Well, it's certainly a great privilege and it's always a pleasure to be back in Oxford when I don't have to deal with all the politics. And Phillips introduction was very kind and being in this new building is he's got some mixed feelings. The Centre for Mathematical Biology was in the lovely old 18th century house, one side of St Giles, and what I thought I would talk about are really three things that are among a few that I've been involved in.
But I assumed that sitting here you were able to read this. I mean, which is really a it's a nice description of three headed monsters. And he got a name from Oxford. I mean, it's I don't know what the university was like then, but he's actually a well-known person. The monsters I'm going to talk about are just a little different. So what I what I wanted to do if if this is going to work. Yeah. So. Not that I'm a great fan of Benjamin Rush.
I mean, he was one of the rebels that signed the Declaration of Independence, but he was a doctor who is best known for his work on the yellow fever in Philadelphia in the 18th century. But what I do like is what he said about anything associated with medicine. That knowledge is of little use when confined to mere speculation. He ended up getting yellow fever himself and he treated himself. And I've had the radical way. And he was accused of killing thousands in this way, but I'm not quite sure.
It was a very prejudiced time. So let me let me start off then. About the embryos and the whole business about deformations. It's been it's been a subject that's fascinated people for a really millennia. And what I wanted to describe was really a theory that could actually be tested experimentally, which is why I. Do you need that one? That both of them. Does that help? Okay, fine. That George Oster came and spent six months.
He's from Berkeley, came and spent six months in the Centre for Math Biology. And we try to think about how things develop other than genes. Because I'm afraid I have this view that for genes it's a close subject. We still won't know how to build a chicken because they don't do the work, they just control it. And so we wanted to talk about the work. So if you look at what embryos are roughly the same stage in development, there's a certain similarity.
And these are these actually are pictures taken from somebody called Michael in the 19th century, who was very well known and produced all sorts of really dramatic pictures. And he was a well-known biologist. The only problem was if it didn't quite fit in with what he believed, he just manipulated it a little bit. So so I'm not sure how human that went really is. But if it if you look at them in a sense, it suggests that that really has some fundamental similarity about the production of them.
So I mean that in the case of a salamander, it's got five fingers, you know, we've got five fingers, a pool and baleen, she had six fingers. And you remember the way one of the wives of Henry the eight. The only problem is they cut off the wrong appendage. And the.
So what I wanted to describe then was to see how if we go back to the basics, then there are really kind of two types of cells, these mesenchymal cells and epithelial cells, and the down the right hand side, they kind of bit hang around the skin. So I'm not going to talk about then it's the one that's on the left and these are small cells. But what they do is they have things called Philip Putty that can pull I mean, and they can move around on a tissue.
And and so when they move around, when they move around on their sorry, when they move around on the tissue, they, they form the environment. So that having an effect on each other even though in the sense they're not not touching. And so what really got got us involved in this was a picture that Albert Harris and University in North Carolina sent us that these are actually these are actually the same fibroblast mesenchymal cells that have some sort of sorry, this. That do have a traction.
And you can see these are the things that pull things around. You can see it there and you can see it there. These are tension lines. And so in a sense, it's like these cells move around on these matrices that have got tension. Like it's a bit like a jungle gym. You kind of climb up, you distort the rope, etc., and you can make it either more easier or more difficult or things, things like that. So these are the these are the hill sales that I wanted to talk.
And that's I said that was it's a friend. Sorry. I'm pressing the wrong button. So it was George Oster that I mentioned. So let's start with the scenario and. What one then has is the sales deform, the extracellular matrix, and that's that's the tissue that they call when and but then when they do deform the matrix that can affect the cell division.
And so you start to get a kind of pictorial scenario where they really you can start to see how they influence each other in such a way that the cells might start to form aggregations. And so what all of these things together then end up the same motion this convection does. A random motion happ to taxes is a kind of mechanical thing where in in the simplest way one cell grabs another and moves it along and makes them join some aggregation.
So we've got contact guidance. Chemotaxis is where the sales react to a chemical gradient. And so Chemotaxis usually means that if the concentration of the chemical, the smell goes up that way, then the cells move up that way. And so it's kind of the opposite, the opposite way to diffusion and say the combination of all this is you start to get cell aggregation patterns. And with a lot of the applications filled. Philip Meany was a major collaboration in all this work.
So what we decided to call it was the mechanical theory of morphogenesis. And so really we want something as simple as possible, but not simple enough that you can kind of get some answers to biological problems. So we're going to just have cells. We're going to have a metric density and we're going to have a tissue displacement. And then we want to see if we can if we can actually construct a practical mathematical model and then examine it, then see what happens.
I'm not going to do much mathematics, so it'll just be a couple of sci slaves as a kind of, you know, a token to the mathematicians. But basically what you have for the mathematicians. You end up with three equations because you've got three unknowns the cells, the extracellular matrix and the tissue displacement. Of course, everybody who've worked in this sort of area knows about reaction diffusion theory. Think about reaction.
Diffusion theory is that there are nice equations to work, want to work on if you're a mathematician and that's what you want to do, if you want to actually find the chemical or the modification that's associated with reaction diffusion equations. It was first proposed in 1952 and one was found last year. That really is deferred gratification. And so what? So the next slide, those of you who are not mathematicians, just just read read the English.
The thing is, what an equation does is just try and quantify how things interact. So what you've got is you've got cells that are changing that being convicted, that being influenced by contact guidance, the kind of jungle gym sort of thing. And then the cell division. And so really you just quantify that.
That's all mathematics does with a few parameters. Then you've got the tissue because you want to know how the tissue changes because that's going to affect how the tissue changes into cartilage and things like that. And so. Then that last equation is just how the forces of all balance together. And so you've got all these stresses, external forces, and there's parameters associated with displacement don't need.
In fact, if one needs an equation to explain what's going on, I always feel you don't understand the problem. And so I tend to like just talk about what they're saying. Okay. So what one did what we did was we took these equations and we did some mathematical analysis on them. And what we found was that. If you take just a narrow domain and you put in a bunch of cells and they've got tissue in it, then the cells start doing their thing. They're pulling the tissue, they're moving around.
And what we what we found was if the traction of the cells wasn't big enough, then nothing happened. But if you increase the traction of the cells above a critical value, you started to get spatial patterns and the spatial patterns quit. The black is where you've got the higher density of cells than where where they're just spotted. Then if you change the tracking even more, what you end up with is you start to get more complex spatial patterns. So. We then? It depends on the geometry.
And so if you have a large domain. So this is just the background before we talk about animals. If you've just got a small domain, all you can have is a small aggregation of cells. If you make it big, then you can have a whole lot of them. Then if you change the domain and make it different, make it look kind of tapered. What came out in the mathematics was that at the top end up there, you can have more two dimensional patterns, but as it gets near the bottom, all you can have a stapes.
And so if you applied this to see pattern formation and animal coat patterns, what it's saying is you can have a spotted animal with a straight tail, but never, ever the other way around. And so if you end up with an animal with stripes down the back, then it tells you something about what the embryo was when the pattern was laid down. And so there's all sorts of friends and colleagues who've been looking for striped animals with spotted tails, but they've never found them yet.
I'm sure they would let me know. Okay. So. This really led us to think about what are called morphic genetic rules and developmental constrains. And so why do we have only five fingers? And this business of developmental constraints, it's fascinated people for a long time. The classical text is a trait, a treatise on Ted Otto Elegy, written by somebody called Santi Olaf in 1836, and he wrote several volumes on it.
And people have been trying to think what forms them, how did they form, what the reason for them. Can we explain explain how the arrows are the good, the bad. And so what we decided to do was look at the vertebrate limb. And the reason we decided on that is we had some experimental colleagues that were mentioned later who actually did some experiments on our predictions.
But the thing about the vertebrate limb, which is why I show it to the salamanders, is that one really has to start thinking about evolutionary and developmental biology. Which leads to the theoretical question is how does Natural Selection act on Developmental Pilgrim's? Because that's really how how we change. And that's, in other words, how how do we create the limb?
But what what is lacking? What was lacking, we thought, was a view of this morphological evolution that takes into account the developmental mechanism that produces the pattern that must be it must be connected. So. What we thought was needed. We had to go beyond the level of observation and try and come up with a mechanistic explanation of how they were all formed.
And so this is what we ended up with, is a theory of limb, morphogenesis, and the central idea is associated with developmental constraints. Why are some morphologies not found? And etc. So as I said, we talked about the basic lib. So that's what that's what the basic line looks like. There's the humerus radius and ulna, and they start to they start to bifurcate.
And so if you look at the development and the lot of some of this work was done with with chick, chick embryos because there was for a long time, people were manipulating chick embryos, cutting it off and seeing what happens, adding little bits on and so on. And our major figure in this was Louis Wolpert, a friend for many years who didn't believe a word of this theory. And he kept doing very clever experiments to show it was wrong, but he never made it.
But, you know, we've been friends ever since. Yeah. Anyway, if you think about it, at the basic level, when you've got it, the humerus is a single aggregation. If you take a cross-section, then it goes to the radius and owner and then you get the carpal Koppel's there and then you get the digits. But of course, that doesn't tell us how they're formed. And that was what we had to start looking at.
And so we wanted to look at more for genetic rules. So what we did was we took we we took an area where. The parameters and the cell traction was such that you got a single aggregation. Then we let the limb develop and asylum developed, limb developed. What we found was it bifurcated. In other words, there was enough space and enough cells that you could actually get not just one bone, but you could get two of them.
Then we did it even more. And we found, really surprisingly, that you didn't get three things coming out. What you got is one of them kind of elongated and you got another aggregation. And we found doing. If you if you do the cost section at the end, these are the cell densities. And so what we what we found is we couldn't generate complicated things. And so this suggested that if you get complicated things. That's what you've got to play with. And so.
That's really what we looked at, the sequential development in the salamander limb. And this is what we found solving the equations. And this is actually how the limb develops in the salamander. In other words, all you've got of these three rules that you can play with and you get aggregations, etc., and you can build up really a complicated limb solely by aggregations, bifurcation and condensation. And of course.
That's how we suggested the salamander limb developed that you get first of all, you get a bifurcation then as is is is a segment of things. And what you do is you generate these the pulls. And of course, that's still all that's still all hypothetical. So we had a friend pick Albert who was interested in mechanisms and some other colleagues who said, Well, have we ever thought of a Proteus? Never, ever had. No, I had no idea of what that was.
And so it's what it is, is an example of Peter Morph ism, which is a slow developmental rate, which means you have fewer number of cells. And so therefore if one has looks at a Proteus, then what you get, if you've got a reduction, you've got fewer digits, car pulls and tarsal elements. And so when you do that, what what really we're seeing is fewer sales give changes in the number of condensation slim limb verifications. And you get a different you get a different limb altogether.
Going back to us. That is mere speculation. So what we did is we convinced Parallel Bear and Neil Shubin to do some some experiments. And we took this salamander, which you saw at the beginning. And what what they did was that they gave these mitotic inhibitors which inhibit the duplication of cells. And if you look at what. We predicted this is a normal limb, the one on the left, and this is the one where they had treated it with colchicine.
And there were a few a number of cells. And so when you have a few digits, they were not enough to have the condensation on the bifurcation. And so. If one looks at a normal limb and then you look at fossil records, one finds that in the fossil records, salamanders actually had fewer limb buds, a few, a few of digits. And this and that you see in the top is an example of the natural variation of P dwarfism.
And what we predicted was decrease the number of cells and what you will get and this is what we predicted and that was what was found experimentally. So if one then starts to think, you know, what does that really imply? I mean, what it does is, one, is moving evolution backwards. And so the salamanders that I'll bear and shoot. Shubin Got it. Actually, every evolutionary forms of the salamander limb, bud, salamander, limb.
And so ten ontologies, really, it was looking for developmental constraints. And one developmental constraint, the obvious one that out into enough cells. And so the whole business about Tier two colleges is really, as I said, it's fascinated people for a very long time. Now, what the big name in the early 19th century was STOCKARD, who wrote the classic paper around 1922, 1921, and where he discussed the paper about nearly 200 pages long.
And so what one's got, of course, are twins, conjoined twins, the bizarre one of the man with the eye thing coming out come out of his stomach. And this is an example of scale and complexity where this hand actually has more fingers than normal. And so. When one looks at the tickling and this is work by my friend Lewis Wolpert. And the thing about the thing about biologists that I found kind of funny. The the the chick has three digits.
And I think if you went into biology, you'd say, let's call them one, two, three. Biologists call them two, three, four. It's not because they can't count. Well, maybe sometimes. But then you see the it's because they think that there were 5 to 5 digits, because that's what the salamander had. Okay. So. So what he did was that he did a chicken, a graft, so that there were more cells. And so he took. He believed in gradients for pattern formation.
But so he took. This is where the chemical kind of starts. He says to effect pattern. So he took it from one chick limb and stuck it in there. And what happened was you get to a chick, the embryo chick with two limbs. And so really at. What one really is finding from a theoretical modelling point of view and experimental is the same. Verification is really that there are developmental constraints.
What we found was we tried we tried for a very long time to see if one could get something that was a single aggregation that goes into two or a single one that goes into three. We could never find it. And looking at the, you know, all the literature, etc., we never, ever found a tri fixation. And one of the nice things, one of the nice things about Oxford is its high table. I don't mean just the food you meet, you meet a lot of people.
And I always like to find out what people were working on, etc. And at this time, somebody said, What were you doing? I said, Well, I just, you know, trying to look at these developmental constraints and we find there's no time for creation. So three headed monsters. And they said. If I can find one. Will you bet a nice bottle of wine? I confidently said. Of course. And anyway, he. Then I found this one, which is a deceitfulness. It's a skeleton. 19th century of one. He produced this one.
But before the end of the before we agreed that we'd settle on a bottle of wine, I said, But if you do find one. This is how it would have happened. That there would be a bifurcation and then there would be another bifurcation. And that's exactly what happened. But they called them tie fly. Okay. And so if you look in just to any of the literature of anything, you do find two headed, two headed monsters are really very common.
Fish particularly so. Then there's a frog which was a and then but the one I think I like best was the chicken. And that the these are these are actually real ones. So the question is, are there any other three headed monsters? Well, there's a lot in in the ancient classical literature, Cerberus is the dog that gods [INAUDIBLE] for Hades. And that is really this fascinated people for centuries.
And this is Pinelli who dog and these three headed dogs are always fierce looking the always just terrible because they are supposed to prevent people getting into [INAUDIBLE] in rescuing the people in [INAUDIBLE]. So they want the dogs to look really fierce. Then I came across in the archaeological museum and Kate see are the nice looking dogs, three headed dogs. But that's that is the only one. However, that may not be three headed dogs, but there are certainly lots of seven headed ones.
And this actually comes from something called the Cabinet of Natural Curiosities that was done by someone called Alberta Sabre in Oak Island about 1750. It's it's an incredible book full of lovely, lovely pictures. Okay. Let me let me now move on to. It's a somewhat less, less cheery thing is I want to talk about brain tumours. And this is work that got started with Buster Alva, who was the head of pathology, and he came to see me to see if I could try and model brain tumours.
And he must have been about 80 at the time. And he had been in the medical world since he was 28 and he had worked on brain tumours specifically called glioma blast tumours. These are sometimes are called grade for tumours. They are the most serious tumour, brain tumour you can get and nobody survives. And so I said to him, how did he know of any case of anyone surviving from when he started?
And he said. Well, that is one possible case, but it's controversial and that was over a period of 50 years. And now there is still no and I want to show why. Why these brain tumours are so are so difficult and so dangerous. And the people that got involved. I always like to get my students involved. These are they're all Ph.D. students of mine who I got involved. But their topic for the thesis was always different to some of these extra, I suppose, kind of curricular things.
Okay, so what do we want to do with the mathematical model? We want. I mean, if it can be used for clinical research, then you're just trying to get grants, writing grant proposals with mathematics or something. What we wanted to do was to try and enhance imaging processes. We also wanted to show the inadequacies of current treatment. And of course, I'll come back to it. We want to know when tumours, such tumours actually stopped.
The other aims are we want to be able to estimate life expectancy from detection. But for other reasons I'll come back to we want to quantify patient treatment prior to their use. The treatments are just unbelievably awful. Chemotherapy, radiation and surgery. And I feel particularly strongly about about the last one. We want to know why some patients live longer than others with the same treatment. And of course, the key thing is we want to help to design scientific, scientific tiles.
Under. I think what was encouraging, we have realised every single one of these aims and that's what I want to describe very briefly, but just a little bit of history. The Incas. People have been doing preparations for thousands and thousands of years. This is an Inca skull from the museum in Cusco. And what one sees, there are four surgeries done on this head, and they have all healed.
And. If one becomes much more recent, the Casey Tribe, which is probably the smallest type in Kenya, they do tap in nations outside where the person goes and sits in the chair. And the people that call the OMA back start scraping the skull until they get rid of all the bone to get into underneath the bone where the tissue is. And often that's the time they think to try and. Really get rid of some pressure in the brain, etc.
It's not very medical. What is interesting is the mortality rate is zero to a first approximation of zero. And people did this. I mean, this is why at the beginning of the 19th century, people used then went into the hospitals to have preparations done. Then they were infected just like now, and they died. And so people stopped doing terminations in that way. Let me show you what a brain tumour really looks like and what how brain tumours are detected.
There's usually some something people collapse or at least the speech becomes faulty, whatever. They then have a computer scan or take a CT scan and they've got different words to the magnetic resonance imaging and that at1, one or two and the T two are more expensive. But what they do is they show more of the tumour. And so. All these scans are used.
But what I mean are roughly what a tumour looks like is that you have the cell density is like that in the middle of the cells, the cells die and you get necrosis and the red line is the the more accurate computer scan and you can see more if the tumour. And so really the question we wanted is. How do you make a model of this? And so I remember the first meeting it was with Buster Alvord, and I said, Well, what colleges are?
To me, he said, Well, there are cancer cells of this, this. And he went on and on. And all of these things are important. And so what I what I felt was how could we make the simplest possible model and what are the only things that are absolutely essential and necessary? And so one of the cancer cells, how did it multiply and how did it move around? And so what one woman did was came up with an equation which in words says you can have the rate of change of a tumour cell.
And what's going to contribute to it? Well, they're going to diffuse just like smoke particles, the cells in the tissue of the brain, the red they grey matter and the white matter and the cells diffuse. And and what else they do is they multiply. That's the model. And all one does mathematically is you just quantify it. And so that's just you can think of it as a shorthand way of saying something in words.
But the key thing about the model is there are diffusion coefficients and cancer cells diffuse differently and white matter and then grey matter. And not only that, cells multiply differently. Some go quickly and some don't. But the thing about the thing about brain tumour is that unbelievably irregular. And so what what one has to do is one has to take scans of the pain of a real brain. And so, although that is it can be a complicated model, which is actually what we used.
What I thought was if you take this tumour and you take all these scans and we get the volume of it, why don't we think that it's just a it's just a sphere. And if we then have two scans, we can then calculate the diffusion of the cells and how quickly they multiply. If each time we do these scans, we take the tumour and make it into a sphere just quantitatively to begin with.
We don't do that later with with real patients. Anyway, you end up with an equation which is a first year mathematics sort of thing. It really it is a very it's just a basic equation. And you can write down the solution. And the solution is just that. The number of cancer cells depends on how many you started with. They grow exponentially at a given rate.
This is that the radius of the, the tumour of the tumour of the spherical tumour and that's a diffusion coefficient and that's the time that so you end up with a simple equation that you know, people in people at school can actually plot. So let me get some facts now. When the radius of a tumour is about 1515 millimetres, that is when it is typically detected. And so the solution of the equation says that the diameter of the tumour is going to grow.
This comes from the simple fine is going to grow at four times the time, times the square root of the multiplication, times the diffusion. That's another of these things where that's just a premise. So what one has to do is one then has to say, Well, how quickly will it go? Well, you just divide by T and you get the velocity. And so you get an expression for the velocity. Unbelievably simple one that anyone can actually do. But of course, that's once again, that's speculation.
So what we did is we got data. From 27. But they are low grade tumours. And the reason that we don't have many for high grade tumours is because people with these glioma blasts almost never, ever. Well, practically never let them go unlimited. They always want some treatment. And so but the one the one the one case that we do have, it is also a straight line, and that comes from Casper. Okay. But as I said, that's a hypothetical thing. So let's look at what a real brain looks like.
These are cross-sections. And this is this is a web page that's put out by the Montreal Neurological Institute. It's public. Anybody can use it. And so this is a typical cross-section of the brains that we worked with. And so this was an example of a real case. This tumour was diagnosed in 2000. And this is what it looked like with scanning. With scanning? Not mathematically. In July 2001. This is it with autopsy.
That is what the autopsy looked like. That is what came from the numerical simulation of the model equation for the real brain. In fact, this patient's brain and that was what we started with and this is the post-mortem, uh, scanning. And so really what one, what one's getting out of this very simple basic model is. This is a typical example. The problem with imaging is that you can't see it. The images aren't really sophisticated enough.
And that black line is the most sophisticated imaging technique available. No. And that's all it can see. And so the tumour has already gone. It's really gone all over the brain. The threshold of detection is really enormous number of cells and it really one would think one would see them know this. I'm not sure if this is this will work. This is a typical development of what a tumour looks like. No, I don't. Let me see if I can get this. Sorry. It's the. It's the Oxford thing.
Anyway, let me go on to. We tried this morning and it didn't work and we tried this afternoon. It did work. I suppose that's the way the cookie crumbles anyway. Let me show you this. This is this is a movie that does work. But the other one was more for parameter values of a real. Sorry, but this. And what you do is you start to see and this is the tumours developing. Imaging cannot see it until you see the black line. And this is. And so it's up to about 125 days.
You don't see anything for something like two thirds of the growth of the tumour. Now it starts to grow. And so really it is so it. This is a more accurate picture of the thing that is quite sophisticated. Magnetic resonance imaging can find, and that is what the model shows based on that patient's. Diffusion of the cancer cells and the multiplication of the cancer cells.
And so what one can do, and it really is somewhat depressingly accurate, but this is what one can do, is when a person is discovered with a tumour and this is the actual patient, the tumour grew. And now what? Subtotal resection is just the medical world for surgical removal. And the thing about the thing about surgeons is they want to. They want to actually operate. And so what with this patient, they tried to cut out the tumour. And then. But the tumour started to grow again.
Then the patient had chemotherapy and radiation and then they can't keep it on forever. Then the tumour started to grow again and when it got roughly up to a diameter of 3050 millimetres, the patient, the patient died. And what one could estimate is if they had done nothing, how much longer would this patient have lived without surgery? So the question is how long will the patient live it? What sort of life is it? And a lot of us have had know all sorts of people who have had this sort of thing.
And one sees how really what a terrible life one can have. Okay. So. There was one woman, 72 year old, who came to the hospital and she was diagnosed with a tumour and she said, I don't want anything done. And so that I must admit that I mean, I learned about this.
I thought maybe that was the right decision. And so what she went through all the tests, the kind of tests about whether the brain was affecting things like a touch your left eye with your little finger of the right hand for up until two months before she died. She lived almost a normal life except for this hanging over her. And so really it is. You know, it really is one of these very difficult questions.
Okay. Let me now come to asking when the tumour stopped and you think, well, you know, why do you care for the tumour stopped? Well. If one. If you think about cell phones and nobody wants to think about cell phones. And it was a long time ago a friend in Paris, a botanist who actually what he did was he took a plant that goes about this high. And as it just when it had got all the branches and the leaves, he took a cell phone and subjected it to 2 hours of cell phone radiation.
He then followed the plant development. The branches and the leaves were totally distorted by the time it should have been normal. He's a fellow of the French Academy. He tried to get those paper published. He could never get it published because the journals that he submitted it to were frightened that the cell phone people would sue them.
So what I would recommend is that next time you buy a cell phone, read the small print, and they say there is no evidence that this will cause any effect to brain tumours. However, if and it's what you read after that. However, you know, thus there is the possibility. Well, two years ago. A person. Then in Washington, a doctor. Vasso had 11 year old children and they took a cell phone and they put it against the children. The parents agreed the cell phone was against it here for 10 minutes.
They then did CAT scans of the brain and they found there was an increase of glucose. Practically all over the brain. Now, nobody has actually been able to see this cancel their. But those of you who are sorry to be evangelical, but those of you who use cell phones should use an earpiece, because I think that that it must be inevitable that if there is radiation, all one needs to remember the curies with radiation.
So one of the questions that I thought one should answer is if somebody is developed with a brain tumour, when did it start? And so what one can do is you run the imaging backwards, the model backwards, and you will get an estimate of when the tumour started. Okay. Just just very quickly, I'll give you data on 57 patients had surgery. It's all data, but it's still relevant. 58 patients were also diagnosed and they were not had surgery.
And the effect of this was that those with with surgery lived seven weeks longer. But as I said, what sort of life did they have? So V.S. is the removal and what we can do with the model is we can carry out virtual some surgery. And. And so what we that's exactly what we this is a typical diagnosis. But if one takes that as what is observed, we know where the tumour has gone from the model and that's where it's gone.
And so what, what surgeons do is that they look at the, the cat scan of the tumour and they then cut out a little bit more. Then what happens is it can be three months, six months later, the tumour. Lo and behold, has regrown. Not only that, it's multifocal. And this has been a kind of it was one of these just mysteries. Of course, it's going to be multifocal. It's all over the place already. And so it's it's why surgery can never work. That's what the surgeon observes.
That's really where it is. And so what one can do then is you can actually work out surgery for a given patient. And this is a this is a specific patient you can work out when it gets to the diagnostic stage and when when that when they die, you can get a simple formula for how long they're going to live depending on the parameters of the of their tumour.
Once again, we wanted to compare it with data. And the red marks of what we predicted with different patients whom theoretically we injected with a tumour that had the same parameter values of diffusion and growth as the echo. So as the actual people whose data was collected. Well it's all, it's all very, very depressing about brain tumours. The problem about people with brain tumour just to go and they talk to people in the end up talking to a neurosurgeon.
And neurosurgeons always want to try and cut it out. You know, the difference between a neurosurgeon and God is that God doesn't think he's a neurosurgeon. And so it's really. And so what I was giving a lecture to a bunch of a bunch of doctors once and afterwards one of them said, you know, I said to him, I mean, you do brain surgeon. And I said, why do you do it? When you know the patient is not going to survive and it can be a miserable life afterwards.
It's really sad. If I don't do it, somebody else will. That's what it's about, I think. Not always. No, I mustn't be. But anyway, to conclude this thing. It shows these brain tumours are unpredictable. They really are. And the model suggests new way to study them. Still doesn't see what we can do about them. But unless we know how the how they spread, we're not going to know how to how to actually cure them. We can now estimate really when the tumour started.
Not only that, we can actually quantify the how the how it grows and spreads. The. It's an explanation of why some patients live longer than others. The values of the diffusion and the cell multiplication.
A cancer cell multiplication. A different. Not only that, we find children, young children live longer because it turns out that the parameters associated with the growth of their tumours are such that the tumour is more contained and so it means that they can actually live longer with radiation which kills more of the cells. But the end result is unfortunately the same.
We can quantify the effects and the side effects before the patient has any treatment and that really is, I think is something that helps anyway. We are now currently gathering a large number of patients with different grade tumours and there's a kind of small clinical study going on. So let me finish with only the Mona Vale again, and I think it was amazingly practical man.
He was the surgeon to two kings, etc. And as you saw, he had a low opinion of people, but he did like surgery and he said it's a cure, were preferable, more noble, more perfect, more necessary and more lucrative. That was in the 14th century. And okay, let me talk about something a little different just to defeat the.
I'll do this quickly. A collaborator was a main collaborator with John Gottman, a clinical psychologist, who wrote to me when I was in Oxford, still in Oxford, and said he wondered if I could maybe try and make a mathematical model about marital interaction. And I thought it was a pile of rubbish and ended up in Washington. We had lunch together and I got kind of hooked. And the think about divorces that are millions a year. And that is now a huge, enormous legal industry.
With lawyers who specialise in divorce of couples who come from different countries. So, for example, if you live in Texas, it's better to be a man. If you live in France and the man has been having an affair, then it's kind of half and half. There are some countries where the man can get a divorce and the wife doesn't even know it until she gets the papers. It is unbelievably complicated. And so once again, I got a bunch of my students.
Uh, Jane. Jane White, Jane Snow in Bath, Rebecca Tyson, Cliff in Canada, and John Gottman. After the results of all this retired air late and started up a clinic for marital therapy. Yeah. Okay. So what do we do? Well, first of all, the people that we got for this study were people who applied for a marriage certificate in King County and in the state of Washington near Seattle. They were invited to partake in the study associated with marital interaction and divorce prediction.
And all they had to do was come. To the department and sit opposite each other and be filmed talking about the problem of contention. Absolutely made no difference what it was. I mean, food, sex, in-laws, laws, you name it, it made absolutely no difference. Okay. So what we did then was we scored what each of them said and there is a scoring system and you assigned an integer between plus four and minus four and.
This. What you then did is you counted the positive things and took away the negative things. Each time one of them spoke and this was data was gathered mainly by psychology students who sat behind a one way mirror and the couple knew they were being filmed. And usually within about a minute they'd forgotten everything except what they were arguing about. And so these six students, it was amazing how close they were in getting the same numbers.
And so what you get each time they speak it, you get a number. So you end up drawing a curve of a kind of Dow Jones average. I gave a talk about this and then somebody in Australia sent sent me a cartoon and that they wanted to show whether this test will show whether or not they're going to get divorced. And they had to choose a topic that was really big. Okay. Yeah. This is what the scoring system, the scoring system is. And it is amazingly easy to assess the numbers.
They're also associated with visual things about whether it's anger, contempt, humour, etc. And this is the 16 code system. Unchained. And so what one gets is you get a kind of Dow Jones average of the conversation. So in this case there, that's zero. And it ends up it becomes progressively more more positive. And so it's roughly a 5 to 1 ratio of positive to negative. It's a very stable marriage, that one on the other hand. And so that is a ratio of about 0.8 positive, positive to one negative.
And this couple actually did end up getting getting divorced. This is the data between 1992 or 2004. We actually studied 700 couples and we analysed the thing. And what we found is the number of the different marriage types were very limited. There were only five of them. And what we predicted was whether they would get divorced, stay married, either happily or unhappily. Then every every 1 to 2 years, we sent out a questionnaire and they were asked to complete it.
We then compared the predictions with what the facts were. So this isn't statistics in the sense that you do a small study and you then expand it. These are real numbers that we got. Well, prediction of which couples would get divorced was actually 94% accurate. It was even better than that in the sense that those we predicted would definitely get divorced. We were 100% correct.
It was some who predicted we'd be staying married unhappily ended up getting divorced, and most of them actually got divorced within four years. Of course, we couldn't possibly tell them anything about the predictions, but okay, this is for the therapy. We had to develop something that would show people who did not have any kind of background. They just knew they were having a terrible time. And so they are showing a picture about the numbers that they would be showing later.
And so we got a couple of actors to show it. And this stayed in this quadrant. They're both happy. And then that one, that boy of that one, the wife is happy. And so what you get when you get from the data is something that you can actually show has a specific meaning for how they are interacting. And so just again, I don't know if this movie will work. Let me see. This is a typical this is typical data from one couple. And and the numbers look as if they're going all over the place.
And after one had seen about 25 of these couples. We could look at this data and say. You're going to get divorced. But of course, that's no good. One has to actually be able to quantify it. So let me briefly describe the model. Justin It's an intuitive model, and this says that the husband can influence the wife and this is the score that I describe after the husband speaks.
That's how she felt before. And that's something I'll mention just later, is how they interpret their view of the marriage. And so you can put this into you can put this into a simple mathematical model. And that's what you get. And what we want to do is evaluate all these parameters A and R, and this means if R is about one, that means the wife isn't very interested in changing her mind. If R is near zero, then it means that she can be influenced by what the husband says.
And so you have that equation and this influence function is how we actually describe marriages. And so. That is typical data. The husband when the husband says something. This is the influence that we were able to evaluate from the data obtained from the 15 minute conversation. So you can put a curve, a kind of best fit curve, but that's very complicated. And so we decided, let's not bother with that. Let's just take near the orange and two straight lines.
Of course, what is really curious that totally surprised us is here. The husband is being incredibly positive. The influence on his wife is negative. In other words, she's starting to think, what's he trying on until. But the fact that this happens happened with the vast majority of these 700 couples, it just was a somewhat surprising phenomenon. And these these are the how we quantify the marriage.
And these are just two examples of the wife's influence and the husband and the husband's influence and wife. And so this is a typical thing that if the husband says something, put something sorry. If the husband says something positive, it has a positive effect. If you say something, it has a negative effect, but less negative. That is very common in in in marriages.
And so these are two examples of the influence numbers for what are conflict avoiding couples that when they say something negative. A the the influence is much less than if they say something positive. And these are numbers, as I said, that actually come from the data. And so that's a theoretical conflict, avoiding couple. They have little effect on each other on the negative side. So then.
Then this. When we looked at the 700 couples, really with only five types of marriage, three stable ones and two unstable ones. And the characteristics are the stable, the volatile ones. Some of them were stable, but generally they were unstable. You know, they have heated arguments, romantic, etc. But then the ones that really were stable but those that were calmer, intimate, shared experience rather than individuality, and they were the ones that really stayed married.
The avoidance was also stable. They didn't want to argue when one of them was being negative. They tried to pacify them. The hostile ones are when you get mixed marriages. And hostile, detached as the husband gets wildly excited and the wife just kind of keeps calm and said, Well, maybe you're right, and so on. These marriages really don't last. So basically what one's got is we have these things, we have the data and we evaluated the parameters.
In the marriages part of the time, they didn't influence each other at all. Then you end up with a model equation and you say, What is the steady state? And so you can just say so. In other words, they don't vary. Everything is the same. You solve it. You get that, you solve it, and the uninfluenced, steady state is just a parameter over one minus on one on one has to be less than one for mathematicians because it's a discrete equation.
These are the parameters that we evaluated. You get the same for the husband. If he is less than zero and B is less than zero, they're almost certainly going to get divorced. It is it was that simple. So one uninfluenced, steady state has to be positive. And this inertia things are, as I say, the fog is near one. It means a pretty rigid in what they believe. Now what one can do is I don't want to do any of the math, but from these equations you can get whatev what are called null claims.
And these are just lines that if you vary. H and W along these lines you have a constant. It's kind of constant. So you have one of these lines for the wife. You have one for the husband where the interact of the steady state of the marriage. So in this case, the black line is a stable, steady state, and that is in the first quadrant. There's another steady state which is unstable.
And so if the ever end up in this third quadrant really getting irritated with each other very quickly, the end up leaving there, it's unstable. And they move up to a stable, steady state. So what one can do with this is this. This is crucially dependent on the uninfluenced, steady state of the wife and of the husband. So suppose the husband gets a little more or less happy with the whole situation. Then what happened? I mean, the happy state. Here you've got this.
It's a thing in the bodily, you know that that's a happy marriage. I don't know what they're doing in the bushes, but anyway. But if the husband becomes less satisfied with the marriage, then he moves his steady state, moves over into the negative region, and that influences the steady state. And so instead of really being very positive, the husband is not very happy with it. And but the wife still putting up with it if the husband becomes even more unhappy with that.
What you end up with is. There's no steady state. And so it depends how the I mean, it can either be, you know, the result can be some, you know, at, I don't know, sort of, you know, a delightful mutual, you know, sort of orgy or some kind of Calvinist [INAUDIBLE]. And but basically what one gets in this very simple model is you can actually quantify certain things about a marriage. Of course, there. I don't know who was it that said, never go to bed angry, stay up and fight.
I don't know who said some. I think it was some American comedian, but I think it's a probably Groucho Marx. His comment, he said, you know, it it's inevitable. Marriage always interrupts and influences and affects romance because if you have a romance, your wife is certainly going to interfere. And so it's a so there's all sorts of great, nice points about this. Okay. So just last few slides is the. I did it with straight lines. But the real numbers of these lines are really quite clear.
They come from the data, the complicated, and that is a typical, uh, stable marriage. And if they start anyway of the conversation negatively, it always ends up at the positive, steady state. Very stable marriage. In this case, the stable states are negative and it doesn't matter where they start, they always end up there. Well, that the data that I showed you, that couple, the steady state was in the third quadrant and they actually got divorced.
So what can one do about therapy? Just to finish. This is where the couple end. The steady state of their interaction was negative. And. The thing about this is that they are both negative. The probability is high for divorce, and in fact, that's exactly what happened to them. On the other hand, this couple, the wife was a bit positive about it.
The husband was it. And so they actually, uh, that really is a couple where it is a candidate for therapy and they actually had therapy and this is what the score was beforehand. The data centre, they started talking and the wife became more negative. The husband was negative all the time. And that. They then had therapy lasted for. Yeah, I can't remember exact two months of something and they then did the test again and this steady state was in the first quadrant.
We then actually did gather the data of them and this is the data. I mean, it might not be your community, but at least at least the you know, one of them is positive. But I think what was interesting is if one of them is positive, then it can be a candidate. So what have we actually gained from all this? Well, therapy is one thing that they are showing the data. They're showing the videotape. They're showing the predictions.
And the only thing they couldn't understand was explained specifically to them by the clinical psychologist. And and this therapy is based. Uh, almost entirely on this very simple model. And so give me it. As I said, that these are the these are the results. It's a new language for discussing psychology. And why don't you down here, as I remember long time friend Christopher Seaman, who most of you will know he I remember talking to him. He talked to me about really the social sciences.
And I thought at the time it was a load of rubbish. And he was absolutely right. I mean, he applied it, did different things, but the energy so it we can use it for that. It's a rationale. We found there were only five types of marriages. A stable ones have got matched interaction style and unstable ones have have the other kind. But basically the last thing I want to cite I want to really show is I quoted Benjamin Rush. In fact, Montaigne had a much bigger influence.
And a lot of what I have done in this work, in this field, we are, as he says, I'm not afraid to converse it. He would offer a candle to St Michael and if needed, another to a dragon. And I think in a sense that's what applied mathematics is. The reason I wanted to show this thing. This is a portrait in the Bodleian. A different view really came from, I thought the only pure mathematician I could quote was be somebody in the 19th century.
And as you can read it, we find the primitive source of rationality. Mathematicians must turn to biologists, you know, how can they otherwise can they carry out their experiments if they don't know any mathematics? Well, when one does interdisciplinary work, it's rather difficult to have that attitude. And I thought this picture really kind of summed up what a lot of people who have worked in this field.
It's a certain kind of arrogance. And I showed this to a friend, an Australian friend, who then sent me that picture. But but basically what what it is, it's I really think if one keeps the mathematics as simple as possible so that you can explain it to people who are not scientists, then maybe one got a chance of actually doing something. I suppose the trouble with mathematicians is sort of 99% of them give the rest a bad name.
And so maybe quite. So I think that is probably the appropriate place to start. Thank you very much.