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Using Probabilistic Logic and the Principle of Maximum Entropy for the
Analysis of Clinical Brain Tumor Data
Julian Varghese1 Christoph Beierle1 Nico Potyka1 Gabriele Kern-Isberner2
1
FernUniversität in Hagen, Germany, 2 Technische Universität Dortmund, Germany
Abstract
Dealing with uncertainty that is inherently present in
any medical domain, is one of the major challenges when
designing a medical decision support system. We demonstrate how probabilistic logic can be used to design medical knowledge bases at the example of analysing clinical
brain tumor data. We use ME CoRe, a system implementing
probabilistic conditional logic, to create a knowledge base
BT that contains medical knowledge originating from both
statistical data as well as from medical experts. Any incomplete or unspecified knowledge is completed by ME CoRe in
an information-theoretically optimal way by employing the
principle of maximum entropy. BT is evaluated with respect
to a series of queries regarding diagnosis and prognosis,
using a real documented patient case.
1
Introduction
Knowledge-based systems encode human expert knowledge in a computer-readable way. In this way, they are
aiming at simulating human decision-making processes and
providing decision support for experts in the field. A particular challenge for medical expert systems is that they
invariably have to deal with uncertain knowledge. This
aspect was already addressed in MYCIN, one of the first
and most popular medical expert systems, which can help
physicians to establish the proper diagnosis and therapy
for patients with infectious disease problems [15]. To deal
with uncertain knowledge, rules contain additional uncertainty factors. However, MYCIN’s uncertainty factors are
a heuristic approach, lacking any well-founded theory explaining its semantics. Another prominent example of a
medical knowledge-based system is Pathfinder [4], which
assists pathologists in the diagnosis of diseases in lymph
nodes. It uses Bayesian networks for knowledge representation, which are based on a comprehensive probabilistic
theory [7]. However, a disadvantage from a knowledge engineering perspective is that Bayesian networks are based
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on complete conditional probability distributions (CPDs),
while often only parts of the CPDs are known. E.g., if we
know a certain disease D causes a certain symptom S with
probability x, we have P (S|D) = x. However, to obtain
a complete CPD, we have to define also P (S|D), i.e., the
conditional probability of the symptom given that the disease is not present. Whereas the former probability can often be obtained from statistical data or can be estimated by
a physician, the latter probability is usually harder to obtain.
The principle of maximum entropy (ME principle) [11, 6]
offers an alternative way to develop a probability-based expert system. Conditionals like P (S|D) = x can be regarded as constraints for a probability distribution. Among
all probability distributions satisfying all given constraints,
the ME principle chooses the unique probability distribution
P ∗ that is the most unbiased one. Hence, the incomplete
information represented by the given conditionals is completed to a full probability distribution over the domain of
interest in an information-theoretically optimal way [11, 6],
and P ∗ can be used to answer arbitrary probabilistic-logical
queries. SPIRIT [14] and ME CoRe [3] are software systems
providing implementations of this framework.
Here, we report on a case study showing how probabilistic logic and the principle of maximum entropy can be used
to model and reason about knowledge in a medical domain
[17]. We look at the analysis of clinical brain tumor data and
represent some core knowledge from this area in a knowledge base BT, using ME CoRe for constructing BT and illustrating ME CoRe’s capabilities of modelling uncertain and
incomplete information. Due to lack of space, for details of
ME CoRe and its underlying methodology we refer to [1].
2 Brain Tumors
In this paper, we use the term brain tumor to refer to intracranial tumors which are tumorous neoplasms localized
in the brain or its meningeal tissues. Two major clinical and
neurophysiological problems are caused by a growing brain
tumor process. One is the local infiltration of tumor tissue
which destroys closely spaced brain tissue. Another one is
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caused by the increase of global intracranial pressure leading to a comprehensive brain damage. This is due to the fact
that the cranium can be seen as a rigid box, since after birth
the cranial fontanels start to ossify leaving the whole brain
with very limited pressure releasing openings.
Clinical Relevance The prevalence of brain tumors is
about 50:100.000 in the middle European region [13]. The
incidence is about 1:10.000 per year. There are two age
peaks, one within the range of 40 years and 70 years, another one within the childhood. Noteworthy, in childhood,
brain tumors are the second most common tumor entity after
leukemia. While adult patients mostly suffer from gliomas,
meningeomas and metastases of other primary tumors, children mostly suffer from medulloblastomas, cerebellar astrozytomas and ependymomas [2].
Guiding symptoms are neurological failures; the brain
tumor itself is mostly confirmed by medical imaging
through CT/MRI-Scans. A histopathological tissue analysis secures the diagnosis and the exact classification-type
and the grading of the brain tumor. Depending on the exact tumor type, the treatment consists of surgical removement and/or chemotherapy. In rare cases of very small tumor sizes and probable benignity surgery can be avoided
if repeated medical imaging over the next months show no
malignant potential.
Classification The international WHO classification [8]
lists all types of known brain tumors according to the origin
of the tumor tissue which is found out through histopathology. Subtypes are listed as refined tumor types according to
cell-analysis such as determining the tumor-grading based
on morphological aspects of neoplastic cells. Cell nucleus
polymorphisms, atypical mitosis, vessel proliferation and
necrosis are such morphologies which indicate the malignancy of the tumor type. Based on that a tumor grading
is classified in four grades as follows: Grade 1 – benign,
Grade 2 – semi-benign, Grade 3 – semi-malignant, Grade 4
– malignant. The grading represents the malignancy potential of the tumor tissue independent from its current infiltration size.
3 Modelling Brain Tumor Knowledge
Variables BT uses the nine propositional variables as listed
in Table 1 together with their possible values. The variable
diagnosis indicates the diagnosis. Its domain corresponds
to the most common brain tumor types like gliomas and
meningiomas [12]. As the probability for certain values like
the type of the tumor depends on the age of the patient, the
variable age classifies patients with respect to three groups.
The first group contains patients with age lower-or-equal
twenty, the second group contains patients with age between
twenty and eighty and the third group contains patients with
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diagnosis – pilocytic-astrocytoma, diffuse-astrocytoma,
anaplastic-astrocytoma, glioblastoma, oligodendroglioma,
ependymoma, meningeoma, medulloblastoma,
cranialnerve-tumor, metastatic-tumor, other
age – le20, 20to80, ge80
warningSymptoms – true, false
malignancy – 1, 2, 3, 4, other
icpSymptoms – true, false
ASA – 1, 2, 3, 4
therapy – conservative, surgical, none
complication – 1, 2, 3
prognosis – very good, good, intermediate, poor, very poor
Table 1. Variables and their values
age greater-or-equal eighty. The variable warningSymptoms
indicates whether certain warning symptoms like perceptual
disturbances or unusual pain in the head can be observed.
Given results of a magnetic resonance tomography
(MRT), the variable malignancy corresponds to the assumed
malignancy of the tumor with respect to the WHO grading
system [9]. A higher index corresponds to a higher malignancy. The variable icpSymptoms indicates whether MRT
results provide symptoms for intracranial pressure (ICP).
The preoperative physical fitness of patients is evaluated
by the ASA (American Society of Anesthesiologists) physical status classification system represented by the variable
ASA. It is associated with perioperative risks. The higher
the value, the higher the risk. We regard only the first four
states, as treatment of a brain tumor is of low priority for a
higher value. The variable therapy denotes a possible therapy. We distinguish a conservative therapy without surgery,
surgery, or no therapy at all.
Possible complications during an inpatient stay are expressed by three degrees by the variable complication. We
distinguish between three stages, higher values correspond
to more serious complications: 1 – no complications or
minor, completely reversible complications like temporary
pain after surgery; 2 – medium or heavy complications with
uncertain reversibility like neurological or other functional
disorders; 3 – life-threatening complications like serious internal bleeding or neurological deficits at the risk of brain
death. Finally, prognosis indicates the expected health of
the patient after inpatient stay.
Rules The ME framework and its support by ME CoRe
allow the definition of flexible conditional rules that can
express statistical facts as well as subjective expert knowledge. Table 2 shows some empirical frequencies of certain brain tumor types collected from different published
sources [2, 5, 10, 16]. By defining our knowledge base appropriately, all given probabilities can be integrated in our
knowledge state. E.g., for adults the diagnosis meningeoma
has a relative frequency of 20%. We can model this empiriCBMS 2013
diagnosis
glioma
- glioblastoma
- pilocytic-astrocytoma
- diffuse-astrocytoma
- anaplastic-astrocytoma
- oligodendroglioma
- ependymoma
meningeoma
medulloblastoma
cranialnerve-tumor
metastatic-tumor
other
Adults
50%
15%
unspecified
10%
10%
10%
4%
20%
7%
7%
10%
unspecified
Children
48%
unspecified
35%
unspecified
unspecified
unspecified
8%
unspecified
25%
unspecified
unspecified
unspecified
Table 2. Empirical brain tumor frequencies
cal observation in ME CoRe by the conditional
(diagnosis = meningeoma | !(age = le20))[0.2].
(1)
where ! is negation. As gliomas appear very frequently
they are further specialized in BT. Table 2 shows that for
adults gliomas have a relative frequency of 50%. Using
(diagnosis = glioma) as an abbreviation for the expression
(diagnosis in {pilocytic-astrocytoma, diffuse-astrocytoma,
anaplastic-astrocytoma, glioblastoma, oligodendroglioma,
ependymoma}), we express the relative frequency of
gliomas in ME CoRe as:
(diagnosis = glioma | !(age = le20))[0.5].
(2)
Analogously, all other probabilities explicitly given in Table 2 induce a corresponding conditional. Additionally, BT
contains probabilistic facts like (age = le20)[0.15] reflecting the age distribution in Germany in the year 2009.
Note, however, that there are some missing frequencies in Table 2. The missing knowledge is completed in
an information-theoretically optimal way by employing the
ME principle, thus by being as unbiased as possible with
respect to each diagnosis with unspecified probability.
Besides available statistical data, another important
knowledge source is the clinical expert knowledge of a
physician. For example, for adults, Table 2 tells us that the
most frequently appearing malignant tumor type is glioblastoma, but no information is provided about its probability
given specific symptoms. An experienced physician working with brain tumor patients might state the following conditionals expressing his expert beliefs about the probability
of a glioblastoma given various observations (where gliob
abbreviates diagnosis = glioblastoma):
(gliob | !(age = le20) ∧ warningSymptoms)[0.2],
(3)
(gliob | !(age = le20) ∧ icpSymptoms)[0.2],
(4)
(gliob | !(age = le20) ∧ (malignancy = 4))[0.4],
(5)
(gliob | !(age = le20) ∧ (malignancy = 3))[0.1],
(6)
(gliob | !(age = le20) ∧ (malignancy = 2))[0.05], (7)
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(gliob | !(age = le20) ∧ (malignancy = 1))[0.01]. (8)
Taking into account only Table 2, the probability for
glioblastoma is 15%. Therefore, given the respective preconditions, rules (3) - (5) increase the probability, whereas
rules (6) - (8) decrease it. ME CoRe allows the smooth and
easy integration of statistical and such expert knowledge
since rules like (3) - (8) can be added directly to BT. Altogether, BT contains 107 conditionals, and its overall semantics is well-defined by the ME principle.
4 Working with BT
We illustrate the use of BT with ME CoRe on the basis
of a real documented patient case. Our patient is 80 years
old, with full consciousness, and he is experiencing amnesic
aphasia since he is complaining about not being able to remember simple words on some occasions. He noticed a
weakness in elevating his right foot and diffuse tingling sensations in the right lower limb. The physical neurological
examination confirmed that and showed furthermore dilated
fixed pupils at both sides. No cranial nerve disorders or further physical neuropathological findings. The MRT reveals
an irregularly formed mass with inhomogeneous and cystic
contrast enhancement localized in the temporal lobe. The
mass is surrounded by a perifocal edema, causing a constriction of the left lateral ventricle and a midline shift. Preexisting illnesses: arterial hypertension. Thus, for our patient the variable warningSymptoms must be set to true, and
we can query the probability of a diagnosis as follows:
BT.query((diagnosis | (age = ge80) ∧
(9)
warningSymptoms)).
(diagnosis = other)[0.0111011134]
(diagnosis = pilocytic-astrocytoma)[0.04996003353]
(diagnosis = diffuse-astrocytoma)[0.09828440105]
(diagnosis = anaplastic-astrocytoma)[0.10608281855]
(diagnosis = glioblastoma)[0.22388719219]
(diagnosis = oligodendroglioma)[0.08626747218]
(diagnosis = ependymoma)[0.03960979843]
(diagnosis = meningeoma)[0.15623784532]
(diagnosis = medulloblastoma)[0.06524103534]
(diagnosis = cranialnerve-tumor)[0.05725318378]
(diagnosis = metastatic-tumor)[0.10607510624]
These results are reasonable from a medical point of view.
In particular, given the observed symptoms, glioblastoma is
indeed the most probable diagnosis from a neurosurgeon’s
perspective. Subsequent MRT results for the patient provided evidence for a malignant tumor and for intracranial
pressure. When working with ME CoRe, this can be taking
into account by adding (malignancy = 4) and icpSymptoms
as additional premises to to the query (9), yielding:
BT.query((diagnosis | (age = ge80) ∧
(10)
warningSymptoms ∧(malignancy = 4) ∧ icpSymptoms).
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Query (10) results in modified probabilities for diagnosis. In
particular, with (diagnosis = glioblastoma)[0.5562571052],
glioblastoma now dominates clearly, having an increased
probability of more than 55%, which is again very plausible
from a medical expert point of view.
As the symptoms indicated a malignant tumor, it was surgically removed. Neuropathological results proved the diagnosis glioblastoma. After the operation, amnesic aphasia
changed for the worse. The remaining neurological deficiencies remained unaffected. The new evidence can be expressed in ME CoRe using (diagnosis = glioblastoma), expressing the diagnostic findings, and (therapy = surgical)
and (complication = 2). We can ask ME CoRe for the probability of post-operative complications when taking into account this evidence with the query
BT.query((complication | (diagnosis = glioblastoma) ∧
(age = ge80) ∧ warningSymptoms ∧ (malignancy = 4)
∧ icpSymptoms ∧ (therapy = surgical))).
for which we obtain the following results:
(complication = 1)[0.00113586771]
(11)
(complication = 2)[0.32888273171]
(12)
(complication = 3)[0.66998140057]
(13)
While complications of grade 2 or 3 are rare in general, the
evidence caused ME CoRe to rise the probabilities for these
types of complications considerably. For the given patient,
there was indeed a complication of grade two, for which
ME CoRe determined a probability of 33%. From a clinical
perspective, the probabilities for complication computed by
ME CoRe is an adequate warning; however, from a medical
point of view, the probability for (complication = 3) is too
pessimistic, since compared to similar patient-risk constellations, life-threatening complications are frequent, but less
than 50%, indicating that a more fine-grained modelling of
complications would be adequate. Further types of queries
for BT are reported in [17]. Asking ME CoRe for the expected health of the patient after inpatient stay, returned a
very realistic prognosis from a medical point of view [17].
5
Conclusion
From a knowledge representation point of view, dealing
with uncertainty that is inherently present in any medical
domain, is one of the major challenges when designing a
medical decision support system. Combining logic with
well-established concepts of probability theory provides a
framework for modelling uncertain knowledge and for reasoning about it. In this paper, we reported on a case study
in which we used probabilistic logic and the principle of
maximum entropy in a medical domain. We showed how to
model clinical brain tumor data and medical export knowledge from this area with probabilistic conditionals. The employed principle of maximum entropy completes any miss404
ing or unknown probabilities to a full probability distribution in an information-theoretically optimal way. Our current work includes extending BT by a more fine-grained
modelling, involving more variables and rules, and further
evaluating BT with respect to additional real world examples.
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