=Paper=
{{Paper
|id=Vol-1218/bmaw2014_paper_9
|storemode=property
|title=Dynamic Bayesian Network Modeling of Vascularization in Engineered Tissues
|pdfUrl=https://ceur-ws.org/Vol-1218/bmaw2014_paper_9.pdf
|volume=Vol-1218
|dblpUrl=https://dblp.org/rec/conf/uai/KomurluSB14
}}
==Dynamic Bayesian Network Modeling of Vascularization in Engineered Tissues==
https://ceur-ws.org/Vol-1218/bmaw2014_paper_9.pdf
Dynamic Bayesian Network Modeling of Vascularization in
Engineered Tissues
Caner Komurlu Jinjian Shao Mustafa Bilgic
Computer Science Department Computer Science Department Computer Science Department
Illinois Institute of Technology Illinois Institute of Technology Illinois Institute of Technology
Chicago, IL, 60616 Chicago, IL, 60616 Chicago, IL, 60616
ckomurlu@hawk.iit.edu jshao3@hawk.iit.edu mbilgic@iit.edu
Abstract depths of the tissue, and form connections to allow
blood circulation.
In this paper, we present a dynamic Bayesian The formation of new blood vessels are triggered and
network (DBN) approach to modeling vascu- a↵ected by growth factors that are released by dis-
larization in engineered tissues. Injuries and tressed cells that are far from the existing blood ves-
diseases can cause significant tissue loss to sels. When these growth factors reach existing blood
the degree where the body is unable to heal vessels, they sprout new branches and these branches
itself. Tissue engineering aims to replace the “search” for the distressed cells by following the gradi-
lost tissue through use of stem cells and bio- ent of the growth factor. This process, however, is
materials. For tissue cells to multiply and stochastic for at least two reasons: i) even though
migrate, they need to be close to blood ves- growth factors are the main ingredients for causing
sels, and hence proper vascularization of the sprouts, they are not the only elements that a↵ect vas-
tissue is an essential component of the en- cularization, and ii) the growth factors are increasingly
gineering process. We model vascularization more uniformly distributed as they go further away
through a DBN whose structure and parame- from the distressed cells, and hence the gradient is al-
ters are elicited from experts. The DBN pro- most uniform, hindering the capability of the blood
vides spatial and temporal probabilistic rea- vessel finding its way correctly.
soning, enabling tissue engineers to test sen-
This inherent stochasticity in the vascularization pro-
sitivity of vascularization to various factors
cess, the spatial nature of the tissue, and the temporal
and gain useful insights into the vasculariza-
aspect of the vascularization make temporal graphical
tion process. We present initial results in this
models a great fit for reasoning with uncertainty in
paper and then discuss a number of future re-
vascularization. In this paper, we present a dynamic
search problems and challenges.
Bayesian network (DBN) for modeling vascularization
in engineered tissues. We elicit the structure of the
DBN from tissue engineering experts and we experi-
1 INTRODUCTION ment with various parameter settings to provide fur-
ther insights into the vascularization process. Because
People lose tissue due to accidents, medical opera- this is a first and novel application of DBNs to tissue
tions, treatments, and illnesses. While some organs, engineering, it avails itself to many interesting future
e.g. liver, can replace the lost tissue most cannot espe- research directions and challenges.
cially when the damage is too severe. For these kinds
Our contributions in this paper include:
of tissue damages, the lost tissue can be replaced by
engineering a new tissue through stem cells and bio-
materials [18]. • We present a novel application of DBNs to vascu-
larization in engineered tissues
An essential process for engineering a healthy tissue
is the proper vascularization (formation of new blood • We present initial results and insights, where we
vessels) of the tissue, as the tissue cells need to be experiment with various parameter settings, and
close to the blood vessels both to discharge their waste
and to receive nutrition and oxygen. The blood ves- • We discuss several future research challenges and
sels need to spread out in the tissue, invade into the opportunities in detail.
89
The rest of the paper is organized as follows: in Sec-
tion 2, we provide a brief background on tissue engi-
neering and vascularization. In Section 3, we describe
our DBN model for vascularization. We present our
experimental setup and results in Section 4. In Sec-
tion 5, we briefly discuss related work. We then dis-
cuss future research directions and challenges in detail
in Section 6, and then conclude.
2 BACKGROUND
In this section, we first provide a brief background on
tissue engineering and vascularization and then discuss
briefly why dynamic Bayesian networks (DBNs) are a
good fit for modeling vascularization.
People lose tissue due to accidents, treatments, and Figure 1: Illustration of vascularization, including the
illnesses. Some organs, e.g. liver, can replace the lost tip cells (active cells), the fixed cells (stalk cells), and
tissue while others cannot. Sometimes, the damage anastomosis. [19]
can be so severe that the body cannot heal itself. For
example, bones can heal after smooth fractures. Yet,
some fractures damage bone body so severely that the
bone cannot regenerate. For these kinds of damages, Vascularization is a key process in tissue development.
the lost tissue can be replaced by engineering a new When cells that are emitting VEGF cannot be reached
tissue through stem cells and biomaterials. in time by the new blood vessels, the cells first fall
in hypoxia (i.e., lack of Oxygen) and then start dy-
Stem cells are generic types of cells that have the abil-
ing. Hence the formation of healthy tissue depends on
ity to replicate and transform to any tissue. Stem
appropriate vascularization; the blood vessels need to
cells, like all other cells, need to be close enough to the
spread out in the newly-formed tissue, invade into the
blood vessels so that they can forward their biological
depth, and need to form connections to allow blood
wastes to the vessels and they can be fed with nutri-
circulation.
tion and oxygen carried by the blood vessels. When
a tissue is engineered through replication and trans- Though it is well-known that the VEGF is a major
formation of stem and tissue cells, there is no existing contributor to sprouting of new blood vessels and that
blood vessel web in the environment; the only blood the tip of the blood vessel typically follows the gradient
vessels available are the original vessels located at the of the VEGF, there are still unknown factors that af-
edges, ready to sprout and progress to the depths of fect vascularization. Moreover, the VEGF distribution
the newly-formed tissue. becomes more uniform as we get further away from
the source of the emission and hence the gradient does
The stem cells that do not have access to blood vessels
not necessarily point to the distressed cell. Therefore,
will not be able to discharge waste and receive nutri-
given our knowledge of the VEGF distribution the en-
tion and oxygen. In such cases, a cell starts signaling
vironment, the blood vessels do not necessarily follow
about its needs by means of emitting chemicals called
a deterministic path; they also do a bit of exploration.
vascular endothelial growth factor (VEGF). VEGF dif-
This is where the uncertainty reasoning capabilities of
fuses and disperses in the environment. When it con-
probabilistic graphical models become handy for mod-
tacts a blood vessel, it triggers a new sprout of blood
eling vascularization.
vessel towards the source of emission. The tip of these
new sprouts typically follow the gradient of the VEGF In this paper, we model the vascularization process
to find the distressed cell. During this process, the through dynamic Bayesian networks (DBNs) to enable
newly-formed blood vessel can also branch and sprout tissue engineering researchers to reason with spatial
new blood vessels. When the branches meet with other and temporal growth of blood vessels. With the help
branches, they merge (this process is called anastomo- of DBNs, the researchers can formulate and query the
sis) and a blood circulation through the new vessel DBNs and try a number of parameter settings, without
starts. The blood circulation helps nearby stem and the need to experiment with every one of them in the
tissue cells, which then stop emitting growth factors. lab. This process allows the researchers to gain further
This event is called angiogenesis or vascularization. insights and formulate new in-vivo (on animals) and
Please see Figure 1 for an illustration of this process. in-vitro (on glass) experiments.
90
3 APPROACH
In this section, we describe our DBN model for vas- 𝐿 𝐿( )
cularization. We made a number of assumptions to
simplify the model. In this model, we assume a 2D
structure, whereas in real-life scenarios, the tissue ob-
viously has a 3D structure. In this 2D structure, which
is illustrated in Figure 2, as also assumed in [1], we as-
sume that the blood vessel grows bottom-up towards
north. Therefore, the status of a location at time t
𝐿 𝐿 𝐿
depends on: i) its status at time t, and ii) the statuses
of its south neighbors at t.
t t+1
Figure 3: A two-time slice representation of the DBN.
A location at a time t + 1 has four parents: itself at
( )
time t and its lower neighbors at time t.
𝐿 𝐿
AC
( ) ( ) ( )
𝐿( )( ) 𝐿 ( ) 𝐿( )( ) 𝐿( 𝐿 ( 𝐿(
)( ) ) )( ) T F
t t+1 PAC PSC PE SC
Figure 2: The tissue grid. Each cell of the grid rep- T F
resents a location, which can be Empty, or can be oc-
cupied with an Active Cell or Stalk Cell. Each PAC PSC PE
location is represented as a random variable in DBN.
To simplify the notation, when we refer to a generic
location Ltxy , we will drop the subscripts and hence
simply use Lt , and when we refer to its neighbors Figure 4: The CPD for P (L(t+1) |Lt , LtSW , LtS , LtSE ).
at its south Lt(x 1)(y 1) , Ltx(y 1) , and Lt(x+1)(y 1) we
will simply use LtSW , LtS , and LtSE , corresponding to
neighbors at south west, south, and south east, respec- • The tip of a blood vessel (AC) at time
tively. We illustrate the relevant 2-time slice dynamic t becomes the body (SC) at time t + 1.
Bayesian network in Figure 3. That is P L(t+1) |Lt = AC, LtSW , LtS , LtSE =
Each location on the 2D grid is a random variable, P L(t+1) |Lt = AC = h✏, 1 2✏, ✏i, where ✏ is a
representing whether that location is Empty, or occu- small noise parameter.
pied by a blood vessel cell. Blood vessel cells are two
• A Stalk Cell at time t either continues
types: the tip of a blood vessel that has the potential to
to remain a Stalk Cell at time t + 1 or
grow (henceforth called an Active Cell) or the body
it might become Active Cell with probabil-
of the blood vessel (henceforth called the Stalk Cell).
ity to sprout a new blood vessel branch.
Therefore, the domain of random variable is [Active
That is, P L(t+1) |Lt = SC, LtSW , LtS , LtSE =
Cell, Stalk Cell, Empty], abbreviated henceforth as
[AC, SC, E]. P L(t+1) |Lt = SC = h , 1 ✏, ✏i. We refer
to as the sprout possibility.
We model the conditional probability distribution,
(CPD), P L(t+1) |Lt , LtSW , LtS , LtSE as a tree CPD as • An Empty location at time t will remain Empty
illustrated in Figure 4. To give a simple overview, at time t + 1 if none of its SW, S, or SE neigh-
at each step in time, an Active Cell elongates and bors are Active Cell at time t; if there is an
moves into a nearby Empty location, forming the body Active Cell at one or more of those neighbor-
of the blood vessel (i.e., Stalk Cell) in the process. ing locations at time t, one of them might elon-
The transitions are: gate to this Empty location at time t + 1. The
91
probability of that an Empty location being oc- grid over three time slices. Then, we present results
cupied by an Active Cell at time t + 1 is mod- on a bit larger scale, 9 ⇥ 9, over nine time slices. Fi-
eled as a Noisy-OR of its neighboring locations. nally, we present a framework where we quantify the
That is P L(t+1) = AC|Lt = E, LtSW , LtS , LtSE uncertainty over the predictions on the last time slice
is a Noisy-OR of LtSW , LtS , LtSE , with parameters and discuss how it is a↵ected by the growth patterns
0 , SW , S , and SE , where 0 is leak param- and sprout possibilities.
eter, and SW , S , and SE corresponds to the
For inference, in the 3⇥3 case, we used exact inference.
possibility that an Active Cell elongates in the
For the 9 ⇥ 9 case, we used forward sampling. Note
NE, N, or NW direction.1 The magnitude of SW ,
that we are able to use forward sampling in our settings
S , and SE are determined by the VEGF gradi-
because we provide the initial condition (all locations
ent. We refer to various configurations of the
at time t = 0) as evidence and compute probabilities
parameters as the growth patterns.
for the remaining time slices.
4 EXPERIMENTAL SETUP, 4.1 Detailed Results for 3 ⇥ 3
RESULTS, AND INSIGHTS
In this toy setting, we provide the evidence for the
In this section, we describe the experiments we per- initial configuration of the experiment, i.e., we provide
formed using various settings for the growth pattern evidence for all locations for time t = 0, and compute
( ) and sprout ( ) parameters. In all the experiments probabilities for all locations for all future time slices.
to follow, we set the noise ✏ and the leak 0 parameters That is, we compute P (L1 , L2 |L0 ), where Lt denotes
to 0.01. For the growth pattern, we present results for all locations at time t. For t = 0, we provide the
two settings: evidence as follows: the middle of the bottom row is
set as the tip of the blood vessel (i.e, L0x=1,y=0 = AC)
and the rest of the locations are set as Empty. Figure 5
• straight-growth: h SW , S , SE i =
illustrates this setting.
h0.01, 0.98, 0.01i. For this pattern, the blood
vessel follows a straight line, growing towards
north. E E E
• uniform-growth: h SW , S , SE i = h 13 , 13 , 13 i.
For this pattern, the blood vessel has equal chance E E E
of growing towards north, north west, or north
east. E AC E
For the sprout possibility, that is a Stalk Cell turn-
ing into an Active Cell, we present results for two Figure 5: The initial configuration for the 3 ⇥ 3 grid.
settings:
The straight-growth results are presented in Figures
6 and 7, and uniform-growth results are presented in
• seldom-sprout: = 0.01. For this setting, the Figures 8 and 9.
Stalk Cell has very small chance (probability of
0.01) of becoming an Active Cell in the next The simplest setting where the blood vessel grows in
time step. a straight path and that does not sprout at all (Fig-
ure 6) is fairly straightforward to analyze. The tip of
• always-sprout: = 0.98. For this setting, the the blood vessel migrates one location towards north at
Stalk Cell has 0.98 probability of becoming ac- each step, forming the body of the vessel along the pro-
tive in the next step. This is quite an unrealistic cess. This setting, therefore, serves as a sanity check.
setting; we present it only for didactic purposes.
In the next setting, which is presented in Fig-
ure 7, we keep the growth pattern the same
We present results for four possible configurations: the (straight-growth) but increase the sprout possibility
cross-product of the growth patterns and sprout pos- to 0.98 (always-sprout). In this setting, the blood
sibilities. We first provide detailed results on a 3 ⇥ 3 vessel grows towards north as expected. Unlike the
1
Note that SW denotes the probability that an Active seldom-sprout case, however, a Stalk Cell at time
Cell at the SW of an Empty location will move to this t = 1 became active at time t = 2.
Empty location; hence SW denotes the possibility that an
Active Cell at SW moves in the NE direction to occupy Next, we present results for the uniform-growth
an Empty location. cases. In this setting, the blood vessel has uniform
92
.01 .01 .01 .04 .95 .04 .01 .01 .01 .04 .95 .04
AC .02 .98 .02 .02 .01 .02 AC .02 .98 .02 .02 .01 .02
.01 .01 .01 .01 .01 .01 .01 .01 .01 .02 .96 .02
.00 .00 .00 .01 .01 .01 .00 .00 .00 .01 .01 .01
SC .00 .00 .00 .02 .96 .02 SC .00 .00 .00 .02 .96 .02
.00 .98 .00 .02 .97 .02 .00 .98 .00 .02 .02 .02
t=1 t=2 t=1 t=2
Figure 6: straight-growth, seldom-sprout. Figure 7: straight-growth, always-sprout. The
This is the most straightforward setting where the blood vessel grows towards north. A location that
blood vessel grows one step at a time towards is a Stalk Cell at time t = 1 (Lx=1,y=0 ) becomes
north. Active Cell at time t = 2.
.01 .01 .01 .22 .31 .22 .01 .01 .01 .22 .31 .22
AC .34 .34 .34 .02 .02 .02 AC .34 .34 .34 .02 .02 .02
.01 .01 .01 .01 .01 .01 .01 .01 .01 .02 .96 .02
.00 .00 .00 .01 .01 .01 .00 .00 .00 .01 .01 .01
SC .00 .00 .00 .34 .33 .34 SC .00 .00 .00 .33 .33 .33
.00 .98 .00 .02 .97 .02 .00 .98 .00 .02 .02 .02
t=1 t=2 t=1 t=2
Figure 8: uniform-growth, seldom-sprout. The Figure 9: uniform-growth, always-sprout. The
blood vessel has equal probability to grow in all blood vessel has equal probability to grow in all
three directions. A Stalk Cell at time t will most three directions. A Stalk Cell will most likely
likely remain as Stalk Cell at t + 1. become Active Cell in the next time slice.
probability of growing towards NW, N, and NE. In becomes an Active Cell at t = 2.
the seldom-sprout case (Figure 8), the Active Cell
These toy experiments provide insights into how the
at t = 0 turned into a Stalk Cell at time t = 1 and
process typically works. Next, we present results for
remained a Stalk Cell at time t = 2. The Active
the 9 ⇥ 9 grid.
Cell, unlike the straight-growth case, has equal
probability of moving in all three directions. In the
last time step, the middle of the top row has higher 4.2 Summary Results for 9 ⇥ 9
probability (.31) than the sides (.22) simply because
the middle location can be reached from more locations Similar to the 3 ⇥ 3 grid, we provide evidence for t = 0
compared to the side locations. The always-sprout case and compute probabilities for the remaining eight
case (Figure 9) is similar except a Stalk Cell at t = 1 time slices. In the initial configuration, the middle
93
.06 .06 .06 .10 .64 .10 .06 .05 .05 .17 .17 .18 .22 .75 .22 .17 .18 .17
.06 .06 .07 .06 .01 .06 .06 .06 .06 .17 .18 .17 .18 .13 .17 .18 .18 .18
.05 .05 .05 .05 .01 .05 .05 .05 .06 .17 .16 .18 .24 .88 .24 .17 .18 .18
.05 .05 .05 .04 .01 .05 .05 .05 .05 .16 .17 .16 .16 .09 .16 .17 .17 .16
Active
.04 .04 .04 .04 .01 .04 .04 .04 .04 .16 .14 .16 .22 .91 .22 .15 .16 .15
Cell
.04 .04 .03 .04 .01 .03 .04 .03 .04 .13 .14 .14 .13 .07 .14 .14 .14 .14
.03 .03 .03 .03 .01 .03 .03 .03 .03 .12 .11 .12 .17 .92 .16 .12 .12 .12
.02 .03 .02 .02 .01 .02 .02 .02 .02 .09 .09 .10 .10 .07 .09 .09 .10 .09
.02 .01 .01 .01 .01 .01 .01 .01 .01 .06 .06 .06 .06 .86 .05 .05 .06 .05
.22 .23 .22 .23 .23 .23 .24 .23 .23 .14 .14 .14 .15 .14 .14 .14 .14 .14
.23 .23 .23 .27 .86 .27 .23 .23 .22 .14 .14 .15 .18 .77 .18 .14 .15 .14
.23 .23 .23 .28 .87 .26 .24 .23 .22 .14 .15 .14 .14 .11 .14 .15 .14 .15
.23 .22 .23 .26 .88 .26 .23 .23 .22 .14 .13 .14 .20 .89 .20 .14 .15 .14
Stalk
.21 .21 .22 .25 .89 .25 .22 .21 .21 .13 .14 .13 .13 .08 .13 .14 .14 .13
Cell
.20 .20 .20 .23 .89 .23 .20 .20 .20 .12 .11 .12 .18 .91 .18 .12 .12 .12
.18 .17 .17 .20 .90 .19 .18 .17 .18 .10 .11 .11 .10 .07 .11 .11 .10 .10
.14 .14 .14 .15 .91 .15 .14 .14 .14 .08 .08 .08 .11 .92 .10 .08 .08 .08
.10 .10 .10 .10 .93 .10 .10 .10 .10 .06 .06 .06 .06 .07 .06 .05 .06 .06
straight-growth – seldom-sprout straight-growth – always-sprout
Figure 10: AC and SC probabilities for the 9 ⇥ 9 grid at the last time slice (t=8) for straight-growth. Left:
seldom-sprout. Right: always-sprout. On the right, it is apparent that the Stalk Cells and Active Cells
alternate.
of the bottom row is set as an Active Cell and the side of Figure 11), the blood vessel can be anywhere
remaining locations are set as Empty. Due to space on the grid, except, as expected, the middle locations
limitations, we present results for only the last time have higher probability. In the always-sprout case
slice, t = 8. The straight-growth case is shown in (the right side of Figure 11), the Stalk Cells and
Figure 10 and the uniform-growth case is shown in Active Cells alternate, as expected. Additionally,
Figure 11. the probabilities for locations being a blood vessel (ei-
ther Stalk to Active) are higher in the always-sprout
In the straight-growth seldom-sprout case (the left
case compared to the seldom-sprout case, again as
side of Figure 10), we see a straight blood vessel for
expected.
the middle of the grid, where every cell of the blood
vessel except the tip is a Stalk Cell and the tip is The results so far have been nothing surprising, but
an Active Cell, as expected. In the always-sprout only confirming our expectations. The value of the
case (the right side of Figure 10), the Stalk Cells and DBNs, however, lies at their capability to reason with
Active Cells alternate, again as expected. spatial and temporal uncertainty as well as their po-
tential for future directions. We discuss one of the
In the uniform-growth seldom-sprout case (the left
94
.04 .06 .07 .08 .09 .08 .07 .06 .04 .17 .22 .24 .26 .26 .25 .24 .22 .17
.03 .04 .03 .04 .04 .04 .04 .04 .03 .16 .20 .21 .20 .20 .22 .21 .20 .16
.03 .04 .03 .04 .03 .04 .04 .03 .03 .21 .30 .34 .40 .40 .38 .34 .29 .21
.03 .04 .03 .03 .03 .03 .03 .03 .03 .15 .19 .18 .17 .17 .17 .19 .19 .15
Active
.03 .03 .03 .03 .03 .03 .03 .03 .03 .22 .37 .51 .60 .63 .59 .50 .36 .21
Cell
.02 .03 .03 .03 .02 .03 .03 .03 .02 .15 .17 .15 .13 .12 .13 .15 .17 .15
.02 .03 .02 .02 .02 .02 .02 .03 .02 .13 .16 .52 .71 .80 .70 .53 .16 .13
.02 .02 .02 .02 .02 .02 .02 .02 .02 .10 .13 .13 .09 .09 .09 .12 .13 .10
.01 .01 .01 .01 .01 .01 .02 .02 .02 .06 .06 .06 .05 .86 .06 .06 .06 .06
.17 .19 .21 .20 .21 .21 .20 .20 .16 .13 .16 .17 .17 .16 .17 .17 .16 .13
.17 .23 .23 .26 .27 .26 .25 .23 .18 .14 .18 .21 .22 .23 .21 .21 .18 .14
.18 .22 .25 .27 .27 .27 .26 .23 .17 .13 .16 .16 .15 .17 .16 .17 .16 .13
.17 .22 .26 .28 .29 .29 .26 .22 .17 .18 .26 .34 .40 .41 .39 .33 .25 .18
Stalk
.17 .22 .26 .30 .31 .30 .26 .21 .17 .12 .15 .15 .14 .13 .14 .15 .15 .12
Cell
.15 .21 .26 .32 .34 .31 .26 .20 .15 .12 .29 .47 .62 .66 .62 .46 .28 .11
.15 .16 .25 .32 .39 .33 .24 .17 .14 .11 .14 .12 .10 .09 .10 .12 .14 .11
.13 .15 .14 .40 .41 .41 .14 .14 .13 .09 .10 .10 .71 .71 .71 .10 .10 .08
.10 .10 .10 .10 .91 .10 .10 .10 .10 .06 .06 .06 .06 .07 .06 .06 .06 .05
uniform-growth – seldom-sprout uniform-growth – always-sprout
Figure 11: AC and SC probabilities for the 9 ⇥ 9 grid at the last time slice (t=8) for uniform-growth. Left:
seldom-sprout. Right: always-sprout. In both cases, the blood vessel can grow uniformly in each direction
and the middle locations have higher probability of having a blood vessel simply because they can be reached
from multiple locations. In the always-sprout case, Stalk Cells and Active Cells alternate.
future directions here supplemented with some pre- periment at time t, what is the uncertainty over LT ?
liminary results, and discuss more future directions in More practically: when is the earliest time we can stop
Section 6. an experiment so that the uncertainty over the last
time slice is below a pre-specified target ?. It is im-
4.3 Quantifying Uncertainty portant to note that when an experiment is stopped,
the researchers dissect the tissue to analyze its proper-
Given an initial condition, L0 , the tissue engineers are ties, such as vascularization, and hence the experiment
interested in the final status of the tissue, LT , where cannot continue beyond that point.
T denotes the final step of the experiment. Because
Given an uncertainty measure, this question can be
real-world experiment take a long time, mostly weeks,
formulated rather straightforwardly using DBNs. Let
they would like to be able to stop an experiment at
U N C P LT |l0 , lt denote the uncertainty over the
time t < T and still be able to reason about time T .
predictions over the last time slice, given the initial
Therefore, they are interested in the following ques-
condition L0 = l0 and the status of the experiment at
tion: given an initial condition L0 , if we stop the ex-
95
time t, Lt = lt . Then, we simply need to find
Uncertainties for Two Growth Patterns
8.00
argmin U N C P LT |l0 , lt < (1)
t 0 unless we stop the experiment. Therefore, we
4.00
need to take an expectation over all possible outcomes
at time t: 3.00
2.00
X 1.00
argmin P Lt = lt |l0 U N C P LT |l0 , lt <
t