models of polyclonal tumor initiation in the mouse intestine
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Models of Polyclonal Tumor Initiation in the Mouse Intestine
Shuang HuangDec. 7th
Fall 2007 Shapiro FellowshipUnder Prof. M. A. Newton and Dr. R. Halberg
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Tumor origin: Monoclonal vs Polyclonal
l Monoclonal- Single initiation event- Normal cell converts to a cancer state- All cells in tumor are descendents of this progenitor.
l Polyclonal- Multiple initiation events contributing to the tumor
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Chimera [Thliveris et al 2005]
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Heterotypic Tumor
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Heterotypic Tumor
l
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My Project
l Develop statistical models for the probability of heterotypic or pure tumors.
l Frequencies of heterotypic (H), blue (B), or white (W) tumors are observable.
l Using such multinomial data, we could compare different models for P(H), P(B), P(W).
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Formation of Polyclonal Tumor
l Two models:RecruitmentSelection
l Data: images showing the chimeric patchwork in mouse intestine.
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Example binary imagel Represents a small section in intestine:
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Recruitment
l Single initiation event at position Xl X may be nonuniformly distributed ~ fl Recruitment distanceδl Model: all cells in
are converted to tumor.(Normal) à (Initiation) à (Recruitment)
( ) { : }disc x y y xδ δ= − ≤
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origin of pure tumor
origin ofheterotypic
tumor
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Recruitment
Events:
Problem: Compute P(B), P(W) and P(H).
c
B=[ Tumor is pure blue ]W=[ Tumor is pure white ]H=(B W)∪
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Recruitment
l DefineThe green part below is :
Obviously, is the part of blue.Similarly, we can define:
( ) { : cells in disc ( ) are pure blue}Bd x xδδ =( )Bd δ
( ) { : cells in disc ( ) are pure white}Wd x xδδ =
(0)Bd
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Computation
l Setl The position X has pdf:l To be a pdf, it requires: l Then
( ) 1{ }c x x is blue= ( ) 0( )
( ) 1c x
f xc x
αβ
== =
( ) ( )1{ ( )} ( ( ))
( ) ( )1{ ( )} ( ( ))
( ) 1 ( ) ( )
B B
W W
P B f x x d dx Area d
P W f x x d dx Area d
P H P B P W
δ β δ
δ α δ
= ∈ = ⋅
= ∈ = ⋅
= − −
∫∫
( (0)) ( (0)) 1W BArea d Area dα β⋅ + ⋅ =
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Computation
( ) 1 ( ) ( )1 (1 ( )) ( (0))
(1 ( )) ( (0))1 ( (0)) ( (0))
( (0)) ( ) ( (0)) ( )( (0)) ( ) ( (0)) ( )
b B
w W
B W
B b W W
B b W W
P H P B P WF Area d
F Area dArea d Area d
Area d F Area d FArea d F Area d F
α δ
β δ
α β
α δ β δ
α δ β δ
= − −= − −
− −
= − −
+ +
= ⋅ ⋅ + ⋅ ⋅
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Result
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Selection
l Multiple initiation eventsl e.g. n=2, at position X,Y ~ fl Conditional onl Model: all cells in
are converted to tumor.(Normal) à (Initiation) à (Selection)
{ }E Y X δ= − =
/ 2 (( ) / 2)disc X Yδ +
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Discussion – Future work
l Combine information of different images.
l Develop models further.
l Compare the prediction of these 2 models in multiple biological context.
l What does the calculation tell us about biology.
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Reference
[1] Andrew T. Thliveris, Richard B.Halberg, Linda Clipson, William F. Dove, Ruth Sullivan, Mary Kay Washington, Stephen Stanhope, and Michael A. Newton. Polyclonality of familial murine adenomas: Analyses of mouse chimeras with low tumor multiplicity suggest short-range interactions. PNAS May10, 2005. vol.102, no.19
[2] Michael A. Newton. On estimating the polyclonal fraction in lineage-marker studies of tumor origin. Biostatistics(2006), 7,4, pp. 503-514
[3] Michael A. Newton, Linda Clipson, Andrew T. Thliveris, and Richard B. Halberg, A Statistical Test of the Hypothesis that Polyclonal Intestinal Tumors Arise by Random Collision of Initiated Clones. Biometrics.