Evaluation of Screening Strategies for Pre-malignant ...€¦ · Evaluation of Screening Strategies...

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

Motivation

Biological Model

Mathematical Model

TSCE Model

MSCE Model

Simulation

Results

1

Motivation

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

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

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

(a) Overview [2]. (b) Removal of polyp [6].

Figure 1: Colonoscopy: screening and intervention.

3

Biological Model

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

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

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

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

Mathematical Model

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

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

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

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

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

Mathematical Model

TSCE Model

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

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

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

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

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

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

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

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

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

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

Mathematical Model

MSCE Model

Goals

• Generalize the results for the TSCE model (Theorem 1) tothe MSCE model

• It should be possible to efficiently sample from theresulting distribution

10

Goals

• Generalize the results for the TSCE model (Theorem 1) tothe MSCE model

• It should be possible to efficiently sample from theresulting distribution

10

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].

11

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

]

12

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 .

13

Figure 4: Schematic representation of the carcinogenesis model [4].

14

Simulation

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

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

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

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

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

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

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

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

16

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.

17

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/

18

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/

18

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

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

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

20

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}

21

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}

21

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∣∣

22

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∣∣

22

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∣∣

22

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)

23

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

24

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.

24

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].

24

Results

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.

25

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

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

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.

28

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

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

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 %

31

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 %

31

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 %

31

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 %

31

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

[6] medicinenet. colonoscopy. 2018. url: https://www.medicinenet.com/colonoscopy/article.htm#whats_new_in_colonoscopy.

[7] Emanuel Parzen. Stochastic processes. SIAM, 1999.

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