I Course
Biomedical Applications of Mathematics
Elements of cardiocirculatory physiology
Roberto Bonmassari
S.C. di Cardiologia
APSS-Ospedale Santa Chiara Trento
Preamble
A Course of Biomedical Applications of Mathematics
……
when the Hospital goes out and
meet the University
I’m not a physiologyst
I am a clinician, a cardiologist
Agenda
Four lessons: 18 - 25 september, 2 – 11 october
•
Aspects of cardiac anathomy
• The cardiac and circolatory function: phisiologic aspects
• Examples of clinical application of a phisiologic application in
a pathologic condition: coronary stenosis and aortic stenosis
The heart is costituted
from 4 chambers:
2 atria (right and left)
2 ventricles (right and left)
Atria
recive blood,
Ventricles
eject blood
POLMONE
Scambio Gassoso
Piccolo Circolo
CUORE
Grande Circolo
ORGANI
Funzione di Pompa
Consumo di Ossigeno
The Heart is a pump
•
The cardiac pump is the ground of the circulation
• The are two circulation system that works in series: systemic
and pulmonary circulation
• The cardiac pump has the primary function of insurance an
adeguate amount of blood flow through the systemic and the
pulmonary vessel bed
• The cardiac pump works with two mechanisms : blood
aspiration and pushing
The Heart is a pump
The cardiac pump produce a mechanical result (the
circulation of the bood) due to the contraction and relaxation
of the muscolar wall of the cardiac chambers (ventriculi and
atria)
•
• But what is the primum movens of the cardiac function?
• Upstrem the mechanical function is necessary the electric
function: the electric excitation
The conduction system: Physiology and pathology
Nodo del
seno
Nodo atrioventricolare
Fascio di His
Branca destra
Branca sinistra
Fibre di Purkinje
The Heart is a pump
•
The cardiac electrical activity is an automatic activity
• Is only marginally influenced by nervous system
• These are the basis of the electro-mechanical coupling
partneship
The cardiac cycle
Arterial Pressure Curve
Fasi
1 contrazione Vs
2 rilasciamento Vs
3 riempimento Vs
Ciclo di Wiggers 1915
Pressure (mm Hg)
Atrial
Systole
120
100
80
60
40
Ventricular
Ejection
Phase
Isovolumetric
Isovolumetric
Contraction
Relaxation
Semi-Lunar
Valve Closes
Semi-Lunar
Valve Opens
AV Valve
Closes
10
0
Ventricular
Filling
1
Arterial Pressure
AV Valve
Opens
2
3
Ventricular Pressure
3
R
Q
S
Atrial
Systole
Approx. Time
0
Electrocardiogram
T
P
0.1
Ventricular
Systole
0.2
0.3
0.4
Diastole
0.5
0.6
0.7
0.8
Ventricular
Ejection
Phase
Atrial
Systole
Isovolumetric
Contraction
Pressure (mm Hg)
120
100
Ventricular
Filling
Isovolumetric
Relaxation
Semi-Lunar
Valve Closes
Semi-Lunar
Valve Opens
Arterial Pressure
80
60
40
AV Valve
Closes
10
AV Valve
Opens
Ventricular Pressure
0
R
Q
S
Atrial
Systole
Approx. Time
0
Electrocardiogram
T
P
0.1
Ventricular
Systole
0.2
0.3
0.4
Diastole
0.5
0.6
Arterial Pressure Curve
0.7
0.8
The cardiac cycle
pressure/volume V sin ratio
Ventricular
Ejection
Phase
Atrial
Ventricular
Systole
Filling
Isovolumetric
Isovolumetric
ContractionRelaxation
Pressure (mm Hg)
120
Semi-Lunar
Valve Closes
Semi-Lunar
Valve Opens
100
Arterial Pressure
80
60
40 AV Valve
10
AV Valve
Opens
Closes
Ventricular Pressure
0
R
T
P
Q
Electrocardiogram
S
Atrial Ventricular Diastole
Systole Systole
Approx. Time
0 0.1 0.20.30.40.50.60.70.8
Ventricular Filling
Arterial Pressure Curve
Ventricular
Ejection
Phase
Atrial
Ventricular
Systole
Filling
Isovolumetric
Isovolumetric
ContractionRelaxation
Pressure (mm Hg)
120
Semi-Lunar
Valve Closes
Semi-Lunar
Valve Opens
100
Arterial Pressure
80
60
40 AV Valve
10
AV Valve
Opens
Closes
Ventricular Pressure
0
R
T
P
Q
Electrocardiogram
S
Atrial Ventricular Diastole
Systole Systole
Approx. Time
0 0.1 0.20.30.40.50.60.70.8
Arterial Pressure Curve
Atrial Systole
Ventricular
Ejection
Phase
Atrial
Ventricular
Systole
Filling
Isovolumetric
Isovolumetric
ContractionRelaxation
Pressure (mm Hg)
120
Semi-Lunar
Valve Closes
Semi-Lunar
Valve Opens
100
Arterial Pressure
80
60
40 AV Valve
10
AV Valve
Opens
Closes
Ventricular Pressure
0
R
T
P
Q
Electrocardiogram
S
Atrial Ventricular Diastole
Systole Systole
Approx. Time
0 0.1 0.20.30.40.50.60.70.8
Isovolumetric Contraction
Arterial Pressure Curve
Ventricular
Ejection
Phase
Atrial
Ventricular
Systole
Filling
Isovolumetric
Isovolumetric
ContractionRelaxation
Pressure (mm Hg)
120
Semi-Lunar
Valve Closes
Semi-Lunar
Valve Opens
100
Arterial Pressure
80
60
40 AV Valve
10
AV Valve
Opens
Closes
Ventricular Pressure
0
R
T
P
Q
Electrocardiogram
S
Atrial Ventricular Diastole
Systole Systole
Approx. Time
0 0.1 0.20.30.40.50.60.70.8
Ventricular Ejection
Arterial Pressure Curve
Ventricular
Ejection
Phase
Atrial
Ventricular
Systole
Filling
Isovolumetric
Isovolumetric
ContractionRelaxation
Pressure (mm Hg)
120
Semi-Lunar
Valve Closes
Semi-Lunar
Valve Opens
100
Arterial Pressure
80
60
40 AV Valve
10
AV Valve
Opens
Closes
Ventricular Pressure
0
R
T
P
Q
Electrocardiogram
S
Atrial Ventricular Diastole
Systole Systole
Approx. Time
0 0.1 0.20.30.40.50.60.70.8
Isovolumetric Relaxation
Arterial Pressure Curve
Left
300
Coronary 200
Blood
Flow
(ml/min) 100
Right
0
Systole
Diastole
Slide courtesy of A.C. Guyton, MD, Textbook of Medical Physiology,
Sixth Edition, 1981 W.B. Saunders Company
How is the Cardiac function?
Cardiac output and pressure
CO= Pr / R= SV x HR
Legenda
1.
2.
3.
4.
Pr = systemic pressure
R = systemic resistance
SV = stroke volume (amount of blood eject every beat)
HR = number of beats per minute
Cardiac function = CO
Determinants
• Cardiac rate
• Inotropic condition = contractility
• Venus return (RV): influenced from neuroumoral factors
(Frank-Starling law)
• Peripheric resistance
=
CARDIAC OUTPUT 4-6 l/min
CARDIAC INDEX 1.6-2.5 l/min/m²
Cardiac output = CO
CO= Pr / R= SV x HR
examples :
1.
2.
3.
4.
5.
Increase of FC -> does not increase GC, decrease of the VOLUME SYSTOLESV (if Rv does not increase)
Increase of RV -> oncreases GC 10-20% (increase of Pa A dx 10 mmHg)
Increase of FC + Increase of RV -> increases GC with = SV
Stirring adrenergethic -> increases RV + increse of the function of the pump
(FC, contract?) = INCREASES GC
Important reduction of FC or alteration of V dx -> increase of Pa A dx =
barrier for the venous comeback
Organs are able to change their flow working on the oppositions; they are able to regulate the
distribution of the CARDIAC RANGE.
Cardiac function:
Frank-Starling mechanism
Cardiac function:
CO-CI & R ; Conduttance= 1/R
diastolica
Difetto di riempimento
sistolica
sistolica
Difetto di espulsione
Cardiac power output
Ventricular function index
Cardiac Power output
=
MAP x CO
=
SW X HR
Cardiac power output
Function of blood circulation
TRANSPORT
• substrates and cathabolism products to and from
organs and tissues
• in a changeable measure and in proportion of her
requirements and necessities
GOAL
• to maintain an optimal composition of the interstitial
fluid necessary for the cellular function
VASO ARTERIOSO
TONACA INTIMA = ENDOTELIO
TONACA MEDIA
TONACA AVVENTIZIA
PLACCA ATEROSCLEROTICA
Distribution of blood flow
• It is necessary to give at any time to organs and tissues
a blood flow distribution based on requirements
• There are two control mechanisms in a close correlation
• Central Autoregulation
• Local Autoregulation
Distribuzion of blood flow
Central Autoregulation
GOAL: TO MAINTAIN COSTANT
perfusion pressure of the organs
(independently from the flow: Fl= P/R)
Mechanism: neuro-ormonal central control on R e Fl
Local Autoregulation
GOAL: TO MAINTAIN ADEQUATE
flow for each organ for the metabolic erquirements
(independently from systemic pressure)
Mechanism: regulation of local vascular resistance due to
metabolism activity
Circulation modelling
•
• ORGANS in parallelo if we consider AORTA with different
metabolic, vascular, anathomic characteristics (f.e. heart,
abdominal argans, muscols, brain…)
• VASCULAR SYSTEM: determinate from a segments sequence
in serie similar in each organ
Arterie – arteriole – capillari – venule – vene
The Vascular System: segments
1) Aorta e great proximal arteries
• Elastic matrix tessue prevalent
HISTOLOGY
• Improve distention of vascular wall during sistole
PHISIOLOGY
• cinetic energy of stroke volume (during systole ) is storaged as elastic
energy released during diastole (partecipate at the diastolic value of BP)
• modulation of suddenly variations of BP : dynamic sistolic reserve
• pressure wave downstream is delayed
• Riduction of elastic properties
PATHOLOGY
• reduction of dynamic sistolic reserve
• increase differential blood pressure even if a costant stroke volume
• increase of afterload of left ventricle
• increase of the rate of propagation of the pressure wave
The Vascular System: segments
2) Muscolar small arteries
• Arteries with a thick muscolar medium tonaca: are the junction betweengreat
arteries and organs and tessues
– Thick medium muscolar tonaca ( high thickness/lumen ratio )
– COSTANT STRESS of the wall (PR x radius/ thickness)
– COSTANT DIAMETER with variations due to neuro-horrmonal control
– Regulatory mechanims : metabolics production and myogenic control
• Play a central role in the vascular resistance control = control and persistance of
an adequate flow to organs and tessues FL= P / R
– Is the principal location of the flow resistance (aortic pressure 100 mmHg,
small arteries pressure 30 mmHg, vein pressure 15 mmHg)
• The pressure gradient between small arteries and veins (about 15 mmHg) permits
– The capillar filtration
– The distal reabsorbtion
The Vascular System: segments
3) Capillars
• In this segments is present the most importanti function of circulation =
SUBSTRATES AND OXIGEN EXCHANGE BETWEEN BLOOD/TESSUES WIT H AN
FOUDAMENTAL ROLE IN OMEOSTASIS OF THE INTERSTITIAL FLUID
– The walls are very smooth , sometimes in certein organs fenestrates
– Little diameter = little transmural stress (Stress= D x P/spessore) with a better tolerance
of the transmural pressure
– There is a significant flow resistance with a fall of pressure (from 30 mmHg to 15 mmHg)
– At the arteriolar portion happen THE FILTRATION at the venular portion THE
REABSORBTION
– In any time not in all THE CAPILLARS there is perfusion: the density of the perfused
capillars is important in terms of exchange between the tessues
– The exchange happen due to:
•Pressure gradients (FLUIDS): hydrostatic pressure (proximal side) and colloido
osmotic pressure (distal side)
•Diffusion: liposoluble gas and yhdrosoluble substances
– The amount of not reabsorbed fluids returns to the systemic circulation through an
alternative circulation : THE LINFATIC SYSTEM = 4-5/L in a day-> 1/4 -1/2 TOTAL
PLASMATIC ALBUMIN
The Vascular System: segments
3) Capillars
CAUSES OF INCREASE OF FILTRATION
1.
Increase of the exchanges surface (due to the increase of number of the perfused
capillars)
2. Increase of the pre-capillar pressure (more diffusion)
3.
Increase of the postcapillar pressure (less reabsorbtion))
4. Increase of the permeability per surface unit (increase of holes)
INTERCAPILLAR DISTANCE =
less distance = more opend capillars = more fast the exchange
The Vascular System: segments
4) Venule and veins
• EMODYNAMIC:
– Pressure = 15 mmHg in orizzontal position
– Prefusion pressure for the veins heart return
• HISTOLOGY:
–Thin wall
– Very strechable
–Muscolar portion very thin
•FUNCTION:
– Return of blood from tissue sto the heart
– Storage of blood able to control the return to the heart
– Influence to the capillar pressure
• LEGS: superficial and deep veins
– Muscolar wall more thick
– Unidirectional valves (distribution of the Hydrostatic pressure in orthostatisms)
Vascular resistance
• Not uniform distribution a long the segments of circulation
1.
2.
3.
4.
Arteries
Small arteries
Capillars
Veins
5%
60%
20%
15%
• Medium vascular resistence-> integrated values of organs and tissues in
parallelo respect the aorta
Vascular resistance
The vascular regulation is a local methabolic process in all organs, a part in
kidney and skin. In fact even if in a denervation condition these organs are
able to matain a adequate vascular tone modificated by methabolic
influences. Flow depend closely by the radius
R = L x h8/ x r4
10% in reduction diameter vessel = 50% in increase resistance
Peripheric distribution of
CO and oxygen consumption
Myocardial ischemia
Fractional Flow Reserve – FFR
Coronary circulation:
anatomic aspects
• Coronary tree = vascular system with dicotomyc
regular ramification with a diameter progressively
smaller
• 2 distinct section in term of anatomic and
functional characteristics
- 1 Epicardic portion
- 2 Microcirculation section
Coronary Circulation: segments
Epicardial segment
Microcirculation)
Coronary circulation:
anatomic aspects
• Epicardic portion (diameter 4-5 - 0.5 mm)
- “Conduttance” vessel : not able to influence the vascular
resistance
- are visible using contrast media
• Microcirculation (diameter < 0.5 mm)
- “ Resistance” vessel site of autoregulation process
- Are able to modify vascular resistance up to 6 times
- Incomplitely visible with contrast media (cause of “myocardial
blush”)
Coronary Circulation
physiologyc aspects
•
•
•
•
•
Coronary Flow = Pressure/Resistance = 300-600 ml/min
5-10% of cardiac output (4-6 l/m)
Can increase up to 5-6 times (Resistance can change up to 6 times)
Pulsed flow (not continue) with a prevalent diastolic component
Cardiac methabolism: exclusively aerobic - O2 dependent with a very
high O2 extraction from the arterial blood (10 ml/100gr/min vs 0.5
ml/100gr/min in the skelectric muscle) and with 30% O2 saturation the
blood of coronary sinus
=
more oxigen demand request more oxigen supply
it is mandatory an increase of coronary flow
it is not possible oxigen extraction
Left
300
Coronary
Blood
Flow
(ml/min)
200
Right
100
0
Systole
Diastole
Slide courtesy of A.C. Guyton, MD, Textbook of Medical Physiology,
Sixth Edition, 1981 W.B. Saunders Company
PLACCA ATEROSCLEROTICA
Physiopathology ischemia
Supply
Demand
MVO2
Supply: stenosi spasmo riserva coronarica
Demand: FC PAO e contrattilità, ipertrofia
Coronary stenosis: physiopathology
concept of coronary flow reserve
• A coronary stenosis (> 40%) determine a reduction in perfusion
pressure without a concomitant flow reduction due to
contemporary microcirculation resistance reduction
FL= P/R, if decrease P and R contemporary = FL remain stable
• Stenosis upper a limit level (80-85% diameter) run out the
dilatation possibilities of the microcirculation: in this condition
every other reduction of pressure means reduction of flow
because the incapacity of reduction of resistance = ischemia
RUN OUT OF THE CORONARY FLOW RESERVE
IVUS - INTERMEDIATE LESION RCA
FFR and CFR: What Do They Investigate?
FFR
CFR
(E TEST NON INVASIVI)
Fractional Flow Reserve (FFR)
Pa
S
Pd
Qmax
FFR = N
Qmax
Pd
=
Pa
During maximal hyperemia
FFR = the ratio of maximal myocardial flow in the stenotic
territory to maximal myocardial flow in that same territory
if the stenosis were absent
FFR: a Flow Index Derived from Pressures
sten
FFR =
Q
N
Q
=
Pd
P
a
Normal Value of Myocardial Fractional Flow Reserve
Pa
Pd
Pd
FFR =
Pa
Normal FFR = 1
Myocardial Fractional Flow Reserve: Definition
S
Qmax
Pa
Pd
Pv
N
= FFR
Qmax
FFR =
Pd / R myo
Pd
=
Pa / R myo
Pa
During Maximal Vasodilatation
Q
Pa
Pv
100
0
Pd
Pv
100 70
0
Pa
Q100
Q70
70
100
Pd
FFR =
Pa
= 0.70
P
R.E. 50-y-old man.
Aborted sudden death.
LV angiogram: mild hypokinesia of the anterior wall.
Coronary Pressure Measurements
1979
2001
Pa
Pd
HYPEREMIA
FFR = Pd /Pa = 56/80 = 0.70
Coronary Pressure Measurements:
Prerequisits
1. Pressure Measuring Guide Wire
2. Maximal Hyperemia
3. FFR instead of
P
Pressure Monitoring Guide
Wires
0.014”
3 cm
Pa = Guiding Cathe
100
50
0
Pd = Pressure Wir
Coronary Pressure Measurements:
Prerequisits
1. Pressure Measuring Guide Wire
2. Maximal Hyperemia
3. FFR instead of
P
Hyperemia - administration
• Hyperemic stimuli
– Intravenous Adenosine
140-160 g/kg/min
– Intracoronary Adenosine
LCA: 20-40 g
RCA: 15-30 g
– Intracoronary Papaverine
LCA: 15 mg
RCA: 10 mg
– Adenosine Triphosphate (ATP) (ic. or iv) (same
dosages as for Adenosine)
ADENOSINE
100
50
0
Influence of Systemic Pressure on Transstenotic Gradient
200
Aortic Pressure = 122 mm Hg
200
P = 70 mmHG
100
Aortic Pressure = 89 mm Hg
P = 49 mmHG
100
Coronary Pressure = 52 mm Hg
0
Coronary Pressure = 40
mm Hg
0
FFR = 52/122 = 0.43
FFR = 40/89 = 0.45
Coronary Pressure Measurements:
Prerequisits
1. Pressure Measuring Guide Wire
2. Maximal Hyperemia
3. FFR instead of
P
Fractional Flow Reserve in Clinical Practice
Baseline
100
Adenosine IC Hyperemia
Pa
Pa
80
60
40
20
PdPd
FFR = Pd /Pa (during hyperemia) = 58/79 = 0.73
0
Fractional Flow Reserve in Clinical Practice
REST
HYPEREMIA
150
112
100
Proximal to
the lesion
58
50
Crossing
the lesion
0
Distal to
the lesion
FFR=58/112=0.52
QCA vs Pressure-derived FFRmyo
1.0
FFRmyo
0.8
0.6
0.4
0.2
20
40
60
Diameter Stenosis (%)
80
100
FFR and CFR: What Do They Investigate?
P
R
R
s
m
FFR
CFR
F=R
Coronary Pressure Measurements
Clinical Applications
1. Diagnostic Setting:
Is this lesion responsible for patient’s complaints?
Should this lesion be revascularized?
2. Interventional Setting:
Is a stent needed after balloon angioplasty?
Is the stent well deployed?
BDB 98/029
Clinical Applications of FFR
1. Before PTCA:
when FFR > 0.75 the prognosis is at least
as good without than with an angioplasty.
when FFR < 0.75 an angioplasty is
justified by a marked symptomatic
improvement following revascularization.
Clinical Applications of FFR
2. After balloon:
when FFR > 0.90 and angio is OK, the longterm outcome after POBA is similar than
what can be expected after additional stent
implantation.
Clinical Applications of FFR
3. After stent:
pullback maneuver : No pressure drop
during hyperemia.
A
FFR = 75 / 101 = 0.74
B
FFR = 90 / 106 = 0.85
C
FFR = 91 / 107 = 0.85
D
FFR = 98 / 100 = 0.98
FFRmyo
Myocardial Fractional Flow Reserve
Myocardial Fractional Flow Reserve
FFRmyo
represents the true fraction of maximum flow
which can still be maintained in spite of the
presence of a stenosis.
It is exactly that index which tells to what extent
a patient is limited by his coronary disease.
FFRmyo
Myocardial Fractional Flow Reserve
FFRmyo
=
Max. myocardial blood flow
in the presence of a stenosis
normal maximum blood flow
In summary
FFRmyo ...
is a lesion specific index
is independent of hemodynamic parameters
has a normal value of 1.0
takes into account collateral flow
has no need for a normal control artery
can be easily obtained: FFRmyo = Pd / Pa
Courtesy of Charles Chan, M.D.
National Heart Center, Singapore
Courtesy of Charles Chan, M.D.
National Heart Center, Singapore
Aortic stenosis
AORTIC VALVE: tricuspid
Valvola normale
Valvola stenotica
Aortic stenosis: pathology
AORTIC STENOSIS: imaging
Aortic stenosis: pressure curves
Acquiring Hemodynamic Data
*Accurate method of measuring CO, especially in patients with low cardiac output.
• O2 consumption measured from metabolic hood or Douglas bag; it can
also be estimated as 3 ml/min/kg or 125 ml/min/m2.
• AVo2 difference calculated from arterial – mixed venous (pulmonary
artery) O2 content, where O2 content = saturation x 1.36 x Hg
Aortic stenosis hemodynamic evaluation
Gorlin equation
CO/(SEP)(HR)
Area in cm² = ---------------------------------------44.3(C)(sq rt of pressure gradient)
Where C = empirical constant
For MV, C = 0.85 (Derived from comparative data)
For AV, TV, and PV, C = 1.0 (Not derived, is assumed based on MV data)
Alternative to the Gorlin Formula
*A simplified formula for the calculation of stenotic cardiac valves proposed by
Hakki et al…Circulation 1981. Tested 100 patients with either AS or MS.
*Based on the observation that the product of HR, SEP or DFP, and the Gorlin
equation constant was nearly the same for all patients measured in the resting
state (pt. not tachycardic). Values of this product were close to 1.0.
*Calculations somewhat comparable………
How measure stenosis severty?
Echocardiography
Pressure gradients
Aortic Stenosis:
valvular area with continuity equation
Continuity equation: equal istantaneal flow through
left ventricular outflow tract (LVOT) and the aortic valve
Flow LVOT = Flow Ao V
VTI AO
VTI
Ao
LVOT
LV
LA
Flow = area (3.14 x (D/2)² x Integral time velocity Doppler (ITV)
Area LVOT (3.14 x (D/2)² x Integral time velocity Doppler (ITV LVOT) = Area Ao V (3.14 x (D/2)² x Integral time velocity Doppler (ITV Ao V)
Aortic Valve area = 3.14 x (D/2)² x (ITV LVOT / ITV Ao V)
Classification of severty of Valve Disease in Adults
Prognostic Importance of Quantitative Ex Doppler
Echocardiography in Asymptomatic Valvular
Aortic Stenosis
All patients who displayed hard events (D or HF) had an AVA 0.75 cm2, an abnormal
exercisetest, and a significant exercise-induced increase in mean transaortic pressure
gradient
Lancellotti et al. Circulation, 2005; 112: I-377
Relationship between CO and Aortic Pressure Gradient over a range of values
for AV area (Based on Gorlin formula)
Discrepancies between Gorlin and
continuity-calculated effective orifice areas
PRESSURE RECOVERY PHENOMENON
JACC 2006; 47: 1241
Since recommended cut-off values for the severity of aortic stenosis are
largely based on the clinical experience with Gorlin-calculated areas, the
use of the inherently lower continuity calculated effective orifice areas will
lead to a systematic overestimation of stenosis severity.