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Spatial Moran Model in Cancer Biology
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0:00
[LECTURER] You give a talk with somebody else's slides, you know, so-- --um, I'm not exactly the height of propriety, but, you know, the, to have other people's jokes, it's very hard to work with them, you know? So-- --yeah in case Mark's wor-- or watching out there, he's been a very good, good post-doc, you know? So, very independent and, you know, he was the one that's really been driving this work, you know? So and we've been doing it in collaboration with people at the Duke Medical School.
0:30
That was one of the nice, you know, attractions from, for coming to Duke, you know? So, other than everybody says, "Oh, you left the snow," you know? So, but was to have the Duke Medical School's
0:44
about a mile away and whereas in at Cornell it's in New York City because, you know, they don't have enough people to experiment with um, in Ithaca. You know, so, all right. So, so Jasmin described the spatial Moran model and the three theorems talk about what the geometry of the pre-malignant field and the malignancy look like. And what we wanted to do was to actually apply this to something. I mean, the words about applications didn't really mean much to the people at analytical and applied
1:19
probability, but I won't get into that further, you know? So a natural context is head and neck squamous cell carcinoma. I mean, this is something in the 1950s when this guy Slaughter was studying this was, it's, it's a cancer where it's a place where you can look at it rather than stuff happening inside your body. But there was a cancer field effect that around the malignancy there was a pre-malignant area, and sometimes you would get a second malignancy at some distance from the first site.
1:57
You know, I'll get back to this. I just-- The talk's out of order in terms of my um, how I talk about it. So then we wanted to do a model that sort of explains this. And you'll see the th-- three regimes in Jasmin's talk addresses this. So, all right.
2:14
So, you know, collaborators, I don't know who the first guy is, but it's Jasmin and Kevin at University of Minnesota.
2:24
Motivation, a clinical problem, you know? So now this picture sort of freaks me out, you know?
2:34
So, so I think it, it dates from the Rolling Stones or something like that. It's like, "Get your ya-yas out." You know? So, I have no idea. Uh, Mark's very f-- um, if he's out there, let me tell an embarrassing story.
2:49
In conjunction with graduate recruiting weekend, you know, there's, we had this math slam where six people give ten-minute talks about their research. You know? So so Mark decided to talk about breast cancer using many pictures of what's her-- Pamela Anderson from Baywatch, you know?
3:09
And this turned out to be a great recruiting tool. But basically on the fact that the grad, director of graduate students jumped up in the middle of his talk and said, "This is disgusting." You know? So, some women were impressed by the fact that, you know he would do that, you know? So-- So so if you're listening Mark, or watching it later, you know? So, next time you'll sh-- come up whether you're sick or not, you know? So-- All right.
3:35
So we're focusing on head and neck squamous cell carcinoma. Um, not because we got a grant from the Miley Cyrus Foundation-- who's concerned what might happen to her tongue, you know? So, but anyway, but it's an interesting thing And we're talking about the things that surgeons are interested in is her surgical resection. You have cancer in your tongue, and if you wanna avoid these things, then stop smoking now, 'cause that's one of the main risk factors here.
4:10
So then you're talking about surgical resection. And so let's see, Stu Karp here or, you know, I ran into him, we'll actually had a nice conversation. He goes, "Well, why would you wanna worry about the size of the resection?" You know, it's like, "Okay, if you don't mind losing your entire tongue, then, you know, you don't have to worry about that." But then, you know-- --If You cut too little, you get a recurrence. Well, obviously you don't wanna take too much.
4:35
So then, you know, the question which we want to address with our research is what size surgical margin should you have? Okay, I should have warned people that-- If you're squeamish, don't look at the next slide. You know? So, so I tell Mark hey, surgeons say, "Ah, come on over. Watch, watch an operation." You know? So it's like five hours they were cutting-- This is the tongue. I don't think it's that big, you know, normal, you know, but in the middle, let's see None.
5:04
So yeah, this sort of black stuff in the tongue is the malignancy. And they, you know, they had this rule of thumb. They go about a centimeter around it. All right. So, and, and Mark was there for about five hours and sort of left when they were still going. So one of the problems with this, there's a frequent recurrence. It's like 20 to 30%, you know? Whoops.
5:33
Oh okay. And it comes, like, one to five years later.
5:37
So here, it's our happy tongue friend who's gonna be with us the entire talk, you know? So, here was the primary cancer. You resected it, and then you didn't take enough so there was a local recurrence nearby.
5:53
Okay. So at this point you wanna go back to Jasmin's talk where we were talking about math here. So, so in Jasmin's talk there were--- The wild-type cells, the premalignant transformation, and then there's the tumor. Yeah, I have no idea what all these slides are for, right? So, and this is out of Jasmine's talk. You know, cancer is a multistage process, you know, so, And one of the things that, you know, I was reacting to your question, you're like, "I thought you were saying, 'Well,
6:29
this is a ridiculous model.'" But it's a ridiculous model 'cause it only has, like, two stages.
6:35
So you have usually a cascade of several stages in colon cancer, which grow slowly and they have lots of information. There are probably four stages. And one of the first work was done by Armitage and Doll in the '50s estimating number of stages from slopes of incidence data and from-- Oh, yeah. Okay, so I have no idea what that was from. So the thing is, the problem is field cancerization. This is a notion that, around the malignancy is a transformed field of premalignant cells. Oh okay.
7:13
Well, that's clever.
7:15
It's just something that's not visible to the surgeon, and it's like, it's not like the tongue is red And then it's orange And then there's this black stuff in the middle. So, you know, these are sort of genetic changes that happen in the cell that don't affect them in a visible way.
7:37
All right, so Jasmine mentioned in this talk is that
7:42
epithelial cells, I mean, not only for our skin, but the lining of various ducts and stuff, the breast cancer ducts, I mean uh, cervix, the uterus, the bladder you know these are tissues that divide a lot, and cells that divide a lot are more prone to the genetic mutations that turn into cancer. And then there are lots of areas where you have potential for field cancerization. I mean, there's an interesting dataset that we looked at, at one point, but got distracted, from bladder cancer when
8:21
they looked at genetic diversity of the cancer.
8:27
Right? So, all right. So, and again, the sort of the questions are, how big a margin? Is there another field present in diagnosis, which is actually probably hard to say, and risk of progression, which is sort of surveillance protocol. And actually, as we talk to the doctors more and more is that, you know, they don't worry about, you know, calculating the risk of progression because you, once you get this once you have one cancer, they're gonna be seeing you every six months or a year
8:59
for the rest of your life. You know? So, okay, so can we develop a mechanistic dynamic model to answer these questions? Or, as is often the case in mathematics, we had the model and we were looking for applications of it.
9:15
Voila!
9:16
All right, so this is a recurrent picture, you know, so-- uh, so we have, as in Jasmine's model, they're gonna be, like, type zeros, which are wild-type cells, ones which are premalignant, and then twos. All right, so many things are done using branching processes. I mean, and, you know, confess to have done that. I mean, something like leukemia, where cells circulate in the bloodstream, that's a natural context. Of course, the leukemic cells are being produced in the bone
9:52
marrow, so there's a little bit of space back there, you know, so here, geometry matters. I just can't get the same enthusiasm that Mark does here. So, but anyway. He also has 35 pairs of shoes, so you know, he has different personality here.
10:09
I have two pairs of shoes, counting multiple, not counting multiplicity, you know? So, all right, so meanwhile, this will teach Mark to show up to conference, you know, so-- So this is an oral mucosa. It's like, if you look really, really closely at your cheek, I guess this is what you would see. Let me go ahead. I don't, I'm not used to pausing, I think. So we're interested down here at the bottom. These guys that are sort of stained purple,
10:40
they're the stem cells. I mean, most of your epithelial tissue is set up like this. These guys down here, which are the stem cells that
10:50
produce all the stuff above it. For a long time, the dogma was that you have stem cells, they always divide into one differentiated cell and one stem cell. And then the differentiated cells divide a limited number of times, and eventually they go out the top. And this sort of picture happens on your skin in your colon, you know, lining your mouth. I mean, down at the bottom you know, have this population that maintains the whole thing.
11:20
And I'm n-- this is, in a sense, an intelligent design because all these guys down here, they're the ones you have to worry about mutations in. If you get a mutation in one of these guys, then it's gonna be up and gone before it does you any damage. You know? So, all right, and then in the last couple years, it was demonstrated by looking at stuff on the mouse, that, you know, sometimes you get division into two, and sometimes you get two differentiated cells.
11:56
So your problem is you go, "Yay, we got critical branching processes coming into the scheme here." And so this is like 80% and these two are 10%, it's a model that, you know, so there, actually, you can sort of see the motor model on the mouse tail if you stain things correctly. So, all right, but that's not gonna play a huge role here, I don't think.
12:20
Whoops, going backwards. Mr. Tongue is back and now it has a square grid laid on it-- --you know, it's like, it's just like most of you when you look in the mirror, you know? So, and so I'm staying over at the Duran and there's this mirror, I guess, maybe that women use for makeup. And I turn around looking and it's like my head's, like, a huge size, you know. But I, but I still didn't see the grid on my tongue here, so-- So we're using this a very simplified model,
12:48
I mean, this square lattice, it sort of depends upon how you want to do your models. You know, when I started talking to Francisco Macor about spatial models and all these cells sort of push each other around. And, you know, in a solid tumor, you know, you get a denser and denser array of sort of tumor cells, which leads to hypoxia and then death of stuff. But, you know, for some reason, if we're doing math, you put these guys on the grid and it's a good first approximation.
13:20
And you know, if we want to prove things like, you know, Whoops. If you're an assistant professor at University of Minnesota Math Department, you might want to prove things, you know. At my age they can't get rid of me, so I don't want to-- so
13:37
unless they cancel my parking permit, you know, so-- Um, all right, so we have this sort of model that Jasmine talked about, you know, zeros, ones and twos. You know, you have your four nearest neighbors, you know, so, All right, so you give birth at rate equal to your fitness, replace a neighbor chosen at random. It's probably not exactly what's happening with cells, but it sort of models the competition, you know, so, movie one, all right, so, Mm-hmm,
14:10
mm-hmm. All right, gotta watch this fast here. All right, so-- No, I don't want to.
14:19
So this is gonna be a simulation of the basic biased voter model. Right?
14:28
So this is type, the type 1s invading the type 0s,
14:33
and they sort of jump back and forth. And this is sort of what the biased voter model looks like. Uh, it's not so sufficiently exciting, I won't see it again, you know, so, all right, so the next one I'll show you twice, if you're good, you know, so-- so this one's gonna be a bigger simulation. It runs really fast, so try to watch it quickly. It's gonna, in fact, that process on a larger length scale, all right?
15:00
And here--
15:04
Whoops. All right, so, got to be ready-- --when I do that, all right? And we hope the cancer doesn't go like that in real time, you know. It's like that, there you are, here's your tongue and it's all gone, you know, so-- All right
15:21
all right.
15:24
All right, so I'm gonna show some with mutations. They sort of go fast. You really can sort of see what's going to happen, you know, you're gonna get two clones and then for this one, it just sort of starts out slow, you know. So we were waiting for time, the time of the first successful type two mutation. Oh, there it is. There, there, there, there's a little one. And surprisingly, the little one gets the mutation first. You know, so, obviously, for in the simulation it's an
15:54
exponential amount of time until the first mutation establishes, and then we get a mutation to type two and then, all right.
16:07
And then one more And then I sleep the rest of the time. This is a simulation that starts with one thing in the middle. I guess you might have got tired of waiting. It has a type two mutation and it has a second one right at the boundary. It's like, "Whoa, what's happening?" All right, so not sure what the parameter values were, but anyway, All
16:31
right, questions about the movies? Everybody's comfy? All right, so there was movie two, sped up process, add mutations, movie three and four. All right, so, so this slide was in Jasmine's talk, you know. So zero is type ones, the type ones start to grow depending upon the value of this gamma. The gamma is sort of intuitively the number of successful type one mutations that you need to get a type two. Much to the fact that it sort of breaks down when gamma equals one.
17:03
I mean, well, we have a gamma equals zero thing, so it's more, mostly, that's why Jasmine gave the more precise interpretation. It's like, how long a time do you need to compare to the waiting time for the first mutation to get, you know, the second one. So when gamma equals zero, everything's going to happen in the first type one. But this is a, a thing where I need, this is gamma sort of in between so I'm gonna need a finite number of these things before something happens.
17:34
And then here, I actually get a second primary tumor in this one, and then in that one. So again, the whole thing of the delineating the three regimes was trying to understand this sort of geometry. So this is a situation where, you know, you would have a primary tumor in a distant field with actually a second primary tumor happening at the same time. But, you know, I mean, these are probably not reasonable parameter values because, you know, you don't want to think, do things with 10 to the
18:06
eighth cells that take years and years, right?
18:10
Right, so this is sort of the cartoon of the mesoscopic model, you know. So I don't know, it's just sort of the referees drove us to this, you know. So
18:22
you know, there was, it still takes, you know, like 30 pages to prove these stupid theorems, but we didn't really want to write 60 pages or something to do these things. So, I mean, it's fairly clear that the Bramson and Griffith Shape Theorem tells you this thing grows and has an asymptotic shape, all right? So, and they don't want to be hassled by worrying about fluctuations about this thing And we ignore, you know, mutations near the edge that might mess this thing up.
18:50
And then I get a second field, and eventually, I get, you know, a primary tumor, and then eventually, I'll get a second primary tumor there. So,
19:02
model analysis.
19:04
All right, so I said this before, I have normal cells, pre-cancerous cells, and malignant cells. I mean, this is a real oversimplification. I mean, in the work we're doing now of more clinical relevance, you have the different phenotypes that the cells go through. You have atypia, dysplasia, carcinoma in situ, and then fully blown full malignancy. And we did that because, in many cases, you don't know what the sequence of steps is, right?
19:38
So in colon cancer, yes, and, but in head and neck squamous cell carcinoma, they tell you some of the genes are involved. And at some point we said, you know, "Why do we need to know this?" I mean, these are phenotypic changes, they happen at certain rates, so let's forget about what gene was, you know, being mutated. And dimension d greater or equal to one, you know, so d equals, d equals one, I'm not sure. There's, you know, maybe breast cancer ducts,
20:09
although they tend to be about 40 cells around. D equals two would be sort of the main one since a lot of epithelial tissues like the edge of the bladder and the cervix are two dimensions. Three dimensions and then, well, what the heck? Ready for general d Unless, you know, we find that somehow there's a relevance of cancer to string theory, you know, so-- All right, so, so we want to do calculations with this. This is why we use the simple model. So here's the start of the timeline.
20:37
So we're in a situation where we think that there are gonna be a number of mutations that lead to the
20:47
final result, all right? So, and something that starts back here is bigger, and in our little universe, these things just blow up like little balloons, all right? So, you know, and then the more recent ones are small. All right? And then we have some idea about the dependence of this on S, you know. So
21:09
and then inside this growing cone, I mean, so, you know, you have these things that in space time they look like this.
21:17
And, you know, when I have enough
21:20
area in these cones, then I'm going to get space time volume in these cones, I'm going to get a mutation.
21:27
And it's a question as what, how long does that have to take? All right, so the theorems are very tiny, which is okay because you don't, probably don't want to read these things anyway, all right? So, the first thing is that when the malignancy happens, what size is the field that it happens in? All right. So here we have this size-biased, have a size-biased pick from the joint distribution of the malignancies and these are, they're initiated at times of Poisson process, so we know
21:59
what the sizes should be. Then I get a size-biased pick, I mean the one I'm getting is proportionate to its base time volume.
22:07
And actually, we were hoping that
22:11
Jim Pittman here had a student named Tran who did some really nice results about size bias picks in a certain setting, and we were hoping she was going to tell us something about our setting but then, their machinery doesn't seem to-- Well, you know, after talking to Tran and to Je-- Jim Pittman, their best advice was, "Why don't you just simulate the thing?" So, okay, so all right, well, anyway. So not that I want to ruin Jim's 65th birthday here or
22:41
something, but, All right, there are formulas. You know, these are gonna be on the test here. So, I mean, basically, you know, you can, at the time of tumor initiation, you can tell what the sizes of the tumor fields are. I mean, and this, you know, looks a little ugly But it's just like, I have a bunch of cones that are started at sizes of, well, times of a Poisson process, and when they have enough space time volume, they generate a mutation and, you know, you spend a couple of months,
23:16
you figure out how to do this and do this calculation. And so it's easy for me to say since Jasmine and Kevin and Mark Reiser did this calculation, you know. So but anyway, And, but one of the points here is that you can do lots of calculations and like this stuck in a loop, you know, something. Uh, maybe Steve will reboot me and get out of it here So I can calculate the number of fields. There are formulas for everything, but, you know,
23:48
this is not, you know, 18th century applied math. It's not like I'm gonna say, "Oh, the mean number is two pi times 17." You know? So but here there are formulas, and they look a little ugly, but you can Okay do the numerical.
24:06
So the key insight, thank you, Mark, getting me back on track here. How do macroscopic and microscopic parameters influence the geometry of the invisible pre-cancer fields? All
24:17
right-Okay. Now let's go back to the clinical issues outlined in the beginning. All right, so Mr. Tongue is back. All right. So how big should the margin be to avoid recurrence from unresected portion of the field? You can tell he's been hanging out with doctors. He-- So-- All right. So excision margin
24:37
zero is not a very good idea, right? I mean, so it takes a while for the thing to come back. This is years, I guess.
24:48
So I have no idea why the units are like this. Zero centimeters. Personally, I would call that two centimeters, but you know, you know me, two centimeters is about the same as zero centimeters. I mean, so four centimeters is a little bit better.
25:07
Six centimeters really does a good job. And try not to think about how big six centimeters is compared to your tongue, you know?
25:17
You know. [AUDIENCE MEMBER] Are they supposed to be, like, actual numbers or?
25:22
[LECTURER] Yeah, I mean there is cli-- Yeah. Yeah, there is clinical data that they did on just, you know, doctors don't want to do math. You know, we had a bunch of people, we did two centimeters, we did four centimeters, and these are the results we got and it's sort of a-- [MODERATOR] How deep are they? How [LECTURER] Deep? I don't [MODERATOR] Know.
25:46
[LECTURER] I think, Over deep. Yeah. There's a picture. Yeah. I think pretty-- I didn't go watch the operation, you know, maybe, maybe it's time to talk to somebody at Stanford Medical School. I would assume that they'd go all the way down, but yeah. [AUDIENCE MEMBER] Is it two centimeters including the tumor or excluding? [LECTURER] No, this is the margin. [AUDIENCE MEMBER] The margin? [LECTURER] Okay. Yeah.
26:10
All right. So other questions? I mean, Yeah, I mean, these are, I think this is consistent with experimental results, and this may be, you know, the source of these numbers. I mean, so it's one of these things that when you get the slides, you go, "Hmm, I don't know."
26:25
[AUDIENCE MEMBER] So [LECTURER] I can tell you multiply by two and multiply by three, but beyond that, I don't know.
26:32
[AUDIENCE MEMBER] This is predictions from the model? [LECTURER] Yeah. [AUDIENCE MEMBER] How? And you parameterize, what's the data that parameterizes it? Like, how do you get the-- I mean, you need, you do it starting by measuring mutation rates and growth rates? Okay,
26:48
[LECTURER] So this is I'm not getting that. Well, okay, so there's a bunch of SEER data on incidence versus time, right? So we have a model that has, um,
26:59
I think I'm doing okay on time. Talk ends at 3:30, right? So does that--They talked with me until 3:30. All right, so in the three-stage model, you have so this is premalignant, this is malignant, and these are, you know, whatever the malignant said, this is like dysplasia, atypia. These things are sort of neutral. So we have a good, you know, from earlier work, we know what in, in sort of Stephen Moseley's thesis, you know, what, you know, times are to the first
27:44
type two, and type twos, because this is neutral, and type twos are coming in at a plus one process. So we feed this end of the front into the work you heard about in the last talk. We can get predictions for
28:01
you know, incidence versus age. And Mark used that to match sort of mutation rates and selection coefficients. So there's, these are all sort of selection ones. This is one plus S1 and this is sort of one plus S2, one plus S2. And so, I mean, sort of the mutation rates, not only do they sort of make sense from a sort of biological point of view, but when you fill in the selection probabilities, you get sort of incidence versus time. And I'm not sure this is the best way to parameterize them
28:39
all, but it's at least done by comparison with, with data.
28:49
If I can just add that there's also a lot of experimental, there's experimental data for some of these S parameters for this particular phenotype yeah, proliferation, staining. [AUDIENCE MEMBER] S is always relative to the sort of-- Yeah. --usual-- Yeah, [LECTURER] That's why I write this. Yeah, that's why I write this. [AUDIENCE MEMBER] Previous. How do you do it? You just put in an amputation to compare the and base it on and post-treatment? [LECTURER] Well, it's Ki67 staining.
29:14
[AUDIENCE MEMBER] Yeah, yeah, but that only shows-- It's a. [LECTURER] Well, it just tells you what fraction of guys that are, you know, in the cell cycle, actively divided, right? So it's not, it's sort of a mild thing of recurrence rate. Yeah, so, all right, but that's-- It's sort of the best you can do. I mean, sort of it's like, you know, a lot of the sort of more population genetics have come up with very low values of selective advantage. And it's also, you can also fit
29:44
it with the sort of what they know about progression.
29:49
[MODERATOR] All right, so other questions? These are good questions. [LECTURER] I [MODERATOR] Mean, all [LECTURER] Right. So
29:56
probability of recurrence as a function of resection margin.
30:04
20 is not, you know, not so good, you know. So
30:10
unless, Let's see what's green. If you're already 90, I mean, who cares, right? So something else is going to get you before that, you know. So I don't like the look of this graph, but anyway, so, If you're 60, well, good, great news. You're going to die of something else before the cancer kills you, you know, but all right. So, you can do these calculations. I mean, this is not something that you know, surgeons really worry about because they don't operate in a world of like,
30:40
"Oh, you're cured, you know, go home." And something happens, call us later. You know, they still look after the patients after their release. All
30:53
right, so-- All right, recurrence, local versus distant. All right, so this is the sort of three regimes of the model, although this one is on a lot larger scale. And again, these parameters, the parameters of simulation don't have a heck of a lot to do with the parameters that you really see in reality. So here, you know, the biased voter model in two dimensions when the size is small has a very fractally boundary. And we saw in Jasmine's talk that the number of
31:30
boundary cells is like, when it has sk over log k roughly times some constant when its size is k. So it says if it was a compact ball, it'd be like square root of k. So this is, you know, sort of, it's very fractally what one would expect that if this thing had grown up quite a bit before the mutation happened, it would look more like a ball. So these are
31:55
probability of local and distant, let's see, I have to go back. [AUDIENCE MEMBER] See what I'm talking [LECTURER] About. Oh, time to local and distant recurrence. This is regime one where the mutation happens in the first malignancy. Here's sort of regime two when there's small number of things, and then, I guess I have to debate some of my coauthor about where-- I mean, this is mathematically a very interesting thing, but this is when there are a large number of mutations before
32:28
you get the first malignancy. I mean, it's an interesting mathematical regime and, you know--Actually, One of the challenges which the referees, didn't really understand is that we have results that go across all these--there are no seams between the results. Of course, I find that often referees misunderstand some of the great points of my papers, so, that's probably how I write them, right? So ironically in this world, the sequel is coming out before
33:02
the first one has been accepted, so it was good that they got me off the team. I actually bailed off the team. And it says Journal of Theoretical Biology, and that's probably, uh, the way Journal of Theoretical Biology works, that should probably al-- probably is already online or something, right?
33:22
So that work is out. And, let's see, ongoing work. So the, these doctors over in the medical school, I mean, they, they're, you know, very excited. We meet with them, you know, about three weeks to a month. Um, Mark's doing a-- He's going to be looking at a lot of sort of pathology data. So you have to do this, you know, there's this Internal Review Board process to get access to the data and he's actually doing that. And, you know, I think-- I forget. Oh, it's Walter Lee is a surgeon,
33:57
Neil Reddy is more of a sort of researcher, but I mean, that's, you know, it's somehow gratifying when you're, you know, we're sitting there having this meeting and this, you know, these guys say, "Well, this has really made us-- the math really made us think about this process in different ways." I mean, so I'm not sure the model is completely parameterized right yet, because then there are some stages in the process that seem to take a really long
34:23
time compared to, you know, it's known from lots of examples that from initial mutation to tumor is like 8 to 10 years, but we seem to have a model that I'm not sure it's parameterized right, that from dysplasia to full-blown cancer is like 30 years, so which would be good, but if it's true. But I'm more worried about it. All right, so again, I sort of, this is the sort of more complicated model, again, I sort of got strayed away from my outline and they went up, they're gonna collect clinical
35:03
data and actually, you have the resected thing and they, the pathologists can look for malignancies in the surrounding tissue. And then again, of course, the, the Holy Grail is to be able to, you know, look at, you know, patient data or imaging of the tongue to be able to say, you know, what what sort of resection margins would be best. Right? So, And Mark loves this slide, I had no idea. Like it's cool, but it's just sort of his vision of what he wants to do, mathematical model
35:44
informing diagnosis, treatment and follow-up. And you know, patient data feeding in from the other side. All right? All right. I didn't do this slide justice, did I? But I can do this one joke. (cartoon music) That's awful. All right, so-- All right.
36:07
Finished with five minutes. [AUDIENCE MEMBER] Yes? Um, do you or Mark know the work of uh, Kristen Swanson? [LECTURER] Yeah. [AUDIENCE MEMBER] Because it's been relevant for the next stage of your project which is trying to use imaging and clinical data in order to compare the outcome of a mathematical model with clinical reality. Obviously, her model has nothing to do with pre-malignant tissue that might be you know, taken care by the surgeon,
36:30
but it still is about what the model predicts that this is the growth and this is staging and we're basically trying to extrapolate what the model predicts so that we can use it to sort of finding the modal nerves allow the surgeons to do a better job next time round 'cause they're working with gliomas and those usually-- Yeah. --can be treated so, and the number of treatments a patient has to go through is usually just all the time until the patient eventually will realize.
36:58
The next thing is so you guys have-- Do I [LECTURER] Get to answer the first one or-- [AUDIENCE MEMBER] Yes, well-- All right. You just give me anything. Well, [LECTURER] I think in her situation, you're dealing with gliomas, so there's a, an initial MRI at diagnosis. There's a second MRI right before surgery. You really have a lot of action in the glioma over that two or three months-- Yeah. --was unfortunately. I know. So I don't think that here, you're gonna have that sort
37:24
of observation window. I think even if you sort of spend, you know, it takes two months to get scheduled for surgery, I don't think a whole lot has happened. But I mean, certainly, you know, looking at surgery and clinical outcome, but Part 2, do you know? [AUDIENCE MEMBER] Part 2 is about the role of this precancerous thesis which is so the surgeon goes and says, "Well, this is what cancer we leave, so just take this with a healthy margin around and then just gonna assume the rest is healthy."
37:50
But and recurrence is supposed to happen from these precancerous tissues that the surgeon forgets about. But another explanation, there's something that some colleagues of mine have been working a lot with is that as the tumor grows, it sort of conditions the microenvironment of the tumor such that if you leave any cell behind, you don't need to leave a substantial amount of tumor cell behind. You just need to leave a few and that's bound to happen. You have this environment that is ready to go,
38:19
so when the tumor cells decide to start growing again, they have all the infrastructure to grow much faster than they were in the beginning. So that could be another explanation for that. Is there something that could--
38:31
[LECTURER] Yeah, I mean, th-- this is you know, it's a type of modeling that, you know, well, we're, I'm, I'm not ready to dive into yet. I mean these sort of hybrid discrete cellular automata where you have cells usually still on the lattice, but they're interacting with
38:51
[AUDIENCE MEMBER] But you could have-- [LECTURER] Glucose and oxygen and hydrogen ion concentrations. That [AUDIENCE MEMBER] Would be more complex. You know, so that-- What? That would be more complex, but you could have so basically most tumors you have tumor cells, but you have also the cells like fibroblasts or macrophages. You don't have to necessarily model chemical. You can just basically have all the-- Oh okay. Sites that are actually tumor cells by the cells that can
39:12
load things for the tumor. [LECTURER] I understand in the next talk we're gonna hear about-- Yeah, that's true. --game theoretic interactions between various types of cells-- Mm-hmm. --by somebody sitting clo-- very close to Dan Fisher at this point. All right. So yeah? [MODERATOR] In the simplified model, can you compute the probability that the primary tumor occurs at the largest compartment-- Mm-hmm. As compared to the second largest and so on
39:39
[LECTURER] Yeah, I think that's among the formulas. Yeah exactly. Is that something, [MODERATOR] That probably doesn't depend on anything right? I mean, it's probably something--
39:46
[LECTURER] Well, no, it's still, [MODERATOR] It's-- It doesn't depend on the parameters. [LECTURER] Well, it's a race. I mean, sort of it's like, you
39:53
know, we have, you know, obviously in the gamma going to zero regime, that probability's one. [MODERATOR] Right. [LECTURER] But I think, you know, in here, it's gonna depend sort of, there's gonna be a lot of variability. I mean, I guess there's some overall average probability, but I don't think this is easy to compute, sort of except sort of numerically. I mean, the way this thing goes is like, you know, okay, I'm gonna look at, you know, the initiation times are a
40:24
Poisson process, which is not a very good Poisson process here. I condition on these things, I know what the volumes are, I know what the joint distribution of this is. I know the relative probabilities of the first thing being in the various things, and I get this expression, which I evaluate numerically. I mean, Jasmine's yeah, find something sort of, Yeah, And so Yeah. I mean, these, you know, these expressions I give, I sort of flashed them by fairly quickly because
40:55
I don't think they're all that informative, you know, but, you know.
40:59
In principle, yes, but in practice, you don't get sort of simple answers that you sometimes get in problems.
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All right, let's thank Jasmine again for her talk. (applause) Uh, we have half an hour before David Bosante's gonna ta-- talk about evolutionary games and, and healthcare a little bit. (background chatter)
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