a simulation study” - CEPACcepac.cheme.cmu.edu/pasi2011/library/bandoni/PASI_2011-Presentation-CV_Modeling-Bando...a simulation study” “Process StSystem Ei iEngineering in Human

“The Cardio‐Respiratory Human System:The Cardio Respiratory Human System: a simulation study”

“P S t E i i i H Ph i l ”“Process System Engineering in Human Physiology”

Elisa Montain, Anibal Blanco, Alberto BandoniPilot Plant of Chemical Engineering, PLAPIQUI (UNS‐CONICET)g g, Q ( )

Bahía Blanca, Argentina

PASI 2011PASI 2011 Process Modeling and Optimization for Energy and Sustainability

S d J l 23 2011 A d R i RJ B ilSaturday, July 23, 2011, Angra dos Reis, RJ, Brazil1PASI 2011 - A. Bandoni

BackgroundBackground TheThe cardiovascularcardiovascular systemsystem (CVS)(CVS) isis responsibleresponsible forfor supplyingsupplying oxygenoxygen

andand nutrientsnutrients toto tissuestissues andand organsorgans.. CVCV diseasesdiseases areare aa majormajor causecause ofof deathdeath inin humanshumans.. ManyMany experimentalexperimental studiesstudies havehave studiedstudied thethe mechanismsmechanisms andand therapytherapy ofof

thethe CVCV diseasesdiseasesthethe CVCV diseasesdiseases TogetherTogether withwith experimentalexperimental approaches,approaches, mathematicalmathematical modelingmodeling hashas

becomebecome aa popularpopular wayway toto analyzeanalyze thethe CVSCVS.. ManyMany modelsmodels havehave beenbeen publishedpublished sincesince thethe preliminarypreliminary andand basicbasic

modelmodel ofof GodinsGodins inin 19591959 ApproachesApproaches includeinclude:: hemodynamichemodynamic modelsmodels ofof thethe vascularvascular system,system, ApproachesApproaches includeinclude:: hemodynamichemodynamic modelsmodels ofof thethe vascularvascular system,system,

distributeddistributed impedanceimpedance andand pulmonarypulmonary arterialarterial stress,stress, lumpedlumped parameterparametermodelsmodels ofof thethe integratedintegrated CVS,CVS, hemodynamichemodynamic monitoringmonitoring models,models, etcetc....InIn thethe lastlast fefe earsears therethere ha eha e beenbeen importantimportant de elopmentsde elopments inin InIn thethe lastlast fewfew yearsyears therethere havehave beenbeen importantimportant developmentsdevelopments ininintegratedintegrated lumpedlumped parameterparameter modelsmodels ofof thethe circulatorycirculatory andand nervousnervouscontrolcontrol systemssystems..

2PASI 2011 - A. Bandoni

A i t i th d i i ki f di l tiA i t i th d i i ki f di l ti

MotivationMotivation Assistance in the decision making of medical practiceAssistance in the decision making of medical practice Diagnosis Diagnosis of diseases of diseases of the of the CVS (coronary CVS (coronary arteries and heart muscles arteries and heart muscles

dysfunctions, dysfunctions, valvularvalvular disorders and pulmonary disease. disorders and pulmonary disease.

ComprehendComprehend the the math. math. concepts and concepts and terms terms defining how defining how CVS CVS system behaves. system behaves.

To teach To teach about about the the complex interactions of the cardiovascular systemcomplex interactions of the cardiovascular system. .

H l t lH l t l ii t t tt t t l il i d t i id t i i d i id i i Help to vascular Help to vascular surgeons in surgeons in treatment treatment planningplanning and to engineers in and to engineers in designing designing better medical devicesbetter medical devices..

A promising integration strategy involves the A promising integration strategy involves the personalization of mathematical personalization of mathematical models models based on biophysical measurementsbased on biophysical measurements..

Analysis of the Analysis of the hemodynamicshemodynamics (blood flow dynamics) of the CVS. (blood flow dynamics) of the CVS.

CCapacityapacity to locate factors that are not directly observableto locate factors that are not directly observable KeyKey role inrole in thethe CCapacity apacity to locate factors that are not directly observable to locate factors that are not directly observable . . Key Key role in role in the the measurement of pump measurement of pump efficiency and tissue stress, to assist treatment decisions.efficiency and tissue stress, to assist treatment decisions.


MotivationMotivation Anesthesia control and drug delivery controlAnesthesia control and drug delivery control: : Control of patient physiological variablesControl of patient physiological variables during intensive care is achieved through during intensive care is achieved through

drug deliverydrug delivery..g yg y

Drug delivery process Drug delivery process depends on the value of the physiological variable under depends on the value of the physiological variable under controlcontrol and on the patient'sand on the patient's conditionconditioncontrolcontrol and on the patient s and on the patient s conditioncondition

Drugs such Drugs such sodium nitroprusside (SNP)sodium nitroprusside (SNP) and and dopamine (DP)dopamine (DP) are normally used for are normally used for regulation ofregulation of Media Arterial Pressure (MAP)Media Arterial Pressure (MAP) oror Cardiac Output (CO)Cardiac Output (CO)regulation of regulation of Media Arterial Pressure (MAP)Media Arterial Pressure (MAP) or or Cardiac Output (CO)Cardiac Output (CO). .

DDoctors octors use their use their discretion to regulate variables discretion to regulate variables that are difficult to quantify in that are difficult to quantify in practice or inferred from other measurements and patient responses to certain practice or inferred from other measurements and patient responses to certain surgical proceduressurgical procedures..

Currently, the drug infusion is done manually or by Currently, the drug infusion is done manually or by programmable pumpsprogrammable pumps. The . The professional is responsible for monitoring the controlled variable (MAP, CO) and professional is responsible for monitoring the controlled variable (MAP, CO) and the drug delivery according to the measurement.the drug delivery according to the measurement.g y gg y g


ObjectivesObjectives Development of an integrated distributed parameter model of

the human cardio respiratory system.

Development of a computational tool to help physicians in the diagnosis of various heart diseasesdiagnosis of various heart diseases.

Study of the drug delivery (SNP, DP, etc.)

The developed model contain the following sub-models:

y g y ( )

p g Circulatory system Baroreceptors Respiratory system Gas transport and distribution in organs Pharmacological effect of drugs on the hemodynamic variables Pharmacological effect of drugs on the hemodynamic variables.


Anatomy andAnatomy andAnatomy and Anatomy and Ph i lPh i lPhysiologyPhysiology


The Cardiovascular System The Cardiovascular System The Cardiovascular System:

It consists of:It consists of:

The heart, which is a muscular pumping device

A closed system of vessels (arteries, veins, andcapillaries).

The HeartThe Heart

The heart is a hollow muscular pump that provides the force necessary toi l t th bl d t ll th ti i th b d th h bl d lcirculate the blood to all the tissues in the body through blood vessels.

The normal adult heart pumps about 5 liters of blood every minutethroughout life.


Heart Anatomy

AortaSSuperior vena cava Pulmonary

truck

Left Atrium

Pulmonary valve

Pulmonary vein

Right atrium

Atrium

Mitralvalve

RightLeft

Tricuspid valve

Aorticvalve

Inferior vena

Right ventricle

Ventricle


Inferior vena cava

Functions of the HeartFunctions of the Heart

Generates blood pressureGenerates blood pressure

Routes bloodRoutes blood Heart separates pulmonary and systemic circulationHeart separates pulmonary and systemic circulation

Ensures oneEnsures one--way blood flowway blood flow Heart valves ensure oneHeart valves ensure one--way flowway flowyy

Regulates blood supplyRegulates blood supplyCh i t ti t d f t h bl d d li tCh i t ti t d f t h bl d d li t Changes in contraction rate and force match blood delivery to Changes in contraction rate and force match blood delivery to changing metabolic needschanging metabolic needs

Most healthy people can increase cardiac output by 300Most healthy people can increase cardiac output by 300––500%500%y p p p yy p p p y

Heart failure is the inability of the heart to provide enough blood flow to Heart failure is the inability of the heart to provide enough blood flow to i t i l t b lii t i l t b limaintain normal metabolismmaintain normal metabolism


Separated bySeparated by

The ChambersThe Chambers Separated by Separated by

InteratrialInteratrial SeptumSeptum InterventricularInterventricular SeptumSeptum

Right AtriumRight AtriumBl d f S i d i f iBl d f S i d i f i d th id th i Blood from Superior and inferior Blood from Superior and inferior venaevenae cavaecavae and the coronary sinusand the coronary sinus

Right VentricleRight Ventricle Receives blood from the right atrium via the right AV valve tricuspidReceives blood from the right atrium via the right AV valve tricuspidReceives blood from the right atrium via the right AV valve, tricuspid Receives blood from the right atrium via the right AV valve, tricuspid

valvevalve Thin wallThin wall

Left AtriumLeft Atrium Receives blood from R and L Pulmonary VeinsReceives blood from R and L Pulmonary Veins

Left VentricleLeft Ventricle Left VentricleLeft Ventricle Receives blood from the Left AV valveReceives blood from the Left AV valve Thick wallThick wall

Pumps to body via Aortic Pumps to body via Aortic SemilunarSemilunar ValveValve10PASI 2011 - A. Bandoni

T f l k h bl d

The ValvesThe Valves Two types of valves: keep the blood

flowing in the correct direction.

Between atria and ventricles:called atrioventricular valves (alsocalled cuspid valves)p )

Bases of the large vessels leavingthe ventricles: called semilunarthe ventricles: called semilunarvalves.

When the ventricles contract atrioventricular valves close to prevent When the ventricles contract, atrioventricular valves close to preventblood from flowing back into the atria.

When the ventricles relax semilunar valves close to prevent blood from When the ventricles relax, semilunar valves close to prevent blood fromflowing back into the ventricles.

V l l i l d bl d R ibl f th h t Vales close passively under blood pressure. Responsible for the heartsounds.


Circulatory SystemCirculatory System


Circulatory SystemCirculatory System

D t d bl d t t Deoxygenated blood returns to the heart via the superior and inferior vena cava, enters the right atrium passes into the rightright atrium, passes into the right ventricle, and from here it is ejected to the pulmonary artery.

Oxygenated blood returning from the lungs enters the left atrium via the pulmonary veins, passes into the left ventricle, and is then ejected to the aorta.


Blood flow pattern through the heartBlood flow pattern through the heart1.1.Blood enters right atrium via the superior Blood enters right atrium via the superior

and inferior and inferior venaevenae cavaecavae

2.2.Passes tricuspid valve into right ventriclePasses tricuspid valve into right ventricle

33 Leaves by passing pulmonaryLeaves by passing pulmonary semilunarsemilunar3.3.Leaves by passing pulmonary Leaves by passing pulmonary semilunarsemilunarvalves into pulmonary trunk and to the lungs valves into pulmonary trunk and to the lungs to be oxygenatedto be oxygenated

4.4.Returns from the lung by way of pulmonary Returns from the lung by way of pulmonary veins into the left atriumveins into the left atrium

5.5.From left atrium past bicuspid valve into left From left atrium past bicuspid valve into left ventricleventricle

6.6.Leaves left ventricle past aortic Leaves left ventricle past aortic semilunarsemilunarvalves into aortavalves into aortavalves into aortavalves into aorta

7.7.Distributed to rest of the bodyDistributed to rest of the body 14PASI 2011 - A. Bandoni

Blood flow pattern through the heartBlood flow pattern through the heart


Blood VesselsBlood Vessels Blood vessels are divided into a pulmonary circuit and systemic circuit.Blood vessels are divided into a pulmonary circuit and systemic circuit. Artery Artery -- vessel that carries blood away from the heart. Usually vessel that carries blood away from the heart. Usually

oxygenated. Exception, pulmonary artery.oxygenated. Exception, pulmonary artery.oxygenated. Exception, pulmonary artery.oxygenated. Exception, pulmonary artery. Vein Vein -- vessel that carries blood towards the heart. Usually vessel that carries blood towards the heart. Usually

deoxygenated. Exception pulmonary veinsdeoxygenated. Exception pulmonary veins Capillary Capillary -- a small blood vessel that allow diffusion of gases, nutrients a small blood vessel that allow diffusion of gases, nutrients

and wastes between plasma and interstitial fluid.and wastes between plasma and interstitial fluid.

Systemic vesselsTransport blood through the body part from left ventricle and back to right atriumg

Pulmonary vesselsTransport blood from right ventricle through lungs and back to leftTransport blood from right ventricle through lungs and back to left atrium

Blood vessels and heart are regulated to ensure blood pressure isBlood vessels and heart are regulated to ensure blood pressure is high enough for blood flow to meet metabolic needs of tissues


The Real ThingThe Real Thing


The Real ThingThe Real Thing


HistoryHistoryHistoryHistory


Mathematical Modelling in PhysiologyMathematical Modelling in Physiology With mathematical models it is possible to simulate almost any kind of With mathematical models it is possible to simulate almost any kind of

phenomena in nature on a computer. phenomena in nature on a computer.

This is a scientific practice of modern science and engineering This is a scientific practice of modern science and engineering ((biology, physiology, medicine, biology, physiology, medicine, climate researchclimate research, ecology, physics, , ecology, physics, chemistry etc )chemistry etc )chemistry, etc.)chemistry, etc.)

Mathematical modeling in medicine and biology has become so important Mathematical modeling in medicine and biology has become so important that this type of research now has its own name: in that this type of research now has its own name: in silicosilico

Mathematical modeling undoubtedly will become the paradigm of scientificMathematical modeling undoubtedly will become the paradigm of scientificMathematical modeling undoubtedly will become the paradigm of scientific Mathematical modeling undoubtedly will become the paradigm of scientific and medical research in the twentyand medical research in the twenty‐‐first century.first century.

In research the ultimate goal is mechanismsIn research the ultimate goal is mechanisms‐‐based models but in realitybased models but in reality In research, the ultimate goal is mechanismsIn research, the ultimate goal is mechanisms‐‐based models, but in reality based models, but in reality models are more often used in a detectivemodels are more often used in a detective‐‐like way to investigate the like way to investigate the consequences of different hypotheses.consequences of different hypotheses.

The mathematics modeling is used as a microscope to unveil information The mathematics modeling is used as a microscope to unveil information about reality, that is otherwise inaccessibleabout reality, that is otherwise inaccessible 20PASI 2011 - A. Bandoni

Heart and Blood Circulation Research HistoryHeart and Blood Circulation Research History

Since the dawn of civilization man has been concerned with theSince the dawn of civilization man has been concerned with the Since the dawn of civilization, man has been concerned with the Since the dawn of civilization, man has been concerned with the understanding of living things.understanding of living things.

I f th t i t di l t ti (I f th t i t di l t ti (N iN i Ji 2697Ji 2697 2597 BC)2597 BC) In one of the most ancient medical treatises (In one of the most ancient medical treatises (NeiNei Jing, 2697Jing, 2697--2597 BC), 2597 BC), blood is mentioned as originating in the heart and distributed in order to blood is mentioned as originating in the heart and distributed in order to return to the starting point.return to the starting point.

Despite widespread knowledge of the anatomy of blood vessels, Greeks Despite widespread knowledge of the anatomy of blood vessels, Greeks were unable to find the start of blood circulation by not knowing the were unable to find the start of blood circulation by not knowing the principle of conservation of mass. principle of conservation of mass.

The Western world had to wait for William Harvey (1578The Western world had to wait for William Harvey (1578--1657) to establish1657) to establish The Western world had to wait for William Harvey (1578The Western world had to wait for William Harvey (1578 1657) to establish 1657) to establish the concept of circulation.the concept of circulation.


Di f th l d i l ti f bl d b Willi HDi f th l d i l ti f bl d b Willi H

HistoryHistory Discovery of the closed circulation of blood by William Harvey Discovery of the closed circulation of blood by William Harvey

(1578(1578‐‐1657). 1657). "De Motu Cordis" ("On the Motion of the Heart and Blood“. Frankfurt, 1628)Stroke volume is 70 ml. per beat and Heart beats 72 times per minute, therefore Cardiac Output

should be 7.258 liters per day

Before 1628, the Before 1628, the GalenicGalenic view of the body prevailed and the concept of view of the body prevailed and the concept of blood circulation was not imaginable. blood circulation was not imaginable.

Galen or Galen or GaleniusGalenius (Greek physician, II century AD), spent most of his (Greek physician, II century AD), spent most of his lifetime observing the human body and its functioning. lifetime observing the human body and its functioning.

Galen believed that the heart acted not as a pump, but rather that it sucked Galen believed that the heart acted not as a pump, but rather that it sucked blood from the veins, that blood flowed from one ventricle to the other of blood from the veins, that blood flowed from one ventricle to the other of the heart through a system of tiny pores of the septumthe heart through a system of tiny pores of the septumthe heart through a system of tiny pores of the septum.the heart through a system of tiny pores of the septum.

Using a simple model, Harvey showed that the amount of blood leaving the Using a simple model, Harvey showed that the amount of blood leaving the h t i i t ld t i bl b b b d b th b d dh t i i t ld t i bl b b b d b th b d dheart in a minute could not conceivably be absorbed by the body and heart in a minute could not conceivably be absorbed by the body and continually replaced by blood made in the liver from continually replaced by blood made in the liver from chylechyle..


HistoryHistory Consequently, this model based evidence established the concept that Consequently, this model based evidence established the concept that

blood must constantly move in a closed circuit, otherwise the arteries and blood must constantly move in a closed circuit, otherwise the arteries and the body would explode under the pressure.the body would explode under the pressure.the body would explode under the pressure. the body would explode under the pressure.

This was discovered about 8 years before the light microscope.This was discovered about 8 years before the light microscope.

The concept or method of using mathematical modeling, as a tool for The concept or method of using mathematical modeling, as a tool for making an inaccessible system accessible or an invisible system visible, making an inaccessible system accessible or an invisible system visible,

f “ ff “ fis therefore being coined as “the mathematical microscope” in honor of is therefore being coined as “the mathematical microscope” in honor of William Harvey.William Harvey.

The mathematical microscopepOttesen (2011)


Th Wi dk lTh Wi dk lThe Windkessel The Windkessel EffectEffect


The Windkessel EffectThe Windkessel Effect The The windkessel windkessel effect is use to describe:effect is use to describe:

• Load faced by the heart in pumping blood through pulmonary or systemic arterial systemsystemic arterial system.

• Relation between blood pressure and blood flow in the aorta or pulmonary artery

Characteristic parameters of CVS such us compliance and peripheral resistance can be described in terms of the Windkessel models, which is

f l iuseful in: • Quantifying the effects of vasodilator or vasoconstrictor drugs. • The development and operation of mechanical heart and heart-lung

himachines.

WindkesselWindkessel: a : a germangerman word that can be translated as air (wind) chamber word that can be translated as air (wind) chamber ((k lk l))((kesselkessel). ).

First description by German physiologist Otto Frank in 1899. p y p y g


The Windkessel EffectThe Windkessel Effect Heart and systemic arterial system similar to a closed hydraulic circuit Heart and systemic arterial system similar to a closed hydraulic circuit

comprised of a water pump connected to a chamber. comprised of a water pump connected to a chamber.

The circuit is filled with water except for a pocket of air in the chamberThe circuit is filled with water except for a pocket of air in the chamber

Arterial compliance

P i h lPeripheral ressistance

As water is pumped into the chamber, the water both compresses the air in As water is pumped into the chamber, the water both compresses the air in the pocket and pushes water out of the chamberthe pocket and pushes water out of the chamberthe pocket and pushes water out of the chamber. . the pocket and pushes water out of the chamber. .


The Windkessel EffectThe Windkessel Effect The compressibility of the air in the pocket simulates the The compressibility of the air in the pocket simulates the elasticitelasticity and y and

extensibilityextensibility of the major artery, as blood is pumped into it by the heart of the major artery, as blood is pumped into it by the heart ventricle. ventricle.

This effect is commonly referred to as arterial This effect is commonly referred to as arterial compliancecompliance. .

The The resistanceresistance water encounters while leaving the Windkessel, simulates water encounters while leaving the Windkessel, simulates the resistance to flow encountered by the blood as it flows through the the resistance to flow encountered by the blood as it flows through the arterial tree from the major arteries to minor arteries to arterioles and toarterial tree from the major arteries to minor arteries to arterioles and toarterial tree from the major arteries, to minor arteries, to arterioles, and to arterial tree from the major arteries, to minor arteries, to arterioles, and to capillaries, due to decreasing vessel diameter. capillaries, due to decreasing vessel diameter.

Thi i t t fl i l f d tThi i t t fl i l f d t i h l i ti h l i t This resistance to flow is commonly referred to as This resistance to flow is commonly referred to as peripheral resistanceperipheral resistance..


Hypotheses:

The Windkessel EffectThe Windkessel EffectHypotheses:

• Unsteady flow. • The pressure diff. across the resistance is a linear function of the flow

trate• The working fluid is incompressible (constant air pressure to volume

ratio)• The flow is constant throughout the ejection phase.

The Windkessel 2-elements considers only the arterial compliance (C) and y p ( )the peripheral resistance (R).

Symbols:Symbols:P : pressure generated by the heart (N.m-2) [mmHg]Q : blood flow in the aorta (m3.s-1) [l.mn-1]R : peripheral resistance (N s m-5) [dyne s cm-5]R : peripheral resistance (N.s.m 5) [dyne. s.cm 5] C : arterial or systemic compliance (m5.N-1) [ml.mmHg-1] t : time [(s)T i d ( )T : period (s) Ts: ejection time (s)


The Windkessel EffectThe Windkessel Effect

Theoretical development of the Windkessel effect

air Q Ts

V(t)P(t)

R

Q (t)Q(t) PQ1(t)Q(t) Pcv tT

Schematic representationSchematic representation of a chamber Systolic phase:

valve in open position

Diastole phase: valve in close

positionposition position


The Windkessel EffectThe Windkessel EffectI S t li h ( l i iti )I - Systolic phase (valve in open position)

sTt 0

Conservation of mass: Qcc: flow to the compliance chamberccoutin QQQ

Thus: Pcv : central venus pressure: (Pcv<< P)(Pcv≅5 mmHg vs. P≅100 mmHg ])dt

dVQQ 1

Hyp.4: Q = Cte. throughout the systolic phase, thus: 1.QRPP cv

Therefore: Compliance (C)dtdP

dPdV

RP

dtdV

RPQ .

Then: or C

tQCRtP

dttdP )()()(

dt

tdPCRtPtQ )(.)()(

CCRdt .dtR



Solution of the differential equation

a) Particular solution (Q = Cte.=0) ).

exp(.)( 1 CRttP

b) Method of variation of parameter ( α1=α1(t) )

CQ

CRtt

CRCRtt

dtd

)

.exp().(

.1)

.exp().( 11

CQ

CRtt

CRdttd

CRt

CRtt

CR

)

.exp().(

.1)()

.exp()

.exp().(.

.1

11

1

Hence: )exp(.)(1 tQtd

Then: 21 )exp(..)( tQRt)

.e p(.

CRCdtThen: 21 )

.exp(..)(

CRQRt



c) The general solution for systolic phase is

tt

)

.exp(.)

.exp(..)( 2 CR

tCRtQRtPs

To determine α2 we can use initial condition P(t=0)=P0 , then α2 = P0-R.Q

QRPPtP .)0( 020

Finally, the pressure waveform for the systolic phase can be written as

).

exp(...)( 0 CRtQRPQRtPs


I Di t li h ( l i l iti )

The Windkessel EffectThe Windkessel EffectI – Diastolic phase (valve in close position) TtTs

air

Following similar reasoning but with Q=Cte.=0

QPdP

V(t)P(t)

CQ

CRP

dtdP

. Q1(t)

...1)exp(03 QRtP

)exp(.)( 3 CR

ttP

With initial condition: P(t=Ts)= Ps(Ts), the solution to the differential equation is:

where ).

p(03 QCR

).

p()( 3 CR

Fi ll th f f th di t li h b ittFinally, the pressure waveform for the diastolic phase can be written as:

)exp(...1)exp()( 0tQRtPtPd

)

.p()

.p()( 0 CR

QCRd


C l t d l

The Windkessel EffectThe Windkessel EffectComplete model

Systolic Phase sTt 0

)exp()( tQRPQRtP

air

V(t)P(t)

).

exp(...)( 0 CRQRPQRtPs

Q1(t)

TtTs

tt

air

V(t)P(t)

Diastolic Phase

).

exp(...1).

exp()( 0 CRtQR

CRtPtPd

Q1(t)

Given:

1)

.exp(

..0CR

T

QRP

s

ordataPandTTCRQ )(

1)

.exp(

..0

CRT

QRPordataPandTTCRQ s )(,,,, 0


Th t R C it i i l i th 2 W b it d t i th “ d”

The Windkessel EffectThe Windkessel EffectThe term R.C it is crucial in the 2-W because it determine the “speed” of the exponential decay. This product is called the “characteristic time”, called

P PP P

P0R.Q

0

t tCase: 0 Case: Case: 0 Case:


The Windkessel EffectThe Windkessel EffectCase: ,0 Hypertension: Ps > 140 mmHg

Pd > 90 mmHg


The Windkessel EffectThe Windkessel EffectThe electrical circuit equivalence

tdPCtPtQ )()()( Basic equation of a 2-element Winkessel model:dt

CR

tQ .)( q

Electric circuit of 2 passive elements: I(t) l t i l t Electric circuit of 2 passive elements: I(t) : electrical currentE(t) : electrical potentialC : capacitance of the capacitor

I(t) I3

R : resistance of the resistorI2

From the Ohm and Kirchhoff laws

dttdEC

RtEtI )(.)()( E(t) C R

I(t) ≡ Q(t) (blood flow)E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance)C ≡ C (arterial compliance)R ≡ R (peripheral resistance)



The 3-element Windkessel model

I(t)R2

I( ) Q(t) (bl d fl )I(t) ≡ Q(t) (blood flow)E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance)

C R1

E(t)R1 ≡ R1 (peripheral resistance

(syst. and pulm.circuits))R2 ≡ R2 (resistance of valves2 2 (

(aortic and pulmonary))

tdPCtPtdERCtIR )()()()(1 1

dtC

RdtRCtI

R)(.)()(..)(.1

21

2

1



The 4-element Windkessel model

I(t)R2

I(t) ≡ Q(t) (blood flow)I(t) Q(t) (blood flow)E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance)R ≡ R (peripheral resistance

C R1E(t)

R1 ≡ R1 (peripheral resistance (syst. and pulm.circuits))

R2 ≡ R2 (resistance of valves(aortic and p lmonar ))

E(t)

(aortic and pulmonary))L ≡ L (inertia of the blood circulation)L

tdPCtPtEdCLtdELCRtIR )()()()()(12

1

dtC

RdtCL

dtRCRtI

R.....)(.1

22

21

2


CompartmentCompartmentCompartment Compartment M d lM d lModelsModels


Compartment ModelsCompartment Models They are used to describe transport material in biological sciencesThey are used to describe transport material in biological sciences They are used to describe transport material in biological sciencesThey are used to describe transport material in biological sciences

A compartment model contains a certain number of compartments, each A compartment model contains a certain number of compartments, each one with a well mixed materialone with a well mixed materialone with a well mixed materialone with a well mixed material

Compartments exchange material following certain rulesCompartments exchange material following certain rules

Material can be stored in the boxes and transported between themMaterial can be stored in the boxes and transported between them

Every compartment has a number of connections entering and leaving it.Every compartment has a number of connections entering and leaving it.

Material can be added from the outside, can be removed or transported.Material can be added from the outside, can be removed or transported.Source

Drain41PASI 2011 - A. Bandoni

Compartment ModelsCompartment ModelsMaterial represent the amount of something that we wish to account forMaterial represent the amount of something that we wish to account for Material represent the amount of something that we wish to account forMaterial represent the amount of something that we wish to account for

To account for the material, the models must fulfill certain conservation To account for the material, the models must fulfill certain conservation laws.laws.

Conservations laws state that the difference between input and outputConservations laws state that the difference between input and output Conservations laws state that the difference between input and output Conservations laws state that the difference between input and output flows amounts how much will be stored.flows amounts how much will be stored.

A compartment model can also represent:A compartment model can also represent: A compartment model can also represent: A compartment model can also represent: Ecological systems (material could be energy and the compartment Ecological systems (material could be energy and the compartment

different species of animals or plants)different species of animals or plants) Physiologic system (material could be oxygen and compartment de Physiologic system (material could be oxygen and compartment de

organs)organs)

Compartment can not be thought as independent. Flow in and out may Compartment can not be thought as independent. Flow in and out may depend on the compartment volumedepend on the compartment volume

Inflow to compartment may depend of outflow of other compartment.Inflow to compartment may depend of outflow of other compartment.42PASI 2011 - A. Bandoni

Compartment ModelsCompartment ModelsState variables depend on each other and on the state of the system as aState variables depend on each other and on the state of the system as a State variables depend on each other and on the state of the system as a State variables depend on each other and on the state of the system as a whole.whole.

The transport in and out is characterized by the flows velocities.The transport in and out is characterized by the flows velocities.

Limitations of the compartment modelLimitations of the compartment model Limitations of the compartment modelLimitations of the compartment model•• Is the system closedIs the system closed. Equation of conservation of mass is correct . Equation of conservation of mass is correct

only if all material added or removed is included in the model. There only if all material added or removed is included in the model. There is some lost of detailed informationis some lost of detailed informationis some lost of detailed information.is some lost of detailed information.

•• Homogeneity assumptionHomogeneity assumption. Not always it is possible to keep this . Not always it is possible to keep this assumption. Then more compartments are needed but also more assumption. Then more compartments are needed but also more information it is requiredinformation it is requiredinformation it is required.information it is required.

•• Accuracy of the balance equationAccuracy of the balance equation. In real physiological system . In real physiological system typically some mass balance are know and other are not.typically some mass balance are know and other are not.R l f th b lR l f th b l N t ll t b d ib dN t ll t b d ib d•• Relevance of the mass balanceRelevance of the mass balance. Not all systems can be described . Not all systems can be described in terms of mass balances.in terms of mass balances.

•• Sensitivity analysisSensitivity analysis. Initial conditions and . Initial conditions and model parameters are model parameters are not not always known precisely.always known precisely.


MathematicalMathematicalMathematical Mathematical M d lM d lModelsModels

Cardiovascular, Respiratory Cardiovascular, Respiratory and Pharmacodynamicand Pharmacodynamicand Pharmacodynamicand Pharmacodynamic


Human Circulatory System ModelHuman Circulatory System Model TheThe historicalhistorical fascinationfascination ofof thethe heartheart hashas lastedlasted forfor manymany centuriescenturies andand

continuescontinues toto attractattract considerableconsiderable attentionattention bothboth theoreticallytheoretically andand clinicallyclinically..

ToTo developdevelop aa physiologicallyphysiologically foundedfounded modelmodel ofof thethe heartheart andand thethevasculature,vasculature, itit isis essentialessential toto havehave aa goodgood modelmodel ofof thethe humanhuman shortshort termtermpress repress re controlcontrol representedrepresented bb thethe baroreceptorbaroreceptor mechanismmechanismpressurepressure controlcontrol representedrepresented byby thethe baroreceptorbaroreceptor mechanismmechanism..

UsingUsing aa lumpedlumped parameterparameter compartmentcompartment model,model, thethe entireentire humanhumanUsingUsing aa lumpedlumped parameterparameter compartmentcompartment model,model, thethe entireentire humanhumancardiovascularcardiovascular systemsystem maymay bebe describeddescribed asas aa networknetwork ofof compliances,compliances,resistancesresistances andand inductancesinductances notnot reflectingreflecting anatomicalanatomical propertiesproperties..

AlthoughAlthough strikinglystrikingly simple,simple, thethe modelmodel givesgives aa veryvery goodgood descriptiondescription ofof thetheinputinput impedanceimpedance ofof thethe arterialarterial systemsystem..

SuchSuch modelsmodels areare valuablevaluable toolstools forfor understandingunderstanding cardiovascularcardiovascular diseasesdiseases(hypertension(hypertension weakweak andand enlargedenlarged heartheart hemorrhageshemorrhages etcetc ))(hypertension,(hypertension, weakweak andand enlargedenlarged heart,heart, hemorrhages,hemorrhages, etcetc..))


Human Circulatory System ModelHuman Circulatory System Model ModelsModels facilitatesfacilitates gettinggetting newnew insightinsight intointo cardiovascularcardiovascular functionsfunctions andand thethe

interactioninteraction withwith otherother systemsystem (central(central nervousnervous system,system, respiratoryrespiratory systems,systems,etcetc..))))

ThisThis typetype ofof modelsmodels cancan bebe reliablereliable andand stable,stable, simplysimply enoughenough toto runrun inin realrealtititimetime..

LumpedLumped cardiovascularcardiovascular modelsmodels areare divideddivided intointo pulsatilepulsatile andand nonnon--pulsatilepulsatile LumpedLumped cardiovascularcardiovascular modelsmodels areare divideddivided intointo pulsatilepulsatile andand nonnon pulsatilepulsatile..

InIn thethe pulsatilepulsatile case,case, thethe heartheart functioningfunctioning isis guidedguided byby aa timetime--varyingvaryingelastanceelastance functionfunction..

AA lumpedlumped pulsatilepulsatile cardiovascularcardiovascular modelmodel thatthat embracesembraces principalprincipal featuresfeatures AA lumpedlumped pulsatilepulsatile cardiovascularcardiovascular modelmodel thatthat embracesembraces principalprincipal featuresfeaturesofof thethe humanhuman circulationcirculation..


Human Circulatory System ModelHuman Circulatory System Model LumpedLumped cardiovascularcardiovascular modelsmodels areare divideddivided intointo pulsatilepulsatile andand nonnon--pulsatilepulsatile..

II thth l till til thth h th t f ti if ti i ii id did d bb titi ii InIn thethe pulsatilepulsatile case,case, thethe heartheart functioningfunctioning isis guidedguided byby aa timetime--varyingvaryingelastanceelastance functionfunction..

AA lumpedlumped pulsatilepulsatile cardiovascularcardiovascular modelmodel thatthat embracesembraces principalprincipal featuresfeaturesofof thethe humanhuman circulationcirculation..


Human Circulatory System ModelHuman Circulatory System Model

Ap3

Pulmonar circulation

Ap2 Vp1

RV

Ap1 Vp2

LA

RA

RV LA

LV Heart

As1Vs2

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As3

As2

Systemic circulation


Human Circulatory System ModelHuman Circulatory System Model

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Human Circulatory System ModelHuman Circulatory System ModelModel of a typical compartment (chamber) of the hemodynamic system

R : ressistance

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Blood output

yelement of a blood chamberpi p0

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Circulatory System Model (Ottesen et al., 2003)• Heart Model

H t it lf 4 h b (2 t i d 2 t i l )o Heart itself: 4 chambers (2 atria and 2 ventricles)o Vascular part

Systemic part: 5 chambers (systemic arteries and veins) Pulmonary part: 5 chambers (arteries and veins)Pulmonary part: 5 chambers (arteries and veins)

• Baroreceptor Modelo Chronotropic effect (on heart rate)o Inotropic effect (on the cardiac contractility)

V l ff t ( t i d i )

Respiratory System Model (Christiansen and Dræby, 1996)• Lung Model

o Vascular effect (on arteries and veins)

Lung Modelo Upper respiratory tracks: 1 chambero Alveoli: 1 chamber

• Gas Transport in Blood Model (O2, CO2, Anesthesia)V l t 5 h bo Vascular part: 5 chambers

o Organs and tissues: 8 compartments Organs compartments: one part of tissue and one part of blood (equilibrium

of the substances distributed by the blood on both sides it is assumed)of the substances distributed by the blood on both sides it is assumed) It is assumed constant blood (VB) and tissue (VT) volumes.

o Capillaries and alveoli: 1 chamber

Ph d i M d lPharmacodynamic Model (Gopinath et al., 1995)• Drug Effect on Hemodynamic Variables Model


The Cardiovascular ModelThe Cardiovascular Model

The Pumping HeartThe Pumping Heart

BasedBased onon anan elastanceelastance modelmodel wherewhere thethe cardiaccardiaccontractioncontraction propertiesproperties ofof thethe twotwo ventriclesventricles arearerepresentingrepresenting byby aa pairpair ofof timetime varyingvarying elastanceelastancerepresentingrepresenting byby aa pairpair ofof timetime--varyingvarying elastanceelastancefunctionsfunctions..

TheThe inertiainertia ofof bloodblood movementsmovements inin thethe ventriclesventricles isis consideredconsidered throughthrough TheThe inertiainertia ofof bloodblood movementsmovements inin thethe ventriclesventricles isis consideredconsidered throughthroughanan inductanceinductance thatthat introduceintroduce aa phasephase shiftshift betweenbetween thethe ventricularventricularpressurepressure andand thethe rootroot aorticaortic pressurepressure..

TheThe viscousviscous propertiesproperties ofof bloodblood inin thethe twotwo atriaatria areare includedincluded bybyventricularventricular fillingfilling resistanceresistancegg


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The Circulatory System ModelThe Circulatory System Model

Single chamber model

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The Baroreceptors ModelThe Baroreceptors Model Baroreceptors (BR) are sensors of mean blood pressure that are located in

the blood vessels of several mammals.

BR nerves are stretch receptors which responds to changes in blood pressure.

BR can send messages to the CNS to increase or decrease total peripheral resistance and cardiac output (CO).

BR act immediately as part of a negative feedback system called the baroreflex, returning mean arterial blood pressure (MAP) to a normal level as soon as there is a changeas soon as there is a change.

BR detect the amount of stretch of the blood vessel walls, and send the signal to the CNS system in response to this stretch.

A hysteresis-like phenomena is observed: the response to a pressure increase is different to the response to a pressure-decrease


The Baroreceptors ModelThe Baroreceptors Model① Increased blood pressure stretched carotid arteries and aorta causing the baroreceptor to increase their basal rate of action potential generation.

② A ti t ti l d t d b② Action potential are conducted by the glossopharyngeal and the vagus nerves to the cardioregulatory and

t t i th d llvasomotor centers in the medulla oblongata.

③ As a result of increased③ As a result of increased stimulation from the baroreceptor, the cardioregulatory center increased parasymphatic stimulation to theparasymphatic stimulation to the heart, which decreases the heart rate.

④ Also, as a result of increased stimulation from the baroreceptor, the④ Also, as a result of increased stimulation from the baroreceptor, the cardiorvascular center decreases sympathetic stimulation to the heart, which decreases heart rate stroke volume. 58PASI 2011 - A. Bandoni

The Baroreceptors ModelThe Baroreceptors Model⑤ The vasomotor center decreases sympathetic stimulation to blood vessels, causing vasodilatation. The vasodilatation along with the decreased heart rate and decreased stroke volume bring the elevated blood pressure back toward normal.

fIf the initial problem were decrease in blood pressure, the activities and effect of baroreceptors, cardiovascular center and vasomotor center would be

it f h t ill t t dopposite of what was illustrated.


The Baroreceptors ModelThe Baroreceptors Model

Heart HBaroreceptor

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The Baroreceptors ModelThe Baroreceptors Model

Afferent sector Efferent sector

SensorsCentral Nervous Eferent

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The Respiratory System ModelThe Respiratory System Model

TheThe respiratoryrespiratory systemsystem isis concernedconcerned withwith thethe transporttransport ofof oxygenoxygenp yp y yy pp ygygbetweenbetween atmosphereatmosphere andand thethe tissuetissue andand organsorgans inin thethe bodybody

OO ii ti lti l t t dt t d bb thth ll dd bl dbl d i iti it OxygenOxygen isis continuouslycontinuously transportedtransported byby thethe lunglung andand bloodblood circuitcircuit..

CarbonCarbon dioxidedioxide isis aa wastewaste productproduct ofof thethe oxidativeoxidative metabolismmetabolism andand isis CarbonCarbon dioxidedioxide isis aa wastewaste productproduct ofof thethe oxidativeoxidative metabolismmetabolism andand isiscarriedcarried byby thethe bloodblood inin thethe oppositeopposite directiondirection


The Respiratory System ModelThe Respiratory System ModelO2 CO2

Ventilation

Atmosphere

Alveoli

Ventilation

O2 CO2

Gas exchange

O2 CO2

L ftRi ht

Pulmonary circulation

Left Heart

Right Heart Gas transport

Systemic circulationO2 CO2

Cell Gas exchange metabolism 63PASI 2011 - A. Bandoni


Lung model: pressureLung model: pressure■ Connect atmosphere (mask)■ Connect atmosphere (mask)

with alveoli trought expressions of gas flow

R0 Alveoli

■ The lung is divided in compartments

R1 R2 RiUpper i

■ In each compartment gas flows are calculated (O2,

R1 R2 Ri

C1 C2 CiC0

airway

Um (t)CO2, Anesthesia)

■ The outputs of the model are: Atmosphere

or i t

Ut (t)

pressure in different sectors, the net volume of air flow, partial pressure of expired air

respiratory mask Muscles

pa t a p essu e o e p ed aand alveoli.


The Respiratory System ModelThe Respiratory System Model Distribution of substances in the organs through bloodDistribution of substances in the organs through blood

AlveolusCapillaryAlveolus

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The Pharmacodynamic ModelThe Pharmacodynamic Model

PharmacologyPharmacology:: thethe history,history, source,source, physicalphysical andand chemicalchemical properties,properties,biochemicalbiochemical andand physiologicalphysiological effect,effect, mechanismsmechanisms ofof action,action, absorption,absorption,distribution,distribution, biotransformationbiotransformation andand excretion,excretion, andand therapeutictherapeutic andand otherotherusesuses ofof drugsdrugs..

PharmacokineticsPharmacokinetics:: absorptionabsorption distributiondistribution metabolismmetabolism andand excretionexcretion ofof PharmacokineticsPharmacokinetics:: absorption,absorption, distribution,distribution, metabolismmetabolism andand excretionexcretion ofofdrugsdrugs..

PharmacodynamicsPharmacodynamics:: biochemicalbiochemical andand physiologicalphysiological effectseffects andand theirtheirmechanismsmechanisms ofof actionaction..



tion

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Concentration of drug in the body as a function of timeConcentration of drug in the body as a function of time


The Pharmacokinetic ModelThe Pharmacokinetic Model

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Drug (intravenous) Affected variable Action

SNP (sodiumSNP (sodium nitroprusside) Peripheral resistance MAP

DP (dopamine) Peripheral resistance, systolic maximum elastance MAP( p ) systolic maximum elastance

PFL (propofol) BIS MAP unconsciousness

Systolic maximum elastance

Peripheral resistance



DPDP andand SNPSNP drugsdrugs areare chosenchosen toto increaseincrease ventricularventricular contractilitycontractility andanddd thth i ti t ff t it i tt bl dbl d flfl ti lti lreducereduce thethe resistanceresistance ofof arteriesarteries toto bloodblood flow,flow, respectivelyrespectively..

PFLPFL isis chosenchosen toto conductconduct unconsciousnessunconsciousness byby measuredmeasured ofof BISBIS PFLPFL isis chosenchosen toto conductconduct unconsciousnessunconsciousness byby measuredmeasured ofof BISBISparameterparameter..

DPDP increasesincreases thethe MAPMAP andand COCO.. SNPSNP decreasesdecreases andand increasesincreases COCOMAPMAP..

Sceneries are simulated by delivering a step of 1μg/kg/min of SNP, DP Sceneries are simulated by delivering a step of 1μg/kg/min of SNP, DP and PFL and registering the dynamic response of the physiological, and PFL and registering the dynamic response of the physiological, pharmacokinetic and pharmacod namic ariablespharmacokinetic and pharmacod namic ariablespharmacokinetic and pharmacodynamic variables. pharmacokinetic and pharmacodynamic variables.


Computational ImplementationComputational Implementation

Model implemented in FortranModel implemented in Fortran

Diff. Diff. EqsEqs. solved with a 4th order . solved with a 4th order RungeRunge--KuttaKutta method. method.

Resolution sequence: Resolution sequence: ((ii) ) the cardiovascular model is solved until to the cardiovascular model is solved until to reach steady state, reach steady state, (ii) (ii) the CO obtained from this model is used in the the CO obtained from this model is used in the breathing model, breathing model, (iii) (iii) the breathing model is solved until to reach steady the breathing model is solved until to reach steady g ,g , ( )( ) g yg ystate.state.

Th d i j ti i i l t dTh d i j ti i i l t d f l f b thi (5 )f l f b thi (5 ) Th thTh th The drug injection is simulated The drug injection is simulated for a cycle of breathing (5 sec.)for a cycle of breathing (5 sec.). Then the . Then the cardiovascular model is fed with the drug concentration cardiovascular model is fed with the drug concentration CdCd to simulate to simulate the the 0.8 sec. a heartbeat0.8 sec. a heartbeat. The updated value of CO is fed back to the . The updated value of CO is fed back to the breathing model.breathing model.



Cd(inhalable)CO2, O2Cd (alveoli)

Cd(inyectable)

Respiratory system

Transport and distribution,

Pharmacokinetics ofsystem Pharmacokinetics of drugs

Qa3, Qp3

Cd (organs)

C di l Ph d i

MAP

Cardiovascular system

Pharmacodynamics of drugsBaroreflex

EmaxRsis

EffEmaxEffRa2EffRa3

EmaxBARORa2BARORa3BARO

Control


Action


Dimensions of the integrated model

Model Var./Eqs. Algebraics

Var./Ecs. Differenctials ParametersAlgebraics Differenctials

Cardiovascular-Respiratory 37 39 53

Respiratory-Pharmacodynamic 60 93 85

Total 97 132 138


ResultsResults


Results: cardio vascular systemResults: cardio vascular system

Wiggers Diagram



Left ventricle and root aortic pressure vs. time

Left ventricular volume vs. time



Outflow of the left ventricleOutflow of the left ventricle

Left ventricular pressure Pressure vs. Volume left ventricle79PASI 2011 - A. Bandoni

Results: Results: baroreflexbaroreflex systemsystem

Heart period vs time Resist sect A of syst arteries vs timeHeart period vs. time Resist. sect. As1 of syst. arteries vs. time

Compliance in sect. Vs1 of sistemicveins vs. time

Unstres. Vol. of sect. Vs1 of sistemic veins vs. time 80PASI 2011 - A. Bandoni

Results: Results: baroreflexbaroreflex systemsystem

Sistolic max. elastance of left ventr.vs. time Comparison of CO vs. time in front of 10 % bleeding, with and without baroreceptor

Comparison of MAP vs. time in front of 10 % bleeding, with and without baroreceptor


Results: gas transportResults: gas transport

Partial pressure of O2 in different compartments of the body

Partial pressure of CO2 in different compartments of the bodycompartments of the body


Results: respiratory systemResults: respiratory systemexpiraciónexpiración

inspiración

Volume vs. Pressue diagram in lungs

Partial pres. profile of CO2 in lung and alveoli.

Partial pres. profile of O2 in lung and alveoli. 83PASI 2011 - A. Bandoni

Results: Results: pharmacodymicpharmacodymic systemsystemEffect of the SNP action 1µg/kg/min

SNP concentration profile at the central arterial compartment


Mean Arterial Pressure, MAP Cardiac Output, CO

Results: pharmacodynamic systemResults: pharmacodynamic system

Effect of the SNP action 1µg/kg/min

Resistance, Ra2 Resistance, Ra3


Results: pharmacodynamic systemResults: pharmacodynamic systemEffect of the DP action 5µg/kg/min

DP concentration profile at the central arterial compartment

86PASI 2011 - A. BandoniMean Arterial Pressure, MAP Cardiac Output, CO

Results: pharmacodynamic systemResults: pharmacodynamic systemEffect of the DP action 5µg/kg/min

Medial arterial resistances

Elastance


Results: pharmacodynamic systemResults: pharmacodynamic systemEffect of the DP action 2, 4, 6, 8 µg/kg/min

Cardiac Index vs. infusion doses (time) Volume Index vs. infusion doses (time)

Systolic and diastolic pressure vs. infusion doses

(time)


Results: pharmacodynamic systemResults: pharmacodynamic systemEffect of the DP action 2, 4, 6, 8 µg/kg/min

Systemic Resistance vs. infusion d (ti )

Cardiac frequency vs. infusion doses (time)doses (time) doses (time)


Effect of the PFL action / /

Results: pharmacodynamic systemResults: pharmacodynamic systemEffect of the PFL action 150µg/kg/min

Mean Arterial Pressure, MAPPFL conc. at the central arterial comp.


Cardiac Output, CO Compliance of sector a1 of systemic arteries

ConclusionsConclusions

DevelopmentDevelopment ofof anan integratedintegrated cardiovascular,cardiovascular,baroreceptor,baroreceptor, respiratory,respiratory, pharmacokineticpharmacokinetic andandpharmacodynamicpharmacodynamic modelmodel..

TheThe effecteffect ofof certaincertain drugsdrugs onon hemodynamichemodynamic variablesvariableswaswas studiedstudied..


Future WorksFuture Works GeneralGeneral modelmodel validationvalidation withwith realreal patientpatient datadata.. CollaborationCollaboration

withwith aa researchresearch groupgroup formedformed byby doctorsdoctors ((FavaloroFavaloro University,University,BsBs AsAs EspañolEspañol HospitalHospital BB BlancaBlanca ArgArg ))BsBs..AsAs.. –– EspañolEspañol Hospital,Hospital, BB.. Blanca,Blanca, ArgArg..))

ModelModel validationvalidation forfor inhalableinhalable anesthesiaanesthesia effectseffects ModelModel validationvalidation forfor inhalableinhalable anesthesiaanesthesia effectseffects..

ModelModel validationvalidation forfor simultaneouslysimultaneously drugsdrugs administrationadministration ModelModel validationvalidation forfor simultaneouslysimultaneously drugsdrugs administrationadministration..

DevelopmentDevelopment ofof aa controlcontrol modelmodel forfor handlinghandling dosedose ofof drugdrug DevelopmentDevelopment ofof aa controlcontrol modelmodel forfor handlinghandling dosedose ofof drugdrugadministrationadministration..

DevelopmentDevelopment ofof aa teachingteaching simulationsimulation modelmodel ofof thethecardiovascularcardiovascular systemsystem ((InstitutoInstituto NacionalNacional dede TecnologíaTecnologíaIndustrial,Industrial, INTI,INTI, BsBs..AsAs..,, ArgArg..))


Basic References:

Cardiovascular Model: Ottesen J., Olufsen M. and Larsen J. Applied Mathematical Models inOttesen J., Olufsen M. and Larsen J. Applied Mathematical Models in

Human Physiology . SIAM, Philadelphia. (2004)

Pharmacodynamic Model:Pharmacodynamic Model: Gopinath R., Bequette B., Roy R. and Kaufman H. Issues in the Design

of a Multirate Model- based Controller for a Nonlinear Drug Infusion System Biotechnol Prog 11 (3) pp 318 32 (1995)System. Biotechnol. Prog. 11 (3), pp 318–32. (1995)

Respiratory Model:Ch i ti T d D b C M d li th R i t S t Christiansen T. and Dræby C. Modeling the Respiratory System Technical. Report IMFUFA, Roskilde University Denmark Text No. 318. (1996)


Other References:

Dua P and Pistikopulos E Modelling and control of drug delivery systems Comp Dua P and Pistikopulos E. Modelling and control of drug delivery systems. Comp. Chem. Eng. 29 pp. 2290-96. (2005)

Montain M Bandoni J y Blanco A Modelado del sistema cardiorespiratorio Montain M, Bandoni J y Blanco A . Modelado del sistema cardiorespiratoriohumano: un estudio de simulación. VI CAIQ (Congreso Argentino de Ing. Química) Mar del Plata 26 al 29 de septiembre (2010).

Rao R, Bequette B and Roy R. Simultaneous regulation of hemodynamic and anesthetic states: a simulation study; Annals of Biomedical Engineering, 28 pp. 71-84. (2000)( )

Dua P, Dua V and Pistikopoulos E. Modelling and mult-parametric control for delivery of anaesthetic agents. Med. Biol. Eng. Comput. 48 543-53. (2010).

Massoud T., G. Eorge, J. Hademenos, W. Young , E. Gao, J. Pile-Spellman and F. Uela. Principles and philosophy of modeling in biomedical research.The FASEB Journal, vol. 12 no. 3, pp.275-285, March 1, 1998.

Ottesen J.T. The Mathematical Microscope ‐ Making the inaccessible accessible. Bi di l d Lif S i S t Bi l V l 2 2011


Biomedical and Life Sciences Systems Biology ‐ Volume 2, 2011.

d“With growing emphasis being placed on the informationprocessing aspects of biomedical investigation, theoretical andexperimental studies assume increasing importance In manyexperimental studies assume increasing importance. In manyinstances, however, there are questions that appear to beunanswerable by present experimental techniques; in such cases,y p p q ; ,models can usefully augment direct scientific experimentation.

Th i l i di f h i ifi h d i h fThe essential ingredient of the scientific method is the use ofmodels. Good modeling is more likely to be achieved by followingthe rules of good thinking However the ideal model cannot bethe rules of good thinking. However, the ideal model cannot beachieved. Partial models, imperfect as they may be, are the onlymeans developed by and available to scientists for understandingp y gthe universe”

Principles and philosophy of modeling in biomedical research.T. Massoud, G. Eorge, J. Hademenos, W. Young , E. Gao, J. Pile-Spellman and F. Uela


(University of California at Los Angeles, Columbia University, University of Dallas)The FASEB Journal, vol. 12 no. 3, pp.275-285, March 1, 1998


Muchas gracias


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dtadQ

11

aQlvQdtadV

lvplapsilaQ 0

laQlQdtladV

2 ladVlaVlaElap ,

1aLdt dt

11,1

1aC

aunVaVap

232

2aR

apapaQ

dt

asplvpsilvL

asplvpdt

lvdQ

21

2aQaQ

dtadV

2

2,22

aCaunVaV

ap

asplvpsilvQ 0

lvQlaQlvdV lvdVlvVtlvElvp ,

31

3aR

vpaspaQ

32

3aQaQ

dtadV

VVlvQlaQdt

lvdlvlvlv ,

mlt

tdtlvQblvV 2,

*

33,3

3aC

aunVaVap

121

1vR

vpvpvQ

1dV 11 VV

tlvEtlvEtlvE max,1min,

tt 2

131

vQaQdtvdV

1

1,11

vCvunVvV

vp

2222 vQvRrapvpvdQ 2dV

httcet

cettcet

tbcetta

t0

02sinsin

2

2222vL

vQvRrapvpdtvdQ 21

2vQvQ

dtvdV

22 vunVvV htcet 10 10 aplvsQRasp 2

2,22vC

vunVvVpv


Cámara derecha del corazón

rvprapsiraL

raQraRrvprapdtradQ

Circulación pulmonar

111211

pLpQpRpppp

dtpdQ

11

pQrvQdtpdV

rvprapsiraQ 0

raQvQdtradV

2 radVraVraErap ,1

1,11

pCpunVpV

pp

2

322

pRpppp

pQ

212

pQpQpdV

2,22

punVpVpp

apprvpsirvL

apprvpdtrvdQ

iQ 0

21 pQpQdt 2

2pCpp

31

3pR

lppsppQ

32

3pQpQ

dtpdV

VVapprvpsirvQ 0

rvQraQdtrvdV

rvdVrvVtrvErvp ,

t

33,3

3pC

punVpVpp

1211

lRlplp

lQ

131 lQpQdt

ldV

11,1

1lC

lunVlVlp

mlt

dtrvQbrvV 2,*

trvEtrvEtrvE max,1min,

1l

22222

lLlQlRlaplp

dtldQ

212 lQlQdtldV

22 lVlV10 pprvpQRapp 2

2,22lC

lunVlVpl

Modelo respiratorio (fracción molar)Modelo respiratorio (fracción molar)

n

i iRipipI

RepmUI

pVpC

Tdt

d1

000

00

000200

0 fffff R

00 x

xI

iipcp

iRiiptUpI

ipiVipiC

Tdt

idfpκ

fff 00

02R

0xx

xI


MAP

MAPsn

1

1

MAP

MAPpn

1

1Barorreceptores

2211 QQ

ccc pcccp

b Qd

Vd1

1

1

iMAPpniMAPsniMAPbi

vCunVpsREHEi ,,,max,

0

21 QQx c pcc

p b

xb

b Qd

Vdt

EiMAPbitix

idttidx

,1

Modelo respiratorio (presión)

OO

OO

cc

cM

22

2

2

0

M

00

2COMM

001

000

CR

n

i dtdpi

iCRpmU

dtdp

iCiRtUipp

dtdpi

0

aaaa

aaaa c

cM

23

904.1806.150909.800005.17806.1278.053554.066943.0

278.0096364.127273.0

xtxt

xt

mU

00122

3 aHaHaH

COaa KNaOHKa ,0Pr,2

PHN OHKKN OHK 5904.105904.057034.0 xt

Modelo de transporte de gases en sangre

00,Pr,0,1 Pr2

HNaOHKKcNaOHKa COaaCOCOa

200Pr,,0 Pr COaCOa cHNaOHKKa

cc

ddd1

Ery

Hb

HbPlaCOEry

Hb

HbEryCO

bCO c

cpH

cc

pHpH 1,,,222pcpcpc

pHpKpHpHc

EryCO

EryCO

EryEry

COpH ,,101,

222

ppppc

pMpMpccpc

pcp

bas

Qz

dd

Vdd

Vdtd

ib

bt

t

cp

b Qd

Vd 11

pHpKpHpHc

EryCO

EryCO

EryEry

COpH ,,101,

222

ppppc 100PASI 2011 - A. Bandoni

pppccp

pAbs

vb

b Qd

Vdt

1

pHpKpHc

PlaCO

PlaCO

Pla

COpH ,101,

222

pppc

pHspHpHEry OEry

pHpK ,06.084.7, 2101log1256 ppp

37

º055.0946.10 C

Tax pp

l c pHpK 10 101log125.6,p

pHspHpHpH OEry ,1035.04.777.019.7,

2pp

78101l1256 pHPla HK aab

baa pc p

Ery

Hb

HbPlaCO

tCO c

cpHcpHc 1,,

22pp

7.810 101log125.6, pHPla pHpK p

2222

, OHbOObO scppH pc

s 1

pppppp 00 5343.0tanh875.1 xxhxxy

pph 53 kPapx /logp t p pyO es

12 pp ah 5.3 kPapx O /log

2p

dpgHbfCO

CO lmmolcx

kPap

cpHa 5/03.007.033.5

log09.04.772.0 2

2

pp

aattaa pc p

222 OO

tO pc p

Modelo farmacodinámico

HbfHiHbCO xxx 28.0174.0386.0

EffkEffEffNdCk

dtdEff

2max1

dtCdEff

BASEaCdtadC PFLa111

d ff

dtRdEff

dtRdEff

sisBARORREffREffdt

sisBAROdRdtsisdR sisSNPsisDP

sisSNPsisDP1

dt

EdEfflvBAROEEEff

dtlvBAROdE

dtlvdE lvDP

lvDPmax

max max1maxmax


Documents

a simulation study” - CEPACcepac.cheme.cmu.edu/pasi2011/library/bandoni/PASI_2011-Presentation-CV_Modeling-Bando...a simulation study” “Process StSystem Ei iEngineering in Human