“The Cardio‐Respiratory Human System: a simulation study” “Process StSystem EEingineeri ng in Human Physi ol ogy”
Elisa Montain, Anibal Blanco, Alberto Bandoni Pilot Plant of Chemical Enggg,ineering, PLAPIQUI (UNS‐CONICET) Bahía Blanca, Argentina PASI 2011 Process Modeling and Optimization for Energy and Sustainability SdSaturday, JlJuly 23, 2011, Angra dos RiReis, RJ, BilBrazil
PASI 2011 - A. Bandoni 1 Background
The cardiovascular system (CVS) is responsible for supplying oxygen and nutrients to tissues and organsorgans.. CV diseases are a major cause of death in humanshumans.. Many experimental studies have studied the mechanisms and therapy of the CV diseases Together with experimental approaches, mathematical modeling has become a popular way to analyze the CVSCVS.. Many models have been published since the preliminary and basic model of Godins in 1959 Approaches include: hemodynamic models of the vascular system, distributed impedance and pulmonary arterial stress, lumped parameter models of the integrated CVS, hemodynamic monitoring models, etc.. etc.. In the last fewwyyears there hav e been important developments in integrated lumped parameter models of the circulatory and nervous control systems. systems.
PASI 2011 - A. Bandoni 2 Motivation AitAssistance ithdiiin the decision maki kifng of medi dilcal practi ce Diagnosis of diseases of the CVS (coronary arteries and heart muscles dysfunctions, valvular disorders and pulmonary disease.
Comprehend the math. concepts and terms defining how CVS system behaves.
To teach about the complex interactions of the cardiovascular system.
HlHelp to vascu lar surgeons iiin tttreatment plilanning andtd to engi neers in diidesigning better medical devices.
A promising integration strategy involves the personalization of mathematical models based on biophysical measurementsmeasurements..
Analysis of the hemodynamics (blood flow dynamics) of the CVS.
Capacity to locate factors that are not directly observable . Key role in the measurement of pump efficiency and tissue stress, to assist treatment decisions. PASI 2011 - A. Bandoni 3 Motivation Anesthesia control and drug delivery control: Control of patient physiological variables during intensive care is achieved through druggy delivery.
Drug delivery process depends on the value of the physiological variable under control and on the patient' s condition
Drugs such sodium nitroprusside (SNP) and dopamine (DP) are normally used for regulation of Media Arterial Pressure (MAP) or Cardiac Output (CO).
DDoctorsoctors use their discretion to regulate variables that are difficult to quantify in practice or inferred from other measurements and patient responses to certain surgical procedures.procedures.
Currently, the drug infusion is done manually or by programmable pumps. The professional is responsible for monitoring the controlled variable (MAP, CO) and the druggy delivery accordin g to the measurement.
PASI 2011 - A. Bandoni 4 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.
Study of the druggy delivery (()SNP, DP, etc.)
The developed model contain the following sub-models: Circulatory system Baroreceptors Respiratory system Gas transport and distribution in organs Pharmacological effect of drugs on the hemodynamic variables.
PASI 2011 - A. Bandoni 5 Anatomy and Physi ol ogy
PASI 2011 - A. Bandoni 6 The Cardiovascular System
The Cardiovascular System:
It consists of:
The heart, which is a muscular pumping device
A closed system of vessels (arteries, veins, and capillaries).
The Heart
. The heart is a hollow muscular pump that provides the force necessary to ciltirculate the bloo d toall the tissues in the bdbody throug h bloo d vessels.
. The normal adult heart pumps about 5 liters of blood every minute throughout life.
PASI 2011 - A. Bandoni 7 Heart Anatomy
Aorta Superior vena cava Pulmonary truck Pulmonary Pulmonary valve vein Left Atrium Right atrium Mitral valve
Tricuspid Aortic valve valve Left Right Ventricle ventricle
Inferior vena cava
PASI 2011 - A. Bandoni 8 Functions of the Heart
Generates blood pressure
Routes blood Heart separates pulmonary and systemic circulation
Ensures one-one-wayway blood flow Heart valves ensure oneone--waway flow
Regulates blood supply Changes i n cont racti on ra te an d force ma tc h bloo d de livery to changing metabolic needs Most healthyyp peo ple can increase cardiac out put b y 300–500%
Heart failure is the inability of the heart to provide enough blood flow to maitiintain norma l me tblitabolism
PASI 2011 - A. Bandoni 9 The Chambers Separated by . Interatrial Septum . Interventricular Septum
Right Atrium . Bloo d from Super ior an d in fer ior venaecavae andthd the coronary s inus Right Ventricle . Receives blood from the right atrium via the right AV valve, tricuspid valve . Thin wall Left Atrium . Receives blood from R and L Pulmonary Veins Left Ventricle . Receives blood from the Left AV valve . Thick wall Pumps to body via Aortic Semilunar Valve PASI 2011 - A. Bandoni 10 The Valves
Two types of valves: keep the bloo d flowing in the correct direction.
Between atria and ventricles: called atrioventricular valves (also called cuspid valves)
Bases of the large vessels leaving the ventricles: called semilunar valves.
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 flowing back into the ventricles.
VlVales close passilivelyunder bloo d pressure. RiblResponsible for the htheart sounds. PASI 2011 - A. Bandoni 11 Circulatory System
PASI 2011 - A. Bandoni 12 Circulatory System
DtdbldDeoxygenated blood rettturns to the heart via the superior and inferior vena cava, enters the 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.
PASI 2011 - A. Bandoni 13 Blood flow pattern through the heart 1.Blood enters right atrium via the superior and inferior venae cavae
2.Passes tricuspid valve into right ventricle
3.Leaves by passing pulmonary semilunar valves into pulmonary trunk and to the lungs to be oxygenated
4.Returns from the lung by way of pulmonary veins into the left atrium
5.From left atrium past bicuspid valve into left ventricle
6.Leaves left ventricle past aortic semilunar valves into aorta
7.Distributed to rest of the body PASI 2011 - A. Bandoni 14 Blood flow pattern through the heart
PASI 2011 - A. Bandoni 15 Blood Vessels Blood vessels are divided into a pulmonary circuit and systemic circuit. Artery - vessel that carries blood away from the heart. Usually oxygenated. Exception, pulmonary artery. Vein - vessel that carries blood towards the heart. Usually deoxygenated. Exception pulmonary veins Capillary - a small blood vessel that allow diffusion of gases, nutrients and wastes between plasma and interstitial fluid.
Systemic vessels Transport blood through the body part from left ventricle and back to right atrium
Pulmonary vessels Transport blood from right ventricle through lungs and back to left atrium
Blood vessels and heart are regulated to ensure blood pressure is high enough for blood flow to meet metabolic needs of tissues PASI 2011 - A. Bandoni 16 The Real Thing
PASI 2011 - A. Bandoni 17 The Real Thing
PASI 2011 - A. Bandoni 18 History
PASI 2011 - A. Bandoni 19 Mathematical Modelling in Physiology
With mathematical models it is possible to simulate almost any kind of phenomena in nature on a computer.
This is a scientific practice of modern science and engineering ((biology,biology, physiology, medicine, climate research,research, ecology, physics, chemistry, etc .)
Mathematical modeling in medicine and biology has become so important that this type of research now has its own name: in silico
Mathematical modeling undoubtedly will become the paradigm of scientific and medical research in the twenty‐first century.
In research, the ultimate goal is mechanisms‐based models, but in reality models are more often used in a detective‐like way to investigate the consequences of different hypotheses.
The mathematics modeling is used as a microscope to unveil information about reality, that is otherwisePASI inaccessible 2011 - A. Bandoni 20 Heart and Blood Circulation Research History
Since the dawn of civilization , man has been concerned with the understanding of living things.
IIfthtitdiltti(n one of the most ancient medical treatises (NNiei Jing, 2697-2597 BC), blood is mentioned as originating in the heart and distributed in order to return to the starting point.
Despite widespread knowledge of the anatomy of blood vessels, Greeks were unable to find the start of blood circulation by not knowing the principle of conservation of mass.
The Western world had to wait for William Harvey (1578-1657) to establish the concept of circulation.
PASI 2011 - A. Bandoni 21 History
Discovery of th e cl osed ci rcul ati on of bl ood b y Willi am Harvey (1578‐1657). "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
Before 1628, the Galenic view of the body prevailed and the concept of blood circulation was not imaginable. should be 7.258 liters per day
Galen or Galenius (Greek physician, II century AD), spent most of his lifetime observing the human body and its functioning.
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 the heart through a system of tiny pores of the septum.
Using a simple model, Harvey showed that the amount of blood leaving the htiitheart in a minute cou ldtld not conce iblbbbdbthbddivably be absorbed by the body and continually replaced by blood made in the liver from chylechyle.. PASI 2011 - A. Bandoni 22 History
Consequently, this model based evidence established the concept that blood must constantly move in a closed circuit, otherwise the arteries and the body would explode under the pressure.
This was discovered about 8 years before the light microscope.
The concept or method of using mathematical modeling, as a tool for making an inaccessible system accessible or an invisible system visible, is theref“fore being coined as “the mathematical microscope” in honor of William Harvey.
The mathematical microscope Ottesen (2011)
PASI 2011 - A. Bandoni 23 The Wi ndk essel Effect
PASI 2011 - A. Bandoni 24 The Windkessel Effect
The windkessel effect is use to describe: • Load faced by the heart in pumping blood through pulmonary or 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 usefliful in: • Quantifying the effects of vasodilator or vasoconstrictor drugs. • The development and operation of mechanical heart and heart-lung machines.
Windkessel: a german word that can be translated as air (wind) chamber (klkessel).
First descrippytion by German pygphysiologist Otto Frank in 1899.
PASI 2011 - A. Bandoni 25 The Windkessel Effect
Heart and systemic arterial system similar to a closed hydraulic circuit comprised of a water pump connected to a chamber.
The circuit is filled with water except for a pocket of air in the chamber
Arterial compliance
PihPeriphera l ressistance
As water is pumped into the chamber, the water both compresses the air in the pocket and pushes water out of the chamber. .
PASI 2011 - A. Bandoni 26 The Windkessel Effect
The compressibility of the air in the pocket simulates the elasticity and extensibility of the major artery, as blood is pumped into it by the heart ventricle.
This effect is commonly referred to as arterial compliance.
The resistance water encounters while leaving the Windkessel, simulates 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 capillaries, due to decreasing vessel diameter.
This resi st ance t o fl ow i s commonl y ref erred t o as perihiphera l res itistance.
PASI 2011 - A. Bandoni 27 The Windkessel Effect Hypotheses: • Unsteady flow. • The pressure diff. across the resistance is a linear function of the flow 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 onlyyp() the arterial compliance (C) and the peripheral resistance (R).
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] C : arterial or systemic compliance (m5.N-1) [ml.mmHg-1] t : time [(s) Tid()T : period (s) Ts: ejection time (s) PASI 2011 - A. Bandoni 28 The Windkessel Effect
Theoretical development of the Windkessel effect
air Q Ts
V(t) P(t)
R
Q(t) Q1(t) Pcv t T Schematic representation of a chamber Systolic phase: Diastole phase: valve in open valve in close position position
PASI 2011 - A. Bandoni 29 The Windkessel Effect I - StlihSystolic phase (l(valve in open pos iti)ition) 0 t Ts
Conservation of mass: Q in Q out Q cc Qcc: flow to the compliance chamber
dV Thus: Q Q Pcv : central venus pressure: (Pcv<< P) 1 dt (Pcv≅5 mmHg vs. P≅100 mmHg ])
Hyp.4: Q = Cte. throughout the systolic phase, thus: P Pcv R.Q1
P dV P dV dP Therefore: Q . Compliance (C) R dt R dP dt
P(t) dP(t) dP(t) P(t) Q(t) Then: Q ( t ) C . or R dt dt R.C C
PASI 2011 - A. Bandoni 30 The Windkessel Effect
Solution of the differential equation
t a) Particular solution (Q = Cte.=0) P(t) .exp( ) 1 R.C
b) Method of variation of parameter ( α1=α1(t) ) d t 1 t Q 1(t).exp( ) 1(t).exp( ) dt R.C R.C R.C C
1 t t d 1(t) 1 t Q . 1(t).exp( ) exp( ) 1(t).exp( ) R.C R.C R.C dt R.C R.C C
d (t) Q t t Hence: 1 .eep(xp( ) Then: (t) R.Q.exp( ) dt C R.C 1 R.C 2
PASI 2011 - A. Bandoni 31 The Windkessel Effect c) The general solution for systolic phase is
t t Ps (t) R.Q.exp( ) 2 .exp( ) R.C R.C
To determine α2 we can use initial condition P(t=0)=P0 , then α2 = P0-R.Q
P(t 0) P0 2 P0 R.Q
Finally, the pressure waveform for the systolic phase can be written as
t P (t) R.Q P R.Q.exp( ) s 0 R.C
PASI 2011 - A. Bandoni 32 The Windkessel Effect I – Diast oli c ph ase (l(valve in c lose pos iti)ition) Ts t T
air Following similar reasoning but with Q=Cte.=0 V(t) P(t)
dP P Q Q (t) dt R.C C 1
With initial condition: P(t=Ts)= Ps(Ts), the solution to the differential equation is: t t P(t) 3.exp( ) where 3 P0 exp( ) 1.R.Q. R.C R.C
Finall y, th e pressure wavef orm f or th e diast oli c ph ase can b e writt en as: t t Pd (t) P0 exp( ) 1.R.Q.exp( ) R.C R.C
PASI 2011 - A. Bandoni 33 The Windkessel Effect CltdlComplete model
Systolic Phase 0 t Ts air
V(t) t P(t) P (t) R.Q P R.Q.exp( ) s 0 R.C Q1(t)
air Diastolic Phase Ts t T
V(t) P(t) t t Pd (t) P0 exp( ) 1.R.Q.exp( ) Q (t) R.C R.C 1
Given: T exp( s ) 1 R.C Q , R , C , T , T and P (data) or P0 R.Q. s 0 T exp( ) 1 PASI 2011 - A. Bandoni R.C 34 The Windkessel Effect The t erm R.C it is cruc ia l i n th e 2 -WbW because it itd det ermi ne th e “ speed” of the exponential decay. This product is called the “characteristic time”, called
P P
R.Q P0
t t Case: 0 Case:
PASI 2011 - A. Bandoni 35 The Windkessel Effect
Case: 0, Hypertension: Ps > 140 mmHg Pd > 90 mmHg
PASI 2011 - A. Bandoni 36 The Windkessel Effect
The electrical circuit equivalence
Basic equation of a 2-element Winkessel model: P(t) dP(t) Q(t) C. R dt
Electric circuit of 2 passive elements: I(t) : eltillectrical curren t I(t) I E(t) : electrical potential 3 C : capacitance of the capacitor R : resistance of the resistor I2 From the Ohm and Kirchhoff laws
R E(t) dE(t) E(t) C I(t) C. R dt
I(t) ≡ Q(t) (blood flow) E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance) R ≡ R (peripheral resistance) PASI 2011 - A. Bandoni 37 The Windkessel Effect
The 3-element Windkessel model
R2 I(t)
I(t) ≡ Q(t) (blood fl ow) E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance) R ≡ R (peripheral resistance R 1 1 E(t) C (syst. and pulm.circuits)) 1 R2 ≡ R2 (resistance of valves (aortic and pulmonary))
R dE(t) P(t) dP(t) 1 1 .I(t) C.R1. C. R2 dt R2 dt PASI 2011 - A. Bandoni 38 The Windkessel Effect
The 4-element Windkessel model
R2 I(t) I(t) ≡ Q(t) (blood flow) E(t) ≡ P(t) (arterial blood pressure) C ≡ C (arterial compliance)
R1 ≡ R1 (peripheral resistance (syst. and pulm.circuits)) E(t) C R1 E(t) R2 ≡ R2 (resistance of valves (aortic and pu lmonary )) L L ≡ L (inertia of the blood circulation)
R L dE(t) d 2E(t) P(t) dP(t) 1 1 .I(t) R .C . L.C. C. 1 2 R2 R2 dt dt R2 dt
PASI 2011 - A. Bandoni 39 Compartment MdlModels
PASI 2011 - A. Bandoni 40 Compartment Models They are used to describe transport material in biological sciences
A compartment model contains a certain number of compartments, each one with a well mixed material
Compartments exchange material following certain rules
Material can be stored in the boxes and transported between them
Every compartment has a number of connections entering and leaving it.
Material can be added from the outside, can be removed or transported. Source
Drain
PASI 2011 - A. Bandoni 41 Compartment Models
Material represent the amount of something that we wish to account for
To account for the material, the models must fulfill certain conservation laws.
Conservations laws state that the difference between input and output flows amounts how much will be stored.
A compartment model can also represent: Ecological systems (material could be energy and the compartment different species of animals or plants) Physiologic system (material could be oxygen and compartment de organs)
Compartment can not be thought as independent. Flow in and out may depend on the compartment volume
Inflow to compartment may depend of outflow of other compartment. PASI 2011 - A. Bandoni 42 Compartment Models
State variables depend on each other and on the state of the system as a whole.
The transport in and out is characterized by the flows velocities.
Limitations of the compartment model • Is the system closed. Equation of conservation of mass is correct only if all material added or removed is included in the model. There is some lost of detailed information. • Homogeneity assumption. Not always it is possible to keep this assumption. Then more compartments are needed but also more information it is required. • Accuracy of the balance equation. In real physiological system typically some mass balance are know and other are not. • RlRelevance o fthf the mass b blalance. NtNot all syst ems can b e d escrib ibded in terms of mass balances. • Sensitivity analysis. Initial conditions and model parameters are not always known precisely.
PASI 2011 - A. Bandoni 43 Mathematical MdlModels Cardiovascular, Respiratory and Pharmacodynamic
PASI 2011 - A. Bandoni 44 Human Circulatory System Model
The historical fascination of the heart has lasted fo r many centuries and continues to attract considerable attention both theoretically and clinically. clinically.
To develop a physiologically founded model of the heart and the vasculature, it is essential to have a good model of the human short term pressu re control represented by the baroreceptor mechanism.
Using a lumped parameter compartment model, the entire human cardiovascular system may be described as aanetworknetwork of compliances, resistances and inductances not reflecting anatomical propertiesproperties..
Although strikingly simple, the model gives aaveryverygood description of the input impedance of the arterial system. system.
Such models are valuable tools for understanding cardiovascular diseases (hypertension, weak and enlarged heart, hemorrhages, etc.)
PASI 2011 - A. Bandoni 45 Human Circulatory System Model
Models facilitates getting new insight into cardiovascular functions and the interaction with other system (central nervous system, respiratory systems, etcetc..)
This type of models can be reliable and stable, simply enough to run in real timeme..
Lumped cardiovascular models are divided into pulsatile and non-pulsatile.
In the pulsatile case, the heart functioning is guided by aatimetime--varyingvarying elastance functionfunction..
A lumped pulsatile cardiovascular model that embraces principal features of the human circulation. circulation.
PASI 2011 - A. Bandoni 46 Human Circulatory System Model
Lumped cardiovascular models are divided into pulsatile and nonnon--pulsatilepulsatile..
In theepupultillsatile case, the hhteart ftiifunctioning iiissguguide d byyaa timeme--varyvaryiiing elastance functionfunction..
A lumped pulsatile cardiovascular model that embraces principal features of the human circulation. circulation.
PASI 2011 - A. Bandoni 47 Human Circulatory System Model
Pulmonar circulation Ap3
Ap2 Vp1
Ap1 Vp2
RV LA LV Heart RA
Vs2 As1
Vs1 As2 Systemic As3 circulation
PASI 2011 - A. Bandoni 48 Human Circulatory System Model
Pp3 Cp3 Vp3 Rp3
Qp2 Ap3 Qp3
Cp2 Pp2 Pl1 Cl2 Rp2 Vp2 Ap2 Vp1 Vl1 Rl2 Qp1 Ql1 Cl2 Cp1 Pl2 P1Pp1 Rl2 Rp1 Ap1 Vp2 Vl2 Vp1 Ll2 Lp1 Qrv PV Ql2 Pla Ela Eminrv Rla Erv(t) Prv Vla Emaxrv RV LA Lla Lrv Vrv Qla MV TV Era Qra Eminlv Pra Plv Elv(t) Rra LV Emaxlv Vra RA Vlv Llv Lra Qv2 AV Qlv Cv2 Ca1 Pv2 Pa1 Rv2 Vs2 As1 Ra1 Vv2 Va1 Lv2 La1 Qv1 Qa1 Cv1 Pv1 Vs1 As2 Pa2 Ca2 Rv1 Vv1 Va2 Ra2
Qa3 As3 Qa2
Pa3 Ca3 Va3 Ra3 PASI 2011 - A. Bandoni 49 Human Circulatory System Model
Model of a typical compartment (chamber) of the hemodynamic system
V0 : volumen at p=0 R : ressistance L : inertia Hemodynamic CliC : compliance element of a p p blood chamber Blood i 0 Blood input output Qin Qout
R L pi p0 Equivalence with an electric Q CV out circuit Qin 0
PASI 2011 - A. Bandoni 50 Circulatory System Model (Ottesen et al., 2003) • Heart Model o Hear t itse lf: 4 c ham bers (2 a tr ia an d 2 ven tr ic les ) o Vascular part Systemic part: 5 chambers (systemic arteries and veins) Pulmonary part: 5 chambers (arteries and veins) • Baroreceptor Model o Chronotropic effect (on heart rate) o Inotropic effect (on the cardiac contractility) o Vlfft(tidi)Vascular effect (on arteries and veins)
Respiratory System Model (Christiansen and Dræby, 1996) • Lung Model o Upper respiratory tracks: 1 chamber o Alveoli: 1 chamber • Gas Transport in Blood Model (O2, CO2, Anesthesia) o VlVascular par t5hbt: 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) It is assumed constant blood (VB) and tissue (VT) volumes. o Capillaries and alveoli: 1 chamber
Pharmacod ynami c M od el (Gopinath et al., 1995) • Drug Effect on Hemodynamic Variables Model PASI 2011 - A. Bandoni 51 The Cardiovascular Model
The Pumping Heart
Based on an elastance model where the cardiac contraction properties of the two ventricles are representing by a pair of time-varying elastance functionsfunctions..
The inertia of blood movements in the ventricles is considered through an inductance that introduce aaphasephaseshift between the ventricular pressure and the root aortic pressure. pressure.
The viscous properties of blood in the two atria are included by ventricular filling resistance
PASI 2011 - A. Bandoni 52 Ql Ql aR L M vp AV p la la lv L p la V lv as
Left LA E E (t) LV Heart la lv
R R Ros a1 La1 Ra2 a3 Rv1 Rv2 Lv2 p p pa3 pas a1 a2 pv1 pv2 Syst. Circ. AA CVi Ca1 Ca2 Ca3 Cv1 Cv2
Qr Qrv aR L TV R PV ra ra p rv Right pra rv pap Heart RA RV Era Erv(t)
PV R R L R R R R L op p1 p1 p2 a3 l1 p l2 l2 pp1 pp2 pp3 pl1 l2 Pulm . PA CVs Circ. Cp1 Cp2 Cp3 Cl1 Cl2
PASI 2011 - A. Bandoni 53 The Pumping Heart
dQ p p R .Q if p p la la lv la la la lv dt Lla
if pla plv Qla 0 dV la Q Q dt l 2 la
t pla Ela .Vla Vd ,la Vlv,b Qlvdt 2ml t* dQlv plv pas Elv t Emin,lv. 1 t Emax,lv.t if plv pas ddt L .t 2..t lv a .sin b .sin , 0 t t if plv pas ce Qlv 0 t tce tce dV 0 , tce t th lv Q Q dt la lv tce 0 1.th plv Elv (t).Vlv Vd ,lv pas R0s.Qlv pas
PASI 2011 - A. Bandoni 54 The Pumping Heart
Elastance model E Emin,lv max,lv
tce
th
.t 2..t a .sin b .sin , 0 t t ce E t E .1t E .t t tce tce lv min,lv max,lv 0 , tce t th
PASI 2011 - A. Bandoni 55 The Circulatory System Model
Single chamber model
pa1 Q1Qa1 dV pa2 a2 Q Q dt a1 a2 Va2 Va2 Vun,a2 pa2 Ca2 Qa2
pa2 pa3 Qa2 Ra2
PASI 2011 - A. Bandoni 56 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.
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
PASI 2011 - A. Bandoni 57 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 AtiAction po tenti tilal are con duc tdbted by the glossopharyngeal and the vagus nerves to the cardioregulatory and vasomottithdlltor centers in the medulla oblongata.
③ As a result of increased stimulation from the baroreceptor, the cardioregulatory center increased parasymphatic stimulation to the heart, which decreases the heart rate.
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.PASI 2011 - A. Bandoni 58 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.
If the initial problem were decrease in blood pressure, the activities and effect of baroreceptors, cardiovascular center and vasomotor center would be opposite of wh at was ill us tra ted .
PASI 2011 - A. Bandoni 59 The Baroreceptors Model
Baroreceptor Heart H system frequency
Systolic Emaxlv, Emaxrv maximum elastance
Cardio- MAP Systemic Ra1, Ra2, Ra3 vascular resistance System arteries Compliance C , C in veins and v1 v2 arteries
Unstressed V , V vol. in syst . unv1 unv2 veins
PASI 2011 - A. Bandoni 60 The Baroreceptors Model
Afferent sector Efferent sector
n MAP Central s x n Eferent i Sensors Nervous np System pathways
1 b ns MAP MAP .n .MAP .n .MAP MAP i i s i p i 1 dxi t 1 b xi t i MAP , i E 1 dt i np MAP MAP 1 i E H, E ,R ,V ,C max ps un v
PASI 2011 - A. Bandoni 61 The Respiratory System Model
The resppyiratory system is concerned with the transport o f oxygen between atmosphere and the tissue and organs in the body
Oxygen iiissconcontinuous ly ttdtransported by the lllung and bloo d ciitircuit.
Carbon dioxide is a waste product of the oxidative metabolism and is carried by the blood in the opposite direction PASI 2011 - A. Bandoni 62 The Respiratory System Model
O2 CO2 Atmosphere
Ventilation
Alveoli O2 CO2 Gas exchange
O2 CO2 Pulmonary circulation
Rig ht LftLeft Heart Gas transport Heart
Systemic circulation
O2 CO2
Cell Gas exchange PASI 2011 - A. Bandoni metabolism 63 The Respiratory System Model
Lung model: pressure ■ Connect atmosphere (mask) with alveoli trought expressions of gas flow R0 Alveoli ■ The lung is divided in Upper compartments R R R airway 1 2 i ■ In each compartment gas C0 C1 C2 Ci Um (t) flows are calculated (O2, CO2, Anesthesia) Atmosphere or Ut (t) ■ The outputs of the model are: respitiratory Muscles pressure in different sectors, mask the net volume of air flow, part ia l p ressu r e of ex pir ed air and alveoli.
PASI 2011 - A. Bandoni 64 The Respiratory System Model
Distribution of substances in the organs through blood
Capillary Alveolus Alveolus
κ.pA κ.p pcp Q.cb (1- λ ).Q.cvs Central CtlCentral Vbcb (p) pv arterial venous Metabolism pas s λ.Q.cvs compartment compartment pli Liver
p ki Kidney Metabolism Viscere cvv phe Heart M M venous - +
compartment pbr Brain Other V c (p) p t t re organs Vbcb (p) Lean venous pco Connective zi.Q.cas zi.Q(p).cb compartment cvl tissue pmu Muscles Adipose tissue venous c p compartment av ad Adipose tissue PASI 2011 - A. Bandoni 65 The Respiratory System Model
Upper Alveoli n dpi airways Um p0 R0. Ci. p0 dp i1 0 dt R0 dt R0.C0 C0 p f0 i C dpi p0 pi Ut Ri i , i 1...n A pcp dt Ri.Ci fi Pressure model
df RT I U p .f f n I p p .f f 0 m 0 e 0 i 0 i 0 2 i1 dt C0 p0 V00 p0 R0 Ri df RT I p U p .f f i 0 t i 0 i κ.p p f 2 cp i i dt Ci .pi V0i .pi Ri 0 x 0 I x x x 0
Molar fractions model PASI 2011 - A. Bandoni 66 The Pharmacodynamic Model
Pharmacology::thethehistory, source, physical and chemical properties, biochemical and physiological effect, mechanisms of action, absorption, distribution, biotransformation and excretion, and therapeutic and other uses of drugs. drugs.
Pharmacokinetics: absorption, distribution, metabolism and excretion of drugsdrugs..
Pharmacodynamics: biochemical and physiological effects and their mechanisms of action. action.
PASI 2011 - A. Bandoni 67 The Pharmacodynamic Model ion tt ncentra rug Co DD
Time Concentration of drug in the body as a function of time
PASI 2011 - A. Bandoni 68 The Pharmacokinetic Model
1 dp dct dcb Vt Vb ziQcas cb p M p M p dt dp dp Pressure balance 0 M c CO2 ithin the organs O2 M M O 2 c M 0 O2 O2 0 caa M aa aa caa
1 dp dcb Pressure balance in Vb Q1 cvs cb p p A p dt dp the capillaries
1 dp dc Q c Q c Pressure balance in V b Q c c p 1 1 2 2 b x b c x the compartments dt dp Q1 Q2
PASI 2011 - A. Bandoni 69 Eff . 70 2 k PFL Eff 1 a C sisSNP dt t R d max dEff dEff . Eff . N d BASE 1 C a . sisDP lvDP 1 R t C k d max t E d 1 dEff a dt dEff dt Eff) dC dEff ± (1 Eff) BARO sisBARO ± v l BARO R (1 max maxBARO E E
sisBARO = isSNP s R
R PASI 2011 - 2011 PASI Bandoni A. max ff lvDP E R=
d Eff max C E Eff isDP s Drug effect R ff 1 Eff system 1 sisBARO maxBARO E R t lvBARO Cardiovascular d The Pharmacodynamic Model max t d sisBARO dE dR Baro- receptors lv t sis max t d d dR dE MAP The Pharmacodynamic Model
Drug (intravenous) Affected variable Action SNP (sodium Peripheral resistance MAP nitroprusside) Peripheral resistance, DP ((pdopamine ) MAP systolic maximum elastance MAP PFL (propofol) BIS unconsciousness
Systolic maximum elastance
Peripheral resistance
PASI 2011 - A. Bandoni 71 The Pharmacodynamic Model
DP and SNP drugs are chosen to increase ventricular contractility and reduce theeresresitistance of artiteries to bloo d flow, respec tive lyy..
PFL is chosen to conduct unconsciousness by measured of BIS parameterparameter..
DP increases the MAP and CO. CO.SNPSNP decreases and increases CO MAPMAP..
Sceneries are simulated by delivering a step of 1μg/kg/min of SNP, DP and PFL and registering the dynamic response of the physiological, pharmacokinetic and pharmacody namic v ariables.
PASI 2011 - A. Bandoni 72 Computational Implementation
Model implemented in Fortran
Diff. EqsEqs.. solved with a 4th order RungeRunge--KuttaKutta method.
Resolution sequence: ((ii)) the cardiovascular model is solved until to reach steady state, (ii) the CO obtained from this model is used in the breathingg, model, (()iii) the breathinggy model is solved until to reach steady state.
The drug in jec tion is s imu la te d fflfbthi(5)or a cycle of breathing (5 sec.). Then th e cardiovascular model is fed with the drug concentration Cd to simulate the 0.8 sec. a heartbeat. The updated value of CO is fed back to the breathing model.
PASI 2011 - A. Bandoni 73 Computational Implementation
CO2, O2 Cd(inyectable) Cd(inhalable) Cd (alveoli)
Transport and Respiratory distribution, system Pharmacokinetics of drugs Qa3, Qp3
Cd (organs)
CdiCardiovascul ar Pharmacod ynami cs Baroreflex system of drugs MAP
Emax EmaxBARO EffEmax Rsis Ra2BARO EffRa2 Ra3BARO EffRa3 Control Action
PASI 2011 - A. Bandoni 74 Computational Implementation
Dimensions of the integrated model
Var./Eqs. Var./Ecs. Model Parameters Algebraics Differenctials
Cardiovascular-Respiratory 37 39 53
Respiratory-Pharmacodynamic 60 93 85
Total 97 132 138
PASI 2011 - A. Bandoni 75 Results
PASI 2011 - A. Bandoni 76 Results: cardio vascular system
Wiggers Diagram
PASI 2011 - A. Bandoni 77 Results: cardio vascular system
Left ventricle and root aortic Left ventricular volume pressure vs. time vs. time
PASI 2011 - A. Bandoni 78 Results: cardio vascular system
Outflow of the left ventricle
Left ventricular pressure Pressure vs. Volume left ventricle PASI 2011 - A. Bandoni 79 Results: baroreflex system
Heart period vs. time Resist. sect. As1 of syst . arteries vs. time
Compliance in sect. Vs1 of sistemic Unstres. Vol. of sect. Vs1 of veins vs. time PASI 2011 - A. Bandoni sistemic veins vs. time 80 Results: baroreflex 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
PASI 2011 - A. Bandoni 81 Results: gas transport
Partial pressure of O2 in different compartments of the body
Partial pressure of CO2 in different compartments of the body
PASI 2011 - A. Bandoni 82 Results: respiratory system expiración
inspiración
Volume vs. Pressue diagram in lungs
Partial pres. profile of Partial pres. profile of O2 in CO2 in lung and alveoli. PASI 2011 - A. Bandoni lung and alveoli. 83 Results: pharmacodymic system Effect of the SNP action 1µg/kg/min
SNP concentration profile at the central arterial compartment
Mean Arterial Pressure, MAP Cardiac Output, CO PASI 2011 - A. Bandoni 84 Results: pharmacodynamic system
Effect of the SNP action 1µg/kg/min
Resistance, Ra2 Resistance, Ra3
PASI 2011 - A. Bandoni 85 Results: pharmacodynamic system
Effect of the DP action 5µg/kg/min
DP concentration profile at the central arterial compartment
Mean Arterial Pressure, MAP Cardiac Output, CO PASI 2011 - A. Bandoni 86 Results: pharmacodynamic system
Effect of the DP action 5µg/kg/min
Medial arterial resistances
Elastance
PASI 2011 - A. Bandoni 87 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)
PASI 2011 - A. Bandoni 88 Results: pharmacodynamic system Effect of the DP action 2, 4, 6, 8 µg/kg/min
Systemic Resistance vs. infusion Cardiac frequency vs. infusion d(ti)doses (time) doses (time)
PASI 2011 - A. Bandoni 89 Results: pharmacodynamic system Effect of the PFL action 150µg///kg/min
PFL conc. at the central arterial comp. Mean Arterial Pressure, MAP
Cardiac Output, CO Compliance of sector a1 of systemic arteries
PASI 2011 - A. Bandoni 90 Conclusions
Development of an integrated cardiovascular, baroreceptor, respiratory, pharmacokinetic and pharmacodynamic modelmodel..
The effect of certain drugs on hemodynamic variables was studiedstudied..
PASI 2011 - A. Bandoni 91 Future Works
General model validation with real patient data. data.Collaboration with a research group formed by doctors ((FavaloroFavaloroUniversity, Bs.As. – Español Hospital, B. Blanca, Arg.)
Model validation for inhalable anesthesia effects.
Model validation for simultaneously drugs administration.
Development of a control model for handling dose of drug administrationadministration..
Development of a teaching simulation model of the cardiovascular system ((InstitutoInstitutoNacional de Tecnología Industrial, INTI, Bs. Bs.Ass..,, Argrg..))
PASI 2011 - A. Bandoni 92 Basic References:
Cardiovascular Model: . Ottesen J., Olufsen M. and Larsen J. Applied Mathematical Models in Human Physiology . SIAM, Philadelphia. (2004)
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)
Respiratory Model: . Chr is tiansen T. andDd Dræ by C. MdliModeling the Resp ira tory Sys tem Technical. Report IMFUFA, Roskilde University Denmark Text No. 318. (1996)
PASI 2011 - A. Bandoni 93 Other References:
. 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 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. Biomedi cal an d Life SiSciences Sys tems Bio logy ‐ VlVolume 2, 2011.
PASI 2011 - A. Bandoni 94 “With growing emphasis being placed on the information processing aspects of biomedical investigation, theoretical and experimental studies assume increasing importance. In many instances, however, there are questions that appear to be unanswerable by present experimental techniques; in such cases, models can usefully augment direct scientific experimentation.
Theessentilial idiingredient of thesciifiientificmethdhod istheuseof models. Good modeling is more likely to be achieved by following 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 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 (University of California at Los Angeles, Columbia University, University of Dallas) The FASEB Journal, vol. 12 no. 3, pp.275-285, March 1, 1998 PASI 2011 - A. Bandoni 95 PASI 2011 - A. Bandoni 96 Muchas gracias
PASI 2011 - A. Bandoni 97 Cámara izquierda del corazón Circulación sistémica dQ p p R Q la la lv la la si p p la lv dQ p p R Q dt Lla a1 a1 a2 a1 a1 dVa1 Qlv Qa1 dt La1 dt Qla 0 si pla plv Va1 Vun,a1 pa2 pa3 p Qa2 dVla a1 R Ql2 Qla pla Ela Vla Vd,la Ca1 a2 dt
dVa2 Va2 Vun,a2 dQlv plv pas Qa1 Qa2 pa2 si plv pas dt Ca2 dt Llv
Qlv 0 si plv pas pas pv1 dVa3 Qa3 Qa2 Qa3 Ra3 dt dVlv Qla Qlv plv Elv tVlv Vd,lv dt Va3 Vun,a3 pv1 pv2 pa3 Qv1 C Rv1 t a3 Vlv,b Qlvdt 2ml t* dVv1 Vv1 Vun,v1 Qa3 Qv1 pv1 dt C Elv t Emin,lv 1 t Emax, lvt v1
t 2t dVv2 dQv2 pv2 praRv2Qv2 a sin b sin 0 t t Q Q ce v1 v2 dt L t tce tce dt v2 0 tce t th
Vv2 Vun,v2 t t p R sQ p pv2 ce 0 1 h as 0 lv a1 Cv2
PASI 2011 - A. Bandoni 98 Cámara derecha del corazón Circulación pulmonar dQra pra prv RraQra dQp1 p p1 p p2Rp1Qp1 dVp1 si pra prv Qrv Qp1 dt Lra dt Lp1 dt
Qra 0 si pra prv Vp1 Vun, p1 p p2 p p3 p p1 Qp2 C p1 Rp2 dVra Qv2 Qra p E V V V V dt ra ra ra d,ra dVp2 p2 un, p2 Qp1 Qp2 p p2 dt C p2 dQrv prv pap si prv pap p ps pl1 dVp3 dt Lrv Qp3 Qp2 Qp3 Rp3 dt Qrv 0 si prv pap Vp3 Vun, p3 pl1 pl2 p p3 Ql1 C R dVrv p3 l1 Qra Qrv prv ErvtVrv Vd,rv dt dVl1 Vl1 Vun,l1 t Qp3 Ql1 pl1 dt Cl1 Vrv,b Qrvdt 2ml * t dVl2 dQl2 pl2 plaRl2Ql2 Ql1 Ql2 dt dt Ll2 Ervt Emin,rv1 t Emax, rvt Vl2 Vun,l2 p R pQ p pl2 ap 0 rv p1 Cl2
Modelo respiratorio (fracción molar)
df R T I U p f f n I p p f f 0 m 0 e 0 i 0 i 0 dt 2 R0 i 1 Ri C 0 p0 V00 p0 0 x 0 I x df RT I p U p f f x x 0 i 0 t i 0 i κ p cp pifi dt C p 2 V p Ri i i 0i i PASI 2011 - A. Bandoni 99 Barorreceptores 1 1 ns MAP n p MAP 1 MAP Q c Q c MAP 1 1 1 2 2 dp dcb 1 c x V Qc c p Q Q b x b 1 2 dt dp b i MAP ins MAP in p MAP i i E H , Emax ,R ps ,Vun ,Cv 0 dxi t 1 b xi t MAP , i E dt i M i c CO2 O2 M M O M 0 2 c Modelo respiratorio (presión) O2 O2 0 caa M aa n dpi aa caa Um p0 R0 Ci dp dt dpi p p U 0 i1 0 i t dt R0C0 dt RiCi 3 2 H a2 H a1H a0 0 0.27273 1.96364t 0 x 0.278 a K NaOH K 0.66943 0.53554t 0.278 x 1.806 2 a,Pr 0 a,CO Um 17.00005 8.50909t 1.806 x 1.904 a K NOHNaOH c K K NOHNaOH H PPr 1 a,CO 0 CO2 a,Pr a,CO 0 0 0.57034 0.05904t 1.904 x 5 a K K NaOH H Pr c Modelo de transporte de gases en sangre 0 a,CO a,Pr 0 0 CO2 1 b Ery c Pla c dp dct dcb Hb Hb cCO p, pH cCO p, pH cCO p, pH 1 Vt Vb ziQ cas cb p M p M p 2 2 Ery 2 Ery dt dp dp cHb cHb Ery Ery pH Ery p, pH pK Ery p, pH 1 c p, pH p 110 CO2 CO2 cCO 2 dp dcb Vb Q1 cvs cb p p A p Ery Ery dt dp c Ery p, pH Ery p 110pH p, pH pK p, pH CO2 CO2 cCO 2 PASI 2011 - A. Bandoni 100 Pla Pla pH pK Pla p, pH c p, pH p 110 T CO2 CO2 cCO 2 x0 p 1.946 a p 0.055 37 ºC pH Ery p, pH 7.840.06s p, pH Ery O2 c pK p, pH 6.125 log10 110 ct p, pH c Plla p, pH 1 Hb CO2 CO2 Ery cHb Ery pH p, pH 7.19 0.77pH 7.4 0.0351 sO p, pH 2 b caa p b paa Pla pH 8.7 pK p, pH 6.125 log10 110 yp 1.875 xp x p hptanh0.5343xp x p cb p, pH p c s 0 0 O2 O2 O2 Hb O2 1 s t O2 yp hp 3.5 ap xp logp / kPa c p p 1 e O2 aa t aa p CO2 mmol ap 0.72 pHcCO p 7.4 0 .09log 0.070.03xHbf cdpg / 5 t 2 5.33kPa l c p p O2 O2 O2
0.386xHbCO0.174xHi 0.28xHbf
Modelo farmacodinámico
dEff dC dEffC k C N Eff Eff k Eff a1 C a1PFL 1 d max 2 a1BASE dt dt dt
dR dR dEffR dEffR sis sisBARO 1 Eff Eff R sisDP sisSNP RsisDP RsisSNP sisBARO dt dt dt dt
dffdEff dEmaxlv dEmaxlvBARO EmaxlvDP 1 EffE EmaxlvBARO dt dt maxlvDP dt PASI 2011 - A. Bandoni 101