

"No systematic method is yet known by which one can tell whether two knots are the same." - 'Solvable and Unsolvable Problems', Science News 31, pp 7-23 (1954)

An algorithm that determines whether a knot is unknotted was first given by Wolfgang Haken in 1961. There are now many different algorithms that solve this problem, using a wide variety of techniques from low-dimensional topology and geometry. However, most of these algorithms are extremely slow for complicated knots and a famous unresolved question is whether there exists an algorithm that runs in polynomial time, which means that its running time is a polynomial p(n) that depends on the number of crossings n.

‘A lot of people have thought about this question ... but this has been a very hard question to resolve.’ (William Thurston, 2011)

In a remarkable Gordian tour-de-force, Oxford Mathematician Marc Lackenby has created an algorithm that determines whether a knot is the unknot in n^{c log(n)} steps, for some constant c, which is known as quasi-polynomial time. This is only slightly slower than polynomial time, and represents a significant advance over what previously was known. Marc outlined his algorithm in a seminar at University of California, Davis on 2 February and a recording of his talk is available. You can also view his slides for that talk. Marc also spoke to BBC World Service about his work and its wider applications.

Figure below: two very tangled diagrams that are both actually the unknot.




Tuesday, 2 February 2021

The Oxford Online Maths Club - online fun (and Maths) every Thursday

So what really happens every Thursday at 4.30pm UK time?

Is the rumour true that it can both improve your maths AND be a place to hang out with like-minded people and have a bit of a laugh?

We have managed to get hold of this video which reveals all.

Have a watch and join the club





Sunday, 31 January 2021

What's on Your Mind - 60 seconds inside the mind of an Oxford Mathematician

So what is on the mind of a mathematician, and specifically an Oxford Mathematician? Always their research? Or maybe nothing of the sort?

Our #WhatsonYourMind films take us inside those minds, young and less young, for 60 seconds. There is a lot going on, including the search for beauty, patterns in biology and data, the puzzle of parked cars in London streets, the damage caused by mathematical conferences, and the difficulties of teaching maths to the young.

The first series, a compilation of the first 13 films, is out now (see below).





Thursday, 28 January 2021

Ambient and intrinsic geometry of Teichmüller spaces

Oxford Mathematician Vladimir Markovic talks about his research into intrinsic geometry of Teichmüller Spaces.

"Geometry and topology are concerned with the world of shapes and forms, geography and maps, loops and knots. The main objects of study are finite dimensional manifolds. Two dimensional manifolds are called surfaces. Our purpose is to describe their shapes (geometry) and their form (topology). A closed surface is determined by its genus which is the number of holes it wraps around in space. This classification only takes into account the form of the surface (the topology of the surface) and not its shape (the geometry of the surface).

On the other hand, a closed surface $\Sigma_g$ of genus $g$ can be endowed with many Riemannian metrics (which yield different geometries on $\Sigma_{g}$). A suitable quotient of the collection of all such metrics is called the Moduli space $\mathcal{M}_{g}$ of Riemann surfaces (or simply the Moduli space). The Moduli space is itself a Kähler manifold of dimension $3g-3$. The Teichmüller space $\mathcal{T}_{g}$ is the universal covering of $\mathcal{M}_{g}$.

Theorem 1 (Bers). $\mathcal{T}_g$ is biholomorphic to a bounded domain in $\mathbb{C}^{3g-3+n}$.

The Moduli and Teichmüller space are equipped with the Kobayashi $d_K$, Carathéodory $d_C$, and Teichmüller $d_\mathcal{T}$ intrinsic geometries. The goal is to study these metrics and relate them to each other. One of the most important results in Teichmüller theory is the theorem of Royden that $d_{\mathcal{T}}\equiv d_K$. On the other hand, Yau proved that the $d_{\mathcal{T}}$ and $d_C$ are proportional.

Question 1. Does the identity $d_T \equiv d_K \equiv d_C$ hold on $\mathcal{T}_g$?

Isometries of the hyperbolic plane ${\mathbb{H}}^{2}$ into $\mathcal{T}_{g}$ (with respect to $d_{\mathcal{T}}$) are called Teichmüller discs. Every two points in $\mathcal{T}_{g,n}$ lie on the unique Teichmüller disc (complex geodesic). The identity $d_C\equiv d_{\mathcal{T}}$ holds on a Teichmüller disc if and only if $\mathcal{T}_{g}$ holomorphically retracts onto this disc. The following theorem provides a positive answer to Question 1 in an important special case.

Theorem 2 (Kra,McMullen). The Teichmüller space $\mathcal{T}_{g}$ holomorphically retracts onto every Teichmüller disc determined by a holomorphic quadratic differential with even order zeroes.

In general, the answer turned out to be negative.

Theorem 3 (Markovic) $\mathcal{T}_{g}$ does not holomorphically retract onto at least one Teichmüller disc. In particular, $d_C \neq d_{\mathcal{T}}$ on $\mathcal{T}_{g}$, for every $g\geq 2$.


Theorem 4 $\mathcal{T}_{g}$ does not holomorphically retract onto a random Teichmüller disc.

In the 80s Siu conjectured that $\mathcal{T}_{g}$ is not biholomorphic to a convex domain in $\mathbb{C}^{3g-3}$. The previous theorem yields the solution to the Siu's conjecture.

Theorem 5 (Markovic). $\mathcal{T}_g$ is not biholomorphic to a convex domain in $\mathbb{C}^{3g−3}$

It remains to classify holomorphic retracts of $\mathcal{T}_g$.

Conjecture $\mathcal{T}_g$ holomorphically retracts onto a Teichmüller disc $\tau^{\varphi}$ if and only if all zeroes of the $\varphi$ are of even order."

Image: surfaces of genus zero, one, two, and three.


I Kra, the Carathéodory metric on abelian Teichmüller disks, Journal Analyse Math. 40 (1981), 129-143 (1982)

V. Markovic, Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases, Duke Math. J. 167 (2018), no. 3, 497-535

C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, Journal Amer. Math. Soc. 16 , no. 4, 857-885 (2003)

We have nearly 200 Oxford Mathematics Case Studies on our website. Take a look.


Tuesday, 26 January 2021

Oxford Mathematics Online Public Lecture: Spacetime Singularities - Roger Penrose, Dennis Lehmkuhl and Melvyn Bragg

Oxford Mathematics Online Public Lecture in Partnership with Wadham College celebrating Roger Penrose's Nobel Prize

Spacetime Singularities - Roger Penrose, Dennis Lehmkuhl and Melvyn Bragg
Tuesday 16 February 2021

Dennis Lehmkuhl: From Schwarzschild’s singularity and Hadamard’s catastrophe to Penrose’s trapped surfaces
Roger Penrose: Spacetime singularities - to be or not to be?
Roger Penrose & Melvyn Bragg: In conversation

What are spacetime singularities? Do they exist in nature or are they artefacts of our theoretical reasoning? Most importantly, if we accept the general theory of relativity, our best theory of space, time, and gravity, do we then also have to accept the existence of spacetime singularities?

In this special lecture, Sir Roger Penrose, 2020 Nobel Laureate for Physics, will give an extended version of his Nobel Prize Lecture, describing his path to the first general singularity theorem of general relativity, and to the ideas that sprung from this theorem, notably the basis for the existence of Black Holes. He will be introduced by Dennis Lehmkuhl whose talk will describe how the concept of a spacetime singularity developed prior to Roger's work, in work by Einstein and others, and how much of a game changer the first singularity theorem really was.

The lectures will be followed by an interview with Roger by Melvyn Bragg.

Roger Penrose is the 2020 Nobel Laureate for Physics and Emeritus Rouse Ball Professor in Oxford; Dennis Lehmkuhl is Lichtenberg Professor of History and Philosophy of Physics at the University of Bonn and one of the Editors of Albert Einstein's Collected Papers: Melvyn Bragg is a broadcaster and author best known for his work as editor and presenter of the South Bank Show and In Our Time.

Watch online (no need to register - and the lecture will stay up on all channels afterwards):
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The Oxford Mathematics Public Lecture are generously supported by XTX Markets