MODELING BLOOD FLOW IN THE CARDIOVASCULAR SYSTEM …msolufse/CardiovascularModels325.pdf · MODELING BLOOD FLOW IN THE CARDIOVASCULAR SYSTEM MA325 – Spring 2013 Department of Mathematics,

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)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )




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)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )




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)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )




Pulmonary

Vstroke ! CdiastolePvQ = FVstroke = FCdiastolePvQ = KPv

Systemic

Psv

Ppv

Compartment Model

(1'4'8)

(1.4.e)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )



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)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )



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


§  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


§  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)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )



§  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?


(1'4'8)

(1.4.e)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )



§  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


§  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)










i.

i

hncontrolled








I


FJISI

i (1 .5 .1 )

(1 '5 '2 )




RnQp=Ppu-Pnu




RrQr=Pru-Pru











Vu : C"uP"u'

V"u : GtP"t,

Vou : CpuPpu,

V o r : C p ' P p u '



(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 )



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

Documents

MODELING BLOOD FLOW IN THE CARDIOVASCULAR SYSTEM …msolufse/CardiovascularModels325.pdf · MODELING BLOOD FLOW IN THE CARDIOVASCULAR SYSTEM MA325 – Spring 2013 Department of Mathematics,