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The Cardio Respiratory Human System: The Cardio Respiratory Human System: a simulation study Process System Engineering in Human Physiology P S t E i i i H Ph i l Elisa Montain, Anibal Blanco, Alberto Bandoni Pilot


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“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 Bandoni Pilot Plant of Chemical Engineering, PLAPIQUI (UNS‐CONICET) g g, Q ( ) Bahía Blanca, Argentina

PASI 2011 PASI 2011

Process Modeling and Optimization for Energy and Sustainability S d J l 23 2011 A d R i RJ B il Saturday, July 23, 2011, Angra dos Reis, RJ, Brazil

1 PASI 2011 - A. Bandoni

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Background Background

The The cardiovascular cardiovascular system system (CVS) (CVS) is is responsible responsible for for supplying supplying oxygen

  • xygen

and and nutrients nutrients to to tissues tissues and and organs

  • rgans.

.

CV CV diseases diseases are are a a major major cause cause of

  • f death

death in in humans humans. .

Many Many experimental experimental studies studies have have studied studied the the mechanisms mechanisms and and therapy therapy of

  • f

the the CV CV diseases diseases the the CV CV diseases diseases

Together Together with with experimental experimental approaches, approaches, mathematical mathematical modeling modeling has has become become a a popular popular way way to to analyze analyze the the CVS CVS. .

Many Many models models have have been been published published since since the the preliminary preliminary and and basic basic model model of

  • f Godins

Godins in in 1959 1959

Approaches Approaches include include: hemodynamic hemodynamic models models of

  • f the

the vascular vascular system, system,

Approaches Approaches include include: hemodynamic hemodynamic models models of

  • f the

the vascular vascular system, system, distributed distributed impedance impedance and and pulmonary pulmonary arterial arterial stress, stress, lumped lumped parameter parameter models models of

  • f the

the integrated integrated CVS, CVS, hemodynamic hemodynamic monitoring monitoring models, models, etc etc.. .. In In the the last last fe fe ears ears there there ha e ha e been been important important de elopments de elopments in in

In In the the last last few few years years there there have have been been important important developments developments in in integrated integrated lumped lumped parameter parameter models models of

  • f the

the circulatory circulatory and and nervous nervous control control systems systems. .

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

Motivation Motivation

 Assistance in the decision making of medical practice Assistance in the decision making of medical practice

Diagnosis Diagnosis of diseases

  • f diseases of the
  • f the CVS (coronary

CVS (coronary arteries and heart muscles arteries and heart muscles dysfunctions, dysfunctions, valvular valvular disorders and pulmonary disease. disorders and pulmonary disease.

Comprehend Comprehend 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 system complex interactions of the cardiovascular system. . H l t l H l t l i i t t t t t t l i l i d t i i d t i i d i i d i i

Help to vascular Help to vascular surgeons in surgeons in treatment treatment planning planning and to engineers in and to engineers in designing designing better medical devices better 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 measurements based on biophysical measurements. .

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

Capacity 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

Capacity 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.

3 PASI 2011 - A. Bandoni

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Motivation Motivation

 Anesthesia control and drug delivery control Anesthesia control and drug delivery control: :

Control of patient physiological variables Control of patient physiological variables during intensive care is achieved through during intensive care is achieved through drug delivery drug delivery. g y g 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 control control and on the patient's and on the patient's condition condition control control and on the patient s and on the patient s condition condition

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 of regulation of Media Arterial Pressure (MAP) Media Arterial Pressure (MAP) or

  • r Cardiac Output (CO)

Cardiac Output (CO) regulation of regulation of Media Arterial Pressure (MAP) Media Arterial Pressure (MAP) or

  • r Cardiac Output (CO)

Cardiac Output (CO). .

Doctors

  • ctors 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 procedures surgical procedures. .

Currently, the drug infusion is done manually or by Currently, the drug infusion is done manually or by programmable pumps programmable 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 g g y g

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Objectives Objectives

 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 diseases diagnosis 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.

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Anatomy and Anatomy and Anatomy and Anatomy and Ph i l Ph i l Physiology Physiology

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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, and capillaries). The Heart The Heart

  • The heart is a hollow muscular pump that provides the force necessary to

i l t th bl d t ll th ti i th b d th h bl d l circulate the blood to all the tissues in the body through blood vessels.

  • The normal adult heart pumps about 5 liters of blood every minute

throughout life.

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Heart Anatomy

Aorta Superior vena cava Pulmonary truck Left Atrium Pulmonary valve Pulmonary vein Right atrium Atrium Mitral valve Right Left Tricuspid valve Aortic valve Inferior vena Right ventricle Ventricle

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Inferior vena cava

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Functions of the Heart Functions of the Heart

Generates blood pressure Generates blood pressure

Routes blood Routes blood

 Heart separates pulmonary and systemic circulation

Heart separates pulmonary and systemic circulation

Ensures one Ensures one-

  • way blood flow

way blood flow

 Heart valves ensure one

Heart valves ensure one-

  • way flow

way flow y

Regulates blood supply Regulates blood supply Ch i t ti t d f t h bl d d li t Ch 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 needs changing metabolic needs

 Most healthy people can increase cardiac output by 300

Most healthy people can increase cardiac output by 300–500% 500% y p p p y y 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 li i t i l t b li maintain normal metabolism maintain normal metabolism

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 Separated by

Separated by

The Chambers The Chambers

 Separated by

Separated by

  • Interatrial

Interatrial Septum Septum

  • Interventricular

Interventricular Septum Septum

 Right Atrium

Right Atrium Bl d f S i d i f i Bl d f S i d i f i d th i d th i

  • Blood from Superior and inferior

Blood from Superior and inferior venae venae cavae cavae and the coronary sinus and the coronary sinus

 Right Ventricle

Right Ventricle

  • Receives blood from the right atrium via the right AV valve tricuspid

Receives blood from the right atrium via the right AV valve tricuspid Receives blood from the right atrium via the right AV valve, tricuspid Receives blood from the right atrium via the right AV valve, tricuspid valve valve

  • Thin wall

Thin wall

 Left Atrium

Left Atrium

  • Receives blood from R and L Pulmonary Veins

Receives blood from R and L Pulmonary Veins

 Left Ventricle

Left Ventricle

 Left Ventricle

Left Ventricle

  • Receives blood from the Left AV valve

Receives blood from the Left AV valve

  • Thick wall

Thick wall

  • Pumps to body via Aortic

Pumps to body via Aortic Semilunar Semilunar Valve Valve

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T f l k h bl d

The Valves The Valves

 Two types of valves: keep the blood

flowing in the correct direction.

 Between

atria and ventricles: called atrioventricular valves (also called cuspid valves) p )

 Bases of the large vessels leaving

the ventricles: called semilunar the ventricles: called semilunar valves.

 When the ventricles contract

atrioventricular valves close to prevent

 When the ventricles contract, atrioventricular valves close to prevent

blood 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 from

flowing 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 heart sounds.

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Circulatory System Circulatory System

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Circulatory System Circulatory 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 right right 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.

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Blood flow pattern through the heart Blood flow pattern through the heart

1. 1.Blood enters right atrium via the superior Blood enters right atrium via the superior and inferior and inferior venae venae cavae cavae 2. 2.Passes tricuspid valve into right ventricle Passes tricuspid valve into right ventricle 3 Leaves by passing pulmonary Leaves by passing pulmonary semilunar semilunar 3. 3.Leaves by passing pulmonary Leaves by passing pulmonary semilunar semilunar valves into pulmonary trunk and to the lungs valves into pulmonary trunk and to the lungs to be oxygenated to be oxygenated 4. 4.Returns from the lung by way of pulmonary Returns from the lung by way of pulmonary veins into the left atrium veins into the left atrium 5. 5.From left atrium past bicuspid valve into left From left atrium past bicuspid valve into left ventricle ventricle 6. 6.Leaves left ventricle past aortic Leaves left ventricle past aortic semilunar semilunar valves into aorta valves into aorta valves into aorta valves into aorta 7. 7.Distributed to rest of the body Distributed to rest of the body

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Blood flow pattern through the heart Blood flow pattern through the heart

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Blood Vessels Blood 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

  • xygenated. Exception, pulmonary artery.
  • xygenated. Exception, pulmonary artery.
  • xygenated. Exception, pulmonary artery.
  • xygenated. Exception, pulmonary artery.

Vein Vein -

  • vessel that carries blood towards the heart. Usually

vessel that carries blood towards the heart. Usually

  • deoxygenated. Exception pulmonary veins
  • deoxygenated. 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 vessels Transport blood through the body part from left ventricle and back to right atrium g Pulmonary vessels Transport blood from right ventricle through lungs and back to left Transport blood from right ventricle through lungs and back to left atrium Blood vessels and heart are regulated to ensure blood pressure is Blood vessels and heart are regulated to ensure blood pressure is high enough for blood flow to meet metabolic needs of tissues

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The Real Thing The Real Thing

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The Real Thing The Real Thing

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History History History History

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Mathematical Modelling in Physiology Mathematical 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 research climate 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 silico silico

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

In research the ultimate goal is mechanisms In research the ultimate goal is mechanisms‐based models but in reality based models but in reality

In research, the ultimate goal is mechanisms In research, the ultimate goal is mechanisms‐based models, but in reality based models, but in reality models are more often used in a detective models 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 inaccessible about reality, that is otherwise inaccessible

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Heart and Blood Circulation Research History Heart and Blood Circulation Research History

Since 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 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 i N i Ji 2697 Ji 2697 2597 BC) 2597 BC)

In one of the most ancient medical treatises ( In one of the most ancient medical treatises (Nei Nei Jing, 2697 Jing, 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 (1578 The Western world had to wait for William Harvey (1578-1657) to establish 1657) to establish

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

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SLIDE 22

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

History History

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 Galenic Galenic 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 Galenius Galenius (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 septum the heart through a system of tiny pores of the septum the 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 d h t i i t ld t i bl b b b d b th b d d heart 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 chyle chyle. .

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SLIDE 23

History History

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 “ f f “ f is 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 microscope p Ottesen (2011)

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SLIDE 24

Th Wi dk l Th Wi dk l The Windkessel The Windkessel Effect Effect

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SLIDE 25

The Windkessel Effect The 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 system systemic 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 i useful in:

  • Quantifying the effects of vasodilator or vasoconstrictor drugs.
  • The development and operation of mechanical heart and heart-lung

hi machines.

Windkessel Windkessel: a : a german german word that can be translated as air (wind) chamber word that can be translated as air (wind) chamber (k l k l) (kessel kessel). ).

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

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SLIDE 26

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

Arterial compliance P i h l Peripheral 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 chamber the pocket and pushes water out of the chamber the pocket and pushes water out of the chamber. . the pocket and pushes water out of the chamber. .

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SLIDE 27

The Windkessel Effect The Windkessel Effect

The compressibility of the air in the pocket simulates the The compressibility of the air in the pocket simulates the elasticit elasticity and y and extensibility extensibility of the major artery, as blood is pumped into it by the heart

  • f 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 compliance compliance. .

The The resistance resistance 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 to arterial 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 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 t Thi i t t fl i l f d t i h l i t i h l i t

This resistance to flow is commonly referred to as This resistance to flow is commonly referred to as peripheral resistance peripheral resistance.

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SLIDE 28

Hypotheses:

The Windkessel Effect The Windkessel Effect

Hypotheses:

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

t rate

  • 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)

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The Windkessel Effect The Windkessel Effect

Theoretical development of the Windkessel effect air Q T

s

V(t) P(t) R Q (t) Q(t) P Q1(t) Q(t) Pcv t T Schematic representation Schematic representation

  • f a chamber

Systolic phase: valve in open position Diastole phase: valve in close position position position

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The Windkessel Effect The Windkessel Effect

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

s

T t  

Conservation of mass: Qcc: flow to the compliance chamber

cc

  • ut

in

Q Q Q  

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

dt dV Q Q  

1

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

1

.Q R P P

cv 

Therefore: Compliance (C)

dt dP dP dV R P dt dV R P Q .    

Then: or

C t Q C R t P dt t dP ) ( ) ( ) (   dt t dP C R t P t Q ) ( . ) ( ) (   C C R dt . dt R

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SLIDE 31

The Windkessel Effect The Windkessel Effect

Solution of the differential equation a) Particular solution (Q = Cte.=0)

) . exp( . ) (

1

C R t t P   

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

C Q C R t t C R C R t t dt d                 ) . exp( ). ( . 1 ) . exp( ). (

1 1

  C Q C R t t C R dt t d C R t C R t t C R              ) . exp( ). ( . 1 ) ( ) . exp( ) . exp( ). ( . . 1

1 1 1

    

Hence:

) exp( . ) (

1

t Q t d   

Then:

2 1

) exp( . . ) (      t Q R t ) . e p( . C R C dt

Then:

2 1

) . exp( . . ) (    C R Q R t

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SLIDE 32

The Windkessel Effect The Windkessel Effect

c) The general solution for systolic phase is

t t     ) . exp( . ) . exp( . . ) (

2

C R t C R t Q R t P

s

          

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

Q R P P t P . ) (

2

     

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

 

) . exp( . . . ) ( C R t Q R P Q R t P

s

   

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SLIDE 33

I Di t li h ( l i l iti )

The Windkessel Effect The Windkessel Effect

I – Diastolic phase (valve in close position)

T t T

s

 

air

Following similar reasoning but with Q=Cte.=0

Q P dP

V(t) P(t)

C Q C R P dt dP   .

Q1(t)

. . . 1 ) exp(

3

Q R t P            ) exp( . ) (

3

C R t t P   

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

) . p(

3

Q C R     ) . p( ) (

3

C R

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

) exp( . . . 1 ) exp( ) ( t Q R t P t P

d

                 ) . p( ) . p( ) ( C R Q C R

d

       

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SLIDE 34

C l t d l

The Windkessel Effect The Windkessel Effect

Complete model Systolic Phase

s

T t  

 

) exp( ) ( t Q R P Q R t P    

air V(t) P(t)

 

) . exp( . . . ) ( C R Q R P Q R t P

s

   

Q1(t)

T t T

s

  t t    

air V(t) P(t)

Diastolic Phase

) . exp( . . . 1 ) . exp( ) ( C R t Q R C R t P t P

d

                

Q1(t)

Given:

          1 ) . exp( . . C R T Q R P

s

  • r

data P and T T C R Q ) (

      1 ) . exp( . . C R T Q R P

  • r

data P and T T C R Q

s

) ( , , , ,

34 PASI 2011 - A. Bandoni

slide-35
SLIDE 35

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

The Windkessel Effect The Windkessel Effect

The term R.C it is crucial in the 2-W because it determine the “speed”

  • f the exponential decay. This product is called the “characteristic

time”, called  P P P P P0 R.Q t t Case:

 

Case:

  

Case:

Case: 

35 PASI 2011 - A. Bandoni

slide-36
SLIDE 36

The Windkessel Effect The Windkessel Effect

Case:

     ,

Hypertension: Ps > 140 mmHg Pd > 90 mmHg

36 PASI 2011 - A. Bandoni

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SLIDE 37

The Windkessel Effect The Windkessel Effect

The electrical circuit equivalence

t dP C t P t Q ) ( ) ( ) ( 

 Basic equation of a 2-element Winkessel model:

dt C R t Q . ) (  

q  Electric circuit of 2 passive elements: I(t) l t i l t  Electric circuit of 2 passive elements: I(t) : electrical current E(t) : electrical potential C : capacitance of the capacitor

I(t) I3

R : resistance of the resistor

I2

From the Ohm and Kirchhoff laws

dt t dE C R t E t I ) ( . ) ( ) (   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)

37 PASI 2011 - A. Bandoni

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SLIDE 38

The Windkessel Effect The Windkessel Effect

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 R

1

E(t) R1 ≡ R1 (peripheral resistance (syst. and pulm.circuits)) R2 ≡ R2 (resistance of valves

2 2 (

(aortic and pulmonary))

t dP C t P t dE R C t I R ) ( ) ( ) ( ) ( 1

1

         dt C R dt R C t I R ) ( . ) ( ) ( . . ) ( . 1

2 1 2 1

         

38 PASI 2011 - A. Bandoni

slide-39
SLIDE 39

The Windkessel Effect The Windkessel Effect

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 R1 E(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

t dP C t P t E d C L t dE L C R t I R ) ( ) ( ) ( ) ( ) ( 1

2 1

                  dt C R dt C L dt R C R t I R . . . . . ) ( . 1

2 2 2 1 2

                 

39 PASI 2011 - A. Bandoni

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SLIDE 40

Compartment Compartment Compartment Compartment M d l M d l Models Models

40 PASI 2011 - A. Bandoni

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SLIDE 41

Compartment Models Compartment Models

They are used to describe transport material in biological sciences They are used to describe transport material in biological sciences

They are used to describe transport material in biological sciences They 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

  • ne with a well mixed material
  • ne with a well mixed material
  • ne with a well mixed material
  • ne with a well mixed material

Compartments exchange material following certain rules Compartments exchange material following certain rules

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

41 PASI 2011 - A. Bandoni

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SLIDE 42

Compartment Models Compartment Models

Material represent the amount of something that we wish to account for Material represent the amount of something that we wish to account for

Material represent the amount of something that we wish to account for Material 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 output Conservations 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

  • rgans)
  • rgans)

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 volume depend on the compartment volume

Inflow to compartment may depend of outflow of other compartment. Inflow to compartment may depend of outflow of other compartment.

42 PASI 2011 - A. Bandoni

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SLIDE 43

Compartment Models Compartment Models

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

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 model Limitations of the compartment model

Limitations of the compartment model Limitations of the compartment model

  • Is the system closed

Is the system closed. Equation of conservation of mass is correct . Equation of conservation of mass is correct

  • nly if all material added or removed is included in the model. There
  • nly if all material added or removed is included in the model. There

is some lost of detailed information is some lost of detailed information is some lost of detailed information. is some lost of detailed information.

  • Homogeneity assumption

Homogeneity 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 required information it is required information it is required. information it is required.

  • Accuracy of the balance equation

Accuracy 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 l R l f th b l N t ll t b d ib d N t ll t b d ib d

  • Relevance of the mass balance

Relevance 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 analysis

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

43 PASI 2011 - A. Bandoni

slide-44
SLIDE 44

Mathematical Mathematical Mathematical Mathematical M d l M d l Models Models

Cardiovascular, Respiratory Cardiovascular, Respiratory and Pharmacodynamic and Pharmacodynamic and Pharmacodynamic and Pharmacodynamic

44 PASI 2011 - A. Bandoni

slide-45
SLIDE 45

Human Circulatory System Model Human Circulatory System Model

The The historical historical fascination fascination of

  • f the

the heart heart has has lasted lasted for for many many centuries centuries and and continues continues to to attract attract considerable considerable attention attention both both theoretically theoretically and and clinically clinically. .

To To develop develop a a physiologically physiologically founded founded model model of

  • f the

the heart heart and and the the vasculature, vasculature, it it is is essential essential to to have have a a good good model model of

  • f the

the human human short short term term press re press re control control represented represented b the the baroreceptor baroreceptor mechanism mechanism pressure pressure control control represented represented by by the the baroreceptor baroreceptor mechanism mechanism.

Using Using a lumped lumped parameter parameter compartment compartment model, model, the the entire entire human human Using Using a lumped lumped parameter parameter compartment compartment model, model, the the entire entire human human cardiovascular cardiovascular system system may may be be described described as as a a network network of

  • f compliances,

compliances, resistances resistances and and inductances inductances not not reflecting reflecting anatomical anatomical properties properties. .

Although Although strikingly strikingly simple, simple, the the model model gives gives a a very very good good description description of

  • f the

the input input impedance impedance of

  • f the

the arterial arterial system system. .

Such Such models models are are valuable valuable tools tools for for understanding understanding cardiovascular cardiovascular diseases diseases (hypertension (hypertension weak weak and and enlarged enlarged heart heart hemorrhages hemorrhages etc etc ) (hypertension, (hypertension, weak weak and and enlarged enlarged heart, heart, hemorrhages, hemorrhages, etc etc.)

45 PASI 2011 - A. Bandoni

slide-46
SLIDE 46

Human Circulatory System Model Human Circulatory System Model

Models Models facilitates facilitates getting getting new new insight insight into into cardiovascular cardiovascular functions functions and and the the interaction interaction with with other

  • ther system

system (central (central nervous nervous system, system, respiratory respiratory systems, systems, etc etc. .)

This This type type of

  • f models

models can can be be reliable reliable and and stable, stable, simply simply enough enough to to run run in in real real ti ti time time. .

Lumped Lumped cardiovascular cardiovascular models models are are divided divided into into pulsatile pulsatile and and non non-pulsatile pulsatile

Lumped Lumped cardiovascular cardiovascular models models are are divided divided into into pulsatile pulsatile and and non non pulsatile pulsatile.

In In the the pulsatile pulsatile case, case, the the heart heart functioning functioning is is guided guided by by a a time time-

  • varying

varying elastance elastance function function. .

A lumped lumped pulsatile pulsatile cardiovascular cardiovascular model model that that embraces embraces principal principal features features

A lumped lumped pulsatile pulsatile cardiovascular cardiovascular model model that that embraces embraces principal principal features features

  • f
  • f the

the human human circulation circulation. .

46 PASI 2011 - A. Bandoni

slide-47
SLIDE 47

Human Circulatory System Model Human Circulatory System Model

Lumped Lumped cardiovascular cardiovascular models models are are divided divided into into pulsatile pulsatile and and non non-

  • pulsatile

pulsatile. . I th th l til l til th th h t h t f ti i f ti i i i id d id d b ti ti i i

In In the the pulsatile pulsatile case, case, the the heart heart functioning functioning is is guided guided by by a a time time-

  • varying

varying elastance elastance function function. .

A lumped lumped pulsatile pulsatile cardiovascular cardiovascular model model that that embraces embraces principal principal features features

  • f
  • f the

the human human circulation circulation. .

47 PASI 2011 - A. Bandoni

slide-48
SLIDE 48

Human Circulatory System Model Human Circulatory System Model

Ap3

Pulmonar circulation

Ap2 Vp1

RV

Ap1 Vp2

LA RA RV LA LV Heart

As1 Vs2 Vs1 As3 As2

Systemic circulation

48 PASI 2011 - A. Bandoni

slide-49
SLIDE 49

Human Circulatory System Model Human Circulatory System Model

Ap3

Qp3 Qp2 Pp3 Vp3 Cp3 Rp3

Ap2 Vp1

Ql1 Qp1 Pl2 Cl2 Pl1 Vl1 Cl2 Rl2 P 1 Pp2 Vp2 Cp2 Rp2 Cp1

RV

Ap1 Vp2

LA

Ql2 Qrv PV Pl2 Vl2 Rl2 Ll2 Pla Vla Ela Rla Lla Prv Erv(t) Pp1 Vp1 Cp1 Rp1 Lp1 Eminrv Emaxrv

RA RV LV

Qv2 Qlv Qla Qra MV AV TV Lla Plv Vlv Elv(t) Llv Eminlv Emaxlv Pra Vra Era Rra Lra Vrv Lrv Emaxrv

As1 Vs2

Qv2 Qv1 Qa1 Qlv AV Pa1 Va1 Ca1 Ra1 La1 Pv2 Vv2 Cv2 Rv2 Lv2

Vs1 As3 As2

Qa3 Qa2 Ca2 Ra2 Pv1 Vv1 Cv1 Rv1 Pa2 Va2 Pa3 Va3 Ca3 Ra3

49 PASI 2011 - A. Bandoni

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SLIDE 50

Human Circulatory System Model Human Circulatory System Model

Model of a typical compartment (chamber) of the hemodynamic system

R : ressistance C li V0 : volumen at p=0 Hemodynamic L : inertia C : compliance Blood input Blood

  • utput

y element of a blood chamber pi p0 Qin input

  • utput

Qout pi Q p0 R L Equivalence with an electric Qout C V0 Qin circuit

50 PASI 2011 - A. Bandoni

slide-51
SLIDE 51

Circulatory System Model (Ottesen et al., 2003)

  • Heart Model

H t it lf 4 h b (2 t i d 2 t i l )

  • Heart itself: 4 chambers (2 atria and 2 ventricles)
  • 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 Model
  • Chronotropic effect (on heart rate)
  • Inotropic effect (on the cardiac contractility)

V l ff t ( t i d i )

Respiratory System Model (Christiansen and Dræby, 1996)

  • Lung Model
  • Vascular effect (on arteries and veins)

Lung Model

  • Upper respiratory tracks: 1 chamber
  • Alveoli: 1 chamber
  • Gas Transport in Blood Model (O2, CO2, Anesthesia)

V l t 5 h b

  • Vascular part: 5 chambers
  • Organs and tissues: 8 compartments

 Organs compartments: one part of tissue and one part of blood (equilibrium

  • f the substances distributed by the blood on both sides it is assumed)
  • f the substances distributed by the blood on both sides it is assumed)

 It is assumed constant blood (VB) and tissue (VT) volumes.

  • Capillaries and alveoli: 1 chamber

Ph d i M d l Pharmacodynamic Model (Gopinath et al., 1995)

  • Drug Effect on Hemodynamic Variables Model

51 PASI 2011 - A. Bandoni

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SLIDE 52

The Cardiovascular Model The Cardiovascular Model

The Pumping Heart The Pumping Heart

Based Based on

  • n an

an elastance elastance model model where where the the cardiac cardiac contraction contraction properties properties of

  • f the

the two two ventricles ventricles are are representing representing by by a pair pair of

  • f time

time varying varying elastance elastance representing representing by by a pair pair of

  • f time

time-varying varying elastance elastance functions functions. .

The The inertia inertia of

  • f blood

blood movements movements in in the the ventricles ventricles is is considered considered through through

The The inertia inertia of

  • f blood

blood movements movements in in the the ventricles ventricles is is considered considered through through an an inductance inductance that that introduce introduce a a phase phase shift shift between between the the ventricular ventricular pressure pressure and and the the root root aortic aortic pressure pressure. .

The The viscous viscous properties properties of

  • f blood

blood in in the the two two atria atria are are included included by by ventricular ventricular filling filling resistance resistance g

52 PASI 2011 - A. Bandoni

slide-53
SLIDE 53

Ql

a

Ql

v

Rla Lla Llv pas E plv pla M V AV

LV LA

Left

Ela Elv(t)

LV LA

Left Heart

Ros

Ra1

Ra2

Ra3

Rv1 Rv2 Lv2 La1 pas pa1 pa2 pa3 pv1 pv2

AA CVi

Syst.

Ca1 Ca2 Ca3 Cv1 Cv2

AA CVi

Circ.

Qr

a

Rra Lra Rrv pap Qrv TV PV prv pra

Right Heart

Era Erv(t)

RV RA

Rop Rp1 Rp2 Ra3 Rl1 Rl2 Ll2 Lp1 pp1 pp2 pp3 pl1 pl2 PV

PA CVs

Pulm .

Cp1 Cp2 Cp3 Cl1 Cl2

PA CVs

Circ. 53 PASI 2011 - A. Bandoni

slide-54
SLIDE 54

The Pumping Heart The Pumping Heart

la la la lv la la

L Q R p p dt dQ .   

lv la

p p if  if

la

Q Q dV  

la

Q

lv la

p p if 

la l

Q Q dt  

2

 

la d la la la

V V E p

,

.   ml dt Q V

t

2   

as lv lv

L p p d dQ  

as lv

p p if  ml dt Q V

t lv b lv

2

*

,

   

       

t E t E t E

lv lv lv

  . 1 .

max, min,

  

lv

L dt 

lv

Q dV

as lv

p p if 

 

       

ce ce ce

t t t t b t t a t , . . 2 sin . . sin .   

  lv la lv

Q Q dt dV  

 

V V t E p ) (  

   

h ce

t t t ,

h ce

t t .

1

   

 

lv d lv lv lv

V V t E p

,

). (  

as lv s as

p Q R p   .

54 PASI 2011 - A. Bandoni

slide-55
SLIDE 55

The Pumping Heart The Pumping Heart

Elastance model

Emax,lv Emin,lv tce

ce

th

       

t E t E t E

lv lv lv

  . 1 .

max, min,

    

          

h ce ce ce ce

t t t t t t t b t t a t , , . . 2 sin . . sin .   

 

h ce

55 PASI 2011 - A. Bandoni

slide-56
SLIDE 56

The Circulatory System Model The Circulatory System Model

Single chamber model

Q 1

pa1

dV

Qa1

pa2 Va2

2 1 2 a a a

Q Q dt dV   V V 

Qa2

2 2 , 2 2 a a un a a

C V V p  p p

2 3 2 2 a a a a

R p p Q  

56 PASI 2011 - A. Bandoni

slide-57
SLIDE 57

The Baroreceptors Model The 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 change as 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

57 PASI 2011 - A. Bandoni

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SLIDE 58

The Baroreceptors Model The Baroreceptors Model

① 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 b ② Action potential are conducted by the glossopharyngeal and the vagus nerves to the cardioregulatory and t t i th d ll vasomotor centers in the medulla

  • blongata.

③ As a result of increased ③ As a result of increased stimulation from the baroreceptor, the cardioregulatory center increased parasymphatic stimulation to the parasymphatic stimulation to the heart, which decreases the heart rate. ④ Also, as a result of increased stimulation from the baroreceptor, the ④ 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.

58 PASI 2011 - A. Bandoni

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SLIDE 59

The Baroreceptors Model The Baroreceptors Model

⑤ 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. f If 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 d

  • pposite of what was illustrated.

59 PASI 2011 - A. Bandoni

slide-60
SLIDE 60

The Baroreceptors Model The Baroreceptors Model

Heart

H

Baroreceptor system

frequency Systolic maximum

Emaxlv, Emaxrv

system

Cardio- vascular MAP

maximum elastance Systemic resistance

Ra1, Ra2, Ra3 vascular System

Compliance in veins and

Cv1, Cv2

resistance arteries in veins and arteries Unstressed vol in syst

Vunv1, Vunv2

  • vol. in syst.

veins

60 PASI 2011 - A. Bandoni

slide-61
SLIDE 61

The Baroreceptors Model The Baroreceptors Model

Afferent sector Efferent sector

Sensors Central Nervous Eferent th

MAP n ns np xi

Sensors System pathways

 

   MAP MAP ns 1

     

i p i s i b i

MAP n MAP n MAP        . . . .         MAP 1

     

 

E i MAP t x dt t dx

b i i i

    , 1 

 

 

        MAP MAP np 1 1

 

C V R E H E i  

 

dt

i

     

 

v un ps

C V R E H E i , , , ,

max

 

61 PASI 2011 - A. Bandoni

slide-62
SLIDE 62

The Respiratory System Model The Respiratory System Model

The The respiratory respiratory system system is is concerned concerned with with the the transport transport of

  • f oxygen
  • xygen

p y p y y p yg yg between between atmosphere atmosphere and and the the tissue tissue and and organs

  • rgans in

in the the body body O i i ti l ti l t t d t t d b th th l l d bl d bl d i it i it

Oxygen Oxygen is is continuously continuously transported transported by by the the lung lung and and blood blood circuit circuit.

Carbon Carbon dioxide dioxide is is a waste waste product product of

  • f the

the oxidative

  • xidative metabolism

metabolism and and is is

Carbon Carbon dioxide dioxide is is a waste waste product product of

  • f the

the oxidative

  • xidative metabolism

metabolism and and is is carried carried by by the the blood blood in in the the opposite

  • pposite direction

direction

62 PASI 2011 - A. Bandoni

slide-63
SLIDE 63

The Respiratory System Model The Respiratory System Model

O2 CO2 Ventilation Atmosphere Alveoli Ventilation O2 CO2 Gas exchange O2 CO2

L ft Ri ht

Pulmonary circulation

Left Heart Right Heart

Gas transport Systemic circulation O2 CO2 Cell Gas exchange metabolism

63 PASI 2011 - A. Bandoni

slide-64
SLIDE 64

The Respiratory System Model The Respiratory System Model

Lung model: pressure Lung 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 Ri Upper i

■ In each compartment gas flows are calculated (O2,

R1 R2 Ri C1 C2 Ci C0 airway

Um (t)

CO2, Anesthesia) ■ The outputs of the model are:

Atmosphere

  • r

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 a and alveoli.

64 PASI 2011 - A. Bandoni

slide-65
SLIDE 65

The Respiratory System Model The Respiratory System Model

Distribution of substances in the organs through blood Distribution of substances in the organs through blood

Alveolus Capillary

Alveolus

V (p) κ.pA κ.p

Q.cb (1- λ ).Q.cvs

pcp

C t l

Vbcb (p)

Q

b

( ) Q

vs

λ.Q.cvs pli pas pv

s

Central venous compartment Central arterial compartment Liver Metabolism cvv

pki

phe Kidney Heart Viscere venous

M- M+

Metabolism pbr pre Brain Other

  • rgans

venous compartment

Vtct (p) Vbcb (p)

cvl pco pmu Connective tissue Muscles Lean venous compartment Adi

Vbcb (p)

zi.Q.cas zi.Q(p).cb Adipose tissue cav pad Adipose tissue venous compartment

65 PASI 2011 - A. Bandoni

slide-66
SLIDE 66

The Respiratory System Model The Respiratory System Model

p0 R Upper airways Alveoli

1

. . dt dpi C R p U dp

n i i m

 

  

pi R0 C0 RiCi p f0

0.C

R dt 

A t i i

n i U p p d dp ... 1 ,    

Pressure model

pcp fi

A i i C

R dt , .

     

              

 

  n i i i e m

R p p I R p U I p V p C T dt d

1 2

. . f f f f f R     

i

i

R R p V p C dt

1 00

  

 

            

 i i cp i i t i

p R p U p I V C T dt d f p κ f f f . .

2

R

 

   

i i cp i i i i i

R p V p C dt . .

2

 

   

x x I

Molar fractions model

 

  

x x x

66 PASI 2011 - A. Bandoni

slide-67
SLIDE 67

The Pharmacodynamic Model The Pharmacodynamic Model

Pharmacology Pharmacology: the the history, history, source, source, physical physical and and chemical chemical properties, properties, biochemical biochemical and and physiological physiological effect, effect, mechanisms mechanisms of

  • f action,

action, absorption, absorption, distribution, distribution, biotransformation biotransformation and and excretion, excretion, and and therapeutic therapeutic and and other

  • ther

uses uses of

  • f drugs

drugs. . Pharmacokinetics Pharmacokinetics: absorption absorption distribution distribution metabolism metabolism and and excretion excretion of

  • f

Pharmacokinetics Pharmacokinetics: absorption, absorption, distribution, distribution, metabolism metabolism and and excretion excretion of

  • f

drugs drugs. .

Pharmacodynamics Pharmacodynamics: biochemical biochemical and and physiological physiological effects effects and and their their mechanisms mechanisms of

  • f action

action. .

67 PASI 2011 - A. Bandoni

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SLIDE 68

The Pharmacodynamic Model The Pharmacodynamic Model

tion ncentrat Drug Co D Time Concentration of drug in the body as a function of time Concentration of drug in the body as a function of time

68 PASI 2011 - A. Bandoni

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SLIDE 69

The Pharmacokinetic Model The Pharmacokinetic Model

         

p M p M p c c c c p

b  

           Q z d V d V d

i b b t t 1

Pressure balance i th

          M

         

p M p M p c c p p

b as  

        Q z d V d V dt

i b t

in the organs

           

 aa O O O O

c M c c M 

2 2 2 2

M           

2

CO

M M     

aa aa aa

c M 

Pressure balance in the capillaries          

p p p c c p c p

A b s

            

 

v b b

Q d d V dt d 1

1

Pressure balance in the compartments

2 2 1 1

Q Q Q Q

x

   c c c

     

p c c p c p

b

         

 x b b

Q d d V dt d

1

69 PASI 2011 - A. Bandoni

p

2 1

Q Q 

p   d dt

slide-70
SLIDE 70

C

The Pharmacodynamic Model The Pharmacodynamic Model

Drug effect

Cd

Baro- receptors

MAP EmaxBARO RsisBARO Emax= EmaxBARO(1±Eff) R= RsisBARO(1±Eff)

receptors

 

Eff k Eff Eff C k dt dEff

N d

. . .

2 max 1

  

Cardiovascular system

         dt dEff C dt dC

PFL a

C BASE a a

1

.

1 1

 

dEff E Eff dE dE

lvDP

E l BARO E lvBARO lv

max

max max

1   

 

dt E Eff dt dt

lvDP

lvBARO Emax max

1  

 

           dEff dEff R Eff Eff dR dR

sisSNP sisDP

R R sisBARO R R sisBARO sis

1

 

      dt dt ff ff dt dt

sisSNP sisDP

sisBARO R R

70 PASI 2011 - A. Bandoni

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SLIDE 71

The Pharmacodynamic Model The Pharmacodynamic Model

Drug (intravenous) Affected variable Action SNP (sodium SNP (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

71 PASI 2011 - A. Bandoni

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SLIDE 72

The Pharmacodynamic Model The Pharmacodynamic Model

DP DP and and SNP SNP drugs drugs are are chosen chosen to to increase increase ventricular ventricular contractility contractility and and d th th i t i t f t i t i t bl d bl d fl fl ti l ti l reduce reduce the the resistance resistance of

  • f arteries

arteries to to blood blood flow, flow, respectively respectively. .

PFL PFL is is chosen chosen to to conduct conduct unconsciousness unconsciousness by by measured measured of

  • f BIS

BIS

PFL PFL is is chosen chosen to to conduct conduct unconsciousness unconsciousness by by measured measured of

  • f BIS

BIS parameter parameter. .

DP DP increases increases the the MAP MAP and and CO CO. . SNP SNP decreases decreases and and increases increases CO CO MAP MAP. .

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 ariables pharmacokinetic and pharmacod namic ariables pharmacokinetic and pharmacodynamic variables. pharmacokinetic and pharmacodynamic variables.

72 PASI 2011 - A. Bandoni

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SLIDE 73

Computational Implementation Computational Implementation

 Model implemented in Fortran

Model implemented in Fortran

 Diff.

  • Diff. Eqs
  • Eqs. solved with a 4th order

. solved with a 4th order Runge Runge-

  • Kutta

Kutta method. method.

 Resolution sequence:

Resolution sequence: (i i) ) 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 y g y state. state. Th d i j ti i i l t d Th d i j ti i i l t d f l f b thi (5 ) f l f b thi (5 ) Th th Th 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 Cd Cd to simulate to simulate the the 0.8 sec. a heartbeat 0.8 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.

73 PASI 2011 - A. Bandoni

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SLIDE 74

Computational Implementation Computational Implementation

Cd(inhalable) CO2, O2 Cd (alveoli) Cd(inyectable)

Respiratory system Transport and distribution, Pharmacokinetics of system Pharmacokinetics of drugs

Qa3, Qp3 Cd (organs)

C di l Ph d i

MAP

Cardiovascular system Pharmacodynamics

  • f drugs

Baroreflex

Emax Rsis EffEmax EffRa2 EffRa3 EmaxBARO Ra2BARO Ra3BARO

Control

74 PASI 2011 - A. Bandoni

Action

slide-75
SLIDE 75

Computational Implementation Computational Implementation

Dimensions of the integrated model

Model Var./Eqs. Algebraics Var./Ecs. Differenctials Parameters Algebraics Differenctials Cardiovascular-Respiratory 37 39 53 Respiratory-Pharmacodynamic 60 93 85 Total 97 132 138

75 PASI 2011 - A. Bandoni

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SLIDE 76

Results Results

76 PASI 2011 - A. Bandoni

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SLIDE 77

Results: cardio vascular system Results: cardio vascular system

Wiggers Diagram

77 PASI 2011 - A. Bandoni

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SLIDE 78

Results: cardio vascular system Results: cardio vascular system

Left ventricle and root aortic pressure vs. time Left ventricular volume

  • vs. time

78 PASI 2011 - A. Bandoni

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SLIDE 79

Results: cardio vascular system Results: cardio vascular system

Outflow of the left ventricle Outflow of the left ventricle Left ventricular pressure Pressure vs. Volume left ventricle

79 PASI 2011 - A. Bandoni

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SLIDE 80

Results: Results: baroreflex baroreflex system system

Heart period vs time Resist sect A

  • f syst arteries vs time

Heart period vs. time

  • Resist. sect. As1 of syst. arteries vs. time

Compliance in sect. Vs1 of sistemic veins vs. time

  • Unstres. Vol. of sect. Vs1 of

sistemic veins vs. time

80 PASI 2011 - A. Bandoni

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SLIDE 81

Results: Results: baroreflex baroreflex system system

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

81 PASI 2011 - A. Bandoni

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SLIDE 82

Results: gas transport Results: gas transport

Partial pressure of O2 in different compartments of the body Partial pressure of CO2 in different compartments of the body compartments of the body

82 PASI 2011 - A. Bandoni

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SLIDE 83

Results: respiratory system Results: respiratory system

expiración expiración 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.

83 PASI 2011 - A. Bandoni

slide-84
SLIDE 84

Results: Results: pharmacodymic pharmacodymic system system

Effect of the SNP action

1µg/kg/min SNP concentration profile at the central arterial compartment

84 PASI 2011 - A. Bandoni

Mean Arterial Pressure, MAP Cardiac Output, CO

slide-85
SLIDE 85

Results: pharmacodynamic system Results: pharmacodynamic system

Effect of the SNP action

1µg/kg/min Resistance, Ra2 Resistance, Ra3

85 PASI 2011 - A. Bandoni

slide-86
SLIDE 86

Results: pharmacodynamic system Results: pharmacodynamic system

Effect of the DP action

5µg/kg/min DP concentration profile at the central arterial compartment

86 PASI 2011 - A. Bandoni

Mean Arterial Pressure, MAP Cardiac Output, CO

slide-87
SLIDE 87

Results: pharmacodynamic system Results: pharmacodynamic system

Effect of the DP action

5µg/kg/min Medial arterial resistances Elastance

87 PASI 2011 - A. Bandoni

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SLIDE 88

Results: pharmacodynamic system Results: pharmacodynamic system Effect 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)

88 PASI 2011 - A. Bandoni

slide-89
SLIDE 89

Results: pharmacodynamic system Results: pharmacodynamic system Effect 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)

89 PASI 2011 - A. Bandoni

slide-90
SLIDE 90

Effect of the PFL action

/ /

Results: pharmacodynamic system Results: pharmacodynamic system

Effect of the PFL action

150µg/kg/min Mean Arterial Pressure, MAP PFL conc. at the central arterial comp.

90 PASI 2011 - A. Bandoni

Cardiac Output, CO Compliance of sector a1 of systemic arteries

slide-91
SLIDE 91

Conclusions Conclusions

 Development

Development

  • f
  • f

an an integrated integrated cardiovascular, cardiovascular, baroreceptor, baroreceptor, respiratory, respiratory, pharmacokinetic pharmacokinetic and and pharmacodynamic pharmacodynamic model model. .

 The

The effect effect of

  • f certain

certain drugs drugs on

  • n hemodynamic

hemodynamic variables variables was was studied studied. .

91 PASI 2011 - A. Bandoni

slide-92
SLIDE 92

Future Works Future Works

 General

General model model validation validation with with real real patient patient data data. . Collaboration Collaboration with with a a research research group group formed formed by by doctors doctors ( (Favaloro Favaloro University, University, Bs Bs As As Español Español Hospital Hospital B Blanca Blanca Arg Arg ) Bs Bs.As

  • As. – Español

Español Hospital, Hospital, B. Blanca, Blanca, Arg Arg.)

 Model

Model validation validation for for inhalable inhalable anesthesia anesthesia effects effects

 Model

Model validation validation for for inhalable inhalable anesthesia anesthesia effects effects.

 Model

Model validation validation for for simultaneously simultaneously drugs drugs administration administration

 Model

Model validation validation for for simultaneously simultaneously drugs drugs administration administration.

 Development

Development of

  • f a control

control model model for for handling handling dose dose of

  • f drug

drug

 Development

Development of

  • f a control

control model model for for handling handling dose dose of

  • f drug

drug administration administration. .

 Development

Development

  • f
  • f

a a teaching teaching simulation simulation model model

  • f
  • f

the the cardiovascular cardiovascular system system ( (Instituto Instituto Nacional Nacional de de Tecnología Tecnología Industrial, Industrial, INTI, INTI, Bs Bs. .As As. ., , Arg Arg. .) )

92 PASI 2011 - A. Bandoni

slide-93
SLIDE 93

Basic References: Cardiovascular Model:

  • Ottesen J., Olufsen M. and Larsen J. Applied Mathematical Models in

Ottesen 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
  • f 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)

93 PASI 2011 - A. Bandoni

slide-94
SLIDE 94

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 cardiorespiratorio

humano: 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

94 PASI 2011 - A. Bandoni

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

slide-95
SLIDE 95

d “With growing emphasis being placed on the information processing aspects of biomedical investigation, theoretical and experimental studies assume increasing importance In many experimental studies assume increasing importance. In many instances, however, there are questions that appear to be unanswerable 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 f The essential ingredient of the scientific method is the use of

  • models. Good modeling is more likely to be achieved by following

the rules of good thinking However the ideal model cannot be the rules of good thinking. However, the ideal model cannot be

  • achieved. Partial models, imperfect as they may be, are the only

means developed by and available to scientists for understanding p y g the 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

95 PASI 2011 - A. Bandoni

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

slide-96
SLIDE 96

96 PASI 2011 - A. Bandoni

slide-97
SLIDE 97

Muchas gracias

97 PASI 2011 - A. Bandoni

slide-98
SLIDE 98

Cámara izquierda del corazón

lv p la p si la L la Q la R lv p la p dt la dQ    

Circulación sistémica

1 1 1 2 1 1 a L a Q a R a p a p dt a dQ    1 1 a Q lv Q dt a dV   lv p la p si la Q   0 la Q l Q dt la dV   2

 

la d V la V la E la p ,   1 a L dt dt 1 1 , 1 1 a C a un V a V a p   2 3 2 2 a R a p a p a Q   dt as p lv p si lv L as p lv p dt lv dQ    2 1 2 a Q a Q dt a dV   2 2 , 2 2 a C a un V a V a p   as p lv p si lv Q   0 lv Q la Q lv dV  

 

lv d V lv V t lv E lv p ,   3 1 3 a R v p as p a Q   3 2 3 a Q a Q dt a dV   V V lv Q la Q dt

 

lv d lv lv lv , ml t t dt lv Q b lv V 2 ,

*

    3 3 , 3 3 a C a un V a V a p   1 2 1 1 v R v p v p v Q   1 dV 1 1 V V 

       

t lv E t lv E t lv E   max, 1 min,     t t 2  1 3 1 v Q a Q dt v dV   1 1 , 1 1 v C v un V v V v p  2 2 2 2 v Q v R ra p v p v dQ  2 dV

 

           h t t ce t ce t t ce t t b ce t t a t 2 sin sin      2 2 2 2 2 v L v Q v R ra p v p dt v dQ   2 1 2 v Q v Q dt v dV   2 2 v un V v V  h t ce t 1     1 a p lv sQ R as p   2 2 , 2 2 v C v un V v V pv 

98 PASI 2011 - A. Bandoni

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SLIDE 99

Cámara derecha del corazón

rv p ra p si ra L ra Q ra R rv p ra p dt ra dQ    

Circulación pulmonar

1 1 1 2 1 1 p L p Q p R p p p p dt p dQ    1 1 p Q rv Q dt p dV   rv p ra p si ra Q   0 ra Q v Q dt ra dV   2

 

ra d V ra V ra E ra p ,   1 1 , 1 1 p C p un V p V p p   2 3 2 2 p R p p p p p Q   2 1 2 p Q p Q p dV   2 , 2 2 p un V p V p p   ap p rv p si rv L ap p rv p dt rv dQ    i Q  2 1 p Q p Q dt 2 2 p C p p 3 1 3 p R l p ps p p Q   3 2 3 p Q p Q dt p dV   V V ap p rv p si rv Q   0 rv Q ra Q dt rv dV  

 

rv d V rv V t rv E rv p ,   t

3 3 , 3 3 p C p un V p V p p   1 2 1 1 l R l p l p l Q   1 3 1 l Q p Q dt l dV   1 1 , 1 1 l C l un V l V l p   ml t dt rv Q b rv V 2 ,

*

   

       

t rv E t rv E t rv E   max, 1 min,    1 l 2 2 2 2 2 l L l Q l R la p l p dt l dQ    2 1 2 l Q l Q dt l dV   2 2 l V l V 1 p p rv pQ R ap p   2 2 , 2 2 l C l un V l V pl  

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

     

                 

n i i R i p i p I R e p m U I p V p C T dt d 1 00 2 f f f f f R

 

     x x I

  

 

               i i p cp i R i i p t U p I i p i V i p i C T dt i d f p κ f f f 2 R

 

    x x x I

99 PASI 2011 - A. Bandoni

slide-100
SLIDE 100

 

         MAP MAP s n 1 1

 

          MAP MAP p n 1 1

Barorreceptores

2 2 1 1

Q Q   c c c

     

p c c c p      

 b

Q d V d

1

        1

        1

     

i MAP p n i MAP s n i MAP b i       

 

v C un V ps R E H E i , , , max ,  

 

 

2 1

Q Q

x

  c

     

p c c p

b

       

x b b

Q d V dt

     

E i MAP b i t i x i dt t i dx           , 1  

Modelo respiratorio (presión)

               

 O O O O

c c c M 

2 2 2 2

M           

2

CO

M M 1 C R n i dt dpi i C R p m U dt dp

 

  

i C i R t U i p p dt dpi   

      

aa aa aa aa

c c M   

     

2 3

               904 . 1 806 . 1 50909 . 8 00005 . 17 806 . 1 278 . 53554 . 66943 . 278 . 96364 . 1 27273 . x t x t x t m U

     

1

2 2 3

   

  

a H a H a H

 

CO a a

K NaOH K a

, Pr , 2

    

 

   

 

P H N OH K K N OH K

      5 904 . 1 05904 . 57034 . x t

Modelo de transporte de gases en sangre

 

 

   

 

, Pr , , 1

Pr

2

H NaOH K K c NaOH K a

CO a a CO CO a

    

   

 

2

Pr , ,

Pr

CO a CO a

c H NaOH K K a   

     

    c c

  d d d

1

     

          

Ery Hb Hb Pla CO Ery Hb Hb Ery CO b CO

c c pH c c pH pH 1 , , ,

2 2 2

p c p c p c

 

   

 

 

pH pK pH pH c Ery CO Ery CO

Ery Ery CO

pH

, ,

10 1 ,

2 2 2

p p

p p c

  

         

p M p M p c c p c p c p

b as  

             Q z d d V d d V dt d

i b b t t

         

c p    

b

Q d V d 1

1

 

   

 

 

pH pK pH pH c Ery CO Ery CO

Ery Ery CO

pH

, ,

10 1 ,

2 2 2

p p

p p c

  

100 PASI 2011 - A. Bandoni

         

p p p c c p p

A b s

            

v b b

Q d V dt 1

slide-101
SLIDE 101

 

 

 

 

pH pK pH c Pla CO Pla CO

Pla CO

pH

,

10 1 ,

2 2 2

p

p p c

  

 

   

 

 

pH s pH pH Ery

O Ery

pH pK

, 06 . 84 . 7 ,

2

10 1 log 125 6

p p

p

 

 

   

          37 º 055 . 946 . 1 C T a x p p

   

   

l

c

 

 

pH pK

10

10 1 log 125 . 6 , p   

     

 

pH s pH pH pH

O Ery

, 1 035 . 4 . 7 77 . 19 . 7 ,

2 p

p       

 

 

7 8

10 1 l 125 6

 pH Pla

H K

 

aa b b aa

p c   p

   

         

Ery Hb Hb Pla CO t CO

c c pH c pH c 1 , ,

2 2

p p

 

 

 

7 . 8 10

10 1 log 125 . 6 ,   

pH Pla

pH pK p

 

2 2 2 2

,

O Hb O O b O

s c p pH    p c

s  1                

p p p p p p 5343 . tanh 875 . 1 x x h x x y     

   

p p h  5 3

 

 

kPa p x / log  p

 

t

 p

 

p y O

e s

  1

2

   

p p a h   5 . 3

 

 

kPa p x

O /

log

2

 p

   

     

dpg Hbf CO CO

l mmol c x kPa p c pH a 5 / 03 . 07 . 33 . 5 log 09 . 4 . 7 72 .

2 2

                      p p

 

aa t t aa

p c   p  

2 2 2

O O t O

p c   p

Modelo farmacodinámico

Hbf Hi HbCO

x x x 28 . 174 . 386 .  

 

Eff k Eff Eff N d C k dt dEff 2 max 1             dt C dEff BASE a C dt a dC

PFL a1

1 1 d ff

 

             dt R dEff dt R dEff sisBARO R R Eff R Eff dt sisBARO dR dt sis dR

sisSNP sisDP sisSNP sisDP

1

 

dt E dEff lvBARO E E Eff dt lvBARO dE dt lv dE

lvDP lvDP max max

max 1 max max   

101 PASI 2011 - A. Bandoni