15 - SN Partial Differential Equations and Applications Webinars
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Join Marie E. Rognes of Simula as she discusses mathematical models and numerical methods for the brain's waterscape across scales—from viewing the brain as a poroelastic medium at the macroscale and zooming in to studying electrical, chemical and mechanical interactions between brain cells at the microscale.
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00:05
So scientists at the similar research laboratories in Oslo,
00:10
so professor at the University of Bergen,
00:13
both of them in Norway.
00:15
Maria is expert on scientific computing,
00:18
particularly for biomedical applications or work,
00:22
has received a lot of.
00:24
Let's say Onaran accolades, including she has.
00:27
She is living up in European Research Council.
00:30
Yes, he's starting ground. She was part of the team
00:34
that won the James Wilkinson Prize for Scientific Software and
00:38
many other words.
00:39
So really, looking forward to your talk Maria,
00:42
which is on the brains numerical waterscape.
00:45
So sounds very exciting. Thanks said thanks Jan for limitation.
00:50
Now, about seven years ago,
00:52
in the fall of 2013,
00:53
I was rather preoccupied with the concept of sleep.
00:58
As some of you have small children,
01:00
perhaps can relate to and I remember sitting in my
01:04
living room couch just next door here and I was
01:07
swiping through newspaper articles and my cell phone with one
01:11
heading caught my interest in particular and having read something
01:16
along the lines of sleep clears the brain of poison.
01:23
And.
01:26
The science underlying this paper that afterwards received massive popular
01:31
interest was the study published in Science in the same
01:35
fall by Mike in Nedergaard and her group.
01:38
She's a well known Danish neuro scientist and an experimentalist.
01:43
And what they have done?
01:46
Was too well. They work with mice.
01:49
And what the first author of this study look?
01:53
See, she had especially spend a large portion of her
01:57
PhD studies teaching mice to fall asleep under a true
02:01
fogtown fogtown fluorescent microscope.
02:03
But prior to making the mice fall asleep under the
02:07
microscope,
02:08
what they do is they inject a small amount of
02:12
so called tracer.
02:13
A fluorescent dye dissolved in a solvent and they injected
02:18
inside the mouse kind of brain environment but outside the
02:23
brain itself.
02:24
And then what they had,
02:25
they had two groups of mice,
02:26
a group that they then.
02:28
Observed falling asleep and one group that my said stayed
02:33
awake and then under this microscope.
02:36
They observe about 200 micro meters.
02:42
Into the brain inside the brain of these mice,
02:45
and they observe how the dye spreads in that region.
02:50
And what they found is really astounding where you can
02:53
see in figure B in this slide.
02:55
Here is that they found that the sleeping mice.
02:58
I have a stye spread much,
03:01
much better. Then in the awake mice of the sleeping,
03:05
my C is spread but was covered here in green
03:08
and figure B and the white mice.
03:10
It only spread what you seen in Orange.
03:14
And this was really they were flummoxed by.
03:17
This finding was completely surprising,
03:19
but very interesting. And I'll get back to this later,
03:23
but it's a question you may ask.
03:25
You know, you might wonder why are we interested in
03:29
die spreading inside the brains of mice.
03:32
While we're of course not really interested in die spreading
03:36
inside the brain in mice,
03:37
but we are immensely interested in understanding how molecules move
03:42
and spread inside and out of the human brain.
03:45
This is because of their Association with neurodegenerative diseases like
03:50
Alzheimer's disease like Parkinson's disease.
03:53
Like to mess with Lewy bodies.
03:55
All of these are the mansion type diseases that are
03:59
associated with abnormal aggregations of proteins inside the brain tissue
04:04
inside the cells or in the extracellular space.
04:08
And this is what this illustration shows.
04:11
This shows two types of abnormal protein aggregations.
04:15
The blue represent the protein tangles of the protein Tau,
04:19
which tends to tangle together inside the NER.
04:23
The neurons, brains, nerve cells,
04:25
cell bodies, and the Brown lumps represent circle,
04:29
emulate beta plucks. Which clumped together in the extracellular space.
04:35
And these protein accumulations are believed to be the cause
04:39
of massive neuronal death,
04:41
an eventual cognitive decline as seen in many forms sentiment.
04:47
Now these proteins are protein fragments,
04:50
Tau and beta amyloid. There are natural byproduct of our
04:54
metabolism as we eat,
04:56
think, sleep. We produce these mataba lick waste product and
05:01
so their presence in natural.
05:03
But this accumulation is a natural right seems to.
05:08
And these are diseases where we really we really do
05:13
not understand.
05:14
The origins of these diseases we do not understand how
05:18
to treat them.
05:19
But one of the hypothesis is that perhaps it could
05:22
be something wrong with the natural transport of these proteins
05:26
out of the brain.
05:27
Something that goes wrong with the clearance of the brain.
05:31
We don't, we don't know,
05:33
but this is one of many viable paths Tim Esque.
05:39
So This is why we are so interested in these
05:42
modes of solid transferred and how die moves inside and
05:46
out of brains.
05:47
And the idea is that understanding the transport mode in
05:51
brain tissue could be key in understanding many neurodegenerative diseases.
05:57
And in some speculate, even that if if these solids
06:00
move faster in the brain during sleep,
06:03
perhaps that is really the purpose of sleep.
06:06
That's the purpose of sleep is to clean up from
06:09
each day's party in the brain.
06:14
But if I now and this this is this kind
06:18
of topic,
06:19
is what I refer just the brain's waterscape?
06:24
How fluids, transport, move and flow?
06:29
Through and around the brain and in its environment.
06:35
But if we assume out of the brain for a
06:37
minute,
06:38
we can think about how does this work in the
06:41
rest of the body.
06:42
How does surplus fluid, together with metabolic waste dead cells,
06:46
bacteria, etc drained from the tissue?
06:49
Well, in the rest of the body we have the
06:52
lymphatic system.
06:53
The lymphatic system is.
06:58
Composed of lymph vessels and lymph nodes.
07:03
These vessels are filled with lymph a clear fluid.
07:08
And it drains the tissue of access fluids and other
07:12
waste product.
07:13
Now the lymphatic system has a lot of functions,
07:17
but this this is one of them.
07:20
And the lymphatic system was essentially characterized already in,
07:24
I would say the 17th century classicly and you can
07:28
see many classical illustrations of these.
07:32
Um and all of their often beautifully drawn,
07:35
any enough can see how they extend up here along
07:40
the neck and on the outside of the break.
07:44
Now I just wanted to know before I continue I
07:47
want to mention a fun fact.
07:49
So the word length, which is this clear fluid,
07:52
comes really from the word lymphatic,
07:55
who was the Roman goddess for clear spring water and
07:59
the Greek nymphs come from the same word.
08:02
But if we return to the point.
08:05
The lymph vessels extend through our entire body except in
08:11
the break.
08:12
There are no lymphatic vessels in the brain.
08:16
What there is is that there expecting emphatic vessels extend
08:21
up on the neck.
08:22
And around the skull they also extend into the circle
08:27
Durham membrane.
08:28
So let me just pass to give you a kind
08:31
of a brain anatomy 101.
08:33
So you have the brain when you think of this
08:36
kind of wobbly,
08:36
wobbly bit that I would think you must've read something
08:40
called the brain parent Kima.
08:41
And it's covered by three membranes,
08:43
so thin sheets you have the innermost circle dpia membrane.
08:47
Then you have the arachnoid membrane.
08:51
An on that at most membrane we have the dural
08:54
membrane.
08:55
And in the German brand,
08:57
there are indeed lymphatic vessels.
09:00
Now, these were originally discovered,
09:03
I think, by Polhamus. Connie,
09:05
around 1787.
09:07
The illustrated, these, and characterized these blood vessels in the
09:12
dura,
09:12
so The Alchemist membrane surrounding the brain.
09:15
Uh, but then this information was rather forgotten and actually
09:19
ridiculed.
09:20
So in later in the 18th and 19th.
09:22
Hundreds he was ridiculed for having seen lymphatic vessels,
09:27
where they do not exist.
09:30
But this magnetic missiles were rediscovered in two quite massively
09:37
popular papers from 2015 in 2016 where they rediscovered these.
09:43
Phatic vessels in the dural membrane,
09:46
and they also showed how they're connected to the CSF.
09:51
The cerebrospinal fluid that surrounds the brain.
09:56
In a layer between the PF,
09:58
the innermost membrane and the arachnoid membrane.
10:02
There's a space between those two membrane.
10:05
It's called the subarachnoid space.
10:08
So this CSF is cleared.
10:10
It has different roots, but in particular it's cleared through
10:15
the dural lymphatic vessels in the door,
10:18
but they do not extend inside the brain power kyma
10:24
itself.
10:24
So the question remains.
10:29
What is the lymphatic system and what place role of
10:32
the lymphatic system in the brain,
10:35
parent time? And that's essentially kind of the big question
10:39
that we are trying to address.
10:43
Because you know, our brain spent about at least 20%
10:47
of our metabolism. If we're thinking heavily,
10:50
it's much more so. It would be very strange if
10:54
the brain this delicate can organ does not have any
10:58
system for waste clearance.
11:00
But it, but it does not seem to have any
11:03
lymph vessels.
11:03
At least no one has been found,
11:05
so it it? It raised this question,
11:08
what could play the role of the lymphatic system?
11:12
And the even more fascinating answers into this question is
11:16
that that nobody knows.
11:18
There's there's really no consensus on this topic.
11:23
But Fortunately there are some theories.
11:26
This is a very active feelin in basic Physiology these
11:30
days,
11:31
and there's a lot of controversy and lot of discussions.
11:35
So something I think everyone agrees on is that their
11:39
diffusion does payroll.
11:41
It would be hard to not be so solids.
11:44
Do you move from kind of protein fragments that are
11:47
produced in this House will move through the tissue and
11:52
somehow out via diffusion.
11:54
Everyone agrees on this. But the question is if there
11:58
exists also any other modes of transport.
12:02
For instance, any connection based routes.
12:07
Where, for instance, one could imagine that CSF,
12:10
the Cerebro spinal fluid that mainly occupies the space outside
12:15
the brain,
12:15
could also move into the brain.
12:18
Along so called para vascular pathways,
12:21
the Paradise Club pathways are Spade hypothesize to be spaces
12:26
surrounding the blood vessels,
12:28
arteries and veins. The small arterioles,
12:32
venules and down, at the capillary level.
12:36
That extend into the brain.
12:38
And it's hypothesized that these paravesical space is kind of
12:42
actors,
12:42
highways for fluid movement in the brain.
12:47
So in a way, one can imagine that these para
12:50
vascular spaces surrounding the Blood Vessels Act as the lymphatic
12:54
vessels of the brain.
12:56
And some people argue that there is indeed.
12:59
Fluid flow.
13:02
Through the parasca spaces, maybe through the tissue itself.
13:07
But the different groups very much arguing whether this exists
13:12
at all and what the directionality would be.
13:15
Whether there are things solutes are transported in on the
13:20
arterial side,
13:21
or whether there are transported out on the arterial side.
13:26
So as a mathematician, this is a.
13:28
This is a puzzling and fun topic to to enter
13:32
into our adventure into because there's so little that's known
13:37
and there's so little consensus.
13:40
Then we decided to start addressing this topic.
13:45
And I think I think we're making some good headway
13:49
because part of why there's so much disagreement and so
13:52
little consensus is because experimental research on the brain is
13:57
very hot. Phone number reasons are practically,
14:00
it's hard because the brain is.
14:02
Enclosed in the skull and in sports,
14:05
it seems like a clear almost a closed system.
14:09
Inside the body. So the moment you manipulated moment you
14:15
inject fluid or inject solids.
14:18
You're tampering with the system the moment you open anything
14:22
up.
14:22
In order to measure something,
14:24
you're also tampering with the system.
14:26
So for this reason it's practically difficult to perform experiments
14:31
on the brain.
14:31
And of course there's also this magic ethical aspect,
14:35
in particular with humans.
14:38
What type of studies you can perform?
14:41
So this is where mathematics is.
14:44
We're hoping to play a role and we have a
14:47
number of research projects focusing on developing mathematical models for
14:54
the brain sworder scope.
14:56
We are trying to make better a new numerical methods
15:00
simulation software and really try to bring better physiological understanding
15:07
using mathematics as an annual investigation.
15:10
Here's my lovely team, so we're talking about 15 to
15:16
20.
15:16
People at similar research lab in Oslo and at the
15:21
University of Oslo.
15:22
Much of the work I'm presenting now is is thanks
15:26
to them.
15:27
So let's dig in an I wanted to first start
15:31
by saying something about what more about what are these
15:36
modes of solid transport in the break.
15:39
So we have a very I would say.
15:45
Interesting collaboration with new radiologist and neurosurgeons at also University
15:51
Hospital in particular gating Stamp Act is still Ada.
15:55
And they're really using cutting edge.
15:58
Kinda clinical techniques for imaging how solutes and Bolton move
16:04
in the human brain.
16:05
And what they do? They've developed a new methodology called
16:09
the pathetic Magnetic Resonance Imaging glymphatic MRI where they inject
16:14
a fluorescent tracer in the spinal down in the spinal
16:18
cord and they see how it moves up and into
16:22
the brain.
16:23
And what they find is that if you inject this
16:26
fluorescent kind of signal molecule,
16:29
but tracer in this battleground,
16:31
it will move up. A power through the neck and
16:35
it will also distribute in the entire break.
16:38
This is these findings are kind of the counter to
16:43
all what's been believed clinically up until about 10 years
16:50
ago.
16:51
That the tracer would. It's been believed that the brain
16:55
is kind of.
16:55
It's prevalent so it would not spread in there,
16:58
but we do really just say yes,
17:00
that residue spread into the brain from the CSF,
17:03
but the commissions observe is that they say it's unlikely
17:07
the diffusion alone explains this brain wide distribution,
17:11
but we wanted to set out to see to what
17:14
extent is this unlikely.
17:16
And we in particular we wanted to see what are
17:19
likely tracer distributions given the uncertainty that exists.
17:23
As I mentioned in you know,
17:25
since there's so much disagreement or what type of connection
17:29
pathways that could be there.
17:30
If they are there and also with the diffusion coefficients
17:35
and so on.
17:35
So we wanted to set out to see what unlikely
17:38
tracer distributions.
17:39
So we took the based on the clinical data.
17:43
We ran on numerical simulations over these.
17:47
A number of sample brains for remodeled different stochastic diffusion
17:52
and velocity fields,
17:54
and we computed number of quantity of interest in order
17:58
to estimate the kind of expected values as well as
18:02
the variance.
18:03
So let's look at this in a bit more detail.
18:06
What did we find? We started by modeling tracer evolution
18:11
with the standard diffusion connection reaction equation.
18:15
So, and we considered the parent Kima only,
18:18
So what we think what we think of as the
18:21
brain,
18:22
at least what I think of as the brain,
18:24
and we were interested in finding the solid concentration C
18:29
depended on time,
18:30
XT space, aksan regiments, Omega such that this diffusion connection
18:36
reaction equation.
18:37
Would hold with a convective velocity V.
18:42
And with an effective diffusion coefficient D star.
18:45
Throughout in this talk are their reaction term will be
18:51
0.
18:51
And we modeled the tracer spread via implicit boundary conditions.
18:55
We assume that the tracer would spread up through the
18:59
subarachnoid space.
19:01
SAS will always be in the subarachnoid space.
19:03
It's not Scandinavian Airlines, it's always subarachnoid space,
19:07
which is the space between the innermost number and appear
19:10
on the next membrane direct memory.
19:12
So that's where the CSF lives,
19:14
and we assume that we prescribe how it spreads there
19:18
based on the clinical images.
19:20
And then what we model and compute is how it
19:24
from their spreads into the parent Kima.
19:27
We started with a kind of a homogeneous constant diffusion
19:31
coefficient D start corresponding to the molecule there using ado
19:36
patrol of about 10 to the minus 10 meters per
19:39
second squared as a baseline model.
19:41
So this is a value from from the literature,
19:45
but it's quite uncertain. But we started with this as
19:49
a baseline.
19:50
And what we do you can see here or snapshots
19:54
goes to be a continuous representation here,
19:57
because he snapshot had a habit spreads after one hour,
20:01
8 hours and 24 hours.
20:02
We see even after 24 hours,
20:04
tracer flowing by diffusion only has penetrated substantially into the
20:09
Gray matter,
20:10
but not at all into these deep central regions.
20:18
So first I think we wanted to address is what
20:21
was the effect of the uncertainty in how large this
20:24
diffusion coefficient is.
20:26
So we included a stochastic diffusion coefficient.
20:32
Where you can see the probability distribution for this here
20:37
on the right and we allowed it to very two
20:40
to three order be two to three orders from two
20:43
to three times larger than two to three times smaller
20:47
essentially.
20:48
And then we computed the.
20:50
So we drew samples from this and computed the expected
20:54
and confidence intervals for the amount of tracer in the
20:59
Gray matter in the white matter.
21:01
So I would define this was that your brains essentially
21:05
are composed of Gray matter and white matter,
21:09
and roughly speaking the Gray matter,
21:11
at least the cortical Gray matter,
21:14
is a region kind of on the close to this,
21:17
the outer surface and the white matter is deeper inside
21:22
the brick.
21:23
Roughly so when we define the amount of Treasury Gray
21:26
matter,
21:26
we just compute the total amounts of the interval of
21:29
the concentration over the Gray matter,
21:32
and similarly for the white matter.
21:36
And what we did find was that the diffusion coefficient
21:40
magnitude did create substantial variation in all outfits.
21:45
The expected amount of. Tracer in greymatter by diffusion alone
21:50
would peak around 15 hours,
21:53
which is in in quite good agreement with the clinical
21:57
observations.
21:58
But there was an the expected amount of tracer in
22:02
the white matter would still be increasing.
22:05
It's been very far from from having reached a peak
22:09
after 24 hours.
22:13
We also investigated the effect of uncertainty in diffusion coefficient.
22:19
And in the field. So effect of heterogeneity but do
22:26
not find.
22:27
See noteworthy differences based on that.
22:33
So without a condom as our baseline,
22:36
we also wanted to model.
22:38
I can potential convective velocity field to see how would
22:44
the results change by adding.
22:47
Lost, it feels and what we did is that we
22:50
want to model the one of the hypothesis for convective
22:54
flow in the this paravesical space is called the glymphatic
22:59
hypothesis.
23:01
Now the glymphatic hypothesis. The idea is that.
23:07
Cerebral spinal fluid makes us with interstitial fluid,
23:11
which is essentially the fluid in the extracellular space is
23:16
running the brain,
23:17
and it flows in along para arterial spaces.
23:20
So Paravesical space is running the arteries.
23:24
It flows through the extracellular space and it flows out
23:28
on the venous side.
23:30
This is the glymphatic hypothesis.
23:33
And so we wanted to try to model is in
23:35
this large can of microscale level,
23:37
and we said it was essentially these three points.
23:41
We said that the velocity field correlation Lane should be
23:44
comperable to the mean distance between materials and manuals.
23:48
We said the party real impermanence pays are equally likely
23:51
at any point in space,
23:53
so the expected value of each of the last component
23:56
should be 0,
23:56
and we assume that we have conservation of mass.
24:00
So that the versions of you there,
24:02
sorry, we should be 0 and we should do Smith.
24:06
And then we defined I've lost field that essentially satisfied
24:10
these stipulations,
24:11
such as the expected value of the magnitude of loss
24:16
T.
24:16
This average last year about zero point 17 micro meters
24:20
per second,
24:21
which matches experimental findings and then we allowed for three
24:25
times larger than 10 times smaller values in this field.
24:31
So this is what we set up as our kind
24:34
of our first sub in Phatic model.
24:37
We use the diffusion coefficient as before.
24:40
The baseline diffusion coefficient. And we computed the amount of
24:45
tracer in the white and Gray matter.
24:48
But we also computed the average concentration into small regions.
24:53
One region in the Gray matter and one in the
24:56
white matter,
24:57
and this total amount that we call big Q little
25:01
at the average concentration we denote by small Q team.
25:06
And the results are shown here.
25:09
In a we have small QG and a small QWNB
25:12
an I think the the main point is that in
25:15
this game phatic circulation as we modeled here does not
25:20
enhance the global tracer influx at all.
25:24
In grey or white matter compared to pure diffusion.
25:29
You see, as minor increase in.
25:34
In this model. So which is the blue line compared
25:36
to the baseline model,
25:38
which is the dotted orange line in the average concentration
25:41
in Gray matter?
25:44
The Nile, but nothing in this global tracer quantities.
25:48
And we also saw very low variability in all outfits.
25:54
So our conclusion was that this is clearly not kind
25:58
of enough to see any enhanced global tracer influx.
26:06
No. In addition to this small,
26:10
shorter scale, so kind of at the arterial and venular
26:16
level.
26:17
Velocity Field it's also been observed that there seems to
26:21
be some large scale directionality associated with the respiratory and
26:27
cardiac pulse propagation and reported,
26:30
so we were interested in see well if you could
26:33
also include this large scale directionality in our model with
26:38
that enhance tracer transport.
26:40
So we made an additional model to look at the
26:43
effect of lymphatic circulation with directionality.
26:48
So in addition to our stochastic model.
26:51
For the velocity field, just representing the small scale.
26:56
Movement we added a kind of a deterministic directional field
27:01
V.
27:01
Do that represent some kind of this is illustrated here
27:05
and is based on on the experiments for the cardiac
27:09
and respiratory pulse.
27:14
And what we did find now is that with this
27:16
circulation we have directionality.
27:18
We do see that the expected amount of tracer in
27:22
the white matter increases.
27:24
Erin Abby, we show the amount to tracer in Gray
27:27
matter.
27:28
Amanda Tracer in white matter the average concentration in Gray
27:32
matter and average concentration in white matter.
27:36
The Gray bars are the 99%
27:38
to 3% confidence intervals for the baseline model.
27:42
She thinks no, not the best lamb,
27:44
but the the one where we have us to castec
27:48
diffusion constant.
27:49
So that's the orange one.
27:51
The blue one is this.
27:55
Kind of a small scale Glymphatic model V1 and the
27:59
red one is this new one where we also have
28:01
directionality.
28:05
The average trace concentration in the in one of those
28:08
small grey region is larger than for diffusion,
28:12
but that's the only kind of quantity of interest.
28:15
But we do see something that's different with very high
28:18
confidence,
28:19
so that the confidence intervals extend outside that of the
28:23
model D1.
28:27
So it's sunrise will be so low is that these
28:29
uncertainty in the diffusion parameters substantially impacted all of our
28:34
output quantities.
28:35
We did find that diffusion matched the clinical observations in
28:39
the Gray matter,
28:40
and this is kind of in the outermost regions.
28:43
Well in fact, but it was not sufficient to transport
28:46
trace it deep into the parent climbers can see from
28:49
later time points the glymphatic blasty S as it's been
28:53
proposed originally, did not increase transport in into any other
28:58
regions considered unless it's been augmented by a flow field
29:02
with a large scale directionality.
29:05
So in short, diffusion seems almost sufficient to explain the
29:09
clinical observations,
29:10
but not quite. But there needs to be some kind
29:13
of large scale directionality in the flow field to have
29:16
it affect.
29:20
So this brings us to the next question,
29:22
namely, if there is such a large scale convective flow
29:26
field,
29:26
something needs to drive it.
29:29
And what could that be?
29:38
So it's well known that your brain is not a
29:42
completely still organ,
29:44
it pulse it's. So during systole blood flows up and
29:49
into the brain and the brain expands and shifts a
29:53
bit and it pushes this cerebral spinal fluid that surrounds
29:58
it down into the spinal canal.
30:02
And this and this reverses during day asking.
30:05
This of this he said goes back up and venous
30:09
blood is pushed out of the brain.
30:11
So the brain is a pulsar type Hogan and this
30:14
can be in images.
30:16
This can be in views or using measurements.
30:19
This can be observed in several ways.
30:21
It can be observed in the so called intracranial pressure
30:26
I CP.
30:27
And it can be observed in measurements of the CSF
30:32
flow.
30:32
So if one it puts a pressure catheter inside the
30:35
brain environment.
30:36
So for instance, here it's inside the ventricles.
30:40
Bitter fluid filled cavities inside the brain.
30:43
We will observe this positive pressure that pulsates with the
30:47
cardiac cycle.
30:48
An depending on whether it's patients,
30:51
a human, this he, patients or non patients,
30:54
you will see a rise in time,
30:57
so this is time here on the X axis and
31:00
the pressure on the Y axis.
31:02
You'll see about four to 8 millimeters of Mercury.
31:06
Change each cardiac cycle.
31:10
What is not well known is how this time kind
31:13
of chases pulsatile aspie distributes in space.
31:16
There's quite a bit of discussion on this,
31:18
but it's just very few measurements.
31:21
But I will get back to that very shortly.
31:23
But if you measure this AICP,
31:25
you will see a very clear kind of cardiac component
31:29
ever,
31:29
one peak every.
31:32
Kind of beat, but if you instead as many others
31:36
do not very many measure these pressures but many others
31:41
look at the flow of CSF.
31:43
In some of these regions,
31:45
aqueduct or down here, we'll see.
31:50
Lots of the flow, or maybe the flux,
31:52
the volume movement of CSF as you see here on
31:56
the right.
31:57
And here the main peaks are not cardiac pics.
32:01
These are respiratory piece and the cardiac waves are only
32:07
these kind of ripples on the surface.
32:10
So this is created a bit of a.
32:13
A conundrum for the conditions in that,
32:16
or at least the discussion point with it asking the
32:20
question,
32:21
what is really this dominant pulsation affect?
32:24
Is it the cardiac signal,
32:26
or is it the respiratory signal that is?
32:29
Important to model and to take into account.
32:32
And as is, there's desperate.
32:34
Finding some. See clearly say that the it's so much
32:38
more clear in the cardiac one,
32:40
while others say the respiratory one is much more clear.
32:43
This is led to a better discussion on this,
32:46
so this is another point where we thought that modeling
32:50
could help in answering and reconciling this finding.
32:55
So what we we did was that we had access
32:58
to quite unique data set of.
33:00
Long term internal pressure measurements where they put in different
33:06
patients where they put one sensor inside the ventricles and
33:10
one sensors subdural E so inside the dura.
33:14
And some of them were even inside the brain,
33:16
parent Kima. And they simultaneously mastered pressures.
33:20
I think only one group in the world more or
33:23
less has have done this in humans and you'll get
33:26
this pressure recordings that you see examples up here.
33:29
You see a zoom in where you see the orange
33:32
and the blue are different the different locations.
33:35
And you see they match very closely,
33:38
but that there perhaps is a small difference between the
33:42
two locations.
33:43
Down here I plotted the pressure difference.
33:46
The difference between the two locations.
33:49
You see that it's it's brother noisy,
33:52
but if you zoom in on it,
33:54
you see that there does seem to be a bit
33:57
of a difference.
33:59
And we modeled this difference,
34:01
extracted the kind of the for her components,
34:04
and found two main components.
34:06
There's a cardiac and respiratory component.
34:12
And we found that this was about,
34:15
you know, the pressure difference we've seen is of the
34:20
order,
34:20
maybe of amplitude up to between 0.2 and 0.4 millimeters
34:26
of Mercury.
34:27
So that seems to be the peak pressure difference between
34:32
these points in the brain.
34:34
Overtime will be also could do was to plug this
34:39
into another Stokes simulation of the flow in.
34:45
In for instance. So here we have the cerebral aqueduct,
34:49
which is one of the kind of fluid pathways that
34:53
are open and clear in the brain environment.
34:56
And we did find that this pressure gradient of around
35:00
0 up to 0.2 millimeters of Mercury can drive flow
35:04
of them lost in magnitude that they do see in
35:07
the UC MRI investigations. So that's I think that's very,
35:12
very promising, because it indicates that there are there are
35:16
very small,
35:16
but there are pressure gradients that can drive fluid movement
35:20
of the right magnitudes.
35:22
But we also could see was that it could also
35:25
reconcile this conundrum for the physiologist that these car deck
35:29
dominated pressure gradients.
35:31
If you plug this into Internet,
35:33
there still looks and it compute the resulting flow and
35:38
the resulting flow volumes.
35:40
Then what is cardiac dominated from the pressure will ever
35:44
induce respiratory volume dominated flow volumes,
35:47
so these are three different patients here,
35:50
and even though the original pressure influx clearly cardiac dominated
35:55
in almost sincere of the respiration,
35:58
will see that the respirations affect is very noticeable in
36:02
the computer flow leaves.
36:04
So this kind of increase makes the pressure and volume
36:07
measurements.
36:11
So I think I'll I'll now turn to ask,
36:14
so we've said that diffusion is almost enough,
36:17
but maybe not quite enough.
36:20
Now we've identified that there are.
36:23
Pressure gradients large enough to drive some flow to address
36:27
some convective flow of the right magnitudes in the brain,
36:31
but if it should brain flow in the brain,
36:33
how would it indeed flow?
36:36
So we view from a modeling perspective the brain as
36:40
a multi porous and elastic medium.
36:43
So the brain is really it's.
36:45
It's clearly elastic. It's very soft.
36:47
It's almost surprisingly soft, they say.
36:51
And it's also permeated by a number of fluid networks.
36:55
Partly the blood vessels, partly para vascular spaces if these
37:01
exist.
37:01
But also the extracellular space.
37:04
So the space between cells forms a network that's filled
37:08
with cerebral spinal.
37:10
Sorry interstitial fluid is water like fluid.
37:17
And to model this, it's typical to use bills questions
37:21
be also questions is set a time dependent linear partial
37:26
differential equation describe displacement flow in a pluralistic medium.
37:31
So the questions reads as follows.
37:34
We find the displacement you and the pressure P /
37:38
a domain Omega in time surity such a balance of
37:42
momentum and mass Holtz.
37:44
So here we have the divergent of two mu times
37:47
the linearized strain epsilon,
37:49
AU plus Lambda times diversity.
37:52
You minus the pressure stuff.
37:54
So here you have the balance on momentum.
37:57
He had a balance of mass,
37:58
so the coefficient C zero times the time derivative of
38:03
P plus divergance attempted to re you minus.
38:07
Divergents of K which is the hydraulic conductivity.
38:11
Great into P is equal to surface a force Jesus
38:15
prescribed source.
38:17
And I put it the reason why I think feels
38:21
question so fun is because in different regimes they produce
38:26
two other equations,
38:28
but in a sometimes a bit and kind of non
38:32
non trivial way.
38:34
And so there's two regimes that we're particularly interested in
38:37
looking at.
38:38
One is the lostorage, an incompressible regime.
38:43
This is when Lambda Ghost Infinity and that case see
38:47
zero.
38:48
This coefficient here is 0.
38:50
Then the Divergency view becomes zero and the system decouples
38:56
into a Darcy type system.
38:58
And analysis detect cyst.
39:03
Another case is when you are in a low storage.
39:06
An impermeable regime, also very relevant in the brain.
39:09
Depending on what your model and this is when.
39:13
The probability or conducted the hydraulic connections goes to 0K,
39:19
in which case the questions reduce almost to Stokes.
39:25
It should be a udot here,
39:27
but after time discretization it becomes almost looks.
39:32
So these relations to Dar sells distance Stokes points at
39:37
the questions are relevant when discretizing bills equations.
39:43
So with that, let's just take a bit of a
39:46
aside tour and we were.
39:47
If you're looking at this conversation so Bill equation,
39:51
we would like to be need to know a bit
39:54
about stability of cell point problems.
39:58
So the general form of saddle point problems in variational
40:03
form takes this abstract form.
40:05
So if you're interested in finding and variable U&P such
40:10
that this system here holds.
40:12
Um?
40:14
A number of conditions need to be satisfied in order
40:20
for solutions to valuable post.
40:23
So if we take the example of if we take
40:25
those equations and discretizing time and set,
40:28
say, the time step to one.
40:31
We also questions will fit this abstract form.
40:34
With a U, the other sister Beth view the divergent.
40:39
See you and this perturbation term corresponding to the storage
40:46
and the permeability.
40:48
An abstract telepoint problems are well posed.
40:51
If a BNC, these personal forms are continuously is positive
40:56
definite and exists.
40:57
Parameters Alpha and beta such that these conditions The Legend
41:03
Scale published Kubert See conditions hold.
41:06
And it's well known if you try discretizations that will
41:10
not satisfy the discrete quivalence of these stability conditions,
41:14
you will often get solutions that do absolutely not make
41:18
sense.
41:18
In this illustration, I think I've discretized Darcy equations using
41:23
a stable discretization on the right.
41:25
We get a reasonable looking solution and an unstable discretization
41:29
on the left.
41:30
We just have a kind of a staggered checkerboard.
41:34
But that is absolutely does not make sense.
41:39
So this brings us down to but.
41:42
But for real, in the case where K goes to
41:45
zero,
41:46
what are the right stability criteria in this case?
41:51
It's now we're looking at a version of bills.
41:54
The question where we introduced an additional variable,
41:57
the flux set and I'm listed in in variational form,
42:01
so it's now listed here.
42:02
Here's our variational form of the momentum equation equation for
42:06
the flux in terms of the pressure.
42:09
So the flux set is defined as the gradient said,
42:12
and here's the conservation of mass equation.
42:16
Now Rodrigo and Colors in 2018 introduced the concept of
42:20
Stokes pure stability.
42:22
They showed that discrete spaces.
42:26
Our so called Stokes Bule stable and give good discretizations
42:30
if this elastic form is better than coercive,
42:34
and the pairing of human cues so the pairing for
42:37
the velocity and the pressure.
42:39
Sorry the displacement and pressure space is still stable.
42:44
And the pairing for the flux and the pressure space
42:48
is Darcy stable.
42:50
Now if you go back to look at our links
42:53
to the.
42:57
A different equations, perhaps this becomes.
43:01
The link to Stokes equation becomes very clear.
43:05
The link to Dalasi equations analysis Tuloso seems reasonable.
43:12
And this is been a very nice in effect in
43:16
Vail designing this Critter Station.
43:19
As for ya'll equations in the limits as K goes
43:22
to 0.
43:26
However, what we did fine and what?
43:28
For that matter other server observed before as well is
43:32
that even if you use discretizations that are not classically
43:36
Darcy stable?
43:37
So here for instance, I've used the discretization that does
43:41
not satisfy the darcee stability condition.
43:44
And here I used the discretion station that do they
43:47
still they still seem fine.
43:49
And it turns out that you can define a minimal
43:52
version of Circle Stokes pure stability,
43:54
where instead of the Dorset stability you assume that the
43:58
diverges of the flux versus included in the pressure space.
44:02
And we can show that these solutions that satisfy these
44:07
smaller conditions will indeed converge as expected.
44:11
The reason for this is at all,
44:14
I think, important in our view,
44:17
is because finding spaces that are Darcis table in the
44:22
limit as.
44:23
Kate goes to serial is very tricky and and in
44:28
fact more you need to introduce waited moves for this.
44:35
Um? I will wrap up very soon.
44:39
I want you to say a little bit about the
44:42
incompressible case as well.
44:45
So we're also working with in the brain teacher.
44:48
I think most it's assumed that it's it's nearly incompressible
44:52
sisters.
44:53
So much fluid water like fluids involved.
44:57
So we are also working.
44:58
I'm interested in in cases where the Lum Dallas District
45:03
property goes to Infinity.
45:05
And to model the brain tissue,
45:07
we've worked externally, generalizing biasa questions to include more fluid
45:12
networks,
45:13
to include bladex. Different plant networks are trees Daniel,
45:17
scapulars and so on. In addition to extracellular space.
45:22
You said so called multiple network poor elistist equations.
45:26
And they've been introduced for brain tissue by Talia Mantikas
45:30
about a decade ago,
45:31
and that we've started studying numerically now.
45:36
I think I will not go into much detail,
45:39
I just want to say that the standard formulation of
45:42
these equations suffers from kind of the typical locking and
45:47
loss of convergence in the incompressible limit.
45:50
But we can introduce a total pressure variational formulation and
45:55
prove that this will restore the optimal convergence human in
46:00
the incompressible limit.
46:04
So with that I think I'll I'll wrap it up.
46:08
I've included a number of references here and I would
46:13
like to thank my collaborators and send the core message
46:19
that.
46:19
Mathematical models can give new insight into Physiology and medicine,
46:24
but also that the human brain,
46:26
I would say, gives an extraordinary rich setting for mathematics
46:31
and numerics,
46:32
and I encourage more mathematicians to dig it.
46:35
OK, thank you. Thank you very much Maria for this
46:39
very interesting lecture.
46:41
We have time for a couple of questions,
46:44
maybe just to start the discussion.
46:46
I can start by asking a question.
46:49
So what is the what is the perspective here?
46:52
So you assume that the fluid flow inside the brain
46:55
satisfies some complicated formal Tapio equations and then that provides
47:00
you the velocity,
47:02
the bulk velocity and then you do tracer transport with
47:06
respect to this velocity and some diffusion.
47:09
And you hope to recover the.
47:12
The, let's say the tracer concentration correctly is that is
47:17
that the vision or somehow the link was at least
47:20
to me a little bit unclear,
47:22
right?
47:25
What you're pointing at is much much of the vision
47:29
rate,
47:30
but.
47:35
At first step is to see if we can use
47:38
these multiple network.
47:40
Equations to say something reasonable about pressures and the resulting
47:46
velocities and the resulting displacements in the brain at all.
47:52
This is this is unclear,
47:54
but I think it's first.
47:56
It's very illuminating too.
48:00
To see what the then our numerical simulations tell us
48:04
and how we can relate it in terms of,
48:07
for instance displacements, elastic displacements in terms of how would
48:13
you make sense of these.
48:15
I can average blood pressures.
48:18
Can we compare the intracranial pressures that we computed with
48:23
into Grendel pressures in the brain?
48:25
And also couple it to the fluids in the brain
48:29
environment.
48:30
Where much more I would say is known.
48:33
So this is kind of the validation.
48:35
This kind of knowledge integration and validation is the first
48:40
step.
48:40
Because I think that.
48:43
This forces us to try to and everyone to kind
48:47
of bring the knowledge of the different clinical measurements and
48:53
the Physiology together.
48:55
I don't think we're there yet,
48:57
but once we're there then one can start making predictions
49:01
about how these different changes in the permeability how different
49:05
regional changes.
49:06
For instance as result of a stroke or if you
49:09
have a subarachnoid bleeding,
49:11
and so on. How this could affect the flow patterns?
49:14
Yes, OK, thank you. So there is a question,
49:17
but on pens in the treasure in the trace diffusion
49:20
numerical study for the case that the external boundary conditions
49:25
are no flux.
49:25
There is the question is,
49:27
are these no flux external boundary conditions or not?
49:31
Right so.
49:34
On the ventricle, so that in the list,
49:37
roughly speaking to boundaries, there's the ventricles.
49:41
Then we just say no flux,
49:43
but in the time simulation time we've simulated.
49:48
The tracer hasn't reached there yet either,
49:51
so I don't think it would affect much on the
49:54
outer region where we assume it's in touch with the
49:58
CSF where the tracer is.
50:00
Injected we use this.
50:03
Implicit type boundary conditions so that we can serve mass.
50:06
Yes, so I think that's the last part of the
50:09
question,
50:09
right? So the total amount of traces.
50:12
Disconsolate yes, so it does.
50:14
Then there's a question about the validity of Navier Stokes
50:17
simulations.
50:18
What are the scales for the boundary conditions?
50:21
In particular, is it a valid approach or the particle
50:24
based method be more appropriate?
50:26
So we've used Navier Stokes for the larger.
50:31
Could have fluid compartments, and now I'm in compartment really
50:36
in open spaces of the brain.
50:38
So these for instance in the subarachnoid space and in
50:42
the aqueduct.
50:43
And these are compartments where I would say now we're
50:46
Stokes or Stokes even are reasonable assumptions.
50:50
These are relatively big in comparison to everything else.
50:54
We have definitely not used these for para vascular spaces,
50:58
for instance, which could be of the order blood vessels
51:01
down to micro meters,
51:02
so the size of red blood vessels at the capillary
51:04
level.
51:07
Maybe I have another question,
51:09
so in this let's say the first part of your
51:12
dog.
51:12
You had this, you had a bulk sort of convective
51:16
velocity,
51:16
and you also had a small scale convective velocity and
51:20
your message was the small scale convexity.
51:23
Convective velocity didn't really matter,
51:26
so that is that correct?
51:27
So if I completely neglect it,
51:30
put it to 0, do we?
51:31
Do we still see the same effects,
51:33
or is that it? Does it still act at a
51:37
small scale?
51:38
Yeah, so I guess it does not have an effect
51:40
on the large scale influx,
51:42
so this global influx you'll see an effect on the
51:46
small scale.
51:47
Up
51:49
there are no further questions.
51:51
We are a bit short of time today,
51:52
so let's thank Maria for a very interesting talk.
51:55
And yeah, yeah, and you want to say something about
51:59
the next lecture.
52:00
Yeah, well, first of all,
52:01
thanks again to both Mary and sit also you for
52:04
checking the talk is really interesting.
52:06
One thing today.
52:09
And thanks everybody for joining you up for next week,
52:12
which will be actually our last session webinar session this
52:16
year.
52:16
It will be marketed as a Wolf wolfram.
52:18
Actually I posted in the chat before I get on
52:20
my feet,
52:21
models and pedestrian dynamics. So again,
52:23
I think an interesting application of keys are quite different
52:26
one,
52:27
but. I guess it's kind of the goal to show
52:30
the variety of PDE research.
52:32
Yeah, so thanks everyone for joining,
52:35
so feel free to join again next week,
52:38
same time, same place and feel free to check out
52:42
our recordings on the website of the webinar,
52:45
OK?
52:47
And yeah, it's nice use it.
52:49
Thank you very much, Marie and.
52:52
See you all, hopefully again in the future.
52:54
Bye bye.