1 00:00:09,830 --> 00:00:17,240 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. 2 00:00:17,750 --> 00:00:27,229 And Phillips introduction was very kind and being in this new building is he's got some mixed feelings. 3 00:00:27,230 --> 00:00:34,040 The Centre for Mathematical Biology was in the lovely old 18th century house, one side of St Giles, 4 00:00:34,670 --> 00:00:44,240 and what I thought I would talk about are really three things that are among a few that I've been involved in. 5 00:00:44,540 --> 00:00:48,860 But I assumed that sitting here you were able to read this. 6 00:00:49,310 --> 00:00:55,310 I mean, which is really a it's a nice description of three headed monsters. 7 00:00:55,610 --> 00:01:05,000 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. 8 00:01:06,980 --> 00:01:15,530 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. 9 00:01:15,980 --> 00:01:24,330 Yeah. So. Not that I'm a great fan of Benjamin Rush. 10 00:01:24,570 --> 00:01:28,530 I mean, he was one of the rebels that signed the Declaration of Independence, 11 00:01:28,530 --> 00:01:38,190 but he was a doctor who is best known for his work on the yellow fever in Philadelphia in the 18th century. 12 00:01:38,700 --> 00:01:47,010 But what I do like is what he said about anything associated with medicine. 13 00:01:47,370 --> 00:01:52,920 That knowledge is of little use when confined to mere speculation. 14 00:01:54,600 --> 00:01:58,110 He ended up getting yellow fever himself and he treated himself. 15 00:01:58,110 --> 00:02:08,070 And I've had the radical way. And he was accused of killing thousands in this way, but I'm not quite sure. 16 00:02:08,190 --> 00:02:15,300 It was a very prejudiced time. So let me let me start off then. 17 00:02:17,220 --> 00:02:24,120 About the embryos and the whole business about deformations. 18 00:02:26,340 --> 00:02:32,280 It's been it's been a subject that's fascinated people for a really millennia. 19 00:02:32,850 --> 00:02:43,440 And what I wanted to describe was really a theory that could actually be tested experimentally, which is why I. 20 00:02:46,260 --> 00:02:49,260 Do you need that one? That both of them. Does that help? 21 00:02:50,050 --> 00:02:57,590 Okay, fine. That George Oster came and spent six months. 22 00:02:57,600 --> 00:03:02,250 He's from Berkeley, came and spent six months in the Centre for Math Biology. 23 00:03:02,640 --> 00:03:08,320 And we try to think about how things develop other than genes. 24 00:03:08,880 --> 00:03:14,010 Because I'm afraid I have this view that for genes it's a close subject. 25 00:03:14,010 --> 00:03:19,700 We still won't know how to build a chicken because they don't do the work, they just control it. 26 00:03:20,010 --> 00:03:31,560 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. 27 00:03:32,190 --> 00:03:40,230 And these are these actually are pictures taken from somebody called Michael in the 19th century, 28 00:03:40,530 --> 00:03:46,440 who was very well known and produced all sorts of really dramatic pictures. 29 00:03:47,430 --> 00:03:56,310 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. 30 00:03:56,820 --> 00:04:00,930 So so I'm not sure how human that went really is. 31 00:04:01,770 --> 00:04:12,090 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. 32 00:04:12,660 --> 00:04:24,150 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. 33 00:04:25,110 --> 00:04:28,680 And you remember the way one of the wives of Henry the eight. 34 00:04:28,980 --> 00:04:33,630 The only problem is they cut off the wrong appendage. And the. 35 00:04:33,870 --> 00:04:47,669 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, 36 00:04:47,670 --> 00:04:59,400 these mesenchymal cells and epithelial cells, and the down the right hand side, they kind of bit hang around the skin. 37 00:04:59,670 --> 00:05:06,030 So I'm not going to talk about then it's the one that's on the left and these are small cells. 38 00:05:06,420 --> 00:05:14,940 But what they do is they have things called Philip Putty that can pull I mean, and they can move around on a tissue. 39 00:05:16,030 --> 00:05:26,020 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. 40 00:05:26,890 --> 00:05:33,040 So that having an effect on each other even though in the sense they're not not touching. 41 00:05:34,180 --> 00:05:46,989 And so what really got got us involved in this was a picture that Albert Harris and University in North Carolina sent 42 00:05:46,990 --> 00:05:57,780 us that these are actually these are actually the same fibroblast mesenchymal cells that have some sort of sorry, 43 00:05:57,790 --> 00:06:01,490 this. That do have a traction. 44 00:06:01,700 --> 00:06:04,730 And you can see these are the things that pull things around. 45 00:06:05,930 --> 00:06:09,110 You can see it there and you can see it there. 46 00:06:09,290 --> 00:06:18,409 These are tension lines. And so in a sense, it's like these cells move around on these matrices that have got tension. 47 00:06:18,410 --> 00:06:20,210 Like it's a bit like a jungle gym. 48 00:06:20,510 --> 00:06:30,380 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. 49 00:06:31,520 --> 00:06:35,759 So these are the these are the hill sales that I wanted to talk. 50 00:06:35,760 --> 00:06:40,400 And that's I said that was it's a friend. Sorry. I'm pressing the wrong button. 51 00:06:43,160 --> 00:06:49,970 So it was George Oster that I mentioned. So let's start with the scenario and. 52 00:06:51,990 --> 00:06:58,100 What one then has is the sales deform, the extracellular matrix, 53 00:06:58,110 --> 00:07:08,370 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. 54 00:07:09,060 --> 00:07:17,969 And so you start to get a kind of pictorial scenario where they really you can start to see 55 00:07:17,970 --> 00:07:24,660 how they influence each other in such a way that the cells might start to form aggregations. 56 00:07:25,260 --> 00:07:33,489 And so what all of these things together then end up the same motion this convection does. 57 00:07:33,490 --> 00:07:40,379 A random motion happ to taxes is a kind of mechanical thing where in in the simplest 58 00:07:40,380 --> 00:07:45,210 way one cell grabs another and moves it along and makes them join some aggregation. 59 00:07:45,900 --> 00:07:53,430 So we've got contact guidance. Chemotaxis is where the sales react to a chemical gradient. 60 00:07:54,120 --> 00:08:06,180 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. 61 00:08:06,990 --> 00:08:18,930 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. 62 00:08:21,060 --> 00:08:24,840 And with a lot of the applications filled. 63 00:08:24,930 --> 00:08:28,740 Philip Meany was a major collaboration in all this work. 64 00:08:29,730 --> 00:08:36,510 So what we decided to call it was the mechanical theory of morphogenesis. 65 00:08:37,950 --> 00:08:48,480 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. 66 00:08:49,050 --> 00:08:58,129 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. 67 00:08:58,130 --> 00:09:08,960 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. 68 00:09:09,560 --> 00:09:18,170 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. 69 00:09:18,980 --> 00:09:24,140 But basically what you have for the mathematicians. 70 00:09:24,380 --> 00:09:35,000 You end up with three equations because you've got three unknowns the cells, the extracellular matrix and the tissue displacement. 71 00:09:36,550 --> 00:09:43,480 Of course, everybody who've worked in this sort of area knows about reaction diffusion theory. 72 00:09:44,140 --> 00:09:45,160 Think about reaction. 73 00:09:45,160 --> 00:09:53,530 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, 74 00:09:54,280 --> 00:10:02,440 if you want to actually find the chemical or the modification that's associated with reaction diffusion equations. 75 00:10:02,920 --> 00:10:08,200 It was first proposed in 1952 and one was found last year. 76 00:10:08,710 --> 00:10:12,850 That really is deferred gratification. And so what? 77 00:10:13,480 --> 00:10:20,980 So the next slide, those of you who are not mathematicians, just just read read the English. 78 00:10:23,830 --> 00:10:31,210 The thing is, what an equation does is just try and quantify how things interact. 79 00:10:31,810 --> 00:10:36,640 So what you've got is you've got cells that are changing that being convicted, 80 00:10:37,570 --> 00:10:43,720 that being influenced by contact guidance, the kind of jungle gym sort of thing. 81 00:10:44,140 --> 00:10:49,410 And then the cell division. And so really you just quantify that. 82 00:10:49,410 --> 00:10:57,790 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 83 00:10:57,790 --> 00:11:03,130 that's going to affect how the tissue changes into cartilage and things like that. 84 00:11:05,340 --> 00:11:11,430 And so. Then that last equation is just how the forces of all balance together. 85 00:11:12,960 --> 00:11:21,220 And so you've got all these stresses, external forces, and there's parameters associated with displacement don't need. 86 00:11:21,720 --> 00:11:27,960 In fact, if one needs an equation to explain what's going on, I always feel you don't understand the problem. 87 00:11:28,350 --> 00:11:33,450 And so I tend to like just talk about what they're saying. 88 00:11:34,350 --> 00:11:41,610 Okay. So what one did what we did was we took these equations and we did some mathematical analysis on them. 89 00:11:42,090 --> 00:11:44,280 And what we found was that. 90 00:11:46,500 --> 00:11:57,600 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. 91 00:11:57,600 --> 00:11:59,760 They're pulling the tissue, they're moving around. 92 00:12:00,360 --> 00:12:08,460 And what we what we found was if the traction of the cells wasn't big enough, then nothing happened. 93 00:12:09,660 --> 00:12:20,350 But if you increase the traction of the cells above a critical value, you started to get spatial patterns and the spatial patterns quit. 94 00:12:20,400 --> 00:12:27,630 The black is where you've got the higher density of cells than where where they're just spotted. 95 00:12:29,040 --> 00:12:37,260 Then if you change the tracking even more, what you end up with is you start to get more complex spatial patterns. 96 00:12:38,920 --> 00:12:44,080 So. We then? It depends on the geometry. 97 00:12:45,070 --> 00:12:51,370 And so if you have a large domain. So this is just the background before we talk about animals. 98 00:12:52,090 --> 00:12:56,290 If you've just got a small domain, all you can have is a small aggregation of cells. 99 00:12:56,560 --> 00:13:08,480 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. 100 00:13:09,140 --> 00:13:15,410 What came out in the mathematics was that at the top end up there, 101 00:13:16,070 --> 00:13:23,510 you can have more two dimensional patterns, but as it gets near the bottom, all you can have a stapes. 102 00:13:24,680 --> 00:13:30,770 And so if you applied this to see pattern formation and animal coat patterns, 103 00:13:31,310 --> 00:13:37,460 what it's saying is you can have a spotted animal with a straight tail, but never, ever the other way around. 104 00:13:39,470 --> 00:13:49,550 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. 105 00:13:50,750 --> 00:13:59,810 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. 106 00:14:00,350 --> 00:14:04,180 I'm sure they would let me know. Okay. 107 00:14:04,190 --> 00:14:15,340 So. This really led us to think about what are called morphic genetic rules and developmental constrains. 108 00:14:16,000 --> 00:14:19,150 And so why do we have only five fingers? 109 00:14:19,750 --> 00:14:26,290 And this business of developmental constraints, it's fascinated people for a long time. 110 00:14:26,680 --> 00:14:39,730 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. 111 00:14:40,240 --> 00:14:48,280 And people have been trying to think what forms them, how did they form, what the reason for them. 112 00:14:48,310 --> 00:14:54,250 Can we explain explain how the arrows are the good, the bad. 113 00:14:54,580 --> 00:14:58,540 And so what we decided to do was look at the vertebrate limb. 114 00:14:59,530 --> 00:15:04,209 And the reason we decided on that is we had some experimental colleagues that 115 00:15:04,210 --> 00:15:09,880 were mentioned later who actually did some experiments on our predictions. 116 00:15:10,870 --> 00:15:15,700 But the thing about the vertebrate limb, which is why I show it to the salamanders, 117 00:15:16,210 --> 00:15:23,680 is that one really has to start thinking about evolutionary and developmental biology. 118 00:15:24,640 --> 00:15:32,170 Which leads to the theoretical question is how does Natural Selection act on Developmental Pilgrim's? 119 00:15:32,620 --> 00:15:38,710 Because that's really how how we change. And that's, in other words, how how do we create the limb? 120 00:15:40,400 --> 00:15:43,460 But what what is lacking? What was lacking, we thought, 121 00:15:43,850 --> 00:15:50,419 was a view of this morphological evolution that takes into account the developmental 122 00:15:50,420 --> 00:15:55,220 mechanism that produces the pattern that must be it must be connected. 123 00:15:56,930 --> 00:15:59,580 So. What we thought was needed. 124 00:15:59,590 --> 00:16:07,690 We had to go beyond the level of observation and try and come up with a mechanistic explanation of how they were all formed. 125 00:16:08,950 --> 00:16:19,480 And so this is what we ended up with, is a theory of limb, morphogenesis, and the central idea is associated with developmental constraints. 126 00:16:20,290 --> 00:16:25,280 Why are some morphologies not found? And etc. 127 00:16:25,370 --> 00:16:31,730 So as I said, we talked about the basic lib. So that's what that's what the basic line looks like. 128 00:16:32,480 --> 00:16:38,690 There's the humerus radius and ulna, and they start to they start to bifurcate. 129 00:16:39,560 --> 00:16:52,790 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, 130 00:16:52,790 --> 00:17:00,560 people were manipulating chick embryos, cutting it off and seeing what happens, adding little bits on and so on. 131 00:17:00,590 --> 00:17:11,270 And our major figure in this was Louis Wolpert, a friend for many years who didn't believe a word of this theory. 132 00:17:11,630 --> 00:17:16,310 And he kept doing very clever experiments to show it was wrong, but he never made it. 133 00:17:16,850 --> 00:17:20,180 But, you know, we've been friends ever since. 134 00:17:20,840 --> 00:17:28,999 Yeah. Anyway, if you think about it, at the basic level, when you've got it, the humerus is a single aggregation. 135 00:17:29,000 --> 00:17:38,510 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. 136 00:17:38,840 --> 00:17:44,420 But of course, that doesn't tell us how they're formed. And that was what we had to start looking at. 137 00:17:45,740 --> 00:17:55,010 And so we wanted to look at more for genetic rules. So what we did was we took we we took an area where. 138 00:17:56,170 --> 00:18:01,600 The parameters and the cell traction was such that you got a single aggregation. 139 00:18:02,290 --> 00:18:07,060 Then we let the limb develop and asylum developed, limb developed. 140 00:18:07,540 --> 00:18:10,570 What we found was it bifurcated. 141 00:18:11,620 --> 00:18:20,110 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. 142 00:18:21,520 --> 00:18:30,280 Then we did it even more. And we found, really surprisingly, that you didn't get three things coming out. 143 00:18:31,180 --> 00:18:36,610 What you got is one of them kind of elongated and you got another aggregation. 144 00:18:37,660 --> 00:18:47,500 And we found doing. If you if you do the cost section at the end, these are the cell densities. 145 00:18:48,100 --> 00:18:52,750 And so what we what we found is we couldn't generate complicated things. 146 00:18:53,590 --> 00:18:58,630 And so this suggested that if you get complicated things. 147 00:19:00,110 --> 00:19:03,330 That's what you've got to play with. And so. 148 00:19:04,940 --> 00:19:11,450 That's really what we looked at, the sequential development in the salamander limb. 149 00:19:12,500 --> 00:19:18,770 And this is what we found solving the equations. 150 00:19:19,670 --> 00:19:24,680 And this is actually how the limb develops in the salamander. 151 00:19:25,430 --> 00:19:35,390 In other words, all you've got of these three rules that you can play with and you get aggregations, etc., 152 00:19:35,960 --> 00:19:47,030 and you can build up really a complicated limb solely by aggregations, bifurcation and condensation. 153 00:19:48,110 --> 00:19:48,680 And of course. 154 00:19:51,690 --> 00:20:05,280 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. 155 00:20:05,790 --> 00:20:10,560 And what you do is you generate these the pulls. 156 00:20:11,920 --> 00:20:16,390 And of course, that's still all that's still all hypothetical. 157 00:20:17,200 --> 00:20:32,800 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? 158 00:20:35,200 --> 00:20:38,620 Never, ever had. No, I had no idea of what that was. 159 00:20:39,040 --> 00:20:49,210 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. 160 00:20:50,380 --> 00:21:02,830 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. 161 00:21:04,160 --> 00:21:18,470 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. 162 00:21:18,710 --> 00:21:22,810 And you get a different you get a different limb altogether. 163 00:21:24,720 --> 00:21:28,380 Going back to us. That is mere speculation. 164 00:21:29,340 --> 00:21:37,590 So what we did is we convinced Parallel Bear and Neil Shubin to do some some experiments. 165 00:21:38,430 --> 00:21:41,340 And we took this salamander, which you saw at the beginning. 166 00:21:42,360 --> 00:21:51,930 And what what they did was that they gave these mitotic inhibitors which inhibit the duplication of cells. 167 00:21:52,950 --> 00:22:05,340 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. 168 00:22:05,550 --> 00:22:17,040 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. 169 00:22:18,570 --> 00:22:27,180 And so. If one looks at a normal limb and then you look at fossil records, 170 00:22:27,660 --> 00:22:38,040 one finds that in the fossil records, salamanders actually had fewer limb buds, a few, a few of digits. 171 00:22:38,580 --> 00:22:46,740 And this and that you see in the top is an example of the natural variation of P dwarfism. 172 00:22:48,910 --> 00:23:00,940 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. 173 00:23:01,870 --> 00:23:06,070 So if one then starts to think, you know, what does that really imply? 174 00:23:06,670 --> 00:23:10,150 I mean, what it does is, one, is moving evolution backwards. 175 00:23:12,040 --> 00:23:17,290 And so the salamanders that I'll bear and shoot. 176 00:23:17,650 --> 00:23:27,070 Shubin Got it. Actually, every evolutionary forms of the salamander limb, bud, salamander, limb. 177 00:23:28,180 --> 00:23:34,900 And so ten ontologies, really, it was looking for developmental constraints. 178 00:23:35,470 --> 00:23:40,030 And one developmental constraint, the obvious one that out into enough cells. 179 00:23:41,470 --> 00:23:50,470 And so the whole business about Tier two colleges is really, as I said, it's fascinated people for a very long time. 180 00:23:51,850 --> 00:24:03,140 Now, what the big name in the early 19th century was STOCKARD, who wrote the classic paper around 1922, 181 00:24:03,190 --> 00:24:10,060 1921, and where he discussed the paper about nearly 200 pages long. 182 00:24:10,570 --> 00:24:22,690 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. 183 00:24:23,270 --> 00:24:34,450 And this is an example of scale and complexity where this hand actually has more fingers than normal. 184 00:24:35,290 --> 00:24:43,820 And so. When one looks at the tickling and this is work by my friend Lewis Wolpert. 185 00:24:44,450 --> 00:24:51,530 And the thing about the thing about biologists that I found kind of funny. 186 00:24:52,010 --> 00:24:55,190 The the the chick has three digits. 187 00:24:56,430 --> 00:25:00,810 And I think if you went into biology, you'd say, let's call them one, two, three. 188 00:25:02,700 --> 00:25:06,450 Biologists call them two, three, four. It's not because they can't count. 189 00:25:06,970 --> 00:25:17,730 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. 190 00:25:18,610 --> 00:25:29,680 Okay. So. So what he did was that he did a chicken, a graft, so that there were more cells. 191 00:25:30,250 --> 00:25:36,600 And so he took. He believed in gradients for pattern formation. 192 00:25:37,020 --> 00:25:42,390 But so he took. This is where the chemical kind of starts. 193 00:25:42,510 --> 00:25:49,710 He says to effect pattern. So he took it from one chick limb and stuck it in there. 194 00:25:51,000 --> 00:25:58,260 And what happened was you get to a chick, the embryo chick with two limbs. 195 00:25:59,370 --> 00:26:10,389 And so really at. What one really is finding from a theoretical modelling point of view and experimental is the same. 196 00:26:10,390 --> 00:26:16,600 Verification is really that there are developmental constraints. 197 00:26:18,010 --> 00:26:25,659 What we found was we tried we tried for a very long time to see if one could get something 198 00:26:25,660 --> 00:26:31,570 that was a single aggregation that goes into two or a single one that goes into three. 199 00:26:32,560 --> 00:26:41,440 We could never find it. And looking at the, you know, all the literature, etc., we never, ever found a tri fixation. 200 00:26:43,430 --> 00:26:50,360 And one of the nice things, one of the nice things about Oxford is its high table. 201 00:26:50,360 --> 00:26:54,320 I don't mean just the food you meet, you meet a lot of people. 202 00:26:54,710 --> 00:26:59,180 And I always like to find out what people were working on, etc. 203 00:26:59,510 --> 00:27:02,780 And at this time, somebody said, What were you doing? 204 00:27:02,810 --> 00:27:12,230 I said, Well, I just, you know, trying to look at these developmental constraints and we find there's no time for creation. 205 00:27:12,240 --> 00:27:17,880 So three headed monsters. And they said. If I can find one. 206 00:27:19,010 --> 00:27:24,110 Will you bet a nice bottle of wine? I confidently said. 207 00:27:24,110 --> 00:27:32,520 Of course. And anyway, he. Then I found this one, which is a deceitfulness. 208 00:27:33,060 --> 00:27:36,950 It's a skeleton. 19th century of one. He produced this one. 209 00:27:38,570 --> 00:27:46,160 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. 210 00:27:47,250 --> 00:27:55,910 This is how it would have happened. That there would be a bifurcation and then there would be another bifurcation. 211 00:27:56,120 --> 00:28:00,260 And that's exactly what happened. But they called them tie fly. 212 00:28:01,760 --> 00:28:13,160 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. 213 00:28:14,590 --> 00:28:25,690 Fish particularly so. Then there's a frog which was a and then but the one I think I like best was the chicken. 214 00:28:27,060 --> 00:28:31,960 And that the these are these are actually real ones. 215 00:28:33,220 --> 00:28:37,840 So the question is, are there any other three headed monsters? 216 00:28:38,530 --> 00:28:49,480 Well, there's a lot in in the ancient classical literature, Cerberus is the dog that gods [INAUDIBLE] for Hades. 217 00:28:50,770 --> 00:28:55,990 And that is really this fascinated people for centuries. 218 00:28:56,290 --> 00:29:06,040 And this is Pinelli who dog and these three headed dogs are always fierce looking the always just terrible 219 00:29:06,370 --> 00:29:14,140 because they are supposed to prevent people getting into [INAUDIBLE] in rescuing the people in [INAUDIBLE]. 220 00:29:14,470 --> 00:29:16,780 So they want the dogs to look really fierce. 221 00:29:18,430 --> 00:29:28,240 Then I came across in the archaeological museum and Kate see are the nice looking dogs, three headed dogs. 222 00:29:28,660 --> 00:29:32,050 But that's that is the only one. 223 00:29:32,710 --> 00:29:38,530 However, that may not be three headed dogs, but there are certainly lots of seven headed ones. 224 00:29:39,190 --> 00:29:45,730 And this actually comes from something called the Cabinet of Natural Curiosities 225 00:29:46,120 --> 00:29:53,140 that was done by someone called Alberta Sabre in Oak Island about 1750. 226 00:29:53,500 --> 00:29:58,570 It's it's an incredible book full of lovely, lovely pictures. 227 00:29:59,560 --> 00:30:10,300 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. 228 00:30:11,620 --> 00:30:21,310 And this is work that got started with Buster Alva, who was the head of pathology, 229 00:30:22,090 --> 00:30:29,740 and he came to see me to see if I could try and model brain tumours. 230 00:30:31,190 --> 00:30:35,540 And he must have been about 80 at the time. 231 00:30:36,230 --> 00:30:46,490 And he had been in the medical world since he was 28 and he had worked on brain tumours specifically called glioma blast tumours. 232 00:30:47,700 --> 00:30:51,210 These are sometimes are called grade for tumours. 233 00:30:51,600 --> 00:30:58,470 They are the most serious tumour, brain tumour you can get and nobody survives. 234 00:30:59,930 --> 00:31:07,430 And so I said to him, how did he know of any case of anyone surviving from when he started? 235 00:31:07,910 --> 00:31:16,030 And he said. Well, that is one possible case, but it's controversial and that was over a period of 50 years. 236 00:31:16,030 --> 00:31:20,260 And now there is still no and I want to show why. 237 00:31:21,620 --> 00:31:26,470 Why these brain tumours are so are so difficult and so dangerous. 238 00:31:27,710 --> 00:31:31,010 And the people that got involved. 239 00:31:31,010 --> 00:31:39,500 I always like to get my students involved. These are they're all Ph.D. students of mine who I got involved. 240 00:31:40,040 --> 00:31:47,570 But their topic for the thesis was always different to some of these extra, I suppose, kind of curricular things. 241 00:31:48,860 --> 00:31:53,450 Okay, so what do we want to do with the mathematical model? We want. 242 00:31:53,450 --> 00:32:04,040 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. 243 00:32:04,790 --> 00:32:09,290 What we wanted to do was to try and enhance imaging processes. 244 00:32:11,650 --> 00:32:16,360 We also wanted to show the inadequacies of current treatment. 245 00:32:17,710 --> 00:32:25,780 And of course, I'll come back to it. We want to know when tumours, such tumours actually stopped. 246 00:32:27,520 --> 00:32:33,910 The other aims are we want to be able to estimate life expectancy from detection. 247 00:32:35,510 --> 00:32:44,870 But for other reasons I'll come back to we want to quantify patient treatment prior to their use. 248 00:32:45,650 --> 00:32:49,880 The treatments are just unbelievably awful. 249 00:32:49,910 --> 00:32:53,720 Chemotherapy, radiation and surgery. 250 00:32:54,260 --> 00:32:57,590 And I feel particularly strongly about about the last one. 251 00:32:58,250 --> 00:33:02,690 We want to know why some patients live longer than others with the same treatment. 252 00:33:04,230 --> 00:33:11,670 And of course, the key thing is we want to help to design scientific, scientific tiles. 253 00:33:13,050 --> 00:33:23,890 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, 254 00:33:24,160 --> 00:33:29,110 but just a little bit of history. The Incas. 255 00:33:29,230 --> 00:33:33,580 People have been doing preparations for thousands and thousands of years. 256 00:33:34,090 --> 00:33:39,310 This is an Inca skull from the museum in Cusco. 257 00:33:39,820 --> 00:33:47,110 And what one sees, there are four surgeries done on this head, and they have all healed. 258 00:33:48,710 --> 00:33:58,220 And. If one becomes much more recent, the Casey Tribe, which is probably the smallest type in Kenya, 259 00:33:58,730 --> 00:34:04,250 they do tap in nations outside where the person goes and sits in the chair. 260 00:34:04,910 --> 00:34:14,990 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. 261 00:34:15,260 --> 00:34:23,440 And often that's the time they think to try and. Really get rid of some pressure in the brain, etc. 262 00:34:23,680 --> 00:34:33,130 It's not very medical. What is interesting is the mortality rate is zero to a first approximation of zero. 263 00:34:33,520 --> 00:34:45,490 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. 264 00:34:46,510 --> 00:34:50,320 Then they were infected just like now, and they died. 265 00:34:51,190 --> 00:34:54,970 And so people stopped doing terminations in that way. 266 00:34:56,050 --> 00:35:03,720 Let me show you what a brain tumour really looks like and what how brain tumours are detected. 267 00:35:03,730 --> 00:35:10,600 There's usually some something people collapse or at least the speech becomes faulty, whatever. 268 00:35:10,990 --> 00:35:25,149 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, 269 00:35:25,150 --> 00:35:30,160 one or two and the T two are more expensive. 270 00:35:30,460 --> 00:35:33,610 But what they do is they show more of the tumour. 271 00:35:35,130 --> 00:35:38,830 And so. All these scans are used. 272 00:35:38,830 --> 00:35:48,430 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, 273 00:35:48,970 --> 00:35:59,140 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. 274 00:36:00,760 --> 00:36:06,400 And so really the question we wanted is. How do you make a model of this? 275 00:36:07,120 --> 00:36:15,790 And so I remember the first meeting it was with Buster Alvord, and I said, Well, what colleges are? 276 00:36:15,820 --> 00:36:20,740 To me, he said, Well, there are cancer cells of this, this. And he went on and on. 277 00:36:21,190 --> 00:36:24,050 And all of these things are important. 278 00:36:24,070 --> 00:36:37,840 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? 279 00:36:38,830 --> 00:36:45,370 And so one of the cancer cells, how did it multiply and how did it move around? 280 00:36:46,240 --> 00:36:57,370 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. 281 00:36:57,730 --> 00:37:06,460 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, 282 00:37:07,450 --> 00:37:11,590 the red they grey matter and the white matter and the cells diffuse. 283 00:37:12,340 --> 00:37:15,880 And and what else they do is they multiply. 284 00:37:16,810 --> 00:37:22,510 That's the model. And all one does mathematically is you just quantify it. 285 00:37:22,870 --> 00:37:29,890 And so that's just you can think of it as a shorthand way of saying something in words. 286 00:37:31,000 --> 00:37:43,960 But the key thing about the model is there are diffusion coefficients and cancer cells diffuse differently and white matter and then grey matter. 287 00:37:45,740 --> 00:37:49,190 And not only that, cells multiply differently. 288 00:37:50,370 --> 00:37:59,610 Some go quickly and some don't. But the thing about the thing about brain tumour is that unbelievably irregular. 289 00:38:00,180 --> 00:38:07,770 And so what what one has to do is one has to take scans of the pain of a real brain. 290 00:38:08,310 --> 00:38:13,980 And so, although that is it can be a complicated model, which is actually what we used. 291 00:38:16,720 --> 00:38:24,130 What I thought was if you take this tumour and you take all these scans and we get the volume of it, 292 00:38:24,970 --> 00:38:29,800 why don't we think that it's just a it's just a sphere. 293 00:38:31,120 --> 00:38:40,780 And if we then have two scans, we can then calculate the diffusion of the cells and how quickly they multiply. 294 00:38:41,290 --> 00:38:50,140 If each time we do these scans, we take the tumour and make it into a sphere just quantitatively to begin with. 295 00:38:50,230 --> 00:39:00,190 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. 296 00:39:00,580 --> 00:39:04,720 It really it is a very it's just a basic equation. 297 00:39:05,350 --> 00:39:09,700 And you can write down the solution. And the solution is just that. 298 00:39:10,150 --> 00:39:14,860 The number of cancer cells depends on how many you started with. 299 00:39:15,310 --> 00:39:18,490 They grow exponentially at a given rate. 300 00:39:19,150 --> 00:39:28,570 This is that the radius of the, the tumour of the tumour of the spherical tumour and that's a diffusion coefficient 301 00:39:28,570 --> 00:39:36,150 and that's the time that so you end up with a simple equation that you know, 302 00:39:36,760 --> 00:39:39,760 people in people at school can actually plot. 303 00:39:41,050 --> 00:39:52,890 So let me get some facts now. When the radius of a tumour is about 1515 millimetres, that is when it is typically detected. 304 00:39:54,340 --> 00:40:03,430 And so the solution of the equation says that the diameter of the tumour is going to grow. 305 00:40:03,460 --> 00:40:12,940 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. 306 00:40:13,810 --> 00:40:17,800 That's another of these things where that's just a premise. 307 00:40:18,250 --> 00:40:23,980 So what one has to do is one then has to say, Well, how quickly will it go? 308 00:40:24,760 --> 00:40:29,950 Well, you just divide by T and you get the velocity. 309 00:40:30,460 --> 00:40:33,730 And so you get an expression for the velocity. 310 00:40:34,240 --> 00:40:38,440 Unbelievably simple one that anyone can actually do. 311 00:40:38,620 --> 00:40:42,130 But of course, that's once again, that's speculation. 312 00:40:42,790 --> 00:40:49,440 So what we did is we got data. From 27. 313 00:40:49,710 --> 00:40:51,660 But they are low grade tumours. 314 00:40:52,920 --> 00:41:02,620 And the reason that we don't have many for high grade tumours is because people with these glioma blasts almost never, ever. 315 00:41:03,330 --> 00:41:08,130 Well, practically never let them go unlimited. 316 00:41:08,460 --> 00:41:19,900 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. 317 00:41:20,290 --> 00:41:24,460 Okay. But as I said, that's a hypothetical thing. 318 00:41:24,760 --> 00:41:28,600 So let's look at what a real brain looks like. 319 00:41:28,630 --> 00:41:37,930 These are cross-sections. And this is this is a web page that's put out by the Montreal Neurological Institute. 320 00:41:38,710 --> 00:41:41,710 It's public. Anybody can use it. 321 00:41:42,190 --> 00:41:48,190 And so this is a typical cross-section of the brains that we worked with. 322 00:41:49,490 --> 00:41:54,530 And so this was an example of a real case. 323 00:41:54,980 --> 00:41:58,040 This tumour was diagnosed in 2000. 324 00:41:59,060 --> 00:42:02,180 And this is what it looked like with scanning. 325 00:42:02,540 --> 00:42:07,040 With scanning? Not mathematically. In July 2001. 326 00:42:09,900 --> 00:42:13,320 This is it with autopsy. 327 00:42:14,370 --> 00:42:27,150 That is what the autopsy looked like. That is what came from the numerical simulation of the model equation for the real brain. 328 00:42:27,510 --> 00:42:37,080 In fact, this patient's brain and that was what we started with and this is the post-mortem, uh, scanning. 329 00:42:38,040 --> 00:42:44,730 And so really what one, what one's getting out of this very simple basic model is. 330 00:42:45,030 --> 00:42:52,380 This is a typical example. The problem with imaging is that you can't see it. 331 00:42:53,220 --> 00:42:57,000 The images aren't really sophisticated enough. 332 00:42:57,360 --> 00:43:04,530 And that black line is the most sophisticated imaging technique available. 333 00:43:05,590 --> 00:43:10,480 No. And that's all it can see. And so the tumour has already gone. 334 00:43:11,710 --> 00:43:13,990 It's really gone all over the brain. 335 00:43:14,830 --> 00:43:25,100 The threshold of detection is really enormous number of cells and it really one would think one would see them know this. 336 00:43:25,280 --> 00:43:33,910 I'm not sure if this is this will work. This is a typical development of what a tumour looks like. 337 00:43:36,660 --> 00:43:44,970 No, I don't. Let me see if I can get this. Sorry. 338 00:43:44,990 --> 00:43:48,120 It's the. It's the Oxford thing. 339 00:43:48,140 --> 00:43:54,290 Anyway, let me go on to. We tried this morning and it didn't work and we tried this afternoon. 340 00:43:54,290 --> 00:43:59,570 It did work. I suppose that's the way the cookie crumbles anyway. 341 00:44:01,850 --> 00:44:05,389 Let me show you this. This is this is a movie that does work. 342 00:44:05,390 --> 00:44:09,680 But the other one was more for parameter values of a real. 343 00:44:12,690 --> 00:44:22,420 Sorry, but this. And what you do is you start to see and this is the tumours developing. 344 00:44:23,770 --> 00:44:27,670 Imaging cannot see it until you see the black line. 345 00:44:29,300 --> 00:44:35,090 And this is. And so it's up to about 125 days. 346 00:44:35,420 --> 00:44:41,360 You don't see anything for something like two thirds of the growth of the tumour. 347 00:44:42,090 --> 00:44:48,140 Now it starts to grow. And so really it is so it. 348 00:44:50,820 --> 00:44:58,260 This is a more accurate picture of the thing that is quite sophisticated. 349 00:44:58,560 --> 00:45:08,550 Magnetic resonance imaging can find, and that is what the model shows based on that patient's. 350 00:45:09,490 --> 00:45:13,720 Diffusion of the cancer cells and the multiplication of the cancer cells. 351 00:45:14,380 --> 00:45:23,710 And so what one can do, and it really is somewhat depressingly accurate, but this is what one can do, 352 00:45:23,950 --> 00:45:33,520 is when a person is discovered with a tumour and this is the actual patient, the tumour grew. 353 00:45:34,090 --> 00:45:43,090 And now what? Subtotal resection is just the medical world for surgical removal. 354 00:45:44,060 --> 00:45:50,230 And the thing about the thing about surgeons is they want to. 355 00:45:50,620 --> 00:45:59,400 They want to actually operate. And so what with this patient, they tried to cut out the tumour. 356 00:46:00,370 --> 00:46:03,820 And then. But the tumour started to grow again. 357 00:46:04,390 --> 00:46:11,560 Then the patient had chemotherapy and radiation and then they can't keep it on forever. 358 00:46:11,920 --> 00:46:25,450 Then the tumour started to grow again and when it got roughly up to a diameter of 3050 millimetres, the patient, the patient died. 359 00:46:26,410 --> 00:46:34,330 And what one could estimate is if they had done nothing, how much longer would this patient have lived without surgery? 360 00:46:34,840 --> 00:46:39,299 So the question is how long will the patient live it? 361 00:46:39,300 --> 00:46:48,400 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. 362 00:46:48,730 --> 00:46:54,430 And one sees how really what a terrible life one can have. 363 00:46:56,220 --> 00:47:10,860 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. 364 00:47:11,970 --> 00:47:18,750 And so that I must admit that I mean, I learned about this. 365 00:47:18,750 --> 00:47:24,990 I thought maybe that was the right decision. And so what she went through all the tests, 366 00:47:25,860 --> 00:47:31,049 the kind of tests about whether the brain was affecting things like a touch your left 367 00:47:31,050 --> 00:47:39,120 eye with your little finger of the right hand for up until two months before she died. 368 00:47:39,390 --> 00:47:43,800 She lived almost a normal life except for this hanging over her. 369 00:47:44,280 --> 00:47:50,720 And so really it is. You know, it really is one of these very difficult questions. 370 00:47:51,680 --> 00:48:02,570 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? 371 00:48:03,230 --> 00:48:13,680 Well. If one. If you think about cell phones and nobody wants to think about cell phones. 372 00:48:14,790 --> 00:48:27,780 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. 373 00:48:28,260 --> 00:48:40,140 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. 374 00:48:41,190 --> 00:48:49,770 He then followed the plant development. The branches and the leaves were totally distorted by the time it should have been normal. 375 00:48:51,060 --> 00:48:55,590 He's a fellow of the French Academy. He tried to get those paper published. 376 00:48:56,040 --> 00:49:05,670 He could never get it published because the journals that he submitted it to were frightened that the cell phone people would sue them. 377 00:49:06,840 --> 00:49:14,550 So what I would recommend is that next time you buy a cell phone, read the small print, 378 00:49:15,060 --> 00:49:21,900 and they say there is no evidence that this will cause any effect to brain tumours. 379 00:49:22,440 --> 00:49:25,560 However, if and it's what you read after that. 380 00:49:25,590 --> 00:49:29,159 However, you know, thus there is the possibility. 381 00:49:29,160 --> 00:49:33,229 Well, two years ago. A person. 382 00:49:33,230 --> 00:49:44,280 Then in Washington, a doctor. Vasso had 11 year old children and they took a cell phone and they put it against the children. 383 00:49:44,520 --> 00:49:49,020 The parents agreed the cell phone was against it here for 10 minutes. 384 00:49:49,980 --> 00:49:56,670 They then did CAT scans of the brain and they found there was an increase of glucose. 385 00:49:57,800 --> 00:50:04,070 Practically all over the brain. Now, nobody has actually been able to see this cancel their. 386 00:50:05,080 --> 00:50:12,400 But those of you who are sorry to be evangelical, but those of you who use cell phones should use an earpiece, 387 00:50:13,720 --> 00:50:24,670 because I think that that it must be inevitable that if there is radiation, all one needs to remember the curies with radiation. 388 00:50:25,030 --> 00:50:32,680 So one of the questions that I thought one should answer is if somebody is developed with a brain tumour, when did it start? 389 00:50:33,310 --> 00:50:43,480 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. 390 00:50:45,380 --> 00:50:53,600 Okay. Just just very quickly, I'll give you data on 57 patients had surgery. 391 00:50:54,590 --> 00:51:04,370 It's all data, but it's still relevant. 58 patients were also diagnosed and they were not had surgery. 392 00:51:05,240 --> 00:51:11,240 And the effect of this was that those with with surgery lived seven weeks longer. 393 00:51:11,990 --> 00:51:15,260 But as I said, what sort of life did they have? 394 00:51:15,830 --> 00:51:22,970 So V.S. is the removal and what we can do with the model is we can carry out virtual some surgery. 395 00:51:24,420 --> 00:51:29,560 And. And so what we that's exactly what we this is a typical diagnosis. 396 00:51:30,340 --> 00:51:39,820 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. 397 00:51:40,150 --> 00:51:50,260 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. 398 00:51:51,010 --> 00:51:56,170 Then what happens is it can be three months, six months later, the tumour. 399 00:51:56,800 --> 00:52:01,390 Lo and behold, has regrown. Not only that, it's multifocal. 400 00:52:01,930 --> 00:52:07,210 And this has been a kind of it was one of these just mysteries. 401 00:52:07,390 --> 00:52:11,050 Of course, it's going to be multifocal. It's all over the place already. 402 00:52:11,950 --> 00:52:15,790 And so it's it's why surgery can never work. 403 00:52:15,940 --> 00:52:18,940 That's what the surgeon observes. 404 00:52:19,660 --> 00:52:28,550 That's really where it is. And so what one can do then is you can actually work out surgery for a given patient. 405 00:52:29,740 --> 00:52:39,250 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, 406 00:52:39,250 --> 00:52:47,830 you can get a simple formula for how long they're going to live depending on the parameters of the of their tumour. 407 00:52:49,380 --> 00:53:02,520 Once again, we wanted to compare it with data. And the red marks of what we predicted with different patients whom theoretically we injected 408 00:53:02,520 --> 00:53:09,570 with a tumour that had the same parameter values of diffusion and growth as the echo. 409 00:53:09,630 --> 00:53:15,270 So as the actual people whose data was collected. 410 00:53:16,930 --> 00:53:22,810 Well it's all, it's all very, very depressing about brain tumours. 411 00:53:23,170 --> 00:53:32,470 The problem about people with brain tumour just to go and they talk to people in the end up talking to a neurosurgeon. 412 00:53:33,750 --> 00:53:38,370 And neurosurgeons always want to try and cut it out. 413 00:53:39,240 --> 00:53:44,370 You know, the difference between a neurosurgeon and God is that God doesn't think he's a neurosurgeon. 414 00:53:45,030 --> 00:53:58,590 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, 415 00:53:59,340 --> 00:54:02,610 you know, I said to him, I mean, you do brain surgeon. 416 00:54:02,620 --> 00:54:10,160 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. 417 00:54:10,190 --> 00:54:14,930 It's really sad. If I don't do it, somebody else will. That's what it's about, I think. 418 00:54:15,230 --> 00:54:21,640 Not always. No, I mustn't be. But anyway, to conclude this thing. 419 00:54:22,010 --> 00:54:27,140 It shows these brain tumours are unpredictable. They really are. 420 00:54:27,620 --> 00:54:30,650 And the model suggests new way to study them. 421 00:54:31,820 --> 00:54:39,680 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. 422 00:54:40,790 --> 00:54:45,110 We can now estimate really when the tumour started. 423 00:54:46,100 --> 00:54:53,870 Not only that, we can actually quantify the how the how it grows and spreads. 424 00:54:55,650 --> 00:55:02,630 The. It's an explanation of why some patients live longer than others. 425 00:55:02,930 --> 00:55:07,100 The values of the diffusion and the cell multiplication. 426 00:55:07,100 --> 00:55:13,679 A cancer cell multiplication. A different. Not only that, we find children, 427 00:55:13,680 --> 00:55:22,499 young children live longer because it turns out that the parameters associated with the growth of their tumours are such that 428 00:55:22,500 --> 00:55:31,020 the tumour is more contained and so it means that they can actually live longer with radiation which kills more of the cells. 429 00:55:31,740 --> 00:55:34,890 But the end result is unfortunately the same. 430 00:55:35,610 --> 00:55:48,990 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. 431 00:55:49,290 --> 00:55:59,520 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. 432 00:56:00,150 --> 00:56:08,850 So let me finish with only the Mona Vale again, and I think it was amazingly practical man. 433 00:56:08,970 --> 00:56:15,870 He was the surgeon to two kings, etc. And as you saw, he had a low opinion of people, 434 00:56:16,470 --> 00:56:29,610 but he did like surgery and he said it's a cure, were preferable, more noble, more perfect, more necessary and more lucrative. 435 00:56:30,720 --> 00:56:40,950 That was in the 14th century. And okay, let me talk about something a little different just to defeat the. 436 00:56:41,130 --> 00:56:55,450 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, 437 00:56:56,080 --> 00:57:04,480 still in Oxford, and said he wondered if I could maybe try and make a mathematical model about marital interaction. 438 00:57:05,710 --> 00:57:10,900 And I thought it was a pile of rubbish and ended up in Washington. 439 00:57:10,900 --> 00:57:22,120 We had lunch together and I got kind of hooked. And the think about divorces that are millions a year. 440 00:57:22,120 --> 00:57:26,560 And that is now a huge, enormous legal industry. 441 00:57:27,500 --> 00:57:34,760 With lawyers who specialise in divorce of couples who come from different countries. 442 00:57:35,810 --> 00:57:40,310 So, for example, if you live in Texas, it's better to be a man. 443 00:57:41,440 --> 00:57:48,100 If you live in France and the man has been having an affair, then it's kind of half and half. 444 00:57:48,370 --> 00:57:55,060 There are some countries where the man can get a divorce and the wife doesn't even know it until she gets the papers. 445 00:57:55,540 --> 00:58:02,410 It is unbelievably complicated. And so once again, I got a bunch of my students. 446 00:58:02,920 --> 00:58:11,170 Uh, Jane. Jane White, Jane Snow in Bath, Rebecca Tyson, Cliff in Canada, and John Gottman. 447 00:58:11,320 --> 00:58:17,710 After the results of all this retired air late and started up a clinic for marital therapy. 448 00:58:18,960 --> 00:58:24,520 Yeah. Okay. So what do we do? Well, first of all, 449 00:58:25,090 --> 00:58:36,370 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. 450 00:58:37,330 --> 00:58:46,510 They were invited to partake in the study associated with marital interaction and divorce prediction. 451 00:58:47,910 --> 00:58:59,130 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. 452 00:59:00,270 --> 00:59:07,950 Absolutely made no difference what it was. I mean, food, sex, in-laws, laws, you name it, it made absolutely no difference. 453 00:59:08,980 --> 00:59:16,030 Okay. So what we did then was we scored what each of them said and there is a scoring 454 00:59:16,030 --> 00:59:23,470 system and you assigned an integer between plus four and minus four and. 455 00:59:24,620 --> 00:59:31,370 This. What you then did is you counted the positive things and took away the negative things. 456 00:59:31,400 --> 00:59:39,170 Each time one of them spoke and this was data was gathered mainly by psychology 457 00:59:39,170 --> 00:59:44,090 students who sat behind a one way mirror and the couple knew they were being filmed. 458 00:59:44,480 --> 00:59:50,450 And usually within about a minute they'd forgotten everything except what they were arguing about. 459 00:59:51,560 --> 00:59:58,220 And so these six students, it was amazing how close they were in getting the same numbers. 460 00:59:59,120 --> 01:00:04,010 And so what you get each time they speak it, you get a number. 461 01:00:04,310 --> 01:00:07,880 So you end up drawing a curve of a kind of Dow Jones average. 462 01:00:08,270 --> 01:00:16,729 I gave a talk about this and then somebody in Australia sent sent me a cartoon and that they 463 01:00:16,730 --> 01:00:21,320 wanted to show whether this test will show whether or not they're going to get divorced. 464 01:00:21,620 --> 01:00:25,250 And they had to choose a topic that was really big. Okay. 465 01:00:25,310 --> 01:00:30,890 Yeah. This is what the scoring system, the scoring system is. 466 01:00:31,310 --> 01:00:35,810 And it is amazingly easy to assess the numbers. 467 01:00:36,380 --> 01:00:42,710 They're also associated with visual things about whether it's anger, contempt, humour, etc. 468 01:00:43,100 --> 01:00:47,330 And this is the 16 code system. 469 01:00:48,480 --> 01:00:55,830 Unchained. And so what one gets is you get a kind of Dow Jones average of the conversation. 470 01:00:57,460 --> 01:01:00,820 So in this case there, that's zero. 471 01:01:01,090 --> 01:01:05,470 And it ends up it becomes progressively more more positive. 472 01:01:05,890 --> 01:01:10,990 And so it's roughly a 5 to 1 ratio of positive to negative. 473 01:01:11,830 --> 01:01:15,190 It's a very stable marriage, that one on the other hand. 474 01:01:18,670 --> 01:01:28,780 And so that is a ratio of about 0.8 positive, positive to one negative. 475 01:01:29,350 --> 01:01:33,310 And this couple actually did end up getting getting divorced. 476 01:01:35,320 --> 01:01:40,750 This is the data between 1992 or 2004. 477 01:01:41,230 --> 01:01:49,070 We actually studied 700 couples and we analysed the thing. 478 01:01:49,090 --> 01:01:54,070 And what we found is the number of the different marriage types were very limited. 479 01:01:54,520 --> 01:02:04,120 There were only five of them. And what we predicted was whether they would get divorced, stay married, either happily or unhappily. 480 01:02:04,690 --> 01:02:12,580 Then every every 1 to 2 years, we sent out a questionnaire and they were asked to complete it. 481 01:02:12,850 --> 01:02:18,430 We then compared the predictions with what the facts were. 482 01:02:18,730 --> 01:02:24,310 So this isn't statistics in the sense that you do a small study and you then expand it. 483 01:02:24,700 --> 01:02:33,930 These are real numbers that we got. Well, prediction of which couples would get divorced was actually 94% accurate. 484 01:02:34,990 --> 01:02:40,270 It was even better than that in the sense that those we predicted would definitely get divorced. 485 01:02:40,270 --> 01:02:42,040 We were 100% correct. 486 01:02:42,730 --> 01:02:52,750 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. 487 01:02:53,020 --> 01:03:03,670 Of course, we couldn't possibly tell them anything about the predictions, but okay, this is for the therapy. 488 01:03:04,030 --> 01:03:11,290 We had to develop something that would show people who did not have any kind of background. 489 01:03:11,560 --> 01:03:19,270 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. 490 01:03:19,970 --> 01:03:25,330 And so we got a couple of actors to show it. And this stayed in this quadrant. 491 01:03:25,810 --> 01:03:33,520 They're both happy. And then that one, that boy of that one, the wife is happy. 492 01:03:35,020 --> 01:03:46,420 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. 493 01:03:47,840 --> 01:03:53,260 And so just again, I don't know if this movie will work. 494 01:03:53,270 --> 01:03:59,420 Let me see. This is a typical this is typical data from one couple. 495 01:04:04,980 --> 01:04:09,300 And and the numbers look as if they're going all over the place. 496 01:04:10,710 --> 01:04:16,410 And after one had seen about 25 of these couples. 497 01:04:17,870 --> 01:04:22,880 We could look at this data and say. You're going to get divorced. 498 01:04:23,030 --> 01:04:28,310 But of course, that's no good. One has to actually be able to quantify it. 499 01:04:29,210 --> 01:04:31,070 So let me briefly describe the model. 500 01:04:32,360 --> 01:04:44,270 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. 501 01:04:44,570 --> 01:04:53,510 That's how she felt before. And that's something I'll mention just later, is how they interpret their view of the marriage. 502 01:04:54,830 --> 01:05:00,290 And so you can put this into you can put this into a simple mathematical model. 503 01:05:01,940 --> 01:05:15,299 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, 504 01:05:15,300 --> 01:05:18,440 that means the wife isn't very interested in changing her mind. 505 01:05:18,980 --> 01:05:24,860 If R is near zero, then it means that she can be influenced by what the husband says. 506 01:05:25,490 --> 01:05:34,820 And so you have that equation and this influence function is how we actually describe marriages. 507 01:05:35,750 --> 01:05:39,030 And so. That is typical data. 508 01:05:40,620 --> 01:05:43,500 The husband when the husband says something. 509 01:05:44,790 --> 01:05:51,420 This is the influence that we were able to evaluate from the data obtained from the 15 minute conversation. 510 01:05:52,740 --> 01:05:57,600 So you can put a curve, a kind of best fit curve, but that's very complicated. 511 01:05:58,050 --> 01:06:04,680 And so we decided, let's not bother with that. Let's just take near the orange and two straight lines. 512 01:06:06,210 --> 01:06:12,030 Of course, what is really curious that totally surprised us is here. 513 01:06:12,030 --> 01:06:17,970 The husband is being incredibly positive. The influence on his wife is negative. 514 01:06:19,380 --> 01:06:24,780 In other words, she's starting to think, what's he trying on until. 515 01:06:25,170 --> 01:06:35,520 But the fact that this happens happened with the vast majority of these 700 couples, it just was a somewhat surprising phenomenon. 516 01:06:36,090 --> 01:06:40,410 And these these are the how we quantify the marriage. 517 01:06:42,000 --> 01:06:48,640 And these are just two examples of the wife's influence and the husband and the husband's influence and wife. 518 01:06:48,990 --> 01:06:57,630 And so this is a typical thing that if the husband says something, put something sorry. 519 01:06:58,650 --> 01:07:02,309 If the husband says something positive, it has a positive effect. 520 01:07:02,310 --> 01:07:06,090 If you say something, it has a negative effect, but less negative. 521 01:07:06,660 --> 01:07:10,470 That is very common in in in marriages. 522 01:07:11,700 --> 01:07:24,360 And so these are two examples of the influence numbers for what are conflict avoiding couples that when they say something negative. 523 01:07:25,360 --> 01:07:30,520 A the the influence is much less than if they say something positive. 524 01:07:30,970 --> 01:07:34,570 And these are numbers, as I said, that actually come from the data. 525 01:07:35,200 --> 01:07:38,830 And so that's a theoretical conflict, avoiding couple. 526 01:07:39,220 --> 01:07:44,240 They have little effect on each other on the negative side. So then. 527 01:07:44,270 --> 01:07:54,480 Then this. When we looked at the 700 couples, really with only five types of marriage, three stable ones and two unstable ones. 528 01:07:55,270 --> 01:08:00,210 And the characteristics are the stable, the volatile ones. 529 01:08:01,380 --> 01:08:05,070 Some of them were stable, but generally they were unstable. 530 01:08:05,520 --> 01:08:09,030 You know, they have heated arguments, romantic, etc. 531 01:08:09,720 --> 01:08:15,330 But then the ones that really were stable but those that were calmer, intimate, 532 01:08:16,020 --> 01:08:24,000 shared experience rather than individuality, and they were the ones that really stayed married. 533 01:08:24,570 --> 01:08:30,600 The avoidance was also stable. They didn't want to argue when one of them was being negative. 534 01:08:30,840 --> 01:08:36,960 They tried to pacify them. The hostile ones are when you get mixed marriages. 535 01:08:37,990 --> 01:08:50,980 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. 536 01:08:51,460 --> 01:09:02,710 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. 537 01:09:04,180 --> 01:09:08,980 In the marriages part of the time, they didn't influence each other at all. 538 01:09:09,550 --> 01:09:14,680 Then you end up with a model equation and you say, What is the steady state? 539 01:09:15,640 --> 01:09:19,810 And so you can just say so. In other words, they don't vary. 540 01:09:19,810 --> 01:09:25,840 Everything is the same. You solve it. You get that, you solve it, and the uninfluenced, 541 01:09:25,840 --> 01:09:36,040 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. 542 01:09:37,320 --> 01:09:41,160 These are the parameters that we evaluated. You get the same for the husband. 543 01:09:42,210 --> 01:09:47,970 If he is less than zero and B is less than zero, they're almost certainly going to get divorced. 544 01:09:48,750 --> 01:09:56,100 It is it was that simple. So one uninfluenced, steady state has to be positive. 545 01:09:58,280 --> 01:10:04,069 And this inertia things are, as I say, the fog is near one. 546 01:10:04,070 --> 01:10:06,380 It means a pretty rigid in what they believe. 547 01:10:08,200 --> 01:10:17,560 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. 548 01:10:18,520 --> 01:10:24,280 And these are just lines that if you vary. H and W along these lines you have a constant. 549 01:10:24,550 --> 01:10:28,480 It's kind of constant. So you have one of these lines for the wife. 550 01:10:29,200 --> 01:10:35,560 You have one for the husband where the interact of the steady state of the marriage. 551 01:10:36,460 --> 01:10:43,720 So in this case, the black line is a stable, steady state, and that is in the first quadrant. 552 01:10:44,410 --> 01:10:47,110 There's another steady state which is unstable. 553 01:10:47,770 --> 01:10:59,440 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. 554 01:10:59,650 --> 01:11:05,120 And they move up to a stable, steady state. So what one can do with this is this. 555 01:11:05,740 --> 01:11:12,250 This is crucially dependent on the uninfluenced, steady state of the wife and of the husband. 556 01:11:12,760 --> 01:11:18,640 So suppose the husband gets a little more or less happy with the whole situation. 557 01:11:19,930 --> 01:11:23,950 Then what happened? I mean, the happy state. Here you've got this. 558 01:11:24,370 --> 01:11:28,690 It's a thing in the bodily, you know that that's a happy marriage. 559 01:11:29,080 --> 01:11:31,240 I don't know what they're doing in the bushes, but anyway. 560 01:11:33,430 --> 01:11:42,040 But if the husband becomes less satisfied with the marriage, then he moves his steady state, 561 01:11:42,370 --> 01:11:48,910 moves over into the negative region, and that influences the steady state. 562 01:11:49,570 --> 01:11:55,010 And so instead of really being very positive, the husband is not very happy with it. 563 01:11:55,810 --> 01:12:02,980 And but the wife still putting up with it if the husband becomes even more unhappy with that. 564 01:12:03,430 --> 01:12:08,280 What you end up with is. There's no steady state. 565 01:12:09,690 --> 01:12:17,579 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, 566 01:12:17,580 --> 01:12:24,840 sort of, you know, a delightful mutual, you know, sort of orgy or some kind of Calvinist [INAUDIBLE]. 567 01:12:25,630 --> 01:12:37,140 And but basically what one gets in this very simple model is you can actually quantify certain things about a marriage. 568 01:12:37,740 --> 01:12:44,430 Of course, there. I don't know who was it that said, never go to bed angry, stay up and fight. 569 01:12:45,120 --> 01:12:51,719 I don't know who said some. I think it was some American comedian, but I think it's a probably Groucho Marx. 570 01:12:51,720 --> 01:12:55,410 His comment, he said, you know, it it's inevitable. 571 01:12:55,410 --> 01:13:04,650 Marriage always interrupts and influences and affects romance because if you have a romance, your wife is certainly going to interfere. 572 01:13:05,160 --> 01:13:09,090 And so it's a so there's all sorts of great, nice points about this. 573 01:13:09,510 --> 01:13:13,980 Okay. So just last few slides is the. 574 01:13:16,050 --> 01:13:21,680 I did it with straight lines. But the real numbers of these lines are really quite clear. 575 01:13:21,690 --> 01:13:28,640 They come from the data, the complicated, and that is a typical, uh, stable marriage. 576 01:13:28,950 --> 01:13:36,360 And if they start anyway of the conversation negatively, it always ends up at the positive, steady state. 577 01:13:37,560 --> 01:13:47,490 Very stable marriage. In this case, the stable states are negative and it doesn't matter where they start, they always end up there. 578 01:13:48,030 --> 01:13:58,080 Well, that the data that I showed you, that couple, the steady state was in the third quadrant and they actually got divorced. 579 01:13:59,540 --> 01:14:06,590 So what can one do about therapy? Just to finish. This is where the couple end. 580 01:14:06,950 --> 01:14:09,950 The steady state of their interaction was negative. 581 01:14:12,080 --> 01:14:16,440 And. The thing about this is that they are both negative. 582 01:14:16,760 --> 01:14:22,340 The probability is high for divorce, and in fact, that's exactly what happened to them. 583 01:14:23,060 --> 01:14:28,640 On the other hand, this couple, the wife was a bit positive about it. 584 01:14:28,910 --> 01:14:32,690 The husband was it. And so they actually, uh, 585 01:14:33,470 --> 01:14:44,960 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. 586 01:14:46,550 --> 01:14:52,050 The data centre, they started talking and the wife became more negative. 587 01:14:52,070 --> 01:14:55,550 The husband was negative all the time. And that. 588 01:14:57,220 --> 01:14:59,860 They then had therapy lasted for. 589 01:15:00,280 --> 01:15:09,760 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. 590 01:15:10,810 --> 01:15:16,600 We then actually did gather the data of them and this is the data. 591 01:15:16,780 --> 01:15:23,650 I mean, it might not be your community, but at least at least the you know, one of them is positive. 592 01:15:25,450 --> 01:15:31,090 But I think what was interesting is if one of them is positive, then it can be a candidate. 593 01:15:31,930 --> 01:15:42,250 So what have we actually gained from all this? Well, therapy is one thing that they are showing the data. 594 01:15:42,280 --> 01:15:45,970 They're showing the videotape. They're showing the predictions. 595 01:15:46,330 --> 01:15:54,790 And the only thing they couldn't understand was explained specifically to them by the clinical psychologist. 596 01:15:56,030 --> 01:15:59,450 And and this therapy is based. 597 01:16:00,480 --> 01:16:04,080 Uh, almost entirely on this very simple model. 598 01:16:05,340 --> 01:16:12,460 And so give me it. As I said, that these are the these are the results. 599 01:16:13,570 --> 01:16:16,690 It's a new language for discussing psychology. 600 01:16:17,290 --> 01:16:29,050 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. 601 01:16:29,200 --> 01:16:33,640 He talked to me about really the social sciences. 602 01:16:34,450 --> 01:16:40,110 And I thought at the time it was a load of rubbish. And he was absolutely right. 603 01:16:40,120 --> 01:16:48,250 I mean, he applied it, did different things, but the energy so it we can use it for that. 604 01:16:48,910 --> 01:16:52,690 It's a rationale. We found there were only five types of marriages. 605 01:16:53,470 --> 01:17:00,070 A stable ones have got matched interaction style and unstable ones have have the other kind. 606 01:17:00,730 --> 01:17:09,850 But basically the last thing I want to cite I want to really show is I quoted Benjamin Rush. 607 01:17:10,510 --> 01:17:15,010 In fact, Montaigne had a much bigger influence. 608 01:17:15,010 --> 01:17:23,480 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. 609 01:17:23,500 --> 01:17:29,110 He would offer a candle to St Michael and if needed, another to a dragon. 610 01:17:29,740 --> 01:17:35,139 And I think in a sense that's what applied mathematics is. The reason I wanted to show this thing. 611 01:17:35,140 --> 01:17:37,060 This is a portrait in the Bodleian. 612 01:17:38,340 --> 01:17:47,970 A different view really came from, I thought the only pure mathematician I could quote was be somebody in the 19th century. 613 01:17:48,870 --> 01:17:53,460 And as you can read it, we find the primitive source of rationality. 614 01:17:54,210 --> 01:18:03,690 Mathematicians must turn to biologists, you know, how can they otherwise can they carry out their experiments if they don't know any mathematics? 615 01:18:03,990 --> 01:18:10,050 Well, when one does interdisciplinary work, it's rather difficult to have that attitude. 616 01:18:10,710 --> 01:18:20,250 And I thought this picture really kind of summed up what a lot of people who have worked in this field. 617 01:18:20,250 --> 01:18:28,140 It's a certain kind of arrogance. And I showed this to a friend, an Australian friend, who then sent me that picture. 618 01:18:29,550 --> 01:18:32,760 But but basically what what it is, 619 01:18:32,790 --> 01:18:43,020 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, 620 01:18:43,350 --> 01:18:46,560 then maybe one got a chance of actually doing something. 621 01:18:46,950 --> 01:18:52,440 I suppose the trouble with mathematicians is sort of 99% of them give the rest a bad name. 622 01:18:53,160 --> 01:18:59,910 And so maybe quite. So I think that is probably the appropriate place to start. 623 01:18:59,970 --> 01:19:00,840 Thank you very much.