# Decision Programming for optimizing the Finnish Colorectal Cancer Screening
Program

Lauri Neuvonen<EMAIL_ADDRESS>Mary Dillon  Eeva Vilkkumaa  Ahti Salo
Maija Jäntti  Sirpa Heinävaara  Department of Information and Service
Management, Aalto University School of Business, Finland  Department of
Mathematics and Systems Analysis, Aalto University School of Science, Finland
The Finnish Cancer Registry

###### Abstract

In Finland colorectal cancer (CRC) incidence rates have steadily increased
over the last decades and as of 2017, CRC is the sixth most common cause of
death. CRC is a crucial concern for the public health of Finland. We optimize
the faecal immunochemical test (FIT) cut-off level for specified target
populations with regard to minimization of cancer prevalence with Decision
Programming, which is a novel approach to solving discrete multi-stage
decision problems under uncertainty. The results present optimal cut-off
levels for Finnish target groups with different colonoscopy capacity
constraints. Finally, the estimated resulting cancer prevalence, the amount of
required colonoscopies and third-party payer costs resulting from found
strategies to the newly implemented Finnish CRC Screening Programme, which
have not previously been published, are presented.

###### keywords:

Decision Programming , Optimization , Colorectal Cancer , Cancer Screening

††journal: European Journal of Operational Research

##  1 Introduction

Colorectal cancer (CRC), also referred to as bowel cancer, is cancer of the
colon and/or rectum. CRC incidence rates have steadily increased over the last
decades with CRC currently being the second most common cancer in adults in
Finland, and the third most common worldwide (Finnish Cancer Registry 2017,
World Health Organization 2018). As of 2017, CRC is the sixth most common
cause of death, whereby it is clearly a crucial concern to public health.

Over 70% of colorectal cancers develop via the adenoma-carcinoma sequence
(Hardy et al. 2000) in which adenomas (i.e., growths on the epithelial tissue)
develop into cancer. This sequence is a slow process that can take from
several years to a decade (Simon 2016). Due to this, early detection and
removal of pre-cancerous adenomas can prevent progression to cancer, making
CRC extremely suitable for population screening. Population screening of CRC
using faecal occult blood testing has long been shown to reduce later stage
CRC incidence and improve mortality rates (Mandel et al. 1993, Kewenter et al.
1994, Kronborg et al. 1996, Mandel et al. 1999). Along with early detection of
adenomas, early diagnosis of cancer improves the probability of successful
treatments and outcomes by providing care at the earliest possible time. This
has an important impact on public health strategy as it helps avoid CRC-
attributed deaths and disabilities, in addition to high treatment and indirect
costs associated with advanced cancer stages (World Health Organization 2017).

In April 2019, Finland established a new CRC screening programme in
volunteering municipalities. The programme will become nationwide in 2022, and
by 2031 all people aged between 56 and 74 will be invited to screen. In the
new programme, the feacal immunochemical test (FIT) replaces the previously
used guaiac-based faecal occult blood test (gFOBT), as the European guidelines
for quality assurance in CRC screening recommend FITs over gFOBTs due to
higher sensitivity levels and increased participation rates (Halloran et al.
2012, Allison et al. 2014, Bretagne et al. 2019). The programme invites entire
age cohorts to screen using a FIT, after which those participants with a
positive FIT result are invited to a colonoscopy.

In the current programme, the FIT cut-off level (hemoglobin level in stool
sample) for a positive result is fixed based on the participant’s sex. In
particular, the Finnish programme uses a cut-off level of 25 mg or ng/g for
females and 70 mg or ng/g for males, regardless of their age. In reality,
however, the interpretation of the result should be different for participants
of different risk profiles (i.e., age, sex, and family history of cancer).
Moreover, because a positive result leads to an invitation to a colonoscopy,
the cut-off level should also take into account the capacity of the
participating municipality to carry out colonoscopies. Consequently, the
programme may not be the most cost-effective approach to reduce Finnish
colorectal cancer incidence and mortality.

Traditionally, cost-effectiveness evaluations of public health programmes,
such as population screening, are assessed through methods of cost-benefit
analysis (e.g. Ellison et al. 2002), cost-utility analysis (e.g. Gupta et al.
2011), or cost-effectiveness analysis (e.g. Dillon et al. 2018). Health
economic methodologies like these typically compare costs and health outcomes
to a baseline strategy (for example, no screening). Quality of life
indicators, such as quality-adjusted life-years (QALYs) and life-years
saved/gained (LYS/LYG), are frequently used to calculate the incremental cost-
effectiveness ratio (ICER) for comparing alternative strategies. In these
analyses, strategies are not optimized in the mathematical sense, but rather
assessed in regard to their dominance. Consequently, such analyses tend to
suggest strategies which are either unaffordable due to a lack of budget
constraint or suboptimal in that (i) the resources could be reallocated to
achieve a better population-level health outcome or (ii) the same health
outcome could be attained with less resources.

To identify optimal strategies within cost-effectiveness analyses, decision
analytic models such as decision trees have been employed (e.g., Hynninen et
al. 2019, Hynninen et al. 2021). These models are typically static in that
they do not account for possible changes in a participant’s state of health
over time, whereby they cannot be readily applied to support the optimization
of screening programmes. To capture time-dependence in a participant’s state
of health, several models utilizing Markov Decision Processes (MDPs) have been
proposed. In MDP models, the participant is assumed to move from one state to
another (e.g., from being healthy to having an adenoma or cancer) with state
transition probabilities that can be affected by carrying out preventive or
treatment actions. In Partially Observable Markov Decision Process (POMDP)
models in particular, the state of a participant may not be perfectly observed
but information regarding this state can be obtained through screening, for
instance. Ayer et al. (2012) build a POMDP model to optimize participant-
specific mammography screening times. Alagoz et al. (2013) provides a tutorial
in optimizing cancer screening using a POMDP approach and Erenay et al. (2014)
develop a finite-horizon POMDP model to determine optimal CRC screening
policies. Whereas, Cevik et al. (2018) present a constrained POMDP model to
study the optimal allocation of limited mammography resources to screen a
population. Lee et al. (2019) optimize the use of limited resources for the
screening of a population for hepatocellular carcinoma by modeling the problem
as a family of restless bandits in which each participant’s disease
progression is assumed to evolve as a POMDP.

POMDP approaches are very useful when the possible deterioration in a
participant’s state of health over time needs to be accounted for.
Nevertheless, POMDP studies often assume fixed interpretations of test results
(i.e., predetermined positivity cut-off levels). In particular, few of the
above studies recognize that the optimal interpretation of test results may
depend on how much resources there are. Moreover, the above studies employ
dynamic programming to solve the optimization problems. For this reason, the
models can only have a single objective such as maximizing the expected health
outcome or expected net benefit, wherein health outcomes are converted into
monetary terms using some parameter such as the willingness to pay threshold
(e.g., Chen et al. 2018). Importantly, focusing on expected outcomes fails to
accommodate risk considerations, which could be highly relevant in the context
of cancer screening.

In this paper, we optimize population screening of CRC in regard to minimizing
cancer prevalence, i.e. share of population with CRC, in the target population
with regard to the maximum number of colonoscopies that can be performed in a
year. Toward this end, we build a two-level optimization model. At the first
level, we find all Pareto optimal screening strategies for each segment of the
population, defined by the participants’ age and sex, maximizing the
probabilities of discovering a cancer or a large growth and minimizing the
probability of performing a colonoscopy. At the second level, we identify the
combination of segment-specific strategies that result in the highest expected
number of detected cancers, given a colonoscopy constraint. In building this
model, we employ Decision Programming (Salo et al. 2022), a novel approach to
solving discrete multi-stage decision problems under uncertainty. In contrast
to earlier approaches to optimizing screening programmes, Decision Programming
(i) finds all Pareto optimal solutions with respect to multiple objectives,
and (ii) permits the use of deterministic and probabilistic risk measures as
objectives or constraints in the model formulation. As a methodology it has
not been applied to the health care context.

The contribution of this paper to existing literature is twofold. First, we
propose defensible decision recommendations for developing the CRC screening
program in Finland. Second, we demonstrate the usefulness of the Decision
Programming methodology in health care context.

The paper is structured as follows: Section 2, Methods, including case study
and model description; Section 3 details the decision programming model
formulation; Section 4 presents the case study results; Section 5 discusses
the model, the results and the conclusions including strengths and limitations
of the method and future research directions.

##  2 Problem overview

The new colorectal cancer screening programme in Finland commenced in April
2019 (see Finnish Cancer Registry 2019 for details). In 2021 all 60-, 62-,
64-, 66- and 68-year-old males and females in volunteering municipalities are
invited to participate in the programme. Presently, twelve out of 311
municipalities in Finland have volunteered to take part in the programme. The
programme will become nationwide in 2022.

A central screening hub mails the screening invitations and FITs, analyzes the
samples, and mails the result letters to participants. The FIT is performed at
home and returned in a pre-paid envelope to the screening hub. Participation
in the screening programme is free of charge for the invitee. Once the
laboratory has analyzed the FIT sample, a written result is given to the
participant via mail. Those with a positive result are required to contact a
screening nurse in their municipality of residence to discuss the need for
further examination. This additional examination is usually a colonoscopy;
however, this may vary between municipalities depending on their standard
procedure. If a growth is found during the additional examination, a sample is
taken and sent to a pathologist for analysis. Once the pathologist’s findings
have been reviewed, a decision on the need for further examination or
treatment is made. If necessary, the participant is referred to surgery.
Treatment may be required at this stage. However, the treatment process is not
within the scope of the screening programme. After the screening period has
ended, i.e. two-years have passed since the invitation, a new invitation is
sent and the next screening period commences. Screening is continued in this
periodic manner as long as the invitee’s age is within the program limits.

The main decisions affecting the outcomes of the screening program include
what population segments are invited, what pre-screening and examination
methods are used. Depending on these selections, a certain number of
examinations will be performed, costs generated and health benefits gained.
However, there’s only a limited amount of colonoscopies that can be performed
in a year. This gives rise to the task of deriving maximum health benefits
from these resources, i.e. maximizing screening efficiency.

This paper develops a model to produce optimal screening strategies for the
Finnish colorectal cancer screening programme. The overall objective is to
minimize the prevalence of cancers while minimizing the total colonoscopies
performed. A screening strategy is defined as a combined selection of
decisions for all target population segments. For this problem, we include the
FIT cut-off level, whether to offer an incentive, and whether to send an
invite as decisions. In the Finnish programme, the FIT cut-off levels are
fixed at 70 $\mu$g/g or 410 ng/ml for males and 25 $\mu$g/g or 150 ng/ml for
females, (see Shaukat and Allen (2015) Table 4.3 for unit conversion). Please
note that the cutoff level stated by the manufacturer is 100 ng/ml (Shaukat
and Allen 2015), however the FIT can measure up to 1000 ng/ml (Sentinel 2021).
The inclusion of an incentive decision in our problem does not reflect the
reality of the programme; however, it may be an option in other countries with
similar programmes.

##  3 Optimisation of the screening programme.

The overall objective is to find a strategy, i.e. a combination of invitation,
FIT cut-off level and incentive decisions for each segment, that minimises
cancer prevalence in the target population with a limited colonoscopy
resource. We model the problem of finding such a strategy via two phases. We
first present an overview of the phases to create context, and later the
details in dedicated sections.

In the first phase, we generate the set of Pareto optimal strategies
$Z^{*}_{g}$ for each sex ($g$) for all periods ($k$), i.e. segments of the
population, that minimizes the expected number of colonoscopies, while
maximizing the expected number of detected growths and cancers in each
screening round. In the second phase, we combine the sex-specific strategies
to create an overall strategy $Z$ that minimizes the prevalence of cancer in
the screened population within a given resource constraint. Figure 1
illustrates our two phased approach.

Figure 1: Schematic description of the multistage optimization approach to
solve the CRC screening optimization problem.

The first phase begins with identifying the set of optimal screening
strategies for those in period $k=1$, i.e. aged 60, of separate sex. Each
period represents one screening round, therefore, in the following period
(i.e. two years later) optimal strategies for those aged 62 are added to the
set. This continues until the final period is reached and we have a set of
optimal screening strategies for both sexes and all ages included in the
screening programme. As the solution at $k$ is dependent on the prevalence
distribution produced by the preceding solution at $k-1$, the segment specific
solutions form a tree structure, each Pareto optimal solution creating a new
branch. We call a path through this tree, i.e. a collection of segment-
specific strategies, one for each period, a _sex-specific strategy_ $Z_{g}$.
From here, the second phase identifies a sex-specific strategy for all-aged
males and for all-aged females from the final set of sex-specific efficient
strategies. This latter phase is a simple selection problem. However, Phase 1
is a relatively complex optimization problem.

###  3.1 Main assumptions

We assume that the sex-specific strategies are stable in the sense that the
same strategy is applied to e.g. 62-year old males regardless of the year the
entered the screening programme. This means that the strategy applied to one
group of people during their whole screening programme is the same that is
applied to the different age groups within one screening period. This
reasonable assumption simplifies the problem of finding sex-specific
strategies significantly. The population segments are assumed to be internally
homogenous, i.e. there’s no difference between what is known about two 62-year
old males before they enter the program.

The transitions from different bowel to other states between beginnings of
periods are assumed to be linear. This means that during a period a portion of
people with normal bowels will develop benign growths, a portion of people
with benign growths will develop large adenomas etc. In addition, we assume
that all participants for whom a abnormal bowel state is identified, are
treated and their bowel state becomes normal. These rather strong assumptions
have been discussed with the Finnish cancer registry and can be deemed
acceptable for the purposes of this paper. However, more refined transition
models could be integrated into this modeling approach.

###  3.2 Segment specific model

The segment specific efficient strategies are generated by combining the
_Modified Augmented Weighted Tchebychev_ (MAWT) norm approach (Holzmann and
Smith 2018) with an Influence Diagram (ID) capturing the screening process for
one participant in a segment. The influence diagram has been modeled in
Decision Programming style (Salo et al. 2022) to allow inclusion of risk
measures. Combining these elements leads to the optimisation problem described
in Equations (6)-(19) that can be solved using the MAWT algorithm described in
Holzmann and Smith (2018). The differences between target segments are
reflected in the parameter values of the IDs. In particular, the probability
of a participant’s bowel to be in a certain state $b$ is the prevalence
$\psi_{g,k,b}$ of that bowel state in the target segment $(g,k)$ the
participant belongs to.

The Finnish screening programme described in the previous section can be
captured by an influence diagram, as shown in Figure 1. An influence diagram
is a discrete acyclic graph constructed of three types of nodes $\mathcal{N}$
represented by sets; decision $\mathcal{D}$, chance $\mathcal{C}$ and values
nodes $\mathcal{V}$, with dependencies shown by directed arcs
$\mathcal{A}\subseteq\\{(i,j)|i,j\in\mathcal{N},i\neq j\\}$. In Figure 1,
decisions in the screening pathway are represented by squares, uncertainties
(or chances) in the screening process are represented by circles, and the
values that are to be optimized are represented by diamonds. Every chance
$j\in\mathcal{C}$ and decision $j\in\mathcal{D}$ node has a finite set of
discrete states $s_{j}\in\mathcal{S}_{j}$. The state $s_{j}\in\mathcal{S}_{j}$
of a given node $j\in\mathcal{C}\cup\mathcal{D}$ represents a chance or
decision alternative, a full list of these alternatives for our problem are
presented below. An arc $(i,j)\in\mathcal{A}$, represented by an arrow,
indicates that node $i$ is the predecessor of node $j$, and that the state at
node $j$, $s_{j}$, is conditionally dependent on the state at the preceding
node, $s_{i}$.

In Figure 1, the decision nodes 1, 2, and 3 correspond to decisions about 1)
what FIT test cut-off value to use to select participants for a colonoscopy in
the given target segment, 2) whether to use an incentive to boost
participation rate among invitees in this segment (specifically, we assume
that an incentive worth 50 euros halves the number of non-returned samples)
and 3) whether to invite the target segment to the screening program. Let us
denote by $s_{j}\in\mathcal{S}_{j},j\in\mathcal{D}$ the alternatives for
decision $j$. Here, the sets $\mathcal{S}_{j}$ of such alternatives are
$\mathcal{S}_{2}=\mathcal{S}_{3}=\\{yes,no\\}$ and
$\mathcal{S}_{1}=\\{10,20,25,40,50\\}\ \mu$g Hg/g of blood in the stool
sample. Let $(g,k)$ with $g\in\mathcal{G}=\\{F,M\\},k\in\\{1,2,3,4,5\\}$ be
the target segment, where $F$ and $M$ refer to female and male, respectively,
and numbers $k$ to the screening period. These periods correspond to a
participant’s age so that period 1 refers to 60-year-olds, period 2 to
62-year-olds etc. The decisions about which alternatives $s_{j}$ (FIT cut-off
level, incentive and invitation) to select for each target segment are modeled
as binary decision variables $z_{g,k}(s_{j})\in\\{0,1\\}$ so that
$z_{g,k}(s_{j})=1$ if and only if alternative $s_{j}$ is selected for segment
$(g,k)$. A full screening strategy $Z$ is a collection of these decision
variables for all target segments and can be written as
$Z=\bigcup_{g,k}Z_{g,k}$ where the segment-specific strategies are defined as
$Z_{g,k}=\bigcup_{j\in D}z_{g,k}(s_{j}),\forall g,k$.

Chance nodes correspond to returning a FIT sample, FIT results, continued
participation, the discovery of polyps and polypectomy in a colonosopy, and
adverse effects from the colonoscopy. In particular, nodes 5 and 8 reveal
information about the participant’s bowel state. Here, we assume that the
bowel state $b\in\mathcal{B}=\\{\mathrm{N,B,L,R}\\}$ of a participant can be
Normal (N), Benign growth (B), Large growth (L) or CRC (R). These bowel states
are reflected at the population level by prevalences
$\psi_{g,k,b}=N_{g,k,b}/N_{g,k}$, where $N_{g,k,b}$ is the number of
participants in segment $(g,k)$ with bowel state $b$, and $N_{g,k}$ is the
total number of participants in segment $(g,k)$. The starting prevalences
$\psi_{F,1,b},\psi_{M,1,b},\forall b\in\mathcal{B}$ as well as the segment-
specific conditional probabilities for chance nodes are obtained from the
literature (please see the Supplementary material for details).

The progression of the prevalences of different bowel states are affected by
two factors: natural progression and screening. The natural progression of
colorectal cancer is reflected by transition probabilities
$\mathbb{T}_{b,b^{\prime}}^{g,k}$, i.e. the probability that the bowel state
of the participant of sex $g$ is $b^{\prime}$ in period $k+1$ given that at
$k$ it was $b$. We assume that this progression follows the adenoma-carcinoma
sequence, meaning the transition through bowel states can be represented by a
linear recurrence relation. Screening, on the other hand, helps decrease the
prevalences of abnormal states in the population, depending on the selected
FIT cut-off level and incentive. In particular, we assume that any benign
growth, large growth, or CRC found during the screening pathway is removed and
that the bowel returns to a normal state. Taken together, the starting
prevalences in screening period $k+1$ can be computed using the following
difference equations:

$\displaystyle\psi_{g,k+1,B}(Z_{g,k})$
$\displaystyle=(\psi_{g,k,B}-\tilde{\psi}_{g,k,B}(Z_{g,k}))(1-\mathbb{T}^{g,k}_{\mathrm{B,L}})+\psi_{g,k,\mathrm{N}}\mathbb{T}^{g,k}_{\mathrm{N,B}}$
(1) $\displaystyle\psi_{g,k+1,\mathrm{L}}(Z_{g,k})$
$\displaystyle=(\psi_{g,k,\mathrm{L}}-\tilde{\psi}_{g,k,\mathrm{L}}(Z_{g,k}))(1-\mathbb{T}^{g,k}_{\mathrm{L,R}})+(\psi_{g,k,\mathrm{B}}-\tilde{\psi}_{g,k,\mathrm{B}}(Z_{g,k}))\mathbb{T}^{g,k}_{\mathrm{B,L}}$
(2) $\displaystyle\psi_{g,k+1,\mathrm{R}}(Z_{g,k})$
$\displaystyle=\psi_{g,k,\mathrm{R}}-\tilde{\psi}_{g,k,\mathrm{R}}(Z_{g,k})+(\psi_{g,k,\mathrm{L}}-\tilde{\psi}_{g,k,\mathrm{L}}(Z_{g,k}))\mathbb{T}^{g,k}_{\mathrm{L,R}}$
(3) $\displaystyle\psi_{g,k+1,\mathrm{N}}(Z_{g,k})$
$\displaystyle=1-\sum_{b\in\\{\mathrm{B,L,R}\\}}\psi_{g,k+1,b}(Z_{g,k})\quad,$
(4)

where $\tilde{\psi}_{g,k,\mathrm{B}}(Z_{g,k})$ stands for the fraction of
participants found to have bowel state $b$ in screening period $k$ as a result
of applying screening strategy $Z_{g,k}$.

Following the notation in Salo et al. (2022), we define an information set
$\mathcal{I}(j)$ as the direct predecessors of a given node $j$, i.e.
$\mathcal{I}(j)=\\{i\in\mathcal{N}|(i,j)\in\mathcal{A}\\}$. For instance, in
Figure 1 the information set of Stage 5 (FIT result) consists of Stage 1 (FIT
cut-off level), and Stage 4 (usable sample is returned). Information states
$s_{\mathcal{I}(j)}\in\mathcal{S}_{\mathcal{I}(j)}$ are defined as the
combinations of states within all nodes of the the information set
$\mathcal{I}(j)$. Relating to our previous example, the information states of
Stage 5 are given as
$s_{\mathcal{I}(5)}\in\mathcal{S}_{\mathcal{I}(5)}=\prod_{i\in\mathcal{I}(5)}\mathcal{S}_{i}=\mathcal{S}_{1}\times\mathcal{S}_{4}$.
For chance nodes, the conditional probability outcome $s_{j}$ depends on these
information states. Specifically, the conditional probability of
$s_{j}\in\mathcal{S}_{j}$ (where $j\in\mathcal{C}$) occurring can be given as
$\mathbb{P}[X_{j}=s_{j}|X_{\mathcal{I}(j)}=s_{\mathcal{I}(j)}]$, where
$X_{\mathcal{I}(j)}$ are the realizations of random variables $X_{i}$ in the
information set $i\in\mathcal{I}(j)$. For example, in Figure 1 the conditional
probabilities for the chance alternatives if returning a usable sample (Stage
4) are dependent on whether a person is invited to screen (Stage 3) and if
they are offered an incentive (Stage 2). Please see A for the conditional
probability tables of each chance node in Figure 1. Whereas, for decision
nodes, the information states detail the information available when making the
decision. For example, in Stage 7, when making the additional examination
decision, the FIT result (Stage 5) and whether participation in the programme
has continued (Stage 6) is available in the information state.

A local decision strategy $Z_{j}$ for node $j\in\mathcal{D}$ is a function
that maps each information state to a decision, i.e.
$Z_{j}:\mathcal{S}_{\mathcal{I}(j)}\mapsto\mathcal{S}_{j}$. Binary decision
variables $z(s_{j}|s_{\mathcal{I}(j)})\in\\{0,1\\}$ denote the decision
alternative $s_{j}\in j\in D$ that defines the local decision strategies. A
global decision strategy $Z_{g,k}$ (for a segment $g,k$) is a set of local
decision strategies that contains one local strategy for each decision node.
An example of a global decision strategy for the ID in Figure 1 could be
$Z_{g,k}=\\{Z_{1}=\textit{50 ,}Z_{2}=\textit{Yes ,}Z_{3}=\textit{Yes}$.

Unlike a decision strategy that solely includes decisions, a path
$s\in\mathcal{S}=\bigtimes_{j=1}^{N}S_{j}$ is defined as the combination of
all chance and decision node alternatives $s_{j}\in S_{j}\ \forall j\in C\cup
D$, in which a global strategy and the value nodes are uniquely defined. For
example, one possible path is:

$s=$(75, No, Yes, Yes, Positive, Yes, Colonoscopy, Benign Adenoma,
Polypectomy, None).

The above path describes a sequence of events in which a person is invited
($s_{3}=$_Yes_) to the screening program without monetary incentive
($s_{2}=$_No_) using a FIT with a cut-off level value of 75 ng/ml
($s_{1}=75$). The invitee returns a usable sample ($s_{4}=$_Yes_) that tests
over 75 ng/ml and is thus scored as positive ($s_{5}=$_Positive_), and contact
is established with the local nurse ($s_{6}=$_Yes_). A further examination is
chosen to be a colonoscopy ($s_{7}=$_Colonoscopy_), the result of which
indicates a benign adenoma in the bowel ($s_{8}=$_Benign Adenoma_). During the
colonoscopy, the growth is removed from the bowel ($s_{9}$=_Polypectomy_) and
no adverse event occurs ($s_{10}$=_None_). The scenario path defines the
direct costs related to the participant’s screening process, together with the
detected cancer or growths, and whether a colonoscopy was administered.

The probability of a scenario path $\pi(s)$ occurring given the chances and
the decisions made during the screening process are modeled as auxiliary
variables. If the scenario path is not compatible with the decision strategy
($Z_{g,k}$) then $\pi(s)=0$. For example, if on scenario path $s^{\prime}$ the
decision alternatives are $s_{1}=50,s_{2}=$No, but the local decision strategy
has binary values that correspond to decision alternatives
$s_{1}=75,s_{2}=$No, then $\pi(s^{\prime})=0$, as the first decision for
$s^{\prime}$ does not match the current strategy. Otherwise, the path
probability is equal to the upper limit, i.e.

$\pi(s)=\begin{cases}p(s)=\prod\limits_{j\in
C}\mathbb{P}(X_{j}=s_{j}|X_{\mathcal{I}(j)}=s_{\mathcal{I}(j)}),\qquad\text{
if }Z_{j}(s_{\mathcal{I}(j)})=s_{j}\forall j\in\mathcal{D}\\\ 0,\hskip
176.407pt\text{ otherwise,}\end{cases}$ (5)

where $X_{j}$ is the random variable associated with chance node $j\in C$ and
$p(s)$ denotes the upper bound for $\pi(s)$.

The optimization problem defined by an influence diagram is both discrete and
linear. Thus, the set of all Pareto efficient (non-dominated) solutions can be
calculated using an _Modified Augmented Weighted Tchebychev_ (MAWT) norm
approach. The original algorithm with proofs is presented in Holzmann and
Smith (2018). We can denote the whole MAWT process as
$(\Xi^{*},\Phi^{*})=\mathrm{MAWT}(ID(sex,k,{\psi}_{k}^{e}))$ where
${\psi}_{k}^{e}$ is the starting prevalence distribution. In the following
model description, the solutions will all be dependent on the MAWT parameters,
i.e. $\xi^{*}=\xi_{k,{\psi}^{e}_{k}}^{sex,*}$ and similarly for all components
and sets. For the sake of avoiding notation clutter, we’ve excluded these
indices from the discussion where possible without sacrificing clarity.

The approach relies on minimizing the MAWT norm generated by the algorithm.
The norm $||\cdot||^{\mathbf{w},\epsilon}$ selects the individual element with
the highest value from the objective vector. The original MOMILP problem
defined by the ID, utopian vector $\phi^{utopia}$, and nadir vector
$\phi^{nadir}$ are given as inputs for the algorithm. The utopian vector is a
weak lower bound for the Pareto optimal objective function values, which is
determined by maximizing each objective individually. The nadir vector is a
strong upper bound over the non-dominated solution set $\Phi$, assuming the
objectives are minimised. The weights $\mathbf{w}$ and parameter $\epsilon$
are computed as in Holzmann and Smith (2018).

We can now formulate the optimisation problem solved by the MAWT algorithm.
This formulation captures the segment specific screening problem description
above as a Multi-objective mixed integer linear problem (MOMILP). The variable
$\mu$ (equation (6)) is the MAWT norm, which is minimised on each iteration
round and constraints (8) are the actual objectives of the Decision
Programming model. Equations (14)-(19) correspond to the Decision Programming
formulation.

$\displaystyle\min.\qquad\mu$ (6) subject to (7) $\displaystyle\mu\geq
w_{O}|O-\phi^{utopia}_{O}|+\epsilon\sum_{O}w_{O}|O-\phi^{utopia}_{O}|,$
$\displaystyle\forall\
O\in\\{\tilde{\Omega},\tilde{N^{Col}},\tilde{\psi}_{B},\tilde{\psi}_{L},\tilde{\psi}_{R}\\}$
(8) $\displaystyle\tilde{\Omega}=\sum_{s}\pi(s)U_{\mathrm{Cost}}(s)$ (9)
$\displaystyle\tilde{N}^{Col}=\sum_{s}\pi(s)U_{\mathrm{PCol.}}(s)$ (10)
$\displaystyle\tilde{\psi}_{B}=\sum_{s}\pi(s)U_{\mathrm{B}}(s)$ (11)
$\displaystyle\tilde{\psi}_{L}=\sum_{s}\pi(s)U_{\mathrm{L}}(s)$ (12)
$\displaystyle\tilde{\psi}_{R}=\sum_{s}\pi(s)U_{\mathrm{R}}(s)$ (13)
$\displaystyle\sum_{s}z(s_{j}|s_{\mathcal{I}(j)})=1,$ $\displaystyle\forall\
j\in\mathcal{D},\ s_{\mathcal{I}(j)}\in\mathcal{S}_{\mathcal{I}(j)}$ (14)
$\displaystyle 0\leq\pi(s)\leq p(s),\qquad$ $\displaystyle\forall\
s\in\mathcal{S}$ (15) $\displaystyle\pi(s)\leq
z(s_{j}|s_{\mathcal{I}(j)}),\qquad$ $\displaystyle\forall\
s\in\mathcal{S},j\in\mathcal{D}$ (16) $\displaystyle\pi(s)\geq
p(s)+\sum_{j\in\mathcal{D}}z(s_{j}|s_{\mathcal{I}(j)})-|\mathcal{D}|,$
$\displaystyle\forall\ s\in\mathcal{S}$ (17) $\displaystyle
z(s_{j}|s_{\mathcal{I}(j)}),\ U_{i},\in\\{0,1\\},$ $\displaystyle\forall\ j\in
D,\ s\in\mathcal{S},i\in\\{\mathrm{PCol.,B,L,R}\\}$ (18)
$\displaystyle\pi(s)\in\mathbb{R}.$ $\displaystyle\forall\ s\in\mathcal{S}$
(19)

Here $z(s_{j}|s_{\mathcal{I}(j)})$ are the actual decision variables that
capture the segment specific strategy. The same strategy is reflected in the
values of path probability variables $\pi(s)$. Values of $U_{i}(s)$ correspond
to the utility node outcomes for the different paths in the ID. These are
defined in A.7. When executed, the MAWT algorithm produces the complete
efficient solution set $\Xi^{*}=\\{\xi^{*}|\xi^{*}\succeq\xi,\
\xi^{*},\xi\in\Xi\\}$, where $\xi=(Z_{g,k},\pi(s))$ (i.e. a tuple of Pareto
optimal variable values) and the non-dominated objective value set
$\Phi^{*}=\\{\tilde{\Omega}(\xi^{*}),\tilde{N}^{col}(\xi^{*}),\tilde{\psi}_{\mathrm{B}}(\xi^{*}),\tilde{\psi}_{\mathrm{L}}(\xi^{*}),\tilde{\psi}_{\mathrm{R}}(\xi^{*})|\xi^{*}\in\Xi^{*}\\}$

###  3.3 Phase 1

In each period of Phase 1 the following steps are taken: (1) Sex- and period-
specific age prevalences of CRC and benign and large adenomas are used as
parameters in the optimization model. (2) The optimization model minimizes
expected costs and expected colonoscopies while maximizing found cancers and
growths in the bowel for that period. (3) The set of Pareto efficient
screening strategies (i.e. paths through the influence diagram) for the
specified period and sex are identified. (4) This set of efficient strategies
is combined with the previous period’s efficient set if it exists. (5) Any
strategies that become inefficient as a result of the merging of sets are
removed. (6) The sex- and period-specific prevalences are updated based on
objective function values, that is, the number of found CRC and benign and
large adenomas, and the natural transition of bowel states in the period
interval. These updated prevalences are used in step (1) of the subsequent
period. Steps (1) to (6) are repeated until all periods are optimized.

### Algorithm

The algorithm used to solve the problem is presented in pseudocode in
Algorithm 1.

Inputs : Process model as Influence Diagram (ID),
$\psi_{F,1,b},\psi_{M,1,b},\forall b\in\mathbf{B}$, screening periods
$k\in\\{1\dots K\\}$

Output : The optimal strategy
$Z^{*}\in\mathcal{E}_{F,K}\times\mathcal{E}_{M,K},\mathcal{E}_{g,k}=\\{e|e\succeq
e^{\prime}\\}$ for each sex $g$ , and the related objective value
$\Psi_{\mathrm{R}}(Z^{*})$.

$\mathcal{E}_{F,k},\mathcal{E}_{M,k}=\varnothing\quad\forall k\in[1\dots K]$ ;

// Initialize set of Efficient solutions as empty sets.

1

2for  _$g\in\\{\mathrm{F,M}\\}$_ do

// First period, $k=1$, handled separately.

3

// Compute Pareto efficient solutions for first period

4

$\mathcal{E}_{g,1}\leftarrow$ MAWT $(ID(g,1),\bar{\psi}_{g,1}))$ ;

// $\bar{\psi}_{g,k}=[\psi_{g,k,b}]_{b\in\mathcal{B}}$

5

6
$\bar{\psi}_{2}^{e^{g}_{1}}\leftarrow\textnormal{{UpdatePrevalences}}(e^{g}_{1},\bar{\psi}_{g,1})\quad\forall
e_{g,1}\in\mathcal{E}_{g,1}$ ;

7

8 $N^{col}_{total}\leftarrow N^{col}(e_{g,1})$

9 for  _$k\in\\{2,\dots,K\\}$_ do

10 for  _$e\in\mathcal{E}_{g,k-1}$_ do

// Compute Pareto efficient solutions for period $k$ based on preceding
strategy $e$

11 $\mathcal{Z}_{g,k}^{*}\leftarrow$ MAWT $(ID(g,k,\bar{\psi}(e)))$;

12

13 for  _$Z_{g,k}\in\mathcal{Z}_{g,k}^{*}$_ do

14

$e_{k}\leftarrow(e,Z_{g,k})$ ;

// Extend $e$ by $Z_{g,k}$ into a new strategy $e_{k}$, covering steps until
$k$

$\mathcal{E}_{g,k}\leftarrow\mathcal{E}_{g,k}\bigcup e_{g,k}$ ;

// Collect found strategies for step k.

15

16
$\bar{\psi}_{g,k}^{e_{g,k}}\leftarrow\textnormal{{UpdatePrevalences}}(e_{g,k},\bar{\psi}_{g,k-1})$
;

17

18 $\bar{\Psi}_{k}^{e_{g,k}}\leftarrow$ TotalPrevalences
($\bar{\psi}_{g,k}^{e_{g,k}},\bar{\Psi}_{k-1}^{e}$);

19

$N^{col,e_{g,k}}_{total}\leftarrow N^{col,e}_{total}+\tilde{N}^{col}(Z_{g,k})$
;

// Add new colonoscopies from $Z_{g,k}$ to preceding colonoscopies.

20

21 if  _$N^{col,e_{g,k}}_{total}\geq N^{col,max}$_ then

22 $\mathcal{E}_{g,k}\leftarrow\mathcal{E}_{g,k}\cap e_{g,k}$

23 end if

24

25 end for

26

27 end for

28
$\mathcal{E}_{g,k}\leftarrow\textnormal{{RemoveDominated}}(\mathcal{E}_{g,k})$

29 end for

30

31 end for

$\mathcal{E}_{K}\leftarrow\textnormal{{RemoveDominated}}(\mathcal{E}_{F,K}\times\mathcal{E}_{M,K})$

Algorithm 1 Phase 1 algorithm

As main inputs, the algorithm requires 1) Influence Diagram (ID) describing
the screening process per one population segment defined by age and sex 2)
MAWT algorithm for computing a Pareto front for one ID 3) Starting prevalences
for the first age segments (one prevalence vector per sex) 3) a prevalence
update function that takes as input a prevalence at period $k$ and a period
specific screening strategy, and outputs an updated prevalence vector for
$k+1$ 5) a function for combining age group specific prevalences into a total
population prevalence. The output is a Pareto efficient set of sex specific
screening strategies, i.e. combinations of period-specific strategies and
their related objective values and supportive information.

The process starts by initialising empty sets of efficient solutions (one per
sex). The sexes are handled separately (line 1). The algorithm then proceeds
to compute strategies, one for each screening segment. The first period is
handled separately, as the process is simpler than for the remaining periods.
A set of Pareto efficient, period specific solutions is computed by
iteratively solving the MAWT version of the Influence Diagram (see section
3.2) until the complete Pareto front is found. These strategies are saved into
efficient solution set $\mathcal{E}_{g,1}$ on line 1. One element in this set
contains a strategy (only for the first period at this point) and the related
objective function values. On line 1 expected prevalences for bowel states
after screening and 2 years of waiting for the next period,
$\bar{\psi}^{z_{1}}_{1}$, are computed for each found strategy. These will act
as starting points for period 2 optimization.

Next, a similar process is performed for periods $k\in[2,\dots,K]$ using the
found solutions $e\in\mathcal{E}^{sex}_{k-1}$ as inputs. For each surviving
previous period solution, a new Pareto front is created by solving the MAWT
problem, now with a prevalence $\bar{\psi}^{e}_{k}$ based on the previous
period efficient strategy $e$ as a starting point. Each of the found
strategies is added to the strategy included in $e$ (row 1), creating a
history of decisions leading up to this decision. Again, the prevalences are
updated as on the first period. An additional step now computes the total
population prevalence for the population consisting of sex specific age groups
corresponding to periods $[1,\dots,k]$. The cancer component in this total
prevalence vector is used as one of the objectives in filtering out Pareto
non-efficient strategies, together with cancer and large adenoma prevalences
of the current age group. If the sum of colonoscopies (computed on row 1) at
this period, with found strategy, does not exceed the maximum number of
coloscopies (row 1), the solution is added to the set of found solutions.

Finally, on row 1 the solutions found at period $k$ are compared in terms of
total cancer prevalence, period $k$ age group cancer and large adenoma
prevalences and required total colonoscopies, and the Pareto inefficient
solutions are removed. Including the different prevalences at this point
ensures that on the next round, the starting points are efficient in terms of
the overall objective of minimising cancer prevalence in the total population
but also that the starting point on the next round $k+1$ is efficient. Large
adenomas are included also in the starting point comparison, as the influence
the generation of new cancers at $k+1$. The algorithm finishes when the
predetermined period $K$ has been reached.

###  3.4 Phase 2

In this phase, a strategy for each age of males and females is selected in
order to minimize CRC prevalence in the entire screening population with
respect to a colonoscopy constraint. In each period of Phase 1 the efficient
solutions of the optimization problem (6) - (19) are combined with the
previous period’s solution set with any now inefficient solutions removed.
This means that in Phase 2, only the solution set from the final period is
relevant as it contains the optimal solutions from all periods. Without a
temporal aspect, the Phase 2 optimization problem is simple and
straightforward to solve.

For this problem we take a utilitarian approach, in that we wish to minimize
the population-level cancer prevalence subject to a population-level resource
constraint. To achieve this, we solve the binary linear programming model:

$\displaystyle\min_{\bar{x}_{\mathrm{F},j},\ \bar{x}_{\mathrm{M},j}}.\quad$
$\displaystyle\Psi_{R}=\frac{N^{R}_{\mathrm{F},j}\bar{x}_{\mathrm{F},j}+N^{R}_{\mathrm{M},j}\bar{x}_{\mathrm{M},j}}{N_{\mathrm{F}}+N_{\mathrm{M}}},$
(20) subject to $\displaystyle
N_{\mathrm{F}}N^{col}_{\mathrm{F},j}\bar{x}_{\mathrm{F},j}+N_{\mathrm{M}}N^{col}_{\mathrm{M},j}\bar{x}_{\mathrm{M},j}\leq
N^{Col,Max},$ (21)
$\displaystyle\bar{x}_{\mathrm{M},j}\in\\{0,1\\},\qquad\forall
j\in\\{1,\dots,J_{M}\\}$ (22)
$\displaystyle\bar{x}_{\mathrm{F},j}\in\\{0,1\\},\qquad\forall
j\in\\{1,\dots,J_{F}\\}$ (23) $\displaystyle\sum_{j}x_{g,j}=1\qquad\forall
g\in\\{\mathrm{F},\mathrm{M}\\}\,$ (24)

where $N$ refers to the population size; $\mathrm{M}$ and $\mathrm{F}$
correspond to males and females; $i$ is the bowel state colorectal cancer; $j$
is a strategy from the set of efficient strategies, sized $J_{M}$ and $J_{F}$
for males and females respectively, identified in the final period of Phase 1;
$N^{col}_{\mathrm{M},j}$ and $N^{col}_{\mathrm{F},j}$ are the expected number
of colonoscopies in strategy $j$ for males and females, respectively; and,
$N^{col,max}$ is the population-level colonoscopy budget constraint. The
binary variables $x_{g,j}$ are used to select one strategy for both sex.

##  4 Results and policy recommendations

In this section, we present optimal screening policies for two situations (i)
when decisions regarding monetary incentives for participation are not
available, and (ii) when these incentives decisions are available. Firstly,
the situation in which no incentives are available reflects the current
Finnish CRC screening programme. These results will provide optimal screening
policy recommendations to aid decision makers when considering the choice of
FIT cut-off level for different age and sex groups. Additionally, the expected
costs of each policy will be presented. To the best of our knowledge, no other
cost analysis for the current Finnish CRC screening programme has been
published in the public domain. Secondly, results and recommendations for a
hypothetical situation in which monetary incentives to participate in
screening are presented, along with their expected costs.

To assess the policy and clinical implications of different colonoscopy
resource constraints are compared to the baseline case of no screening. To
also study the role of the available colonoscopy resources, we solve the
problem in 4 cases, i.e. for 8000, 12000, 16000 and 20000 colonoscopies as the
maximum colonoscopy limit for all 5 groups screened.

###  4.1 Results for no monetary incentive option

Case (max colonoscopies) | Sex | Age 60 | Age 62 | Age 64 | Age 66 | Age 68
---|---|---|---|---|---|---
Case 1 (8 000) | Male | 50 | 50 | 50 | 50 | 50
| Female | - | 40 | 25 | 25 | 25
Case 2 (12 000) | Male | 50 | 25 | 50 | 20 | 20
| Female | 20 | 25 | 20 | 20 | 10
Case 3 (16 000) | Male | 20 | 20 | 20 | 20 | 20
| Female | 10 | 10 | 10 | 10 | 10
Case 4 (20 000) | Male | 10 | 10 | 10 | 10 | 10
| Female | 10 | 10 | 10 | 10 | 10
Table 1: Optimized screening policies with no option of incentive: FIT cut-off
level in $\mu$g/g, no invite represented by -, colonoscopy performed when
positive FIT.

In each case and screening age, the FIT cut-off level used is presented in
Tables 1 and 2. For example, for Case 2 with no option of incentive (Table 1),
the male cut-off level is 50$\mu$g/g and the female is 20$\mu$g/g for age 60.
Whereas, the cut-off level of 250$\mu$g/g is used for both males and females
aged 62.

Figure 2: Cancer prevalence over screening rounds with no option of incentive,
policies given in Table 2

###  4.2 Results for monetary incentive option

The optimal strategies for different cases are in Table 2 and the
corresponding cancer prevalences in Figure 3. In our model the use of
incentives can be beneficial, even at the cost of lower sensitivity due to
higher cut-off levels, which can be seen by comparing these strategies to the
ones where incentives were not available. The benefits of incentives are
apparent in Figure 3 where the total prevalences are lower for the
incentivized strategies, except for case 1 where the availability of
incentives caused the screening to focus heavily on males. This implies that
ways of increasing the participation rate of males could prove valuable for
total cancer prevalence in the population, regardless of the exact method for
achieving such increase.

Case (max colonoscopies) | Sex | Age 60 | Age 62 | Age 64 | Age 66 | Age 68
---|---|---|---|---|---|---
Case 1 (8 000) | Male | 50+i | 50+i | 50+i | 20+i | 20+i
| Female | 50 | - | - | - | 25
Case 2 (12 000) | Male | 50+i | 50+i | 50+i | 40+i | 20+i
| Female | 40 | - | 40+i | 10+i | 10+i
Case 3 (16 000) | Male | 50+i | 20+i | 20+i | 25+i | 20+i
| Female | 20+i | 25+i | 25+i | 10+i | 10+i
Case 4 (20 000) | Male | 20+i | 20+i | 20+i | 20+i | 10+i
| Female | 10+i | 10+i | 10+i | 10+i | 10+i
Table 2: Optimized screening policies with option of incentive (+i): FIT cut-
off level in $\mu$g/g, no invite represented by -, colonoscopy performed when
positive FIT. Figure 3: Cancer prevalence over screening rounds with option of
incentive, policies given in Table 2

##  5 Discussion

In this manuscript we have proposed a novel approach for optimizing a cancer
screening program and applied it to the Finnish colorectal cancer screening
program. The optimization approach combines several different algorithms and
multi-objective optimization to find an optimal solution to a multi-period
screening process captured in IDs, and respective of a colonoscopy constraint.
Given the ID and its related parameter values and a sex-specific starting
prevalences, our approach is able to find an optimal strategy for a 5-period
screening process from among approximately 26 billion possible alternative
strategies. Compared to previous approaches to screening program optimization
documented in literature, one of the main strengths of our approach is the
ability to optimize the strategy over several screening rounds.

We computed results for 4 different colonoscopy resource levels. The different
result sets show a clear differences in selected FIT test cut-off levels and
prioritization between male and female participants, and the potential
benefits from varying the FIT cut-off level between different sex and age
segments, when resources are limited. We also present separate results for
cases where a hypothetical incentive is available, or not. There is a clear
difference in the incentivized results compared to non-incentivized in the
low-resource cases. With incentives available, the results emphasize testing
males with incentives at the cost of screening females.

This study opens up several avenues for future research. Modeling-wise, we
have made some simplifications, on parts of which the model could be improved.
In particular our assumptions regarding the transition probabilities and
healing of participants for whom an abnormal bowel was discovered are
relatively strong. Improvement of the transition model could therefore lead to
more accurate estimates and strategy proposals. Inclusion of mortality as an
objective could also provide important insights into the outcomes of the
model. Due to lack of data, mortality was outside the scope of this study but
the situation is likely to improve as the screening program runs for more
years. The approach we have presented is flexible in the sense that it is also
capable of solving the problem under different types of constraints.

Our approach is computationally quite intensive, as it, in practice, generates
a multi-periodic decision tree to be solved. However, due to the nature of the
algorithm, the main memory requirement is set by the structure of the
influence diagram, i.e. the period-specific model, whereas the addition of
periods increases only the computation time. Separating the branches of the
whole multi-period tree into separate sub-problems opens up the possibility to
solve the in parallel, which could be used to decrease the required
computation time. Additional problem-specific constraints can possibly also be
used to decrease computational time.

##  6 Acknowledgements

The calculations presented above were performed using computer resources
within the Aalto University School of Science “Science-IT” project.

This research has been supported by The Foundation for Economic Education
(grant number 16-9442) and The Paulo Foundation.

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##  Appendix A Conditional probabilities and values in the influence diagram

###  A.1 Conditional probabilities for stage 4: Usable sample returned

The conditional probabilities for returning a sample in good condition are

$\mathbb{P}(X_{4}=s_{4}|X2=s_{2},X_{3}=s_{3}),\quad\forall
s_{4}\in\\{\mathrm{No,Yes}\\},s_{2}\in\\{\mathrm{No,Yes}\\},s_{3}\in\\{\mathrm{No,Yes}\\}.$
(25)

This can be presented as a matrix:

$\mathbb{P}(X_{4}=s_{4}|X2=s_{2},X_{3}=s_{3})=$ *2c cc *2¿c¡ ¿c¡ No & Yes
$s_{2}$ $s_{3}$
[*2c]¿c¡ ¿c¡ 1.0 & 0.0 No No
$1-P^{ret.OK}_{g,a}$ $P^{ret.OK}_{g,a}$ No Yes
1.0 0.0 Yes No
$(1-P^{ret.OK}_{g,a})/2$ $P^{ret.OK}_{g,a}+(1-P^{ret.OK}_{g,a})/2$ Yes Yes

We have assumed that incentivizing the invited persons has the effect of
halving the probability of not returning a sample. The probability
$P^{ret.OK}_{g,a}$ is calculated as

$P^{ret.OK}_{g,a}=P(\mathrm{return}|Age=a,Sex=g)P(\mathrm{SampleOK}).$ (26)

###  A.2 Conditional probabilities for stage 5: FIT test result

The conditional probabilities for chance node $C_{5}$ are

$\mathbb{P}(X_{5}=s_{5}|X_{1}=s_{1},X_{4}=s_{4}),\quad\forall
s_{5}\in\\{NA,-,+\\},s_{1}\in\\{50,75,100,150,250\\},s_{4}\in\\{No,Yes\\}$
(27)

This can be presented as a matrix:

$\mathbb{P}(X_{5}=s_{5}|X_{1}=s_{1},X_{4}=s_{4})=$ *3c cc *5¿c¡ NA & \+ -
$s_{1}$ $s_{4}$
[*3c]¿c¡ ¿c¡ 1.0 & 0.0 0.0 50 No
0.0 $P^{FIT+}_{50}$ $1-P^{FIT+}_{50}$ 50 Yes
1.0 0.0 0.0 75 No
0.0 $P^{FIT+}_{75}$ $1-P^{FIT+}_{75}$ 75 Yes
1.0 0.0 0.0 100 No
0.0 $P^{FIT+}_{100}$ $1-P^{FIT+}_{100}$ 100 Yes
1.0 0.0 0.0 150 No
0.0 $P^{FIT+}_{150}$ $1-P^{FIT+}_{150}$ 150 Yes
1.0 0.0 0.0 250 No
0.0 $P^{FIT+}_{250}$ $1-P^{FIT+}_{250}$ 250 Yes

where the positive result probabilities for FIT threshold levels $l$
$P^{FIT+}_{l}$ are calculated as follows:

$\displaystyle P^{FIT+}_{l}=\sum_{b}P(FIT+|l,b),\quad
b\in\\{\mathrm{Normal,Benign,Large,CRC}\\}$ (28) $\displaystyle
P(FIT+|l,b)=\sigma_{l,b}^{+,FIT}\psi_{b},\quad
b\in\\{\mathrm{Benign,Large,CRC}\\}$ (29) $\displaystyle
P(FIT+|l,\mathrm{Normal})=(1-\sigma_{l,\mathrm{Normal}}^{-,FIT})\psi_{\mathrm{Normal}}.$
(30)

Here $\sigma^{+,FIT}_{l,b}$ is the sensitivity of the FIT test for detecting
bowel states $b$ given threshold $l$ and $\sigma^{-,FIT}_{l,\mathrm{Normal}}$
is the corresponding specificity. The prevalences are segment (sex, age),
period and preceding strategy specific, but for sake of avoiding index
clutter, the information is omitted from the formulation. For situations where
no usable sample was returned, i.e. $s_{4}=\mathrm{No}$, the only possible
outcome for the FIT test result is NA.

###  A.3 Conditional probabilities for stage 6: Continued participation

The conditional probabilities for chance node $C_{6}$ are

$\mathbb{P}(X_{6}=s_{6}|X_{5}=s_{5}),\quad\forall
s_{6}\in\\{No,Yes\\},s_{5}\in\\{\mathrm{NA},+,-\\}$ (31)

which can be presented as a matrix

$\mathbb{P}(X_{6}=s_{6}|X_{5}=s_{5})=$ *2c c *3¿c¡ No & Yes $s_{5}$
[*2c]¿c¡ 1 & 0 NA
$1-P(Yes|Sex,Age)$ $P(Yes|Sex,Age)$ \+
1 0 -

###  A.4 Conditional probabilities for stage 8: Examination result

The conditional probabilities for chance node $C_{8}$ are

$\begin{array}[]{l}\mathbb{P}(X_{8}=s_{8}|X_{1}=s_{1},X_{6}=s_{6},X_{7}=s_{7}),\\\
\forall
s_{8}\in\\{NA,Normal,Benign,Large,CRC\\},s_{1}\in\\{50,75,100,150,250\\},\\\
s_{6}\in\\{No,Yes\\},s_{7}\in\\{No,Yes\\}\end{array}$ (32)

This can be presented as a matrix:
$\mathbb{P}(X_{8}=s_{8}|X_{1}=s_{1},X_{6}=s_{6},X_{7}=s_{7})=$
*5c *3c *8¿c¡ NA & Normal Benign Large CRC $s_{1}$ $s_{6}$ $s_{7}$
[*5c]¿c¡ ¿c¡ ¿c¡ 1.0 & 0.0 0.0 0.0 0.0 50 No No
1.0 0.0 0.0 0.0 0.0 50 No Yes
1.0 0.0 0.0 0.0 0.0 50 Yes No
0.0 $P^{\mathrm{COL,Normal}}_{50}$ $P^{\mathrm{COL,Benign}}_{50}$
$P^{\mathrm{COL,Large}}_{50}$ $P^{\mathrm{COL,CRC}}_{50}$ 50 Yes Yes
1.0 0.0 0.0 0.0 0.0 50 No No
1.0 0.0 0.0 0.0 0.0 50 No Yes
1.0 0.0 0.0 0.0 0.0 50 Yes No
0.0 $P^{\mathrm{COL,Normal}}_{75}$ $P^{\mathrm{COL,Benign}}_{75}$
$P^{\mathrm{COL,Large}}_{75}$ $P^{\mathrm{COL,CRC}}_{75}$ 75 Yes Yes
1.0 0.0 0.0 0.0 0.0 100 No No
1.0 0.0 0.0 0.0 0.0 100 No Yes
1.0 0.0 0.0 0.0 0.0 100 Yes No
0.0 $P^{\mathrm{COL,Normal}}_{100}$ $P^{\mathrm{COL,Benign}}_{100}$
$P^{\mathrm{COL,Large}}_{100}$ $P^{\mathrm{COL,CRC}}_{100}$ 100 Yes Yes
1.0 0.0 0.0 0.0 0.0 150 No No
1.0 0.0 0.0 0.0 0.0 150 No Yes
1.0 0.0 0.0 0.0 0.0 150 Yes No
0.0 $P^{\mathrm{COL,Normal}}_{150}$ $P^{\mathrm{COL,Benign}}_{150}$
$P^{\mathrm{COL,Large}}_{150}$ $P^{\mathrm{COL,CRC}}_{150}$ 150 Yes Yes
1.0 0.0 0.0 0.0 0.0 250 No No
1.0 0.0 0.0 0.0 0.0 250 No Yes
1.0 0.0 0.0 0.0 0.0 250 Yes No
0.0 $P^{\mathrm{COL,Normal}}_{250}$ $P^{\mathrm{COL,Benign}}_{250}$
$P^{\mathrm{COL,Large}}_{250}$ $P^{\mathrm{COL,CRC}}_{250}$ 250 Yes Yes

where the probabilities $P^{COL,r}_{l}$ for colonoscopy results given a
certain FIT level $l$ are calculated as follows:

$\displaystyle
P^{\mathrm{COL},r}_{l}=\sigma^{\mathrm{+,COL}}_{b}P(b|\mathrm{FIT+})$ (33)
$\displaystyle
P(b|\mathrm{FIT+},l)=\frac{\sigma^{\mathrm{+,FIT}}_{l,b}\psi_{b}}{P(\mathrm{FIT+})}$
(34) $\displaystyle
P(\mathrm{FIT+},l)=\sum_{b\neq\mathrm{Normal}}\sigma^{\mathrm{+,FIT}}_{l,b}\psi_{b}+(1-\sigma^{\mathrm{-,FIT}}_{l,\mathrm{Normal}})\psi_{\mathrm{Normal}}.$
(35)

Here we have assumed that the specificity of colonoscopies is perfect, i.e.
$P(+|Normal)=0$. The sensitivities and specificities are acquired from data as
are the prevalences of bowel states in the first period. In further periods,
the prevalences are acquired from optimization results of preceding periods.
The prevalences are segment (sex, age), period and preceding strategy
specific, but for sake of avoiding index clutter, the information is omitted
from the formulation.

###  A.5 Conditional probabilities for stage 9: Polyp found

The conditional probabilities for finding a polyp, and thus leading to a
polypectomy, are

$\mathbb{P}(X_{9}=s_{9}|X8=s_{8}),\quad\forall s_{9}\in\\{\mathrm{No\
result,No\ polyp,Polyp}\\},s_{8}\in\\{\mathrm{NA,Normal,Benign,Large,CRC}\\}.$
(36)

This can be presented as a matrix:

$\mathbb{P}(X_{9}=s_{9}|X_{8}=s_{8})=$ *4c *4¿c¡ No result & No polyp Polyp
$s_{8}$
[*3c] ¿c¡ 1 & 0 0 NA
0 1 0 Normal
0 0 1 Benign
0 0 1 Large
0 0 1 CRC

This means that if any growth is found, the finding of a polyp is also
assumed.

###  A.6 Conditional probabilities for stage 10: Adverse event

The conditional probabilities for an adverse event (bleed or a perforation)
during polypectomy are

$\mathbb{C}_{10}=\mathbb{P}(X_{10}=s_{10}|X9=s_{9}),\quad\forall
s_{10}\in\\{\mathrm{None,Bleed,Perforation}\\},s_{9}\in\\{\mathrm{NA,Nopolyp,Polyp}\\}.$
(37)

This can be presented as a matrix:

$\mathbb{C}_{10}=$ *4c *4¿c¡ No adverse event & Bleed Perforation $s_{9}$
[*3c] ¿c¡ 1 & 0 0 No result
$1-P^{\mathrm{Bleed}}-P^{\mathrm{Perf.(woPoly.)}}$ $P^{\mathrm{Bleed}}$
$P^{\mathrm{Perf.(woPoly.)}}$ No polyp
$1-P^{\mathrm{Bleed}}-P^{\mathrm{Perf.(wPoly.)}}$ $P^{\mathrm{Bleed}}$
$P^{\mathrm{Perf.(wPoly.)}}$ Polyp

These probabilities are acquired from data.

###  A.7 Conditional values for utility nodes

The conditional values for all growth types (Benign, Large, CRC),
${U}_{\mathrm{B}},{U}_{\mathrm{L}},{U}_{\mathrm{R}}$ are simply 1 if such a
growth is found or 0 if not. Similarly, the value for colonoscopies,
${U}_{\mathrm{PCol.}}$, is -1 if a colonoscopy is performed and 0 otherwise.
The negativity of the value is just a stylistic choice to reflect the
’resource consuming’ role of performing a colonoscopy.

The cost value ${U}_{\mathrm{Cost}}$ is conditional to all cost generating
stages. It can be formulated as

$\mathbb{U}_{14}=\Omega|(S_{j}=s_{j})=\sum_{j}\Omega(s_{j}),\quad
j\in\\{2,3,4,7,8,9,10\\}$ (38)

where the individual cost values for stage realizations are derived from data.