Applied Mathematics, 2011, 2, 508-513
doi:10.4236/am.2011.24066 Published Online April 2011 (http://www.SciRP.org/journal/am)
Actively Circulating Volume as a Consequence of
Stochasticity within Microcirculation
Viktor V. Kislukhin
Transonic Systems Inc., Ithaca, USA
Received March 6, 2011; revised March 13, 2011; accepted March 16, 2011
Abstract
It is well established that in the pathology of the cardio-vascular system (CVS) only a portion of the blood
volume (BV) can be in active circulation. This portion of BV is named the actively circulating volume (ACV)
and is evaluated from a monotone decrease of dilution curve produced by an intravascular tracer. In given
paper is presented Markov chain as a math model of the flow of a tracer throughout CVS. The consideration
of CVS as a set of segments with respect to an anatomical structure and assuming the existence for CVS
steady-state condition; leads to the Markov chain of the finite order with constant coefficients. The conclu-
sions of the article are 1) there are open and closed microvessels, such that the switching from open to closed
and back is a stochastic process, 2) if the switching is slow then the ACV, as the volume of heart chambers
and only open for circulation vessels, can be detected.
Keywords: Blood Volume, Actively Circulating Volume, Microcirculation, Vasomotion, Markov Chain,
Math Model of Cardiovascular System
1. Introduction
The importance of knowing BV is commonly accepted.
However, BV is not routinely used in clinics or during
experimental investigations. The primary drawback to
measuring BV is that the mixing time for a tracer can
vary from 2 - 3 min to 30 min [1]. Multiple blood-samp-
ling method had been developed to solve the mixing di-
lemma. However, as stated by Wiggers [2], if mixing of a
tracer requires more then 10 min, the resulting volume is
not the volume responsible for cardiac output and the
distribution of blood pressure. Thus, the concept that only
part of the BV is actively circulating was developed [2],
meaning that the BV separates into ACV and slow circu-
lating volume (SCV). The analysis of blood sample data
is based on a two-compartment representation of BV,
such that within the ACV a tracer mixes instantaneously,
and a slow exchange occurs between ACV and SCV [3,4].
The calculation of ACV is based on the back extrapola-
tion of the indicator’s concentration decay to the time of
injection [4,5]. ACV could be up to 50% of BV [5,6]. In
this paper we address the question: what could be a cause
for the monotone drop of the concentration toward a
steady state concentration (this concentration is used for
BV calculation [1]). To answer this question we exploit
the hypothesis made by Romanovsky in his monograph
“Discrete Markov Chain” [6] that “the movement of
blood particles in a human organism where the heart is
the central point of branching, is a Markov (polycyclic)
process.”
A mathematical description of a tracer passing through
the CVS began with the work of Stephenson [7]. He sug-
gested that a dilution curve could be interpreted as a dis-
tribution of the time it takes for an indicator to pass thr-
ough an organ. His approach was further developed by
Meier & Zierler [8]. They established and generalized
the relationship among mean transit time, flow, and blood
volume based on the interpretation of dilution curves as a
convolution of an input with a distribution of transit time.
Application of operational methods, such as Laplace
transform, to the distribution of time to pass different
segments of the CVS leads to the system of linear equa-
tions for description of the evolution of a tracer through-
out CVS [9]. If time is measured in car-diac-cycles and
the CVS is a finite set of segments with respect to an
anatomical structure then the flow of blood (and a tracer)
can be described by a finite matrix A. The spectral de-
composition of A leads to the conclusion: the concentra-
tion of an intravascular tracer in any systemic artery is
described by the sum of three terms: 1) the steady-state

V. V. KISLUKHIN 509
term corresponding to the complete mixing of a tracer; 2)
the term of damped oscillations corresponding to the first
pass and recirculation waves, and 3) the term of steadily
(exponentially) decreasing items. An analysis of the con-
ditions that enables the monotone decreasing term leads
to the conclusion: the appearance of the SCV is due to
the presence of closed microvessels and, additionally, the
switching of the microvessels from the closed state to the
open state and back is a slow process.
2. Mathematical Model for the Passage of an
Intravascular Tracer
We begin with the assumption: the future trajectory of
any blood particle depends only on the current site of the
particle. A model based on the given assumption is a
Markov chain [10] if it includes the following three com-
ponents: 1) a structure of the CVS, 2) a distribution of a
tracer throughout the CVS, and 3) an operator of the
transition of the tracer throughout the CVS. In detail:
1) A structure of the CVS. It is a set of segments {Sk; k
= 1 ··· N}, such as the heart chambers, conductive vessels
and microvessels. The segments are enumerated. The
numeration starts with the right atrium (RA) designated
as S1. The numeration of other segments follows the rule
that blood flows from the segments with lower subscripts
to segments with the higher subscripts. Only the segments
connected with the RA are exceptions to this rule. Fig-
ure 1 demonstrated the numeration of segments begin-
ning at the left ventricle.
2) Distribution of a tracer throughout the CVS. The
distribution is a vector z(t) = {zk(t), k = 1, ···, N}, where
the kth component of z(t) is the fraction of a tracer within
the Sk at time t. As an initial distribution of a tracer, z(0),
will be taken z1(0) = 1, and for all k > 1 zk(0) = 0, mean-
ing that a tracer is injected into the right atrium at time t
= 0,
3) An operator A = {aij} that provides the transition of
a tracer during one cardiac cycle, where the aij is the
fraction of a tracer within Si that passes during one car-
diac-cycle into Sj: As a result the distribution of the trac-
er at time t, z(t), transforms to the distribution at time t +
1: z(t + 1) = z(t)A, and, recursively:
Figure 1. A possible numeration of the segments beginning
with the left ventricle.


0t t
z
zA (1)
The text-book approach to dealing with (1) is to ex-
pand it through the characteristic numbers, {si}, (the
roots of the equation Det(sA − E) = 0). Thus the dilution
curve recorded in the aorta, zm(t), is the power series of
three components:



1
1
1cos
t
t
mmmii mjjj
zt bbsbst



 
 (2)
where, bm1, {bmi}, {bmj}, and {ωj} are the combinations
of eigenvectors of matrix A [10].
The examination of the components of (2) leads to the
following:
1) The constant, bm1, corresponds to the concentration
of tracer after the mixing has been complited.
2) The second term, with all si real and > 1, is the
steadily decreasing term; it will be connected with the
detection of ACV.
3) The damped oscillating term, with the frequencies
of oscillations {ωj}. All sj are the complex numbers and,
by modulus, > 1.
3. Results
The main conclusion of this article is the consequence of
the statement: if diagonal elements of A, are zero {aii = 0}
then the equation Det(sA – E) = 0 has only one real solu-
tion, s1 = 1. The proof follows from the statement that
two equations Det(sA – E) = 0 and F(s)=1 are equivalent
(see Appendix 1) (F(s) is the generating function of the
first pass throughout CVS). The equation for F(s) is, see
(A2):


1
k
k
Fs ps


 (3)
where pk is the fraction of a tracer that passes through the
CVS (from RA to RA) in k-cardiac-cycles. Since the pk
add to 1, then s = 1 is the only real positive characteristic
number. Taylor decomposition of (3) at s = 1, leads to
the other characteristic numbers. They are

121;1,
k
skiFk 

 .
with


1F

as the mean transit time (MTT) for passage
throughout CVS. Conclusion: thus, zm(t), see (2), has
only damped oscillations around bm1 and the frequencies
of damped oscillations are multiples of


21F 

.
In other words, to have a steadily decreasing term in (2)
we must have non-zero elements on the main diagonal of
A. There are at least four aii > 0, and they correspond to
the heart chambers. However, the mean time to pass any
heart chamber is about 2 - 3 cardio-cycles (in pathologi-
cal enlargement of the heart the time to pass can be up to
20 cm3), thus the heart cannot be the cause of a monotone
drop with the half-time 3 - 9 min. Consequently, there

V. V. KISLUKHIN 510
must be non-heart elements aii > 0. The passage through
such segments can be described as follows: if in the i-
segment a tracer stays for a while, we should have at least
two segments, let them be numbered (i − 1) and (i + 1),
such that the tracer enters i-segment from (i − 1) and lea-
ves to (i + 1)-segment. Formally, from the (i − 1)-seg-
ment a tracer partly enters the i-segment and could partly
enter the (i + 1)-segment, these parts are ai − 1i and ai − 1i + 1,
and will be denoted as  and  = 1 −  . A tracer from i-
segment partly stays for the next cardiac-cycle, and partly
passes to the (i + 1)-segment, these parts are aii and aii + 1,
and are denoted as
 and  = 1 −  . The generating func-
tion to pass such construction is given by (4), see Appen-
dix 2.

;
1
s
vss s
s




 (4)
There are two realizations of the formal construction,
see Figure 2:
1) Required segment (with aii > 0) contains microves-
sels closed for circulation, and (i − 1)-segment contains
perfused microvessels. When closed microvessel beco-
mes open its content passes to (i + 1) segment.
2) Segment (or a group of segments) is a mixing ch-
amber, kind of “peripheral heart”.
The following reasoning is based on the first realize-
tion. The first realization is chosen because 1) it is well
established that practically in all tissues a part of micro-
vessels is closed, and their recruitment is a way to res-
pond to an increase in flow [11,12]; 2) in muscle tissue it
is established that a fraction of ink-containing capillaries
depends on the time of infusion of ink. For 4 sec of the
infusion the fraction of ink-containing capillaries is about
12%, and for 90 sec infusion there are 90% of ink-con-
taining capillaries [13]; 3) despite the presence of a kind
of “peripheral” hearts, they fall far from the needed rela-
tion [volume]/[flow] be 3 - 9 min.
With non zero diagonal elements, the expression for a
generating function for the passage of the CVS should
include 1) the passage through the heart chambers with
the generating function as
4
11
j
j
j
bs
as

, where aj and bj
are residual and ejection fractions of j-heart chamber; 2)
the passage through the systemic and pulmonary conduc-
tive vessels; and 3) the passage through the microcircula-
tion. The (5) gives the combined generating function,
F(s), to pass throughout CVS, where F1(s) includes the
passage of the heart and conductive vessels, and the ex-
pression in the brackets is the generating function of the
passage through microcirculation, p1 + p2 = 1.
 
11322 1
s
FsF spFspFss
s

 


 
 



(5)
Additionally to F(s), given by (5), we introduce the ge-
nerating function, F0(s), to pass CVS if there is no
switching of the state (open/closed) of microvessels (  =
0 and  = 1, Figure 2):
(a)
(b)
Figure 2. Two possible realizations for non-zero elements from main diagonal of A. (a) Schematic for stochastic ex-
change between open and closed microvessels; (b) Schematic for the passing of microvessels as a mixing chamber.

V. V. KISLUKHIN 511
 

011322
F
sFspFspFs  (6)
Thus, our model of CVS has two distinct blood vol-
umes:
1) Total blood volume, BV. By using established by
Meier & Zierler [8] the relationship among mean transit
time, flow, and volume we have:

1BV FSV

, with SV as the stroke volume; (7)
2) A second blood volume, ACV, as the volume of
open for circulation segments of CVS, such as heart
chambers, conductive vessels, and open for flow micro-
vessels. From (6):


1
o
A
CV FSV


. (8)
Now the aim of the article can be formulated as fol-
lows: the volume given by (8) and the volume obtained
by back extrapolation, if monotone decrease of zm(t) ex-
ists are the same. In Appendix 3, there are derivations of
the next parameters:
1) The expression for a concentration after complete
mixing occurs,

111 m
bF

.
 
1
1
1
mm
SV
zb
F
BV
 
; (9)
2) The expression for the real characteristic number,
s2 > 1,
21BV
s
A
CV

 (10)
with ACV as the volume given by (8)
3) The term

22
1
m
bFs

 that is the factor at s2, and
the term

22
1
tt
m
bs Fss

 is responsible for mono-
tone decrease of zm(t). The back extrapolation of zm(t):
12
12
mm
mm
SV SV
bb ACV
A
CVb b
 (11)
By compare (8) and (11) one can conclude that ACV
as the volume of the heart, conductive vessels, and open
microcirculation and ACV obtained by the back extrapo-
lation of zm(t) are the same.
From (10) one has the condition to have a clear mo-
notone decrease of the concentration of the intravascular
tracer toward the steady state (and consequently to have
opportunity to measure ACV): the  should be small,
such as 1/ ~ 3 - 5 min (after cardiac cycles are transfor-
med into minutes).
The volume SCV = BV – ACV with minutes consti-
tuting the mean time of returning to the circulation could
be used as the explanation for the disorder: 1) the bends
from the removal of N2 since nitrogen in the tissue around
microvessels constituent SCV has slow removal; 2) the
urea rebound, since removing of the urea in patients un-
der dialysis treatment from the tissue around of SCV is
delayed [14]. Thus, the appearing of monotone decrease
of zm(t) could be a the sign of microcirculation disorder.
There is a high probability that different parts of the
microcirculation have different characteristics in the
change of the state (open-closed) of microvessels. Con-
sequently, (5) transforms into:
 
113
21
with 1
Kj
jj jj
jj
i
s
FsF spFspFss
s
p

 




 









(12)
This poses the main problem with the traditional me-
thod for obtaining ACV. From Appendix 3 it follows that
different  in (12) lead to different real characteristic
numbers, and the back extrapolation becomes dependent
on the chosen time interval, the phenomena observed in
the measurements of ACV [15].
4. Discussion
The main assumption, that leads to a Markov chain as a
model for the transition of a trace throughout the CVS, is
that every sequence of segments from the aorta to the
right atrium and from the pulmonary trunk to the left
atrium can be presented as a finite set. Two other assum-
ptions are less significant. However, they simplify calcu-
lations: 1) the stability of hemodynamic, meaning that
matrix A is a constant matrix, and 2) the velocity of blood
is the same throughout the cross-section of any vessel.
Since the work of Krogh it has been well established
that the recruitment of microvessels is the leading res-
ponse of the tissue to the demand for nutrients [11,16].
Experiments with ink infusion [13] have demonstrated
that the longer the infusion time the more microvessels
are exposed to infused particles. Thus, the indirect evi-
dence for the involvement of closed microvessels into the
circulation under the steady-state conditions is establi-
shed.
Back extrapolation of a tracer’s decreased concentra-
tion is a way to obtain ACV [5,17]. The experiments and
analysis presented in [15] point out the relatively low
reliability of the back extrapolation, meaning that ACV
depends on the time chosen for the back extrapolation.
Consequently, it is plausible to suggest that different or-
gans have different microcirculatory characteristics, thus
a calculations based on a two-compartment presentation
of CVS can produce low repeatability. The existence of
microvessels that are out of circulation for more than 2 -
3 min should be considered when the rate and dosage of
a drug is chosen.

V. V. KISLUKHIN 512
5. Conclusion
ACV as the volume of heart chambers and only open for
circulation vessels can be detected if the switching pro-
cess is slow.
6. Competing Interests
The author declares that he has no competing interests.
7. References
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[15] T. Schroder, U. Rosler, I. Frerichs, G. Hahn, J. Ennker
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1163.

V. V. KISLUKHIN 513
Appendix 1
The expression for the determinant of the matrix B = sA
– E, if all main diagonal elements of A are zeroes.
To obtain the DetB let take b11= –1. By taking b11 we
are forced to take only the elements from the main di-
agonal, thus the first term of DetB is (–1)N. To get other
terms of DetB let take the second non-zero element, sa12,
of the first row. The choice of next elements follows the
repeatable procedure: 1) if the element saij is chosen, the
next element should be taken from j-row; 2) if in j-row
there is the choice then the closest to the main diagonal
element should be taken. The procedure continues unless
we run into the element ak1. The product of all chosen
elements is 12 231
q
k
aa as. This is the fraction of a
trace that passes CVS by the chosen path for the time in
q-cardiac-cycles. The product becomes the term of DetB
after multiplication by (–1)q + 1, and by all bjj where j are
the numbers of the segments not presented in the given
path. Since all bjj = −1, we have the term of DetB as:
 
1
12 231
11
qNq
q
k
aa as

 (A1)
The (A1) establishes the one-to-one correspondence
between the paths throughout CVS and nonzero elements
of DetB, The sums of all terms of (A1) with the same
time to pass CVS, let it be q, is the fraction of injected
tracer such that passes CVS in q cardiac-cycles. Let de-
note this fraction as pq. With the use of {pq} the equation,
DetB = 0 can be written as:
1
1
Mq
q
q
ps


 (A2)
with the M as the longest path from RA to RA.. By the
definition [10] the left part of (A2) is the generating
function for the first time to pass through the CVS, and
will be denoted as F(s).
Appendix 2
The equations for the evolution of the part of the z(t) =
(···, zi − 1(t), zi(t), zi + 1(t), ···), where subscript i denotes the
non-heart segment of CVS with aii > 0, accordingly to
Figure 1 is:


 
1
11
1
1
iii
iii
ztz tzt
ztzt zt





 (A3)
Multiplying both parts of (A3) by st + 1 and summing
with respect to t, gives the following equation for con-
nection between zi − 1(t), and zi + 1(t) in terms of a gene-
rating function:
 
11 1
1
t
ii i
t
s
Z
szts ssZs
s


 


 




(A4)
where

1
s
vss s
s






 
 is the generating func-
tion for the passage through the segments (i – 1), (i), and
(i + 1).
Appendix 3
The search for real characteristic numbers that are > 1.0.
The equation v(s) = 1 has two solutions sv1 = 1 and


211 v
s



 . Between s1 and s2 there is the pole


11
p
s


 of v(s) and, consequently, of F(s). Since
in the interval (sp, sv2) the F(s) varies from minus infinity
to F(sv2) > 1, F(s) = 1 has the solution in the given inter-
val. The use of Taylor decomposition of F1(s), F2(s), and
the difference for v(s) in the vicinity of sv2 (the difference,
not the derivative, is taken because of the proximity of
the pole of v(s)) leads to the expression for the real cha-
racteristic number >1:
 
 
113 22
2
113 22
11 1
11
11 1
FpFpF BV
s
F
pFp FACV





 

 
   (A5)
The coefficient at s2, in the spectral decomposition of the A, bm2, is


2
1
F
s
 thus
  

 
22
113 22
113 222
22
1
11 1
11 11
1
m
b
FpFpF
FpFpF
pF



 


 




(A6)
The sum b1m + bm2, with b1m given by
 
1
113 22
1
11 1
m
b
FpFpF





  
, is as follows:
   
12
21 1322
11 1
1111
mm
SV
bb
F
Fs FpFpFACV
 
 
 (A7)