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MODELING BLOOD FLOW IN THE CARDIOVASCULAR
SYSTEM MA325 – Spring 2013
Department of Mathematics, North Carolina State University
Mette S Olufsen
NC STATE University
Overview
§ Introduction to cardiovascular dynamics
§ Algebraic model
§ Differential equations model
§ Data
§ Subset selection and parameter estimation
§ Numerical solution with personal data
§ Summary
Before 1628
Pliny the Elder (Roman), Galen (Greek),…;
Two distinct types of blood were thought to exist:
§ “Nutritive blood” was thought to be made by the liver and carried through veins to the organs, where it was consumed
§ “Vital blood” was thought to be made at the heart and pumped through arteries to carry the “vital spirits”
§ Blood was thought to be produced and consumed at the ends of a transport system whereas idea of a circulatory blood system was unthinkable
It was believed that the heart acted not to pump blood, but to suck it in from the veins and that blood flowed through the septum of the heart from one ventricle to the other through a system of tiny pores
Ancient View
Liver
Body tissue
Heart
Lungs
Brain
Stomach
Nutritive blood flow
Vital spirits flow
Air flow
Nutrition
This seemed absurd when compared to the amount of blood in a human body which in most humans is around 5 liters and even more absurd compared to the amount of liquid intake per day (in food and in drinking) which is less than 5 liter per day
William Harvey’s 1615 modeling: Stroke volume is 70 milliliters/beat Heart beats 72 times/minute
Cardiac output of 7,258 liters/day
§ Modeling gave birth to the view that blood must circulate. Harvey successfully announced the discovery of the circulatory cardiovasclar system
§ Harvey discovered the circulation of blood 46 year before the discovery of the light microscope
§ Consequently Harvey changed the view of the world using a simple mathematical model making the inaccessible accessible 1. In 1615 Harvey wrote (hand-written notes) that he was convinced that
blood circulated
2. Anton Van Leeuwenhoek's microscope from 1674 (the first functioning light microscope) was sufficiently strong to make the capillaries visible
3. Marcello Malpighi, made the discovery simultaneously and independently, published his discovery of the capillaries in 1675. He is often credited the discovery of the capillaries by use of the microscope
Modeling
CV System Left side Right side
Heart
Sys Arteries (red) Pulm Arteries (blue)
Sys veins (blue) Pulm veins (red)
Body
Head
Pulmonary circulation
Exchange of O2 (in) and CO2 (out)
Systemic circulation Exchange of O2
(out) and CO2 (in)
Lungs
§ Large Arteries (cm)
§ Small Arteries (mm)
§ Arterioles (100 μm)
§ Capillaries (50 μm)
Systemic Arteries
Arterioles and Capillaries
Pressure and Area
Pressure
Systolic pressure
Pulse pressure
Diastolic pressure
L ventricle Aorta Arteries Arterioles
Vessel area
Veins Capil- laries
Ven oules
Mean pressure
Total vessel area (cm2)
Cardiovascular Modeling
§ Blood pressure p (mmHg) § Force per unit area
§ mmHg relates to the height of a column of mercury that can be supported by the pressure in question
§ Blood flow q (ml/min) or (l/min)
§ Cardiac output is the flow pumped by the heart
§ Blood volume V (ml) or (l)
§ Stressed volume and unstressed volume
Compartment Models
In-flow Out-flow Volume, Concentration, Amount
dVdt
= flow in ! flow out
= qin ! qout
The Reverend Hales demonstrating blood pressure in a horse.
Reproduced from: History of hypertension series. Sandwich, Kent; Pfizer Ltd., 1980.
Why mmHg ?
Pressure - Flow - Volume
q =pin ! pout
RV = C p ! pext( )
pBlood Flow Out (q2)System
C R2R1
Blood Flow In (q1)q
pin pout
pext
Psv=0
Normal Values Variables Systemic Pulmonary Unit
P Psa 100 Ppa 15 mmHg
Psv 2 Ppv 5 mmHg
V Vsa 1 Vpa 0.1 liter
Vsv 3.5 Vpv 0.4 liter
V0 5 liter liter
§ Stroke volume 70 ml/beat
§ Heart rate 80 beats/min
§ Cardiac output 5.6 l/min
Large Arteries and Veins
V = CP
(1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
Left heart Right heart
Systemic
Pulmonary
Systemic and Pulmonary Capillaries
Q =pin ! pout
RRQ = pin ! pout
(1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
Left heart Right heart
Systemic
Pulmonary
Psa Psv
Ppa Ppv
The Heart and veins are pri-
ressure differences
essels, while their
be of resistance is
smallest arteries,
rt but where large
L flow and pressure
Low for changes in
ince we can always- P) lQ. In fact ,
conditions where
/hen R changes, a
on usually involves
re smooth muscles
generated by the
stances circulating
Lcts of metabolism.
lutoregulation.
rmpliance relations
rn to say that the
lssure, but the rela-
vely less compliant
nplicity.
(Pr) ana transfer it
the pump performs
by the pump is the
difference Pz - Pt.
ds on P1 and &.
Lgure 1.3). The left
an outflow (aortic)
v valve is open and
;he left ventricle re-
sentially that ofthe
:5 mmHg. When
es and the outflow
rd into the systemic
l0 mmHg
er these conditions?
Lced by Sagawa and
1.4. The Heart as a Pair of Pumps l 1
DIASTOLE SYSTOLE
Figure 1.3. Cross section of the left side of the heart with the left ventricle in
DIASTOLtr (relaxed) and SYSTOLE (contracted). LA : left atrium, LV : left
ventricle, Ao : Aorta. The numbers in the chambers are pressures in millimeters
of mercury (mmHg). Note that the inflow (mitral) valve is open in diastole and
closed in systole, whereas the outflow (aortic) valve is closed in diastole and open
in systole. Pressures are approximately equal across an open valve, but can be
very unequal, with the downstream pressure higher, across a closed valve. The
valves ensure the unidirectional flow of blood from left atrium to left ventricle to
aorta.
his collaborators: We regard the ventricle as a compliance vessel whose
compliance changes with time. Thus, the ventricle is described by
v ( t ) :Va+C( t )P ( t ) , (1 .4 .1 )
where C(l) is a given function with the qualitative character shown in
Figure 1.4. The important point is that C(t) takes on a small value C.""1o1"
when the ventricle is contracting, and a much larger value C6i."1o1" when
the ventricle is relaxed. For simplicity, we have taken V6 to be independent
of t ime.
Using (1.4.1), we can construct a pressure-volume diagram of the cardiac
cycle (Figure 1.5). For a detailed explanation of this type of diagram, see
the paper by Sagawa et al., cited in Section 1.14. For our purposes, it is
sufficient to note that the maximum volume achieved by the ventricle (at
end-diastole) is given by
V a o : V a * C a i u " t o t " P r ,
while the minimum volume (achieved at end-systole) is given by
Vps :Va*C"v " to t "P r ,
where P. is the pressure in the arteries supplied by the ventricle and P" is
the pressure in the veins that fill it. Thus, the stroke volume is given by
%t.ok : Vso - VpS : Cai."tol"Pt - C"v"tot"Pr. (r.4.4)
(7.4.2)
(1 .4 .3 )
Pressure Volume Loop
V
1.4. The Heart as a Pair of PumPs 13
V = Va + Cryrtote P
Inflow valve closesOutflow valve oPensOutflow valve closesInflow valve oPens
V = Va + Cdiastolc P
vd ves vEp
Figure 1.5' Pressure-volume diagram for either ventricle' Landmarks on the vol-ume axis (V) are Va : "deaJvolume"' i'e'' the volume at zero transmuralpressure, Ves : end-systolic volume, Vno : end-diastolic volume' Landmarkson the pressure (P; a*it are the venous (inflow) pressure P-.'' u.od the arterial(outflow) pressure P"' Since the venous pressure is essentially the same as theatr ialpressure,i tmaybeconsideredtheinf lowpressurefortheventr icle.Slantingl inesradiat ingfromV6describethepressure-volumerelat ionshipsoftheventr i-cre at the end of diastole and at the end of systole; these lines are labeled withtheir equations. At other times, the pressure-volume relationship is indicated byasimilarslantingl ine(notshown)thati iesbetweenthetwolinesshown'R'ectan-gular loop ABCDA ,h;; l;" ptth follo*"d bv the ventricle during the cardiaccycle. On the vertical ntt""n"tjeg and CD)' both valves are closed' so the vol-ume of the ventricle i,
"o"'tut'i' On each of the horizontal branches' one valve
is open: the arterial t""1i"*j "rr"e
along Bc and the atrioventricular (inflow)valve along DA' The pt""t""'ain"'"""" uito" the open valve is here idealized aszero, and the arterial and venous pressures are idealized as constants' The namesof the four phases of ttre cardia"
"y"le are AB : isovolumetric contraction' BC
: ejection, CD : isovolumetric relaxation' DA : filling'
t
rr of time. During diastole'
[igh value C6i6s1.1. 8s t.rre
s the ventricle contracts'
i
. :0. so that the sYsto)icf .rolume
(1.4.5)
lshall use. FinallY, if F
fwe Set cardiac outPut
t
i"" (1'4'6)
t[tho"gtt we shall consider
bfine
(1.4.7)
h of the ventricle' Although
i(which are driven bY the
fier in the thin-walled right
iso K is greater on the right
it ur" "orrtt".ted
to different
spressions for the right and
A: Inflow valve closes B: Outflow valve opens C: Outflow valve closes D: Inflow valve opens
V=Vd+CdiastoleP
Le: and Right Heart (1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
Left heart Right heart
Pulmonary
V =Vd +C(t)P(t)VES =Vd +CsystolePa (Min)
VED =Vd +CdiastolePv (Max)Vstroke =VED !VES = CdiastolePv !CsystolePsVstroke "CdiastolePv
Psv
Ppv
Systemic
Le: and Right Heart (1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
Left heart Right heart
Pulmonary
Vstroke ! CdiastolePvQ = FVstroke = FCdiastolePvQ = KPv
Systemic
Psv
Ppv
Compartment Model
(1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
Variables and Parameters
§ Variables § Pressure (P)
§ Flow (Q)
§ Volume (V)
§ Parameters § Resistance (R)
§ Compliance (C)
§ Heart pumping (K)
Solve Equations Find variables (parameters) § Summary equa?on
Variables: V, P, Q Parameters: K, C, R
Vi = CiPiV0 = Vi!Qi =
1Ri
Pi"1 " Pi+1( )Qh = KL/RPhQ =Qi
(1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
Solving the Equations § The heart
§ Capillaries
Q = KRPsv & Q = KLPpv
Psv =QKR
& Ppv =QKL
Q = 1Rs
Psa ! Psv( ) & Q = 1Rp
Ppa ! Ppv( )
Q = 1Rs
Psa !QKR
"#$
%&'
& Q = 1Rp
Ppa !QKL
"#$
%&'
Psa =Q Rs +1KR
"#$
%&'
& Ppa =Q Rp +1KL
"#$
%&'
Solving the Equations
§ Volume equations V=CP
§ Define Ti
Vsv = CsvPsv =Csv
KR
Q
Vi i = pv, sa, pa
Tsv =Csv
KR
! Vsv = TsvQ
Ti i = pv, sa, pa
Solving the Equations
§ Volume and flow equations revisited
Function of parameters!
Vi = TiQ
V0 =Vsa +Vsv +Vpv +Vpa
V0 = TsaQ +TsvQ +TpvQ +TpaQ
Q = V0Tsa +Tsv +Tpv +Tpa
Solving the Equations
§ Use same approach to predict
Q = V0Tsa +Tsv +Tpv +Tpa
Vi = TiQ = ...
Pi =ViCi
= ...
Normal Parameter Values Systemic Pulmonary Units
R Rs 17.5 Rp 1.79 mmHg min/liter
C Csa 0.01 Cpa .00667 liter/mmHg
Csv 1.75 Cpv 0.08
Right Left Unit
K KR 2.8 KL 1.2 liter/min/mmHg
V V0 5.0 liter
Parameter values can be calculated from solution formulas
Homework (due Monday)
• Solve all equa?ons for V, P and Q as a func?on of the model parameters (R, C, K)
• V: ( Vsa =, Vsv =, Vpa=, Vpv =) • P: (Psa =, Psv =, Ppa =, Ppv =) • Q: (Ql = Qr = Qs = Qp = Q)
• You are expected to write equa?ons and get values using parameters from the table
Balancing the System
(1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
§ Right and left sides (R = L) § If:
KR = KL and Rs = Rp …
§ Then:
Psa = Ppa and Psv = Ppv …
§ Questions: § How are the two sides
coordinated?
§ What mechanisms control the two sides of the heart?
Balancing the System
(1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
§ Suppose KR is suddenly reduced. Then § QR will
§ Net transfer of blood to the systemic circulation
§ Psv will § Ppv will
§ Drive cardiac output back to equality
§ Net transfer of blood persist until the two sides are equal again, but with new volumes
decrease
increase decrease
Balancing the System
§ Volume balances
Vi =TiV0
Tsa +Tsv +Tpv +TpaVp
Vs=Vpa +Vpv
Vsa +Vsv
=Tpa +TpvTsa +Tsv
=Cpa +Cpv
KL
+CpaRp!"#
$%&/ Csa +Csv
KR
+CsaRs!"#
$%&
Balancing the System § The key to the control is the
dependence of cardiac output on venous pressure
§ If cardiac output Q was constant (parameter), then we loose 2 equations but 1 variable. Furthermore
would replace equation for V0.
§ The two sides become arbitrary. WHY?
Vp =Vpa +Vpv
Vs =Vsa +Vsv
(1'4'8)
(1.4.e)
rmed that the Pressurenen the distending Pres-
" but P.t - 4ho.t* and
I chest. In fact, Rho,,* ls
l. and this contributes to
*toti. volume VBn' This
r the chest is oPened dur-
ssume for simPlicitY that
19) without modification'
r i.t E*.rci."t 1'8 through
i.
i
hncontrolled
rave been develoPed above
lation. In the form that we
chanisms that regulate the
ody. In subsequent sections
[he qirculation' indePendentI
Ifrechanisms;r| ' -nodel[n
construct a slmPre rr
I
[ior,, (t"" Figure 1'6): First'
FJISI
i (1 .5 .1 )
(1 '5 '2 )
bemic and PulmonarY arteries
ilt-t" ot" 1r'S'z; instead of
1.5. Mathematical Model of the Uncontrolled Circulation t i )
RnQp=Ppu-Pnu
Vpu = CpuPpu Vpu = CpuPpu
q* = KsP* q" = KlPpv
Vru = CruP.u Vru = CruPru
RrQr=Pru-Pru
Figure 1.6. Equations of the circulation' One equation is shown'for each of the
eight principal elements oiiir" "ir"]rrution.
For .dditiottal equations that relate
these elements to eacn "*h";;;Jil
text' Each element is characterized by one
parameter (K : pump "t"m"i""''
C : compliance' R: resistance) and two or
lhree unknowns (Q:;; ; ,- t- : 'ptessure' 1/: volume)' Subscript notat ion is
s : systemic, p : p"h;;;y, a : arterial) v : venous' L : left' ft : right'
(1.3.3). That is, we neglect V6 in these vessels' This gives the equations
Vu : C"uP"u'
V"u : GtP"t,
Vou : CpuPpu,
V o r : C p ' P p u '
Third, we assume that the systemic and pulmonary
resistance vessels, so that
(1.5.3)
(1.5.4)
(1 .5 .5 )
(1.5.6)
tissues act like
Q " :
Q p :
*",* , - P" ' ) ,
I tro. - Pp').f L p
(1.5.7)
(1 .5 .8 )
At this point, we have an equation for each element of the circulation'Each equation
"ot'tui"' J i*t*'"t"' that characterizes that element' These
External Control § Exercise requires more blood to the legs
Rs
§ Cardiac output rises to support more blood to the system
§ Heart rate increases
§ Systemic arterial pressure goes down
§ External control system to bring arterial pressure back up – baroreceptor regulation
decreases
Sensitivities
§ Use sensitivity analysis to study the impact of changing parameters
§ Sensitivity of the variable Y wrt the parameter X
§ In the limit X and Y going to 0
!YX = ! logY! logX
= logY '" logYlogX '" logX
= log(Y '/Y )log(X '/ X)
!YX = d logYd logX
= dYY/ dXX
Sensitivities
§ Sensitivity of Q wrt Rs
§ Assume Rs is reduced by 50% giving
Rs’ = Rs/2 giving
!QRs =log(Q '/Q)
log Rs2/ Rs
!"#
$%&= log(Q '/Q)
log 12
!"#
$%&
!QRs =! logQ! logRs
= log(Q '/Q)log(Rs '/ Rs )
Sensitivities § How do we calculate Q’ and Q
- Use solution formulas
Q = V0 (Rs )Tsa (Rs )+Tsv (Rs )+Tpv (Rs )+Tpa (Rs )
Q ' = V0 (Rs ')Tsa (Rs ')+Tsv (Rs ')+Tpv (Rs ')+Tpa (Rs ')
Normal Rs’ =Rs/2 Change % Change
Q 5.6 6.2 + 0.6 l/min + 11%
Psa 100 57 - 43 mmHg - 43%
Sensitivities
§ How do we calculate sensitivities
Rs KR … …
-0.15 … … …
0.81 … … …
!QRs =log(Q '/Q)log 0.5( ) =
log 6.2 / 5.6( )log 0.5( ) = !0.1468
!QRs
! PsaRs
Homework (due monday)
• Problems 1.2, 1.3, 1.4, 1.5, 1.10
Recommended