-
Introduction
Pulmonary arterial hypertension (PAH) is a
severe and often deadly disease, originating from
an increase in pulmonary vascular resistance. Its
prevention and treatment are of vital importance
to public health. A group of medical researchers
proposed a calculator for estimating the risk of
dying from PAH, available for a variety of
computing platforms and widely used by health-care
professionals. The PAH Risk Calculator is based
on the Cox’s Proportional Hazard (CPH) Model,
a popular statistical technique used in risk
estimation and survival analysis, based on data from
a thoroughly collected and maintained Registry
to Evaluate Early and Long-term Pulmonary
Arterial Hypertension Disease Management
(REVEAL Registry). In this paper, we propose an
alternative approach to calculating the risk of
PAH that is based on a Bayesian network (BN)
model. Our first step has been to create a BN
model that mimics the CPH model at the
foundation of the current PAH Risk Calculator. The
BN-based calculator reproduces the results of the
current PAH Risk Calculator exactly. Because
Bayesian networks do not require the somewhat
restrictive assumptions of the CPH model and
can readily combine data with expert knowledge,
we expect that our approach will lead to an
improvement over the current calculator. We plan
to (1) learn the parameters of the BN model from
the data captured in the REVEAL Registry, and
(2) enhance the resulting BN model with
medical expert knowledge. We have been
collaborating closely on both tasks with the authors of the
original PAH Risk Calculator.
Pulmonary arterial hypertension (PAH) is a fatal, chronic,
and life-changing disease originating from an increase in
pulmonary vascular resistance, and leading to high blood
pressure in the lung
(Benza et al., 2010; Subias et al., 2010)
.
Patients with PAH suffer from shortness of breath, chest
pain, dizziness, fatigue, and possibly other symptoms
depending on the progression of disease
(Hayes, 2013)
.
Currently, there is no cure for PAH and treatment is often
determined based on the symptoms. With an early diagnosis and
proper treatment, patients’ lives can be extended by five or
more years.
With the long-term goal to characterize the clinical course,
treatment, and predictors of outcomes in patients with PAH
in the United States, a group of medical researchers
established a Registry to Evaluate Early and Long-term
Pulmonary Arterial Hypertension Disease Management
(REVEAL Registry)
(Benza et al., 2010)
. The REVEAL
registry is quite likely the most comprehensive collection of
data of patients suffering from PAH and it has led to
interesting insights improving the diagnosis, prediction, and
treatment of PAH. One of the prominent applications of
the REVEAL Registry is the PAH Risk Calculator
(Benza
et al., 2012)
, a statistical model learned from the REVEAL
Registry data and predicting the survival of patients at risk
for PAH. A computer implementation of the PAH Risk
Calculator is available for a variety of computing
platforms and widely used by health-care professionals (see
http://www.pah-app.com/ for more information).
The PAH Risk Calculator is based on the Cox’s
Proportional Hazard (CPH) model
(Cox, 1972)
, a popular
statistical technique used in risk estimation and survival analysis.
One weakness of this approach is that the underlying model
can be only learned from data and is not readily amenable
to refinement based on expert knowledge. Another possible
weakness is that the CPH model rests on several
assumptions simplifying the interactions between the risk factors
and the disease. While these assumptions are reasonable
and the CPH model has been successfully used for decades,
it is interesting to question them with a possible benefit in
terms of model accuracy.
In this paper, we propose an alternative approach to
calculating the risk of PAH that is based on a Bayesian network
(BN)
(Pearl, 1988)
model. BNs are acyclic directed graphs
in which vertices represent random variables and directed
edges between pairs of vertices capture direct influences
between the variables represented by the vertices. A BN
captures the joint probability distribution among a set of
variables both intuitively and efficiently, modeling
explicitly independences among them. A representation of the
joint probability distribution allows for calculation of
probability distributions that are conditional on a subset of
variables. This typically amounts to calculating the probability
distributions over variables of interest given observations of
other variables (e.g., probability of one-year survival given
a set of observed risk factors). There is a well developed
theory expressing the relationship between causality and
probability and often the structure of a BN is given a causal
interpretation. This is utmost convenient in terms of user
interfaces, notably knowledge acquisition and explanation
of results. The first step in our work has been to create a
BN model that mimics the CPH model at the foundation of
the current PAH Risk Calculator. In this, we use the BN
interpretation of the CPH model proposed by
Kraisangka
and Druzdzel (2014)
. Our BN-based calculator reproduces
the results of the current PAH Risk Calculator exactly.
Because Bayesian networks do not require the assumptions
of the CPH model and can readily combine data with
expert knowledge, we expect that our approach will
eventually lead to an improvement over the current PAH Risk
Calculator. Our mid- to long terms plans include (1)
learning the parameters of the BN model directly from the data
captured in the REVEAL Registry, and (2) enhancing the
resulting BN model with medical expert knowledge. We
are collaborating on both tasks with the team maintaining
the REVEAL Registry and the authors of the original PAH
Risk Calculator.
The remainder of this paper is structured as follows.
Section 2 describes the problem of PAH, the CPH model, and
the PAH Risk Calculator. Sections 3 and 4 describe
application of Bayesian networks to risk estimation and the
proposed BN-based PAH Risk Calculator. Finally, Section 5
describes our conclusions and future work.
2
Pulmonary Arterial Hypertension
This section introduces some facts related to the pulmonary
arterial hypertension (PAH), notably its risk factors, the
Cox’s Proportional Hazard (CPH) model, and the PAH
Risk Calculator based on the CPH model.
Risk can be defined as the rate of an occurrence of a
particular disease or adverse event
(Irvine, 2004)
. Although
PAH can occur at any age, in any races, and any ethnic
background
(Hayes, 2013)
, there are risk factors that make
some people more susceptible. For example, females are
at least two and a half times more susceptible than men to
idiopathic PAH. Recently, medical care professionals
treating PAH have relied on existing patient registries to
understand PAH better. Several risk factors have been identified
and used to develop prognostic models for guiding their
therapeutic decision making. For example, a study based
on the Registry to Evaluate Early and Long-Term
Pulmonary Arterial Hypertension Disease Management
(REVEAL)
(Benza et al., 2010)
extracted several demographic,
functional, laboratory, and hemodynamic parameters
associated with patient survival in PAH
(Benza et al., 2012)
by
means of a multivariate Cox’s proportional hazard model
(CPH) (discussed in more detail in the following section).
By developing a prognosis model, physician can access
a short-term and long-term patient survival in the context
of current treatment and clinical variables
(Benza et al.,
2012)
. Although prognostic tools for patient survival have
improved the quality of predictions, the models are still
imperfect and more research is needed on improving them.
Cox’s Proportional Hazard Model
Hazard is a measure of risk at a small time interval t, which
can be considered as a rate
(Allison, 2010)
. In survival
analysis, the hazard function can be represented by
probability distributions (e.g., exponential distribution) or can
be modeled by regression techniques. The Cox’s
proportional hazard model (CPH)
(Cox, 1972)
is a set of
regression methods used in the assessment of survival based on its
risk factors or explanatory variables. The probability of an
individual surviving beyond time t can be estimated with
respect to a hazard function
(Allison, 2010)
. As defined
originally by
Cox (1972)
, the hazard regression model is
expressed as
(t) =
0 (t) exp 0·X .
(1)
This hazard model is composed of two main parts: the
baseline hazard function, 0 (t), and the set of effect
parameters, 0 · X = 1X1 + 2X2 + ... + nXn . The
baseline hazard function determines the risks at an
underlying level of explanatory variables, i.e., when all
explanatory variables are absent. The s are the coefficients
corresponding to the risk factors, X. According to
Cox (1972)
,
this 0 (t) can be unspecified or can follow any distribution
and be estimated from data.
The application of the CPH model relies on the assumption
that the hazard ratio of two observations is constant over
time
(Cox, 1972)
. For example, a hazard ratio of a group of
PAH patients having renal insufficiency to a group of PAH
without renal insufficiency (control/baseline group) is
estimated as 1.90. This assumption means that patients with
renal insufficiency always have a 90% higher risk for
dying from PAH than patients without renal insufficiency by
Cox’s assumptions. The ratio of two hazards is defined as
:
=
2 (t)
1 (t)
=
exp ( 0X2) .
exp ( 0X1)
If the risk factors X are binary, their value could be
expressed as presence (X = 1) or as absence or baseline
(X = 0) of the risk factor. Once, we know the hazard
ratio of one group toward another group, we can estimate the
survival probability
(Casea et al., 2002)
by
S (t) = S0 (t) .
S0 (t) is the baseline survival probability estimated from
the data, i.e., when all risk factor are absent or at their
baseline value (X = 0) at any time t, while is hazard ratio of
an interested group to the baseline group. In other words,
the survival probability of any patients relative to the
baseline group can be estimated from
S (t) = S0 (t)exp 0·X .
An example of CPH model used as a prognosis model for
PAH patients is from the REVEAL Registry Risk Score
Calculator
(Benza et al., 2012)
. The model, including 19
risk factors, was developed to predict a one-year survival
probability. The main survivor function is
S(t = 1) = S0 (1)exp 0·X
,
where S0(1) is the baseline survivor function of 1 year
(0.9698) and in this equation is the shrinkage coefficient
after model calibration (0.939)
(Benza et al., 2010)
. The
risk factors X (listed in Table 1) included PAH associated
with portal hypertension (APAH-PoPH), PAH associated
with connective tissue disease (APAH-CTD), family
history of PAH (FPAH), modified New York Heart
Association (NYHA)/World Health Organization(WHO)functional
class I, III, and IV, men aged > 60, renal insufficiency,
systolic blood pressure(SBP) < 110 mm Hg, heart rate
> 92 beats per min, mean right atrial pressure (mRAP)
> 20 mm Hg, 6-minute walking distance(6MWD), brain
natriuretic peptide (BNP)> 180 pg/ml, 165 m, brain
natriuretic peptide (BNP), 180 pg/mL, pulmonary vascular
resistance(PVR)> 32 Wood units, percentage predicted
diffusing capacity of lung for carbon monoxide (Dlco)
32%, and presence of pericardial effusion on
echocardiogram. Most of the risk factors were associated with
increasing mortality rate (indicated by positive sign in in
Table 1), while only four factors were associated with
increased one-year survival (indicated by negative sign in
in Table 1).
(2)
(3)
(4)
(5)
Risk factors Xi
APAH-CTD
FPAH
APAH-PoPH
Male >60 years age
Renal insufficiency
NYHA Class I
NYHA Class III
NYHA Class IV
SBP <110 mmHg
Heart Rate >92bmp
6MWD 440 m
6MWD <165 m
BNP <50 pg/ML
BNP >180 pg/ML
Pericardial effusion
% Dlco 80%
% Dlco 32%
mRAP > 20 mmHg
PVR >32 Wood units%
To be able to summarize from the model, patients were
stratified into five risk groups according to their range of
survival probability
(Benza et al., 2010)
including the low
risk group where the predicted 1-year survival probability
> 95%, average risk with 90% to 95% survivals,
moderately high risk with 85% to 90% survivals, high risk with
70% to 85% survival, and very high risk group with
survival probability < 70%.
PAH Calculator
Based on the CPH model, the further application of the
CPH model is in the form of a risk calculator. This
simplified calculator are useful in everyday clinical practice
by helping physicians to decide patient therapies based on
level of risk
(Benza et al., 2012)
. The calculator was
designed from assigning score to variables according to their
hazard ratio. For the risk factors associated with
increasing mortality (positive coefficients), score of two points
were assigned for the risk factors which has their hazard
ratio (exp( )) at least two or more folds, i.e., those with
exp( ) 2 , and one point were assigned for other risk
factors. Risk factors associated with decreasing mortality
(negative coefficients) were assigned a negative score.
Figure 1 shows all risk factors and the interpretation of their
hazard ratio rate.
Figure 2 shows the user interface of the PAH Risk
Calculator. Each risk factor from the CPH model is listed and
mapped with the score. The calculator allows for adding
and subtracting the score based on the data entered for an
individual patient case. To avoid a negative total score, the
base score of 6 is set as a starting score. The total score
is interpreted in the same way as the survival probability
given by the CPH model, i.e., it includes the low risk group
with the score 7, average risk with score = 8, moderately
high risk score = 9, high risk with score between 10 and
11, and very high risk group with score 12. The score,
defined as above, makes it simpler for health care providers
to use than probabilities.
3
Application of Bayesian Networks to Risk
Calculation
An alternative approach to the traditional survival
analysis is the use of Bayesian networks
(Pearl, 1988)
to
estimate risks. Compared to the CPH model and several other
Artificial Intelligence and Machine Learning techniques, a
Bayesian network can model explicitly the structure of the
relationships among explanatory variables with their
probability
(Hanna and Lucas, 2001)
. A Bayesian network can
be built from expert knowledge, available data, or
combination of both. If there exists a probabilistic interpretation of
existing modeling tool, like in case of the CPH model, a BN
model can also be an interpretation of the existing model.
The structure of a Bayesian network can depict a complex
structure of a problem and provide a way to infer posterior
conditional probability distributions, useful for prognosis
and diagnosis, including medical decision support systems
(Husmeier et al., 2005)
.
To estimate risks using Bayesian network, the prognosis
can be created as a static model, i.e., it can predict the
survival at a future point in time. For example, the work
of
Loghmanpour et al. (2015)
focuses on risk assessment
models for patients with the left ventricular assist devices
(LVADs). Bayesian network have been shown to estimate
the risk at various points in time (including 30 days, 90
days, 6 months, 1 year, and 2 years) with accuracy higher
than traditional score-based methods
(Loghmanpour et al.,
2015)
. An alternative, more complex approach could use
dynamic Bayesian networks (DBN), which are an
extension of Bayesian networks modeling time explicitly. van
Gerven et al. (2007) implemented a DBN for prognosis of
patients that suffer from low-grade midgut carcinoid tumor.
Instead of treating risk factors independently at each time
point, the DBN model considered how the state of patient
changed under the influence of choices made by physicians.
This model was shown suitable to temporal nature of
medical problems throughout the course of care and provide
detailed prognostic predictions. However, DBNs requires
additional effort during model construction, for example
expertise to structure of temporal interaction, large amounts
of (complete) data, which translates to time-consuming
efforts
(van Gerven et al., 2007)
.
4
Bayesian Network PAH Risk Calculator
BN Interpretation of the PAH Calculator
The original PAH Risk Calculator uses the hazard ratios
in the CPH model to derive the risk score for the
calculator
(Benza et al., 2012)
. We apply the same approach in
our model. Equation 6 captures the survival probabilities s
given the states of risk factor. We can extract a hazard ratio
of each variable by configuring states of other risk factors
to be absent. For example, the hazard ratio of a risk factor
xj can be estimated from
=
log(P r(s |x¯1, . . . , x¯j 1, xj, x¯j+1, . . . , x¯n))
log(P r(s |x¯1, . . . , x¯j 1, x¯j, x¯j+1, . . . , x¯n))
. (7)
The term log(P r(s |x¯1, . . . , x¯j 1, x¯j, x¯j+1, . . . , x¯n)) is
similar to the baseline survival probability in the CPH
model (S0(1) = 0.9698). Hence, with this equation, we
can track back all hazard ratios.
We use the same criteria as the original PAH Risk
Calculator to convert the hazard rate to the score, i.e., score of
2 indicates at least two-fold increase in risk of mortality
compared to the baseline risk.
Figure 4 shows a screen shot of our prototype of the
Bayesian network risk calculator. The left-hand pane
allows for entering risk factors for a given patient case. The
right-hand pane shows the calculated score and survival
probabilities. Currently, our calculator is a Windows app
running on a local server. The numerical risks that
produced by the BN calculator are identical to those of the
original CPH-based PAH Risk Calculator
(Benza et al.,
2012)
.
learned from the REVEAL Registry data and available in
the literature. To this effect, we used a Bayesian network
interpretation of the CPH model
(Kraisangka and Druzdzel,
2014)
.
Our calculator reproduces the results of the current PAH
Risk Calculator exactly. From this point of view, we have
not yet offered a superior calculator. However, we plan to
refine the calculator by (1) learning the parameters of the
BN model from the data captured in the REVEAL
Registry, and (2) enhancing the resulting BN model with
medical expert knowledge. The extended model will relax the
assumption of the multiplicative character of interactions
between the risk factors and the survival variable. It will
also relax the assumption that the risk ratio is constant over
time. Another direction of our work is allowing risk
variables that are not binary. Instead of having 19 binary risk
factors, we will be able to group those risk factors that
are mutually exclusive, e.g., WHO Group or NYHA/WHO
Functional Class. As a result, we can control the number of
risk factors and reduce complexities of the model. Yet
another direction is allowing dependencies between the risk
factors, something that is not straightforward in the CPH
model. We should be able to refine the Bayesian network
model by using expert knowledge or by training its
elements from available data. The current calculator produces
a patient-specific score based on hazard ratio. Because the
new Bayesian network model will no longer use the
multiplicative CPH model, we plan to create new risk score
criteria based on the probability of survival rather than the
hazard ratio. We have little doubt that with some further
modeling effort we should be able to obtain a superior
calculator in the sense of producing higher accuracy of the risk
estimate than the original CPH-based risk calculator.
Acknowledgements
We acknowledge the support the National Institute of
Health under grant number U01HL101066-01 and the
Faculty of Information and Communication Technology,
Mahidol University, Thailand. Implementation of this work
is based on GeNIe and SMILE, a Bayesian inference
engine available free of charge for academic teaching and
research use at http://www.bayesfusion.com/. While we take
full responsibility for any remaining errors and
shortcomings of this paper, we would like to thank anonymous
reviewers for their valuable suggestions.