









































Abstract: Correctly identifying when a hemorrhagic patient needs immediate medical attention to prevent acute 
hypotensive episode (AHE) is vital in the short- and long-term care, but is often complicated due to the physiological 
responses in the sympathetic and parasympathetic nervous systems that mask symptoms until a significant amount 
of blood loss has occurred. These physiological responses affect the arterial blood pressure waveform, changing 
both dynamics and waveform morphology. Through the use of Markov chain analysis of the arterial blood pressure 
waveform, we first analyzed patient blood pressure waveforms from a challenge dataset published by Computing 
in Cardiology 2009 and the MIMIC III database. Markov chain analysis was applied to 20-second intervals over 
the entirety of a patient’s known acute hypotensive episode. Each interval or segment is one second apart from 
the previous segment with a nineteen second overlap. The mixing rate (2nd largest eigenvalue of the transition 
probability matrix) was determined for all segments. A subset of patients showed a Pearson correlation coefficient 
with shock index (SI), i.e., with the ratio of heart rate and systolic blood pressure, similar to a previous swine study. 
These patients (mean correlation coefficient -0.423 ± 0.32, median -0.352) were found to have been administered 
pressors (vasoconstrictors), compared to patients who were not administered pressors (mean correlation coefficient 
0.392 ± 0.29, median 0.447). Patients were also analyzed based on diagnoses of gastrointestinal bleeding by the ICD-
9 code, and mixing rate results were compared between patients in this subgroup and found to have no significance 
as a metric of predicting acute hypotensive episodes.

Aisthesis      Volume 9,  201837

Second Largest Eigenvalue of the Transition Probability Matrix for 
the Markov Chain Constructed from the Arterial Blood Pressure 
Waveform is Not Correlated to Shock Index in Hemorrhagic 
Human Subjects

by Brett Collar, Marvi Bikak, Paul Brown, Zachary Pitluk, and Mohammad 
Adibuzzaman

Introduction
Hemorrhagic shock is a condition produced by 

rapid and significant loss of intravascular volume, 
which may lead sequentially to hemodynamic 
instability, decreases in oxygen delivery, decreased 
tissue perfusion, cellular hypoxia, organ damage, 
and death. The primary goals are to stop the 
bleeding and to restore circulating blood volume, 
as hemorrhagic shock can rapidly become fatal. 
Resuscitation may well depend on the estimated 
severity of hemorrhage. Hemorrhage results in 
over 80% of operating room deaths after major 
trauma and almost 50% of all deaths in the first 24 
hours of trauma care [1]. Identifying occurrence of 
hemorrhage in a timely manner is the cornerstone of 
medical and surgical management. However, one of 
the major limitations in accurately identifying a state 
of hemorrhage is the poor predictive ability of heart 
rate, mean arterial pressure, and shock index [2]. 
These are the most commonly used physiological 
parameters which help guide clinicians to diagnose 
hemorrhage. In other words,  developing a method 

to identify patients that require immediate medical 
care is of critical importance [2].

Heart rate variability is one such method with the 
potential to identify hemodynamic instability caused 
by hemorrhage [3]. Previous literature suggests that 
reductions in vagal activity assessed with heart rate 
variability or baroreflex sequences may represent 
identifiable early markers of hemorrhage [4]. 
Hypovolemia triggers a simultaneous reduction of 
the parasympathetic nervous system and activation 
of the sympathetic nervous system in an attempt 
to increase heart rate and compensate the drop in 
blood pressure. As a result, clinical signs present 
during early stages of hemorrhage may be ignored 
due to the mean arterial pressure remaining stable. 
Only after a significant loss of blood will a change 
in blood pressure be identified, at which point 
medical interventions could be limited. Clinically, 
abnormal shock index values, defined as the ratio 
of heart rate (HR) to systolic blood pressure (SBP), 
have been demonstrated to portend worse outcomes 
in traumatically injured patients [5]. Markov chain 



Second Largest Eigenvalue of the Transition Probability Matrix 

Aisthesis      Volume 9,  201838

methods may describe changes in hemodynamic 
instability prior to traditional vital signs, potentially 
providing an early indicator of hemorrhage.

A Markov chain is defined as a system with 
different states, where the transition probability from 
one state to the next depends only on the current 
state, the Markov assumption [6]. Regular Markov 
chains have a limit distribution or steady state, 
where the mixing rate of a Markov chain represents 
how fast the system is approaching the steady state. 
Eigenvalues of Markov chains display changes in 
system dynamics that are not captured by other 
nonlinear methods, such as Poincaré plots [4]. These 
eigenvalues of Markov chains represent changes 
in the dynamics of arterial blood pressure (ABP) 
waveforms as the body attempts to compensate for 
hypovolemia. An empirical Markov chain can be 
constructed from ABP recordings. As the system 
dynamics and waveform morphology change, the 
system will approach steady state faster or slower 
and this can be observed through the mixing rate. 
An animal study on hemorrhagic swine showed 
strong correlation with shock index and the mixing 
rate of the arterial blood pressure [7].

We studied the applicability of the method in 
hemorrhagic human subjects from the intensive care 
unit using two different data sets. We hypothesized 
that if a specific subset of patients was found to 
exhibit strong correlations between mixing rate and 
traditional biomarkers, these patients would share 
a demographic detail that explained the behavior. 
Additionally, a larger patient population collectively 
sharing similar clinical diagnoses was analyzed 
to determine whether specific conditions affected 
clinical findings (mean correlation coefficient with SI 
-0.0087 ± 0.19, median 0.0080). We also provide the 
theoretical foundation for the change in the second 
largest eigenvalue from time series data.

Methods
A. Previous Work

A protocol was approved and performed on a 
swine model to test the efficacy of the Markov chain 
mixing rate as a metric for detecting hemorrhage 
prior to noticeable changes in traditional vital signs 
[7]. Immature swine (N=7, female, 37.1 ± 15.1 kg 
(mean ± SD)) were anesthetized and instrumented 
with bilateral catheters in femoral arteries and 

veins. Data were collected during a continuous 
hemorrhage of 10 ml/kg over 30 minutes. Heart rate 
and beat-by-beat blood pressures (systolic, diastolic, 
mean) were calculated from the ABP waveform, with 
shock index (SI) calculated by dividing heart rate by 
systolic blood pressure.

Using a high-frequency (125 Hz) arterial blood 
pressure waveform, an empirical Markov chain 
was created from a 20-second period of data by 
segmenting the range of blood pressures over a 
fixed number of states, each covering an equal range 
of pressures. From there, a transition probability 
matrix was computed as the probability that blood 
pressure will enter any state given only its current 
state. The matrix was normalized by dividing each 
row by the cumulative sum of the row to have a 
probability distribution. Finally, eigenvalues were 
determined from the transpose of the transition 
probability matrix, and the eigenvalue with the 
second largest magnitude was defined as the mixing 
rate. The arterial blood pressure waveform was then 
advanced by one second (125 samples), and the 
process was repeated to find the next Markov chain 
mixing rate. These mixing rates were collected in 
a list and graphed with respect to time to compare 
with traditional vital signs.

Pearson correlation coefficients were determined 
between the mixing rate and each vital sign (heart 
rate and systolic blood pressure). The mixing rate 
and high-frequency ABP waveforms (determined at 
125 Hz) were then smoothed using a moving average 
filter (100 samples window) before computing 
the correlation coefficients. In an anesthetized pig 
model, the Markov chain mixing rate of the ABP 
waveform is strongly correlated with the vital signs 
(mean correlation coefficient with SI -0.889 ± 0.143, 
median -0.95) [7]. The relationship between the 
mixing rate and traditional vital signs suggests that 
this new marker might be an indicator of impending 
hemodynamic imbalance.

B. Computing in Cardiology Challenge Dataset
To evaluate the efficacy of the marker on humans 

subjects, a dataset published by Computing in 
Cardiology Challenge (CinC) 2009 and the Medical 
Information Mart for Intensive Care – III (MIMIC 
III) database was used [8, 9]. One limitation of 
the challenge data set was that it does not have 



Second Largest Eigenvalue of the Transition Probability Matrix 

Aisthesis      Volume 9,  201839

information regarding hemorrhage in the data, and 
acute hypotensive episode was defined as a 30-minute 
period in which mean arterial pressure was less than 
60 mmHg and greater than 10 mmHg for at least 90% 
of the 30-minute period. A total of 30 patient blood 
pressure waveforms were obtained and had mixing 
rates calculated over a period immediately preceding 
a known acute hypotensive episode (AHE).

As shown in Figure 1, ABP waveforms 
corresponding to each episode were downloaded 
over a 70 minute window; the first 10 minutes were 
used as a noncritical observation window to establish 
a baseline, and the AHE of interest occurred ~30 
minutes into the proceeding 60-minute forecast 
window. These data were analyzed using a MATLAB 
script to compute Markov chain mixing rates, and 
Pearson correlation coefficients were determined 
between mixing rate and traditional vital signs 
(HR, ABP). Additionally, the correlation coefficient 
between mixing rate and shock index was calculated 
as a metric of the predictive capabilities of mixing 
rate against clinical triage criteria.

Fig. 1. An example of an arterial blood pressure waveform 
analyzed for Markov chain mixing rates. A 10-min. 
baseline was used, with the AHE occurring approximately 
30 min. into the given 60-min. forecast window.
   
C. GI Bleeding Dataset

To find hemorrhagic patients based on clinical 
diagnosis, and not by AHE (as was the case for the 
challenge data set), a comprehensive search of the 
MIMIC III database was performed in order to 
identify and analyze additional patients suffering 
from acute hypotensive episodes. Using ICD-9 codes 
across the entire MIMIC III database, patients were 
identified based on their respective diagnoses, and 
data for patients with high-frequency waveform 
were recorded. For this analysis, the ICD-9 code 
578.9 (gastrointestinal bleeding) was chosen, as 
GI bleeding was considered to be the most similar 
in behavior to the controlled hemorrhage of the 

previous swine study. However, while clinical data 
may exist for individual patients, it is imperative 
that high-frequency physiological signals, more 
specifically, ABP, are also present in the waveform 
database, as only a subset of the subjects from the 
clinical data have their high frequency waveform data 
in the matched subset [10]. After searching the entire 
MIMIC III database, 938 patients were diagnosed 
with ICD-9 code 578.9, and of those patient records, 
162 contained matching high-frequency waveform 
data.

The MIMIC:SciDB platform (v16.9, Paradigm4, 
Waltham, MA), was created by Paradigm4 in 
collaboration with Regenstrief Center for Healthcare 
Engineering (RCHE) and hosted at RCHE. 
Extensive cleaning of the MIMIC time series data 
was necessary to ensure the integrity of the data and 
flag missing data. MIMIC:SciDB stores the 10 TB 
of time series data in arrays with dimensions [time, 
intensity, patient ID]. Each cell in the 3-dimensional 
array contains a value for ECG, HR, ABP, and SpO2, 
among others. Using SciDB’s streaming API for 
parallel distributed computation, an algorithm was 
applied to a subject’s entire hospital stay waveform to 
identify all episodes of acute hypotension, defined as 
a 30-minute period during which the mean arterial 
pressure was below 60 mmHg for at least 90% of the 
time [11, 12]. An R program was created to locate the 
corresponding high-frequency data and download 
the relevant data over the identified window of 
AHE, including a 30-minute baseline prior to AHE 
onset. Mixing rate analysis was performed for each 
episode, and correlation coefficients were calculated 
between mixing rate and vital signs (HR, ABP), as 
well as against shock index.
   
Results
A. Computing in Cardiology Challenge Dataset

Analysis of the corresponding results (Figure 
2) showed a small subset of patients displayed 
correlation coefficients similar to the previous animal 
study, a result that could possibly be explained by the 
administration of pressors (vasoconstrictors). Of 
the selected cohort, patients who were administered 
pressors accounted for 5 of the 7 largest decreases 
in Markov chain mixing rate, as shown in Table 1 
(see Appendix). Conversely, patients who were 



Second Largest Eigenvalue of the Transition Probability Matrix 

Aisthesis      Volume 9,  201840

not administered pressors accounted for 4 of the 6 
smallest decreases in mixing rate, as shown in Table 
2 (see Appendix). 

The correlation coefficients between the 
mixing rate and the vital signs of patients who 
were administered pressors show a resemblance to 
the results obtained in the previous animal study 
of swine that were administered pressors and 
underwent controlled hemorrhage. The mixing 
rate was inversely correlated with heart rate and 
shock index, and showed a positive correlation to 
systolic blood pressure. In the subset of patients 
who were not administered pressors, virtually none 
of the previous findings hold true. Patients showed 
a negative correlation with systolic blood pressure 
and a positive correlation with shock index, with 
no significant correlation between mixing rate and 
heart rate.

Fig. 2. High-frequency mixing rate of patients provided by 
the CinC Challenge dataset. The blue line is the average of 
all patient data over the course of 60 minutes. Hemorrhage 
starts at time 0 (red vertical line).

B. GI Bleeding Dataset
Of the 162 patients identified in the matched 

subset, 47 were analyzed for heart rate variability. Of 
those, 8 patients were found to suffer from episodes 
of acute hypotension, with a total of 81 hypotensive 
episodes identified within the patient subset. 
However, after analyzing all hypotensive episodes, 
correlation values greater than |±0.50| accounted for 
4 out of 243 measurements (1.6%) between mixing 
rate and HR, SBP, and SI. Additionally, correlations 
values between |±0.25| and |±0.50| accounted for 19 
out of 243 measurements (7.8%) between mixing 
rate and HR, SBP, and SI.

Mathematical Justification 
For the changes in the mixing rate, we 

hypothesized that the mixing rate change is observed 
in the Markov chain because of the changes in the 
density (number of non-zero elements of the matrix 
divided by the total number of elements) and the 
self-transition probability (the summation of the 
probabilities that each state will stay at the same 
state) of the chain. We provide the mathematical 
justification of the hypothesis using Gershgorin 
circle theorem that describes the relationship 
between eigenvalues and the structural properties of 
the transition matrix [13].

Gershgorin Circle Theorem
Let B be an arbitrary matrix. Then the eigenvalues λ 
of B are located in the union of the n disks,

Where bkk is the diagonal element (self-transition 
probability) and bkj is the non-diagonal elements for 
each row (related to density) of the matrix.

For the analysis, from equation (1), we get,

From (2) and (3),

The construction of the Markov chain is such 
that the summation of each row is 1. As a result,
   



Second Largest Eigenvalue of the Transition Probability Matrix 

Aisthesis      Volume 9,  201841

As more states transition to the other states, 
the self-transition probability bkk decreases and  
increases, resulting in a decrease in the lower 
bound of this inequality. This is what we observed 
in the empirical analysis after the hemorrhage. We 
conclude that the lower bound for the second largest 
eigenvalue decreases with progressive hemorrhage 
in anesthetic swine, as captured by the algorithm. 
But the same conclusion did not hold true for 
hemorrhagic human subjects, as defined by our 
study.

Discussion
In the CinC 2009 Challenge dataset, the Markov 

chain mixing rate of the ABP waveform did not 
match the results of the previous animal study. 
However, the mixing rate has a higher correlation 
coefficient with the shock index for the patients 
who were administered pressors, suggesting patients 
administered pressors benefit more from Markov 
chain analysis than those not administered pressors.  
For patients administered pressors, the mixing rate 
had an average correlation of 0.43 with arterial blood 
pressure, -0.194 with heart rate, and -0.423 with 
shock index (5 patients analyzed). Among patients 
not administered pressors (4 patients analyzed), 
the mean correlation coefficient was -0.386 ABP, 
-0.075 with heart rate, and 0.392 with shock index. 
These correlations do not reflect the findings of the 
previous study.

There are limitations for the data sets that we 
used. For the challenge data set, the patients were 
assumed to have an acute hypotensive episode as a 
consequence of hemorrhage, which is not always true. 
Oftentimes, diagnoses of coronary artery disease, 
congestive heart disease, or similar complications 
can lead to symptoms of hypotensive episodes. For 
the GI bleeding data set, the patients were identified 
using ICD-9 code for GI bleeding, and an algorithm 
was applied to define the time for acute hypotensive 
episode from arterial blood pressure data. It is possible 
that a patient experiencing an acute hypotensive 
episode was not having hemorrhage, but rather had 
another clinical diagnosis, such as septic shock, 
hypotension resulting from adrenal insufficiency, or 
as a result of abdominal compartment syndrome. All 
of the aforementioned conditions can be expected in 
a patient with a GI bleeding diagnosis.

The relationship between higher correlation 
coefficients for patients with the administration 
of pressors might be a useful factor to consider for 
future Markov chain analyses. Hypovolemia ensues 
when intravascular plasma volume is considerably 
low. This results in hypotension, which is clinically 
managed by volume administration (e.g., saline 
products or blood products). However, if this does 
not resolve hypotension, pressors are administered, 
increasing the systemic vascular resistance and 
hence the blood pressure. Oftentimes, medications 
administered for anesthesia can result in refractory 
hypotension requiring pressors, meaning that the 
patients who were administered pressors may have 
been experiencing similar symptoms to the previous 
animal study, leading to the comparable decreases 
in mixing rate and correlation coefficients between 
studies.

For the data analyzed of patients diagnosed 
with ICD-9 codes for gastrointestinal bleeding, 
correlation coefficients greater than ±0.50 accounted 
for 1.6% of all vital sign measurements. Correlation 
coefficients between ±0.25 and ±0.50 accounted for 
7.8% of all vital sign measurements. From the data 
currently analyzed within this patient subset, the 
results do not provide enough evidence to support 
the claim that mixing rate is a predictive metric for 
hypotensive events.

References
[1] D. S. Kauvar, R. Lefering, and C. E. Wade, 

“Impact of hemorrhage on trauma outcome: an 
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[2] M. Y. Rady, E. P. Rivers, and R. M. Nowak, 
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[3] J. F. Thayer, S. S. Yamamoto, and J. F. Brosschot, 
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Aisthesis      Volume 9,  201842

[4] S. Merrill, “Markov chains for identifying 
nonlinear dynamics,” Nonlinear Dynamical 
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Real Data, pp. 401-423, 2010.

[5] C. M. Cannon, C. C. Braxton, M. Kling-Smith, 
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[6] S. Merrill and J. Cochran, “Markov chain 
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[7] M. Adibuzzaman, G. C. Kramer, L. Galeotti, 
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[8] “Predicting Acute Hyptotensive Episodes: 
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[9] A. Johnson et al., “MIMIC-III, a freely accessible 
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[10] M. Adibuzzaman, K. Musselman, A. Johnson, 
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[11] J. Lee and R. G. Mark, “An investigation 
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[12] P. Brown and M. Adibuzzaman, ed, 2018.

[13] W. D. James, Applied Numerical Linear Algebra. 
Philadelphia, USA: SIAM, 1997.

   



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Aisthesis      Volume 9,  2018

Table 1. Correlation coefficients between mixing rate and vital signs during hemorrhage in patients who were 
administered pressors.

Table 2. Correlation coefficients between mixing rate and vital signs during hemorrhage in patients who were not 
administered pressors.

Appendix

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