Přepis řeči - COOPERATIVE MAXIMUM LIKELIHOOD ESTIMATION FOR FLUID FLOW DYNAMICS IN BIOSENSOR ARRAYS

0:00:13	using window so maybe you can should we have from the
0:00:16	so
0:00:24	yeah
0:00:38	thank you so uh the title of my talk is cooperative operative maximum likelihood estimation for food flow dynamics you
0:00:45	know by a sensor array and uh
0:00:48	maybe somewhat unlike the previous talks in the section this is actually a real system with built so so we
0:00:53	actually trying to
0:00:55	see how we can model the very complicated system
0:00:58	uh so for so to actually describe you what this real system is and it's kind of quite and uh
0:01:02	interesting thing in it's own right
0:01:04	so uh let's focus first on the diagram of the
0:01:07	right hand side of the slide and
0:01:09	oh what what happened was that a colleague of whose name is bruce cornell uh in
0:01:15	in in the nineteen nineties an actually he's continue to work on that
0:01:18	he's build a a remarkable now don't machine
0:01:21	which can actually be uses as a by sensor and
0:01:24	the goal of our work is to try and mortal that how does that work and how can be used
0:01:29	that to do useful things
0:01:31	so
0:01:32	it's first important to to understand how this thing works because then we can model the dynamics of that
0:01:37	uh
0:01:38	so this by sensors actually build
0:01:41	our of a
0:01:42	synthetic cell number eight i mean all of the know what a cell membrane as it's on to sell and
0:01:47	it can of two lose of fact basic bits score than that that by layer
0:01:50	uh
0:01:51	mathematically the tool is a fact walk around
0:01:55	move around according to a random walk that's that's typically what people to
0:01:58	so what what we do next is they insert
0:02:02	in this slip by there
0:02:04	protein now don't two
0:02:06	these two are very easy to synthesise an approach you make that
0:02:10	oh what you can imagine now is you have two layers moving around according to a random walk
0:02:15	when two two
0:02:17	combine
0:02:18	they form of conducting or and a car can go through
0:02:22	when to to to do not combine there is a car going to
0:02:25	of course are several thousands of such do and so the probability of
0:02:28	a couple of dupes combining is quite large and what you see some sort of car which goes up and
0:02:32	down
0:02:34	so that's really what's shown there
0:02:36	now
0:02:36	that's fairly easy to synthesise that the main idea behind this of course as the next stage where
0:02:42	you attached to the top layer
0:02:44	pacific and bodies which can detect
0:02:47	molecules you wanted to do
0:02:49	so suppose to interesting in to a interested in detecting H I V or H one and one or or
0:02:53	or expose a molecule
0:02:55	you can build an approach you makes flat
0:02:57	specific antibodies which latch on to this
0:03:01	talk talk Q
0:03:02	so what happens is when you target molecule comes
0:03:06	these antibodies bodies score to point to them
0:03:09	and this stock
0:03:10	so what happens is a top player cannot move anymore
0:03:13	they graph
0:03:15	so this changes the dynamics of the system
0:03:17	previously we or on hindered random what what things were moving around
0:03:21	you are on and off
0:03:23	basically
0:03:24	card
0:03:25	now all the top is
0:03:27	stuff
0:03:27	and so the current dramatically decreases one other words that impedance
0:03:31	increases substantially
0:03:33	to this was the by sensor they build he publishes paper and ninety ninety seven in nature
0:03:37	and uh it's it's quite a remarkable sense so because this can detect
0:03:42	a low concentrations
0:03:44	we can detect up a fan till more lower concentrations and if you think about that that's that's pretty surprising
0:03:48	because
0:03:49	oh one more are as as you know from high school chemistry is one have a cat was number roughly
0:03:53	ten to the twenty three a ten to the three molecules in one or water
0:03:57	once stand to model lower
0:03:59	multiply by the by ten to the minus fifty
0:04:01	so what can to the eight
0:04:03	molecules molecule lead or more and that is extremely low
0:04:06	concentration
0:04:07	and that these things work
0:04:09	remarkably we for that
0:04:10	okay so that this is a system that field
0:04:12	oh goal has been to try to see
0:04:15	how can be more the system how can we predict how before
0:04:18	was if we can do that we could possibly fine the system and make it work better
0:04:23	and we can also maybe
0:04:25	extend the system to work in the scenarios
0:04:28	so we actually done a lot of work in this in the past and we have a couple of people
0:04:31	which came out of the transaction that of technology just last year
0:04:34	which dealt with more in the specific system
0:04:37	what wanna talk about today is
0:04:39	just ongoing work is
0:04:41	suppose that you take this by sensor when you build a a of such five sensors
0:04:45	how can you model that
0:04:47	and it turns are be highly nontrivial problem tell you why know
0:04:51	okay so that's for start with the individual electrodes to this is a signal by sensor
0:04:55	what happens is you of the fluid
0:04:57	which is still a word
0:04:58	to this by a the
0:05:00	you know food would delivery system
0:05:02	so you basically have a
0:05:03	some liquid
0:05:05	such as sodium chloride
0:05:07	containing the molecules tools you wish to detect
0:05:10	and that is flowing
0:05:11	how
0:05:12	this by sense
0:05:14	so this equation is the part of the to equation of of fluid flow
0:05:17	it's the parabolic pde
0:05:19	with the diffusion constant and so on
0:05:22	now what happens is where
0:05:25	the molecules in the fluid
0:05:27	and come to this and electrode
0:05:30	the trade off a chemical reaction which is a bunch of non than your or be different role questions
0:05:34	so i here is the concentration
0:05:37	both
0:05:38	the stuff stuff you want estimate
0:05:40	such as a H be you whatever
0:05:42	these are the chemical reactions to trade off
0:05:44	and what you measure eventually a some noisy version
0:05:47	of a specific chemical in this chemical reaction
0:05:50	plus
0:05:51	but
0:05:52	so these are the dynamics of the system
0:05:54	a there pretty dirty for several reasons you see this guy and it's all is pretty nice this is just
0:05:59	a straightforward forward flow P D to still be Q
0:06:03	the back part lot of things we conditions
0:06:05	you see that select lies at the bottom of the food chamber
0:06:08	and this is where the stuff happens
0:06:10	this is where the molecules which which we should detect
0:06:13	re yeah with this by sensor
0:06:16	to give you your measurement which is a increase in impedance
0:06:19	this is a a boundary condition only and one location so it's not smooth
0:06:23	it's not a it it's just a that location
0:06:26	uh so when you have these complicated
0:06:29	do rate it is that your boundary should these of
0:06:31	uh one alignment boundary conditions that and the fairly difficult to deal with
0:06:35	uh in addition you have noise you okay so this this system is is quite sophisticated it are so that
0:06:40	you can construct a very nice models for this and and we've done that of the part
0:06:44	now let's look at what we're trying to do here
0:06:46	so now we have a a rate of such things
0:06:48	see what is really nonstandard standard in this is but you have an array of such a electrode
0:06:53	and particular if your concentration a is very small
0:06:56	you of fluid goes pasta for select role
0:06:59	the electrode graph
0:07:00	some some of the molecules to react with
0:07:03	that means when you measure the system you actually changing the system
0:07:06	and that's likely non standard and signal processing
0:07:08	in most signal processing we do typically menu measure the system you don't change
0:07:13	so as to four flows but here
0:07:15	the first electrode brat the molecules so the second electrode has fewer Q
0:07:20	to detect
0:07:20	so if you placed the second electrode very close to the first select road
0:07:24	there's a depletion layer and no you don't a measure and fig
0:07:27	if you pay a second were very far away from the first electrode
0:07:30	it takes a while for the food to reach their and it means that you're detection times very low
0:07:34	so so actually designing
0:07:36	where you place you electrodes is is
0:07:38	it itself also an interesting problem
0:07:40	so anyway a goal at the stage is per some to be able to model this and come up with
0:07:44	some tractable approximations as to how how this stay
0:07:47	and i one wanna describe some of those two
0:07:49	so you can be this is a problem of saying that given this
0:07:52	fairly complex the
0:07:54	would multiple measuring devices
0:07:56	how do like estimate the concentration at the initial concentration
0:08:00	that's that's what we wish to
0:08:01	to me
0:08:03	so i let's go a little bit of it you should before we start so P D's activity i mean
0:08:07	so the first step you can think of is
0:08:09	can be construct some sort of
0:08:11	time scale approximation to replace this by or we differ to question
0:08:15	in that sense we would simply we have a nonlinear D ease which which is a more tractable problem
0:08:19	then you can deal with the non linear regression and and and a signal processing like that it for
0:08:24	not not not reveal but least it's it's
0:08:26	manageable
0:08:27	okay so the way one can do it is
0:08:29	you can use actually multi time scale dynamics
0:08:32	it turns so that as flow goes by
0:08:35	you see stuff at the top of the chamber which is very far away for the electrode
0:08:39	uh
0:08:40	it very quickly
0:08:41	a G station stationarity in other words
0:08:43	stuff at the top
0:08:45	at of T reaches at infinity
0:08:47	very quick
0:08:49	oh it turns out you can really segment
0:08:51	spatially this
0:08:52	in to several compartment
0:08:55	regions which uh far away from the electrode
0:08:58	are are basically constant you don't need the speedy
0:09:00	that you just have to sort
0:09:02	regions which are very close
0:09:04	of course you have to speed D but you make for the approximation to as described in a minute
0:09:08	so the idea you you to use something called averaging two you with some of you may be familiar with
0:09:12	but when you D with the filters
0:09:14	that
0:09:15	one at time scale where things happen slow and fast
0:09:18	on the slow time scale you can replace the fast i i'd average which in this case is a constant
0:09:23	away from the true
0:09:24	and a man the stuff the electrode you have to be but a bit more careful to to see what
0:09:28	to be done
0:09:29	oh case that's roughly didn't you should another vehicle want to
0:09:31	how you construct such more
0:09:35	so uh this is actually the as that mentioned before you have this fluid would flow and then these are
0:09:39	the equations of the chemical reactions to the sensors
0:09:42	you don't so actually these chemical reactions of cells at two time scales and so you can for the simplified
0:09:46	things
0:09:47	you can actually average job the fast times deal with a so that's good
0:09:51	and as i set a goal is to estimate the concentration at in the in light of this
0:09:55	through chamber
0:09:57	so
0:09:58	this is what we don't to do is we are place this distributed parameter system or P D if you
0:10:02	like by multiple compartment
0:10:04	and that becomes a bunch of more than your own to use it and then you can do of a
0:10:07	question
0:10:08	questions
0:10:12	so uh the way you do this this this idea of using multiple compartment model that is widely studied by
0:10:17	people who to be on with fluid flow chemistry and and they they analyse some fairly complex devices
0:10:23	for that sort of thing
0:10:24	the main idea is this
0:10:25	so if you could can our say a single electrode
0:10:28	what you can do is
0:10:30	i and this is that in and you can actually make this quite rigorous mathematical you uh so
0:10:34	if you know that you P
0:10:36	you could view
0:10:37	heuristic lead a P well with the spatial dimension as an infinite system of all the east but you want
0:10:43	you space
0:10:44	okay
0:10:45	so you to think of a great over space
0:10:47	where at each grade you have a or T E and of course all these all these are interacting you
0:10:52	of infinite
0:10:52	system a to use based
0:10:55	what what you think that show is that the only D which is very close to here
0:10:59	is really the boundary condition
0:11:01	so that's a chemical reaction
0:11:04	the only which is a way from this
0:11:06	is essentially the one which is a power with this chamber here with things happen very fast
0:11:11	and that there's no P D it's just a that because
0:11:14	it's a it's fluid flow
0:11:16	approximation of right
0:11:18	so it's you have to boundary conditions
0:11:20	and this chamber just a bunch of than you these that's that's really that you
0:11:24	so what you and it up with is a two compartment model here
0:11:28	where here you have a constant
0:11:29	here you have a nonlinear only
0:11:32	and you have the boundary condition which is of the all that would be
0:11:35	so you now down to a finite dimensional system
0:11:37	which is which is tractable
0:11:38	now as a said this was a heuristic thing because
0:11:41	to be really uh
0:11:42	is you have to kind of quantized of course this up sub step i learn
0:11:46	and then you have to
0:11:47	proof that the errors are bounded in so
0:11:50	this is a deterministic system so it's not that difficult to do things i mean it's of a stochastic could
0:11:54	be slightly harder
0:11:55	so actually it it's it's not not that difficult if if you assumes so the regularity the system then then
0:11:59	you can do one
0:12:00	okay now so that's the to compartment model which is easy to do
0:12:04	now you can imagine conceptually uh extended this robotic a model
0:12:07	and it's roughly the same thing so in the first compartment is identical to to what you have to you
0:12:12	for the first like true
0:12:13	i'm and stuff was fast post to to the second electronic can you have a to keep up with the
0:12:18	and and of course you have to keep in mind that because the first electrode is grad
0:12:22	some of the and a like
0:12:23	some of the ball was to try to better
0:12:26	the second electrode has different boundary conditions which are vertical boundary conditions you
0:12:31	but it's it's quite easy to take care of conceptual
0:12:33	so that you have it to basically have a whole system of
0:12:37	norm that you're or ease which is conceptually at least easier than dealing with a much more complex as to
0:12:43	so once you
0:12:44	now as i mentioned that i'm not gonna give you the details of these all these it doesn't really serve
0:12:48	any purpose
0:12:49	but just to kind of go back and you roughly what these ordinary we differential equations do
0:12:54	uh
0:12:56	you see they really model the chemical reactions you
0:12:59	the model the movement of these
0:13:03	now i tubes in the
0:13:05	a number eight
0:13:06	they model how these chemical reactions happen
0:13:09	the chemical bonds
0:13:11	and how when things couple
0:13:13	they don't move so really there about seven chemical species it you know which you you model by of order
0:13:18	to control question
0:13:20	these are quite tractable in in fact
0:13:21	the reaction rate constants and the one by by a of so the the
0:13:25	all the parameters of those equations are actually
0:13:29	or we so once to end up with this as a said you know all the reaction rate parameters you
0:13:33	only unknown quantity is your input
0:13:35	and like
0:13:36	the creation which which is what we wish to be
0:13:39	so that was really the main idea uh and
0:13:42	just a can give you now some it you should be how this would work
0:13:46	so if you think about
0:13:47	even if this was a of now a bunch of ordinary differential equations questions or if you like discrete as
0:13:51	a what time it's a bunch of difference equations
0:13:54	a you have this fluid flow coming in here
0:13:57	so as you think that this guy remote say some proportion of the
0:14:02	molecules
0:14:03	see alpha
0:14:05	that means you're signal power here is reduced by but for because you grab some of the guys
0:14:09	so you can think that the second electrode what has a little or signal to noise ratio because you less
0:14:14	signal
0:14:15	now of course the next electrode will be out of the squared and so becomes a geometric matrix V still
0:14:19	it's obvious that you won't get
0:14:22	dramatic improvement as you have multiple electrodes because it dies of as a geometric matrix series
0:14:26	i mean that's gonna a used
0:14:28	even at the linear case now of course is the non linear regression you have to be slightly more careful
0:14:32	uh in any case one could actually
0:14:34	workout out
0:14:35	the asymptotic covariance of this
0:14:37	okay and it's it's a fairly complex question
0:14:39	and one could actually show that
0:14:42	in as you have more more electrodes you your performance that's really from frame
0:14:47	to you'd expect if you and N electrodes everything was
0:14:49	i i D you had no interaction between things you get one of and improved
0:14:53	sure you don't
0:14:54	so that's that's something which
0:14:56	so be the case
0:14:57	uh
0:14:59	okay so this is the actual system be but we've tested this on on
0:15:03	so it to test this several things you can do
0:15:05	you not to run of the real experimental system
0:15:08	and compare it with without a of different look should model that's to kind of to a model that's the
0:15:12	first step
0:15:13	and those that things we've done it quick you come the pa that's that that were
0:15:16	a second stage is how can you actually
0:15:18	shall
0:15:19	if the multi component model is a good approximation because that's the we we want to estimate of constant
0:15:25	and it are actually they work extremely well as
0:15:27	we can see these diagrams you
0:15:29	the actually approximate the
0:15:31	P E extremely well
0:15:33	uh the arrow it use that to be equally between six to eight percent
0:15:37	and even maybe go to
0:15:39	concentrations which are very very low
0:15:41	concentrations of
0:15:42	almost almost
0:15:44	well below an animal are still
0:15:45	so these these that she work quite well
0:15:48	uh and and therefore it means that you can apply standard a question analysis to solve for these concentrations
0:15:54	i want make a few other
0:15:56	comments before finish the all
0:15:57	this is still work in progress i mean it's a very nice to come up with these approximations but you
0:16:02	don't the sort of things we have done which would not please people are rigorous not my
0:16:06	we haven't even shown that this system of equations
0:16:09	has a unique solution it's really down hard to show that
0:16:12	so i mean if you think about P D easy we bad this in a function space like a stop
0:16:16	let's space you wanna show that you it's of a solution
0:16:19	highly nontrivial
0:16:20	so
0:16:21	i mean although the system works other real axis system
0:16:24	sure you you this is really hard
0:16:26	it's something which we working on of the moment
0:16:28	uh
0:16:29	the are the issues are it would be really useful to come up with a nice
0:16:34	approximations
0:16:36	for this pde itself
0:16:39	it's still of just using a multi compartment model we've done
0:16:42	because we still ending up with a bunch of all we different role equations which
0:16:46	we don't have a
0:16:47	so form solution but eventually do you man
0:16:50	we much nice if we can come up with further there were approximations which allows to get
0:16:53	some inside
0:16:55	as to how the system works
0:16:56	so there are still a couple of
0:16:58	a that's to our approach
0:16:59	but i think this is a in the sense that
0:17:02	given the complexity of the system we can model it
0:17:05	it works pretty good
0:17:06	we can approximate it and we can actually estimate using elementary nonlinear linear regression
0:17:11	the sort of concentrations
0:17:12	we want to estimate
0:17:14	so that's really all of wanna say
0:17:16	uh if you're interested in any of the stuff this was the original paper we by a colleague
0:17:20	uh and these are a couple as we did where we dealt with the signal electrode case maybe be model
0:17:24	it that and actually did
0:17:26	a not than you're question on that
0:17:28	oh can thank you very much
0:17:37	yeah right are you know
0:17:42	oh okay if gram so a a really interesting problem uh uh of course um
0:17:46	is a lot of comments i guess for instance with the pay T um
0:17:51	you've got a linear spatial up right the first to terms so there's a brings function that's an an
0:17:57	i i you could generate a one approximations and stuff like that a you've been looking in that direction we
0:18:01	have quite a bit what really kills us is cool
0:18:04	so
0:18:05	is is this
0:18:07	i a gonna this is a horrible but we could dish of the that
0:18:10	this a are could if should which likes to the P
0:18:13	right and and
0:18:15	and this this really is a a a i was not a at the takes the all of
0:18:20	you know the concentration respect to spatial axes and sets to one of the things so then one is thing
0:18:24	of boundary element methods because uh that's the use a to the with that of stuff yeah
0:18:29	i numerically what we done actually is because you have a a solution are obvious so you don't have strong
0:18:33	so should we use a finite element method which takes to functions automatically
0:18:36	but again
0:18:37	oh for our approach is completely and hearing in the sense that we have a solution it works
0:18:42	but from a mathematical point to we we still have even shown any structure proper okay but good of array
0:18:47	element method is different from finite element you i don't know if you familiar menu with a a a a
0:18:53	type find so
0:18:54	a any whites aims to make that might worth looking at
0:18:56	that are thing is on not entirely clear that the regression analysis because the use is i'd time varying quantity
0:19:04	so um
0:19:05	uh
0:19:07	i great so for instance if you nate to
0:19:10	in house some regularization on the a temporal times of the use signals
0:19:15	okay so let's a go to it so a is the concentration which changes over time yes we interested a
0:19:20	zero which is a initial concentration wrong
0:19:23	a a is coupled to these sceptical reactions which are you which you also over time
0:19:28	these are pretty easy to show for example that these are always non-negative and so the they just basic chemical
0:19:33	reactions a can i the only difficulty in this is a non linearity of that
0:19:37	but you've got gotta a construct you of tape right
0:19:40	well one specific um of that is a what be measure in noise
0:19:45	okay i but when you do the regression what do us to my just i it's a so one one
0:19:49	gram use of that
0:19:50	you of taste he is in there must be dealt with a the why something is it so what you
0:19:55	have is but you re braces by multi can pop and model you have a a bunch of or very
0:19:58	different role questions here
0:20:00	and here
0:20:01	which are in you
0:20:02	and the interested in one specific component of you
0:20:05	observed in
0:20:07	which is a concentration of the diners basically
0:20:10	uh
0:20:13	basically
0:20:14	the number of couple guys you
0:20:16	because the cover is proportional to the
0:20:18	no i i understand all that but i guess again but in the middle somewhere in that regression you are
0:20:22	as a function of tape must some have a being with carpet
0:20:25	in order to waste one one this upon it
0:20:27	one can hold diffuse for so i just one where the in a some temporal regularization or or or or
0:20:32	the that of
0:20:33	a us we have applied in such thing so far so are
0:20:36	yeah my help
0:20:37	it it back
0:20:39	or a

COOPERATIVE MAXIMUM LIKELIHOOD ESTIMATION FOR FLUID FLOW DYNAMICS IN BIOSENSOR ARRAYS

Estimation Theory and Methods

Přednášející: Vikram Krishnamurthy, Autoři: Maryam Abolfath-Beygi, Vikram Krishnamurthy, University of British Columbia, Canada