evaluation of screening strategies for pre-malignant ...€¦ · evaluation of screening strategies...
TRANSCRIPT
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Evaluation of Screening Strategies forPre-malignant Lesions using aBiomathematical Approach
Mathematical Modelling Approaches for Cancer MortalityProf. Christina Kuttler, Cristoforo Simonetto, Noemi Castelletti
June 28, 2018
Lukas Köstler
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Motivation
Biological Model
Mathematical Model
TSCE Model
MSCE Model
Simulation
Results
1
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Motivation
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Colorectal Cancer (CRC)
• Estimated deaths in the US in 2018: 50 6301
• Colonoscopies offer a method for screening andintervention
• Individuals often asymptomatic⇒ a biomathematical model can help to choose good
screening strategies
1National Cancer Institute. Cancer Stat Facts: Colorectal Cancer. 2018. url:https://seer.cancer.gov/statfacts/html/colorect.html.
2
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Colorectal Cancer (CRC)
• Estimated deaths in the US in 2018: 50 6301
• Colonoscopies offer a method for screening andintervention
• Individuals often asymptomatic
⇒ a biomathematical model can help to choose goodscreening strategies
1National Cancer Institute. Cancer Stat Facts: Colorectal Cancer. 2018. url:https://seer.cancer.gov/statfacts/html/colorect.html.
2
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Colorectal Cancer (CRC)
• Estimated deaths in the US in 2018: 50 6301
• Colonoscopies offer a method for screening andintervention
• Individuals often asymptomatic⇒ a biomathematical model can help to choose good
screening strategies
1National Cancer Institute. Cancer Stat Facts: Colorectal Cancer. 2018. url:https://seer.cancer.gov/statfacts/html/colorect.html.
2
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(a) Overview [2]. (b) Removal of polyp [6].
Figure 1: Colonoscopy: screening and intervention.
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Biological Model
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Colorectal Cancer Model
• Luebeck & Moolgavkar propose a 4 stage MSCE model [5]• APC gene is a cancer suppressor• Two mutations and one positional effect lead to clonalexpansion
Figure 2: Schematic representation of the carcinogenesis model [4].
4
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Colorectal Cancer Model
• Luebeck & Moolgavkar propose a 4 stage MSCE model [5]• APC gene is a cancer suppressor• Two mutations and one positional effect lead to clonalexpansion
Figure 2: Schematic representation of the carcinogenesis model [4].
4
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Colorectal Cancer Model
• Luebeck & Moolgavkar propose a 4 stage MSCE model [5]• APC gene is a cancer suppressor• Two mutations and one positional effect lead to clonalexpansion
Figure 2: Schematic representation of the carcinogenesis model [4].
4
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Colorectal Cancer Model
• Luebeck & Moolgavkar propose a 4 stage MSCE model [5]• APC gene is a cancer suppressor• Two mutations and one positional effect lead to clonalexpansion
Figure 2: Schematic representation of the carcinogenesis model [4].
4
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Mathematical Model
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Goals
• At screening we only consider individuals withoutmalignant cells
• To evaluate the effect of screening, the size distribution ofpolyps should be known/simulatable
• Need to evaluate hazard/survival function after screeningand different possible interventions, e.g. (in)completeremoval of polyps
⇒ Different screening strategies can be compared againsteach other
5
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Goals
• At screening we only consider individuals withoutmalignant cells
• To evaluate the effect of screening, the size distribution ofpolyps should be known/simulatable
• Need to evaluate hazard/survival function after screeningand different possible interventions, e.g. (in)completeremoval of polyps
⇒ Different screening strategies can be compared againsteach other
5
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Goals
• At screening we only consider individuals withoutmalignant cells
• To evaluate the effect of screening, the size distribution ofpolyps should be known/simulatable
• Need to evaluate hazard/survival function after screeningand different possible interventions, e.g. (in)completeremoval of polyps
⇒ Different screening strategies can be compared againsteach other
5
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Goals
• At screening we only consider individuals withoutmalignant cells
• To evaluate the effect of screening, the size distribution ofpolyps should be known/simulatable
• Need to evaluate hazard/survival function after screeningand different possible interventions, e.g. (in)completeremoval of polyps
⇒ Different screening strategies can be compared againsteach other
5
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Goals
• At screening we only consider individuals withoutmalignant cells
• To evaluate the effect of screening, the size distribution ofpolyps should be known/simulatable
• Need to evaluate hazard/survival function after screeningand different possible interventions, e.g. (in)completeremoval of polyps
⇒ Different screening strategies can be compared againsteach other
5
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Mathematical Model
TSCE Model
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Definitions
Clone All pre-malignant cells that are produced through abirth-death process from one initiated cell. Size Y (u, t),initiation time u.
Polyp All clones that derive from the same APC-/- progenitorcell. Size Y (t).
Z (t) Indicator for clinical cancer, i.e. at least one malignantcell. Z (t) ∈ {0, 1}.
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Definitions
Clone All pre-malignant cells that are produced through abirth-death process from one initiated cell. Size Y (u, t),initiation time u.
Polyp All clones that derive from the same APC-/- progenitorcell. Size Y (t).
Z (t) Indicator for clinical cancer, i.e. at least one malignantcell. Z (t) ∈ {0, 1}.
6
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Definitions
Clone All pre-malignant cells that are produced through abirth-death process from one initiated cell. Size Y (u, t),initiation time u.
Polyp All clones that derive from the same APC-/- progenitorcell. Size Y (t).
Z (t) Indicator for clinical cancer, i.e. at least one malignantcell. Z (t) ∈ {0, 1}.
6
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Clone: Conditional Size Distribution
P∗ [Y (u, t) = n] = Pr [Y (u, t) = n|Z (u, t) = 0, Y (u,u) = 1]
=
ξ (α+ p) (α+ q)
(qe−p(t−u) − pe−q(t−u))
q (α+ p) e−p(t−u) − p (α+ q) e−q(t−u) , n = 0
(1− P∗ [Y (u, t) = 0]) (1− αζ) (αζ)n−1 , n ≥ 1
ξ =e−p(t−u) − e−q(t−u)
(q+ α) e−p(t−u) − (p+ α) e−q(t−u)
{pq
}=
12(− α+ β + µ
{−+
}√(α+ β + µ)2 − 4αβ
)
7
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From Clones to Polyps
The size of a polyp is the sum over the sizes of its clones:
Y (t) =M(t)∑j=1
Y(uj, t
)(1)
where u1, . . . ,uM(t) are the initiation event times of clones.They follow a Poisson process with rate ρ (u) X (u).Remark: The derivations are valid for any positive X. For thispresentation we consider the case of a single APC− /−progenitor cell, i.e. X ≡ 1.
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From Clones to Polyps
The size of a polyp is the sum over the sizes of its clones:
Y (t) =M(t)∑j=1
Y(uj, t
)(1)
where u1, . . . ,uM(t) are the initiation event times of clones.They follow a Poisson process with rate ρ (u) X (u).
Remark: The derivations are valid for any positive X. For thispresentation we consider the case of a single APC− /−progenitor cell, i.e. X ≡ 1.
8
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From Clones to Polyps
The size of a polyp is the sum over the sizes of its clones:
Y (t) =M(t)∑j=1
Y(uj, t
)(1)
where u1, . . . ,uM(t) are the initiation event times of clones.They follow a Poisson process with rate ρ (u) X (u).Remark: The derivations are valid for any positive X. For thispresentation we consider the case of a single APC− /−progenitor cell, i.e. X ≡ 1.
8
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Conditional Polyp Size Distribution
Theorem (1)For n ≥ 0, and Z (t) the indicator for clinical cancer at time t,the size distribution for the number of polyp cells at time tconditioned on no clinical cancer is given by
Pr [Y (t) = n|Z (t) = 0, Y (0) = 0] =Γ (ρX/α+ n)
Γ (n+ 1) Γ (ρX/α) (1− αζ)ρXα (αζ)n .
This is the negative binomial distribution with parametersr = ρX/α and success probability p = 1− αζ .
Remark I: Because the size distribution follows a known,parametric distribution, generating samples, evaluating thePMF, etc. is computationally cheap.Remark II: This is Theorem 1 and Corollary 1 & 2 in [4].
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Conditional Polyp Size Distribution
Theorem (1)For n ≥ 0, and Z (t) the indicator for clinical cancer at time t,the size distribution for the number of polyp cells at time tconditioned on no clinical cancer is given by
Pr [Y (t) = n|Z (t) = 0, Y (0) = 0] =Γ (ρX/α+ n)
Γ (n+ 1) Γ (ρX/α) (1− αζ)ρXα (αζ)n .
This is the negative binomial distribution with parametersr = ρX/α and success probability p = 1− αζ .
Remark I: Because the size distribution follows a known,parametric distribution, generating samples, evaluating thePMF, etc. is computationally cheap.
Remark II: This is Theorem 1 and Corollary 1 & 2 in [4].
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Conditional Polyp Size Distribution
Theorem (1)For n ≥ 0, and Z (t) the indicator for clinical cancer at time t,the size distribution for the number of polyp cells at time tconditioned on no clinical cancer is given by
Pr [Y (t) = n|Z (t) = 0, Y (0) = 0] =Γ (ρX/α+ n)
Γ (n+ 1) Γ (ρX/α) (1− αζ)ρXα (αζ)n .
This is the negative binomial distribution with parametersr = ρX/α and success probability p = 1− αζ .
Remark I: Because the size distribution follows a known,parametric distribution, generating samples, evaluating thePMF, etc. is computationally cheap.Remark II: This is Theorem 1 and Corollary 1 & 2 in [4].
9
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Mathematical Model
MSCE Model
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Goals
• Generalize the results for the TSCE model (Theorem 1) tothe MSCE model
• It should be possible to efficiently sample from theresulting distribution
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Goals
• Generalize the results for the TSCE model (Theorem 1) tothe MSCE model
• It should be possible to efficiently sample from theresulting distribution
10
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Definitions
Let X (t) be the number of normal cells, Y1 (t) , . . . , Yk−2 (t) bethe number of cells in pre-initiation stages, Yk−1 (t) be thetotal number of polyp cells and Yk (t) be the indicator forclinical cancer.
Figure 3: Schematic representation of the MSCE model [4].
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Conditional Size Distribution I
Theorem (2)Let ϕ∗ (y;u, t) be the PGF of the size of a clone born at u ≤ t.Let Ψ∗ (y1, . . . , yk−2, y; t) be the joint PGF of the number of cellsin each stage, conditioned on no clinical cancer at t, then
Ψ∗ (1, . . . , 1, y; t) =
exp[ ∫ t
0µ0 (u1) X (u1) Sk−1 (t− u1)
(exp
[ ∫ t
u1
µ1 (u2) Sk−2 (t− u2)(. . .
(exp
[ ∫ t
uk−2
µk−2 (uk−1) Sk−2 (t− uk−1) (ϕ∗ (y;uk−1, t)− 1)duk−1
]− 1
). . .
−1)du2
]− 1
)du1
]
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Conditional Size Distribution II
Theorem 2 shows that the MSCE model conditioned on noclinical cancer at time t is equivalent to an unconditional MSCEmodel with rates
µ0 (u1) Sk−1 (t− u1) X (u1) ,
µ1 (u2) Sk−2 (t− u2) ,
...µk−2 (uk−1) S1 (t− uk−1) .
Sk−1 (t− u) is the survival function of k− 1 stage MSCE modelstarting with one cell in the first pre-initiation stage at time u,i.e.
X (u) = 0, Y1 (u) = 1, Y2 (u) = 0, . . . Yk (u) = 0 .
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Figure 4: Schematic representation of the carcinogenesis model [4].
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Simulation
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Steps
• By Theorem 2 we know that we have to simulate a k-stage(4 for the example) MSCE model with modified rates
• We simulate non-homogeneous Poisson processes up tothe last pre-initiation stage and then use Theorem 1
• Requirements:• Evaluate survival functions Sk efficiently: ∼ O
(107
)evaluations of S3
• Simulate non-homogeneous Poisson process• Draw samples from negative binomial distribution• Simulate screening/intervention• Calculate hazard functions after screening
15
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Steps
• By Theorem 2 we know that we have to simulate a k-stage(4 for the example) MSCE model with modified rates
• We simulate non-homogeneous Poisson processes up tothe last pre-initiation stage and then use Theorem 1
• Requirements:
• Evaluate survival functions Sk efficiently: ∼ O(107
)evaluations of S3
• Simulate non-homogeneous Poisson process• Draw samples from negative binomial distribution• Simulate screening/intervention• Calculate hazard functions after screening
15
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Steps
• By Theorem 2 we know that we have to simulate a k-stage(4 for the example) MSCE model with modified rates
• We simulate non-homogeneous Poisson processes up tothe last pre-initiation stage and then use Theorem 1
• Requirements:• Evaluate survival functions Sk efficiently: ∼ O
(107
)evaluations of S3
• Simulate non-homogeneous Poisson process• Draw samples from negative binomial distribution• Simulate screening/intervention• Calculate hazard functions after screening
15
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Steps
• By Theorem 2 we know that we have to simulate a k-stage(4 for the example) MSCE model with modified rates
• We simulate non-homogeneous Poisson processes up tothe last pre-initiation stage and then use Theorem 1
• Requirements:• Evaluate survival functions Sk efficiently: ∼ O
(107
)evaluations of S3
• Simulate non-homogeneous Poisson process
• Draw samples from negative binomial distribution• Simulate screening/intervention• Calculate hazard functions after screening
15
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Steps
• By Theorem 2 we know that we have to simulate a k-stage(4 for the example) MSCE model with modified rates
• We simulate non-homogeneous Poisson processes up tothe last pre-initiation stage and then use Theorem 1
• Requirements:• Evaluate survival functions Sk efficiently: ∼ O
(107
)evaluations of S3
• Simulate non-homogeneous Poisson process• Draw samples from negative binomial distribution
• Simulate screening/intervention• Calculate hazard functions after screening
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Steps
• By Theorem 2 we know that we have to simulate a k-stage(4 for the example) MSCE model with modified rates
• We simulate non-homogeneous Poisson processes up tothe last pre-initiation stage and then use Theorem 1
• Requirements:• Evaluate survival functions Sk efficiently: ∼ O
(107
)evaluations of S3
• Simulate non-homogeneous Poisson process• Draw samples from negative binomial distribution• Simulate screening/intervention
• Calculate hazard functions after screening
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Steps
• By Theorem 2 we know that we have to simulate a k-stage(4 for the example) MSCE model with modified rates
• We simulate non-homogeneous Poisson processes up tothe last pre-initiation stage and then use Theorem 1
• Requirements:• Evaluate survival functions Sk efficiently: ∼ O
(107
)evaluations of S3
• Simulate non-homogeneous Poisson process• Draw samples from negative binomial distribution• Simulate screening/intervention• Calculate hazard functions after screening
15
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Survival Functions I
The formulas for the survival functions are:
Sk (t) = exp[ ∫ t
0µ0
(exp
[ ∫ t
u1
µ1(. . .
(exp
[ ∫ t
uk−3
µk−3 (S2 (t− uk−2)− 1)duk−2]− 1
)· · · − 1
)du2
]− 1
)du1
]
S2 (t) =(
q− pqe−pt − pe−qt
)µk−2/α
S1 (t) = 1+ 1α
pq(e−pt − e−qt)
qe−pt − pe−qt
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Survival Functions II
0 10 20 30 40 50
0.999
0.9992
0.9994
0.9996
0.9998
1
Figure 5: Survival functions for 0 ≤ t ≤ 50.
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Survival Functions III
• Multiple evaluations at t1 < t2:∫ t2 =
∫ t1 +∫ t2t1
• When useful use log Sk, Sk − 1, log1p and exp1m• Use cheap but accurate approximation to S3
⇒ Chebyshev polynomials using chebfun1 toolbox
1T. A Driscoll, N. Hale, and L. N. Trefethen. Chebfun Guide. PafnutyPublications, 2014. url: http://www.chebfun.org/docs/guide/
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Survival Functions III
• Multiple evaluations at t1 < t2:∫ t2 =
∫ t1 +∫ t2t1
• When useful use log Sk, Sk − 1, log1p and exp1m• Use cheap but accurate approximation to S3
⇒ Chebyshev polynomials using chebfun1 toolbox
1T. A Driscoll, N. Hale, and L. N. Trefethen. Chebfun Guide. PafnutyPublications, 2014. url: http://www.chebfun.org/docs/guide/
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Survival Functions IV
0 10 20 30 40 50
-8
-6
-4
-2
010
-14
• Maximum Error 2.2 10−16 = eps (1), i.e. accurate tomachine precision
• Evaluation takes O(10−4) s for the direct method and
O(10−7) s for the approximation
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Survival Functions IV
0 10 20 30 40 50
-8
-6
-4
-2
010
-14
• Maximum Error 2.2 10−16 = eps (1), i.e. accurate tomachine precision
• Evaluation takes O(10−4) s for the direct method and
O(10−7) s for the approximation 19
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Non-homogeneous Poisson Process & Negative Binomial Distri-bution
Non-homogeneous Poisson Process
• Standard Problem• One possible method: Thinning, i.e. rejection sampling• Simulate a homogeneous Poisson process with rateλ∞ ≥ ||λ (t)||∞ and accept each occurrence tj withprobability
λ(tj)
λ∞.
Negative Binomial distribution
• Standard Problem• Use MATLAB’s built in methods
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Intervention Methods I
• Perform simulation for i = 1, . . . ,N = 104
individuals/samples• Before screening/intervention at time σ−
Number healthy cells XNumber APC+/- cells N−
2
Number APC-/- cells N−3
Polyp size set N−4
Number polyp cells N−4 =
∣∣N−4∣∣
• After screening/intervention Ai ={X,N+
2i ,N+3i ,N
+4i}
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Intervention Methods I
• Perform simulation for i = 1, . . . ,N = 104
individuals/samples• Before screening/intervention at time σ−
Number healthy cells XNumber APC+/- cells N−
2
Number APC-/- cells N−3
Polyp size set N−4
Number polyp cells N−4 =
∣∣N−4∣∣
• After screening/intervention Ai ={X,N+
2i ,N+3i ,N
+4i}
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Intervention Methods II
Method Description ExampleComplete Remove all polyps
above threshold andassociated APC-/- cells.
N+4 = {5000}
Incomplete Remove all polyps abovethreshold and leaveAPC-/- progenitor cells.
N+4 = {5000, 0}
Realistic Decrease polyp size to10% of threshold andleave APC-/- cells.
N+4 = {5000, 1000}
Table 1: Intervention Methods. For the example: N−3 = 2,
N−4 = {5000, 20000} and the threshold is 104. N+
3 =∣∣N+
4∣∣
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Intervention Methods II
Method Description ExampleComplete Remove all polyps
above threshold andassociated APC-/- cells.
N+4 = {5000}
Incomplete Remove all polyps abovethreshold and leaveAPC-/- progenitor cells.
N+4 = {5000, 0}
Realistic Decrease polyp size to10% of threshold andleave APC-/- cells.
N+4 = {5000, 1000}
Table 1: Intervention Methods. For the example: N−3 = 2,
N−4 = {5000, 20000} and the threshold is 104. N+
3 =∣∣N+
4∣∣
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Intervention Methods II
Method Description ExampleComplete Remove all polyps
above threshold andassociated APC-/- cells.
N+4 = {5000}
Incomplete Remove all polyps abovethreshold and leaveAPC-/- progenitor cells.
N+4 = {5000, 0}
Realistic Decrease polyp size to10% of threshold andleave APC-/- cells.
N+4 = {5000, 1000}
Table 1: Intervention Methods. For the example: N−3 = 2,
N−4 = {5000, 20000} and the threshold is 104. N+
3 =∣∣N+
4∣∣
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After screening
• Separate N = 104 samples into groups, e.g. negativescreen with threshold 103, positive screen with threshold104
• For each group, the average survival and hazard functionsare:
S (t− σ|Ai) = S4 (t− σ)X S3 (t− σ)N+2i S2 (t− σ)N
+3i S1 (t− σ)N
+4i
h (t− σ|Ai) = Xh4 (t− σ) + N+2ih3 (t− σ)
+ N+3ih2 (t− σ) + N+
1ih4 (t− σ)
S (t− σ) ≈ 1N
N∑i=1
S (t− σ|Ai)
h (t− σ) ≈∑
j S (t− σ|Ai)h (t− σ|Ai)∑j S (t− σ|Ai)
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Parameters
The paper [4] uses the following parameters for the simulation:
α = 9X = 108
p = −1.519930× 10−1
q = 3.893446× 10−6
µ0 = µ1 = 1.364459× 10−6
ρ = 6.886327× α
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Parameters
The paper [4] uses the following parameters for the simulation:
α = 9X = 108
p = −1.519930× 10−1
q = 3.893446× 10−6
µ0 = µ1 = 1.364459× 10−6
ρ = 6.886327× α
From literature.
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Parameters
The paper [4] uses the following parameters for the simulation:
α = 9X = 108
p = −1.519930× 10−1
q = 3.893446× 10−6
µ0 = µ1 = 1.364459× 10−6
ρ = 6.886327× α
Estimated from SEER data: white males (1973-2000) [5].
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Results
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APC-/- progenitor cells
Num. APC-/- cells Count Percent0 7893 78.93 %1 1886 18.86 %2 204 2.04 %3 16 0.16 %4 1 0.01 %
Table 2: Distribution of the number of APC-/- progenitor cells forN = 10′000 at age σ = 50 years.
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Polyp size distribution
100
101
102
103
104
105
106
0
0.05
0.1
0.15
0.2
Figure 6: Size distribution of polyps at age σ = 50 years forN = 10′000. Note that one individual might have multiple polyps. 26
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Negative Groups
50 60 70 80 90 10010
-6
10-5
10-4
10-3
10-2
10-1
Figure 7: Hazard after screening at age σ = 50 for negative screeninggroups. Sample size N = 10′000. 27
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Negative vs. Positive Group
50 60 70 80 90 100
10-4
10-3
10-2
10-1
100
101
Figure 8: Hazard after screening at age σ = 50. N = 10′000.
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Positive Groups with complete Intervention
50 60 70 80 90 10010
-6
10-5
10-4
10-3
10-2
10-1
Figure 9: Hazard after screening at age σ = 50 with completeintervention. N = 10′000. 29
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Positive Groups with incomplete Intervention
50 60 70 80 90 10010
-6
10-5
10-4
10-3
10-2
10-1
Figure 10: Hazard after screening at age σ = 50 with incompleteintervention. N = 10′000. 30
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Lifetime CRC Risk
Table 3: Lifetime colorectal cancer (CRC) risk at 80 years for differentscenarios. Screening at σ = 50 years. Sample size N = 10′000.
Scenario Threshold Lifetime RiskBackground 6.57 %
Neg. Screen 105 4.99 %Neg. Screen 104 1.25 %Neg. Screen 103 0.11 %Pos. Screen 105 99.72 %Pos. Screen 104 75.22 %Pos. Screen 103 44.79 %Realistic Intervention 104 → 103 5.36 %Complete Intervention 104 → 0 1.27 %
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Lifetime CRC Risk
Table 3: Lifetime colorectal cancer (CRC) risk at 80 years for differentscenarios. Screening at σ = 50 years. Sample size N = 10′000.
Scenario Threshold Lifetime RiskBackground 6.57 %Neg. Screen 105 4.99 %Neg. Screen 104 1.25 %Neg. Screen 103 0.11 %
Pos. Screen 105 99.72 %Pos. Screen 104 75.22 %Pos. Screen 103 44.79 %Realistic Intervention 104 → 103 5.36 %Complete Intervention 104 → 0 1.27 %
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Lifetime CRC Risk
Table 3: Lifetime colorectal cancer (CRC) risk at 80 years for differentscenarios. Screening at σ = 50 years. Sample size N = 10′000.
Scenario Threshold Lifetime RiskBackground 6.57 %Neg. Screen 105 4.99 %Neg. Screen 104 1.25 %Neg. Screen 103 0.11 %Pos. Screen 105 99.72 %Pos. Screen 104 75.22 %Pos. Screen 103 44.79 %
Realistic Intervention 104 → 103 5.36 %Complete Intervention 104 → 0 1.27 %
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Lifetime CRC Risk
Table 3: Lifetime colorectal cancer (CRC) risk at 80 years for differentscenarios. Screening at σ = 50 years. Sample size N = 10′000.
Scenario Threshold Lifetime RiskBackground 6.57 %Neg. Screen 105 4.99 %Neg. Screen 104 1.25 %Neg. Screen 103 0.11 %Pos. Screen 105 99.72 %Pos. Screen 104 75.22 %Pos. Screen 103 44.79 %Realistic Intervention 104 → 103 5.36 %Complete Intervention 104 → 0 1.27 %
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References
[1] T. A Driscoll, N. Hale, and L. N. Trefethen. Chebfun Guide. Pafnuty Publications,2014. url: http://www.chebfun.org/docs/guide/.
[2] wikipedia Euchiasmuse. Colonoscopia. 2018. url:https://en.wikipedia.org/wiki/Colonoscopy#/media/File:Colonoscopia.jpg.
[3] National Cancer Institute. Cancer Stat Facts: Colorectal Cancer. 2018. url:https://seer.cancer.gov/statfacts/html/colorect.html.
[4] Jihyoun Jeon et al. “Evaluation of screening strategies for pre-malignant lesionsusing a biomathematical approach”. In: Mathematical biosciences 213.1 (2008),pp. 56–70.
[5] E Georg Luebeck and Suresh H Moolgavkar. “Multistage carcinogenesis and theincidence of colorectal cancer”. In: Proceedings of the National Academy ofSciences 99.23 (2002), pp. 15095–15100.
32
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[6] medicinenet. colonoscopy. 2018. url: https://www.medicinenet.com/colonoscopy/article.htm#whats_new_in_colonoscopy.
[7] Emanuel Parzen. Stochastic processes. SIAM, 1999.
33