research article application of mathematical modeling for

Research ArticleApplication of Mathematical Modeling for Simulationand Analysis of Hypoplastic Left Heart Syndrome (HLHS) inPre- and Postsurgery Conditions

Ali Jalali,1 Gerard F. Jones,1 Daniel J. Licht,2 and C. Nataraj1

1Department of Mechanical Engineering, Villanova University, Villanova, PA 19085, USA2Neurovascular Imaging Lab, University of Pennsylvania School of Medicine, Philadelphia, PA 19104, USA

Correspondence should be addressed to Ali Jalali; ali.jalali@villanova.edu

Received 4 September 2014; Accepted 19 February 2015

Academic Editor: Joakim Sundnes

Copyright © 2015 Ali Jalali et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the mathematical modeling of a severe and common congenital defect called hypoplastic left heartsyndrome (HLHS). Surgical approaches are utilized for palliating this heart condition; however, a brain white matter injury calledperiventricular leukomalacia (PVL) occurs with high prevalence at or around the time of surgery, the exact cause of which isnot known presently. Our main goal in this paper is to study the hemodynamic conditions under which HLHS physiology maylead to the occurrence of PVL. A lumped parameter model of the HLHS circulation has been developed integrating diffusionmodeling of oxygen and carbon dioxide concentrations in order to study hemodynamic variables such as pressure, flow, and bloodgas concentration. Results presented include calculations of blood pressures and flow rates in different parts of the circulation.Simulations also show changes in the ratio of pulmonary to systemic blood flow rates when the sizes of the patent ductus arteriosusand atrial septal defect are varied. These changes lead to unbalanced blood circulations and, when combined with low oxygen andcarbondioxide concentrations in arteries, result in poor oxygen delivery to the brain.We stipulate that PVLoccurs as a consequence.

1. Introduction

Hypoplastic left heart syndrome (HLHS) is a congenital heartdefect (CHD) in which the left side of the heart is severelyunderdeveloped. Without early surgical palliation HLHS isuniversally fatal [1]. HLHS results from a failure of the aorticor mitral valve to form. Lack of antegrade flow through thevalves causes insufficient growth of both the left ventricle andthe ascending aortic arch. A typical HLHS heart is comparedwith a normal heart in Figure 1.

Underdevelopment of the left ventricle-aorta complexresulting in critical aortic valve stenosis or aortic valve atresiawith an intact ventricular septum is themost recognized formof HLHS. There are corresponding changes in the right sideof the heart in the case of HLHS. All right sided cardiacstructures are larger than normal including the right atrium,pulmonary artery, and pulmonary valve [2].

As blood returns from the lungs to the left atrium, itmust pass through an atrial septal defect to the right side of

the heart. In cases of HLHS, the right side of the heart mustpump blood to the body through a patent ductus arteriosus(PDA). This maintains fetal parallel circulation where theright ventricle is the only active pump [3]. But, since theductus arteriosus (DA) usually closes within eleven days afterbirth, for an HLHS baby, blood flow is severely compromisedleading to very low circulation and possible death. Hence,the management of neonates with HLHS is very complex.Treatment generally commences with vigorous infusion ofprostaglandin to prevent the PDA from closing. However,reduction in pulmonary resistance after birth results in anunbalanced circulation where most of the blood goes into thepulmonary circulation thereby compromising the systemicoxygen supply. A typical way to treat this disease is shuntsurgery. A shunt is a surgically created connection betweenthe systemic arterial circulation and the pulmonary arteries.Authors in [4] present a good survey of shunt modeling andits application in planning the surgery.

Hindawi Publishing CorporationBioMed Research InternationalVolume 2015, Article ID 987293, 14 pageshttp://dx.doi.org/10.1155/2015/987293

2 BioMed Research International

Normal heart

Hypoplastic leftheart syndrome

(A)(B)

Figure 1: Comparison between normal heart on the left and HLHS heart on the right. (A) is atrial septal defect, (B) is hypoplastic heart, and(C) is patent ductus arteriosus.The left ventricle is very small.The aortic arch is extremely hypoplastic and flow is retrograde. Systemic outputis ductal dependent. Adapted from wikipedia.

The need for a detailed study of the problem of HLHSstems from our recent studies [5–7] to predict the occurrenceand extent of periventricular leukomalacia (PVL) after Nor-wood surgery in neonates. The PVL is a form of white matterbrain injury, characterized by necrosis (more often coagula-tion) of white matter near the lateral cerebral ventricles. Itcan affect newborns and (less commonly) fetuses; prematureinfants are at the greatest risk of the disorder. Affectedindividuals generally exhibitmotor control problems or otherdevelopmental delays, and they often develop cerebral palsyor epilepsy later in life [8]. Despite being full-term infants,PVL is found in more than 50% of the neonates after cardiacsurgery [1, 9]. As there has been a dramatic reduction inmortality rates following surgery for complex CHDs, therehas been an increasing recognition of adverse neurodevel-opmental sequelae in some survivors. Evaluation of childrenfollowing neonatal repair of CHD demonstrates long termneurodevelopmentalmorbidity and learning disability in sur-vivors of these infant heart surgeries. Management strategiesbefore, during, and after surgery including the type of supportduring operation have been implicated as factors in postoper-ative neurodevelopmental dysfunction. Deficiency in oxygencirculation and low carbon dioxide concentrations (𝑃CO

)have also been considered as important factors affecting themorbidity rates of neurodevelopmental dysfunction [6].

In our previous work in the field of PVL prediction, wehave applied computational intelligence techniques to thedata obtained from 59 patients which were collected at theChildren’s Hospital of Philadelphia (CHOP). Computationalintelligence (CI) is a set of nature-inspired computational

methodologies, examples of which include Fuzzy Logic sys-tems, Neural Networks, and Evolutionary Computation. Thedecision trees that we have developed suggest some ranges forcritical hemodynamic parameters, such as oxygen concentra-tion and blood pressure, which predict a high probability forthe occurrence of PVL [5, 10, 11]. But since these numbershave been obtained from a retrospective review of limitedset of subjects, they cannot be used as a general applicablecriterion for all neonates. Hence, it is important to carry outphysiological modeling of the cardiovascular system to beginto understand the underlying causal relationships between aparticular set of parameters with the occurrence of PVL.

In this paper a lumped parameter model of the HLHScirculation has been developed to study blood pressure andflow in different parts of the cardiopulmonary circulationsystem. The diffusion modeling of oxygen (O

2) and carbon

dioxide (CO2) is also included in themodel. Changes induced

in O2and CO

2concentrations and variation of blood flow

rates in different parts of the body due to changes in someimportant model parameters have also been studied.

2. Pressure Flow Model

Several lumped parameter models of the cardiovascular sys-tem have been developed in past research, for instance, [12–14].This paper differs frompreviously developed cardiovascu-lar models in that this model is tailored to the uniquephysiology of the HLHS heart in comparison to the abundantliteraturewhich dealswith the simulation of the normal heart.Furthermore, this model combines blood gas modeling with

BioMed Research International 3

Hypoplastic heart

Pulmonary circulation

Systemic circulation

DASABSVBSLV

Figure 2: Lumped parameter model of HLHS. Model is made up of three parts: (a) hypoplastic left heart, (b) pulmonary circulation, and (c)systemic circulation. Direction of blood flow is shown by arrows. ASD: atrial septal defect; RA: right atrium; LA: left atrium; TV: tricuspidvalve; RV: right ventricle; PV: pulmonary valve; PA: pulmonary artery; PAB: pulmonary arterial bed; PVB: pulmonary venous bed; PDA:patent ductus arteriosus; DA: descending aorta; SAB: systemic arterial bed; SVB: systemic venous bed; SLV: systemic large veins.

the circulation model to represent a more comprehensivemodel of the HLHS physiology. Moreover, by using para-metric analysis in this study we have been able to track thechanges in the heart for before and after surgery conditions.Like all other lumped parametermodels, thismodel is limitedby simplification assumption that it makes, for example,overlooking solid fluid interactions of arteries and blood flowand Newtonian flow assumption. For this work the lumped-parameter approach used to model the HLHS circulation isshown in Figure 2. Each box represents a lumped parametermodel for a complex system of blood vessels and heartcomponents. This model is built based on methodologiespreviously published for the fetal [15] and neonatal cerebralcirculation [16–19] and the Norwood procedure [20, 21]. Thecomplete model is composed of threemain parts: a hypoplas-tic left heart, systemic circulation, and pulmonary circulation.Each compartment is made of resistances, capacitances, andinductances. Resistances are used to model the flow in thearteries and veins, and capacitors are used to represent theelasticity of these vessels. Inductors are not typically used formodeling the flow in the veins because veins do not functionprimarily in a contractile manner and so their inductance

values are considered to be negligible in comparisonwith thatof the arteries.

2.1. Hypoplastic Heart. The heart has been assumed to becomposed of three parts: right atrium (RA), left atrium (LA),and right ventricle (RV). Here, as mentioned earlier, we haveconsidered the most severe case of the hypoplastic heartin which the left ventricle is completely blocked (the mostcommon form of HLHS ismitral atresia/aortic atresia; atresiaimplies no flow through the valve). Hence the left ventricle isnot taken into consideration in the model.

The activity of the heart is modeled following the treat-ment in [22]. For both the atria and the ventricle, the totalpressure is expressed as a nonlinear function of the volumeand cycle-time:

𝑃 (𝑡) = 𝑃𝑠(𝑡) + 𝑃

𝑑(𝑡) , (1)

where𝑃𝑠= 𝛼 (𝑡) 𝐸max (𝑉 (𝑡) − 𝑉

𝑢) ,

𝑃𝑑= (1 − 𝛼 (𝑡)) 𝑃

0(𝑒𝐾𝑒⋅𝑉(𝑡)

− 1) .

𝑃 is the pressure, 𝑉 is volume, 𝑉𝑢is the unstressed volume,

and 𝐸max is the maximum elasticity of the heart wall duringthe heart cycle. The activation function 𝛼 is the driving forceof the heart andmodels the release of Ca2+ which initiates thecontraction of the heart muscle [23].

Since the heart muscle’s contraction is different for systoleand diastole cycles, the activity function is defined by differ-ent differential equations in the systole and diastole. Duringdiastole, the activation function is defined by the following:

𝑑𝛼

𝑑𝑡= −𝐾𝑟𝛼, (3)

where 𝐾𝑟is the relaxation rate. For the systole period it is

defined by the following:

𝑑2𝛼

𝑑2𝑡+ 2𝐾𝑒

𝑑𝛼

𝑑𝑡+ 𝐾2

𝑒𝛼 = 𝐾

𝑒𝛼max, (4)

where 𝐾𝑒is the excitation rate and 𝛼max is the limiting value

of the activation function. The solutions to the above lineardifferential equations yield the following:

𝛼 (𝑡) = 𝛼 (𝑡𝑑) 𝑒−𝐾𝑟(𝑡−𝑡𝑑),

𝛼 (𝑡) = 𝛼max − (𝛼max − 𝛼 (𝑡𝑠)) (1 + 𝐾

𝑒(𝑡 − 𝑡𝑠))

⋅ 𝑒−𝐾𝑒(𝑡−𝑡𝑠),

where 𝑡𝑠is the systole time and 𝑡

𝑑is the onset of diastolewhich

we set equal to zero assuming the heart cycle starts fromonsetof diastole. This assumption is just for convenience and doesnot impact generalization of themodel.The parameter valuesof 𝑡𝑠, 𝐾𝑒, 𝐾𝑟, and 𝛼max are different for ventricles and atria

and are shown in Table 1 [23]. The activation functions arefunctions of time for the ventricle and the atrium and areshown in Figure 3.

The 𝐸max is a function of volume of the ventricle toaccount for the decreasing elastance with increasing volume[20]. Consider

𝐸max = 𝐸1+ 𝐸2(𝑉 (𝑡) − 𝑉

𝑢) , (6)

where 𝐸1is constant; 𝐸

2is a negative-valued constant to

account for decreasing elastance of the ventricle. 𝐸2is zero

for the atria resulting in a constant 𝐸max, consistent with theliterature. Again, 𝑉

𝑢is the unstressed volume of the chamber

which is the volume at zero pressure.This is the 𝑥-intercept of the tangent at the end-systolic

point for the pressure-volume (𝑃-𝑉) curve. For the diastole,the ventricle and atria fill through an exponential 𝑃-𝑉function reflected in the equations below.

Hence, for maximum isometric pressure, we have [22]

𝑃isomax,RA (𝑡) = 𝛼𝑛RA (𝑡) 𝐸1,RA (𝑉RA (𝑡) − 𝑉

𝑢,RA)

+ (1 − 𝛼𝑛RA (𝑡)) 𝑃0,RA (𝑒

𝐾𝑒⋅𝑉RA(𝑡) − 1) ,

𝑃isomax,LA (𝑡) = 𝛼𝑛LA (𝑡) 𝐸1,LA (𝑉LA (𝑡) − 𝑉

𝑢,LA)

+ (1 − 𝛼𝑛LA (𝑡)) 𝑃0,LA (𝑒

𝐾𝑒⋅𝑉LA(𝑡) − 1) ,

Table 1: Heart model parameters.

Parameters ValuesRight atrium

𝐸1, RA 7.35mmHg/mL

𝑃0, RA 0.17mmHg

𝐾𝑒, RA 0.484mL−1

𝑉𝑢, RA 1mL

Left atrium𝐸1, LA 7.35mmHg/mL

𝑃0, LA 0.17mmHg

𝐾𝑒, LA 0.484mL−1

𝑉𝑢, LA 1mL

Right ventricle𝐸1, RV 8.5mmHg/mL

𝐸2, RV −0.042mmHg/mL2

𝑃0, RV 0.9mmHg

𝐾𝑒, RV 0.062mL−1

𝑉𝑢, RV 4mL

ASD𝑅ASD 0.1mmHg⋅s⋅mL−1

Tricuspid valve𝐾Tric 0.0006mmHg⋅s2⋅mL−1

Pulmonary valve𝐾PV 0.0008mmHg⋅s2⋅mL−1

Activation function𝐾𝑒(ventricle) 51

𝐾𝑒(atrium) 75

𝐾𝑟

55𝑡𝑠(ventricle) 0.193 s

𝑡𝑠(atrium) 0.173 s

𝛼max (ventricle) 150𝛼max (atrium) 130

𝑃isomax,RV (𝑡) = 𝛼𝑛RV (𝑡) (𝐸1,RV + 𝐸

2,RV (𝑉RV (𝑡) − 𝑉𝑢,RV))

⋅ (𝑉RV (𝑡) − 𝑉𝑢,RV) + (1 − 𝛼

𝑛RV (𝑡)) 𝑃0,RV

⋅ (𝑒𝐾𝑒⋅𝑉RV(𝑡) − 1) ,

where 𝛼𝑛is normalized 𝛼. The atria and ventricles are

modeled with variable capacitors to account for the time-dependent relationship of pressure with volume and alsoviscous losses. A flow resistance has been introduced betweenthe right and left atria to account for the defect in the wallsof the heart permitting a mixing of blood flow between theatria. In the present work, this mixing has been modeledas an orifice unlike [24] where it was modeled as a simpleresistor.The reason for modeling ASD as a variable nonlinearresistor instead of as a simple linear resistor is because of itsvery small diameter in comparison with other compartmentswhich leads to a nonnegligible nonlinearity. A nonlinear

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1

Time (s)

Heart muscle activation function

VentricleAtrium

𝛼(t

Figure 3: Ventricular and atrial activation functions.

pressure flow relationship (Darcy-Weisbach equation) is usedas follows:

Δ𝑃 (𝑡) = 𝐾ASD ⋅ 𝑄 (𝑡)2. (8)

The tricuspid and the pulmonary valves are also modeledas unidirectional orifices (diodes) and a similar pressure flowrelationship has been used for them. This model representsan idealized situation since, in reality, the tricuspid valve doesleak sometimes and the flow is not completely unidirectional.

2.2. Pulmonary Circulation. The pulmonary circulation isdivided into three compartments: proximal pulmonary arter-ies (PA), pulmonary arterial bed (PAB), and pulmonaryvenous bed (PVB). The PA and PAB are modeled using 𝑅𝐿𝐶and PVB by 𝑅𝐶 circuits. All these circuit elements haveconstant values. Flow enters into the PA from the pulmonaryvalve and the blood flows out to the left atrium (LA). Itshould be noted that the PDA is present in the newbornwhichnormally closes after 5–10 days. To model it, a nonlinearresistor is added between the pulmonary artery and theaorta. The addition of the PDA to the model is anothermajor difference between this model and other previouslydeveloped models; this is especially important in the currentstudy because of our focus on HLHS.

2.3. Systemic Circulation. The systemic circulation is dividedinto four compartments: descending aorta (DA), systemicarterial bed (SAB), systemic venous bed (SVB), and systemiclarge veins (SLV). The DA and SAB are modeled by 𝑅𝐿𝐶 andSVB and SLV are modeled by 𝑅𝐶 circuits. Blood comes infrom the PDA and flows to the right atrium.

2.4. Fluid Flow. For each compartment, time-dependentvariables, such as pressure in the capacitance, are expressedas differential equations and since these equations cannot

QoutQin P1 P2

Figure 4: Simple RLC compartment to describe model. 𝑃 ispressure; 𝐿 (inductance) accounts for blood inertance;𝑅 (resistance)accounts for resistance to blood flow; 𝐶 (capacitance) accounts forcompliance.

be solved analytically, numerical integration is used. Thisapproach calculates solutions with variable time steps toavoid mathematical instability that would lead to divergenceof the solution. For every RLC compartment shown inFigure 4, analysis has been carried out as shown.

The relationship between pressure and volume in thecapacitor leg in Figure 4 can be calculated using the following:

𝑑𝑃 (𝑡) =𝑑𝑉 (𝑡)

𝐶, (9)

where 𝑑𝑉(𝑡) is the change in the compartment volumecalculated from the pressure differential.The pressure change𝑃1− 𝑃2in Figure 4 is given by the following:

𝑃1(𝑡) − 𝑃

2(𝑡) = 𝑅 × 𝑄out + 𝐿

𝑑𝑄out𝑑𝑡

. (10)

Hence, for every compartment, the change in the output flowrate is expressed using the following equation:

𝑑𝑄out𝑑𝑡

=𝑃1(𝑡) − 𝑃

2(𝑡) − 𝑄out × 𝑅

𝐿. (11)

ASD is assumed to be a variable nonlinear resistor; therefore,we can write the relationship between𝑄 and pressure changethrough Darcy-Weisbach equation:

𝑄ASD =

{{{{{{

√𝑃RA − 𝑃LA𝐾ASD

𝑃RA > 𝑃LA

√𝑃LA − 𝑃RA𝐾ASD

𝑃LA > 𝑃RA,

where 𝑃RA and 𝑃LA are the right and left atrial pressures,respectively. By assuming realistic initial values, numericalintegration is carried out for a sufficient number of heartcycles to achieve steady-periodic values for every parameter.The PDA flow rate is estimated from the following:

𝑄PDA =𝑃PA − 𝑃Aorta

𝑅PDA, (13)

where 𝑃PA and 𝑃Aorta are the pulmonary artery and descend-ing aorta pressures, respectively.

2.5. Oxygen and Carbon Dioxide Diffusion Modeling in theVascular System. Theoxygen diffusion takes place from lungsto the alveolar capillaries and then from the blood capillariesto the body tissues. At steady state, the oxygen release anduptake will be equal, and hence we need to model only onediffusion process to find the partial pressures. The processof oxygen diffusion modeling is the same as what has beendone in [20] but carbon dioxide calculation is added inour model. The reason for including CO

2diffusion to the

model is that our previous findings have suggested bloodCO2

concentration to be an important parameter in prediction ofthe PVL occurrence.

On the whole, the oxygen uptake and release equationscan be written as was done in [25]. Consider

𝑄𝑝(𝐶pvO

− 𝐶𝑎O2

) = SVO2,

𝑄𝑠(𝐶𝑎O2

− 𝐶VO2

) = CVO2,

SVO2= CVO

where SVO2is the oxygen uptake rate in the lungs expressed

in mL/min, and it is specified as an input to the system,CVO2is the whole body oxygen consumption, 𝐶pvO

isoxygen concentration in the pulmonary vein, 𝐶

𝑎O2

is oxygenconcentration in the aorta, and 𝐶VO

is oxygen concentrationin the veins. Since the PDA concentration of oxygen in aortaand pulmonary artery is equal, we simply replace oxygencontent in the pulmonary artery with its value in the aorta inthe second equation. It should be noted that this assumptionis only true for the MA (atresia) patients.

The term 𝑄𝑝is the average pulmonary flow obtained by

taking an average of 𝑄ASD (atrial septal defect flows) overmultiple cardiac cycles. There are two reasons. Firstly, understeady state conditions, all the pulmonary venous return flowpasses through the ASD and pulmonary venous flow is equalto pulmonary flow. Hence, on average atrial septal defect flowand pulmonary flow are equal. Secondly, a large fraction of(ideally half) the pulmonary artery flow goes to aorta, andhence, to find the average pulmonary flow, it is more reliableto calculate 𝑄ASD. Similarly, 𝑄

𝑆is the average systemic flow

obtained by taking an average of PDA over the heart cycles.Oxygen exchange in alveoli can be modeled using Hill’s

approach [26]. Oxygen in the blood is determined by theamount of oxygenated hemoglobin (98%) and the oxygendissolved in blood (2%). So, the total volume of blood oxygenconcentration (𝐶

𝑏O2

) depends on the saturated blood oxygen(𝑆𝑏O2

) and the partial pressure of blood oxygen (𝑃𝑏O2

) bythe model from [16]; this is shown in (15). The term 𝛽 isthe concentration of O

2in the blood hemoglobin at 100%

saturation and the term 𝛾 is concentration of dissolved O2.

𝛽 and 𝛾 have units of mL⋅mL−1mmHg−1 and the unit ofconcentration is mL/mL. One has

𝐶𝑏O2

= 𝛽 ⋅ 𝑆𝑏O2

+ 𝛾 ⋅ 𝑃𝑏O2

. (15)

Differentiating (15), we get the following:𝜕𝐶𝑏O2

𝜕𝑡= 𝛽 ⋅

𝜕𝑆𝑏O2

𝜕𝑡+ 𝛾 ⋅

𝜕𝑃𝑏O2

𝜕𝑡. (16)

The relationship between oxygenized hemoglobin anddissolved oxygen is given by the oxygen dissociation curve:

HbO2= 𝑆𝑏O2

(𝑃𝑏O2

/𝑝50)𝑛

1 + (𝑃𝑏O2

/𝑝50)𝑛, (17)

where HbO2is the oxygenized hemoglobin, 𝑝50 is the partial

pressure of oxygen at 50% saturation (𝑆𝑏O2

= 0.5), and 𝑛

is a constant. Differentiating the above equation, we get thefollowing:

𝜕𝑆𝑏O2

𝜕𝑡= (

𝜕𝑆𝑏O2

𝜕𝑃𝑏O2

𝜕𝑡) , (18)

𝜕𝑆𝑏O2

𝜕𝑃𝑏O2

𝑝50

𝑛 × (𝑃𝑏O2

/𝑝50)𝑛−1

(1 + (𝑃𝑏O2

/𝑝50)𝑛

)2. (19)

By combining the above equations, we have the following:

𝜕𝐶𝑏O2

𝜕𝑡=

𝜕𝑃𝑏O2

𝜕𝑡(𝛾 + 𝛽(

𝜕𝑆𝑏O2

𝜕𝑃𝑏O2

)) . (20)

Assuming that oxygen in the capillaries of the lungs is at aconstant partial pressure of 150mmHg and that there is acontinuous oxygen transfer from the lungs to the pulmonaryblood capillaries, if we take the volume in the capillaries asa single unit with a volume 𝑉cap, then the time available forthe transfer of oxygen to blood capillaries can be assumed tobe the time taken for the pulmonary flow to fill the capillaryvolume, 𝑉cap. One has

𝑡 = (

𝑉cap

𝑄𝑝

) , (21)

where𝑄𝑝is the pulmonary flow rate. At any point of time, the

diffusion flux from the lungs to the blood capillaries is givenby the following:

𝐷O2

= 𝐷𝐿O2

(𝑃alv𝑂2

− 𝑃𝑏𝑂2

) , (22)

where 𝐷𝐿O2

is the total lung diffusion capacity for oxygen.The diffusion flux of oxygen 𝐷O

will lead to a change inthe concentration of the volume in the blood capillaries, andhence we could write

𝐷O2

= 𝑉cap𝜕𝐶𝑏O2

𝜕𝑡. (23)

Thus, we have the following:

𝑉cap𝜕𝐶𝑏O2

𝜕𝑡= 𝐷𝐿O2

(𝑃alvO2

− 𝑃𝑏O2

) . (24)

Substituting for 𝜕𝐶𝑏O2

/𝜕𝑡 we get

𝑉cap𝜕𝑃𝑏O2

𝜕𝑡(𝛾 + 𝛽(

𝜕𝑆𝑏O2

𝜕𝑃𝑏O2

)) = 𝐷𝐿O2

(𝑃alvO2

− 𝑃𝑏O2

) . (25)

Hence, at any instant, the change in the partial pressureof oxygen in blood capillaries by diffusion is given by thefollowing:

𝜕𝑃𝑏O2

𝜕𝑡=

𝐷𝐿O2

(𝑃alvO2

− 𝑃𝑏O2

𝑉cap (𝛾 + 𝛽 (𝜕𝑆𝑏O2

/𝜕𝑃𝑏O2

. (26)

Integrating this equation over the limits, by using the numer-ical trapezoidal integration method with a time step of 0.01 s,

𝑃pvO2

𝑃𝑎O2

(𝛾 + 𝛽 (𝜕𝑆𝑏O2

/𝜕𝑃𝑏O2

(𝑃alvO2

− 𝑃𝑏O2

𝜕𝑃𝑏O2

𝐷𝐿O2

𝑉cap∫

𝑉cap/𝑄𝑝

𝜕𝑡,

where𝑃pvO2

is the partial pressure of oxygen in the pulmonaryveins. Since this is a complex function of 𝑃

𝑏𝑂2

, integration iscarried out numerically. An estimated value of 𝑃

𝑎𝑂2

is usedas an initial guess and the value of 𝑃pvO

is calculated fromthe integral. If (27) is not satisfied, an improved value of𝑃𝑎O2

is chosen and integration is repeated. This procedureis continued until (27) is satisfied to the required level ofaccuracy. The values of 𝑃

𝑎O2

and 𝑃pvO2

are checked to satisfythe integral equation and if the left hand side of (27) turnsout to be less (or more) than the right hand side, then theintegration is carried out again with a lower (or higher) value.This process is repeated until the equation is satisfied to therequired level of accuracy. Using the final value of 𝑃

𝑎O2

andoxygen release equation, the value of 𝑃VO

is calculated.The exchange modeling for CO

2has been implemented

using the same approach as we used for oxygen. The onlydifference is that the CO

2is released in the lungs and it is

taken up by the blood from the body tissues. Hence, we have

𝑄𝑝(𝐶𝑎CO2

− 𝐶pvCO2

) = SVCO2,

𝑄𝑠(𝐶VCO

− 𝐶𝑎CO2

) = CVCO2,

SVCO2= CVCO

where SVCO2is the CO

2release rate in the lungs expressed in

mL/min, and it is specified as an input to the system, CVCO2

is the whole body CO2release rate, 𝐶pvCO

is CO2con-

centration in the pulmonary vein, 𝐶𝑎CO2

is CO2concentra-

tion in the aorta, and𝐶VCO2

is CO2concentration in the veins.

Referring to [17, 18] for the relationship of concentrationof CO

2to its partial pressure at a temperature of 37∘C and

pH =7.4 andneglecting the effect of oxygen saturation (whichis reasonable), we get

𝐶𝑏CO2

= 0.009083𝑃𝑏CO2

, (29)

𝜕𝐶𝑏CO2

𝜕𝑡= 0.009083

𝜕𝑃𝑏CO2

𝜕𝑡, (30)

where 𝐶𝑏CO2

is the blood CO2concentration in mL/mL. The

diffusion equation for CO2becomes

𝑉cap𝜕𝐶𝑏CO2

𝜕𝑡= 𝐷𝐿CO2

(𝑃alvCO2

− 𝑃𝑏CO2

) , (31)

where 𝐷𝐿CO2

is diffusion capacity of lung for CO2which is

known and 𝑃alvCO2

is alveolar pressure of CO2, which is set to

41mmHg. Using (30) and (31), and after integration, we have

𝑃pvCO2

𝑃𝑎CO2

0.009083

(𝑃alvCO2

− 𝑃𝑏CO2

𝜕𝑃𝑏O2

𝐷𝐿CO2

𝑉cap∫

𝑉cap/𝑄𝑝

𝜕𝑡. (32)

Equation (32) is numerically integrated with an initial valueof 𝑃𝑎CO2

such that the value of 𝑃pvCO2

satisfies carbon dioxiderelease equation in the lungs. It is iterated until the equationis satisfied with a certain level of accuracy. Again, using thevalue of𝑃

𝑎CO2

and using the equation of carbon dioxide takenup by the blood, we get a solution for 𝑃pvCO

2.6. Parameter Values. The parameter values for the caseof a newborn HLHS patient were taken from a child afterstage 1 circulation with a neoaorta and a Blalock-Taussigshunt replacing the PDA [20]. Since the first step Norwoodprocedure is applied for a child in his/her early few weeks,the parameter values for these infants would be comparable;these values are listed in Tables 1–3. It should be noted that𝑅ASD,𝑅PDA, and𝑅PA are the only variableswhich change post-operatively; we performan analysis by varying the parametersto include a wide range of patients.

The value of the resistance for the pulmonary arteries isalmost a hundred times that which is used in [20]. This isan estimated value to show very severe conditions. This isbecause, for a newborn baby, it is well known that pulmonaryresistance is very high, falls precipitously after the first breath,and continues to fall over the first few weeks of his life.The average body surface area (BSA) for an adult male isaround 1.5m2 and the average surface area for a newbornchild is 0.30m2. Hence a scaling factor of five can be assumedwhile estimating some of the parameters for a child fromthe parameters of an adult. Although this method is widelyused by clinicians in estimating the parameters, it brings someuncertainty to the model; hence, a parametric analysis hasbeen carried out in order to understand the effect of thosevariables on the outputs.

The oxygen consumption rate in an average adult is300mL/min [24] and hence the average oxygen consump-tion rate in a child can be assumed to be approximately60mL/min. The total lung diffusion capacities for O

CO2are 62mL/min (mmHg)−1 and 478mL/min (mmHg)−1,

respectively, for an adult. Considering the scaling factor of 5and the fact that the diffusion capacity is inversely propor-tional to the square of the surface area, we have a diffusioncapacity for a child for O

2and CO

2as 2.5mL/min (mmHg)−1

and 19mL/min (mmHg)−1, respectively. Average CO2release

rate by an average adult is 400mL/min, and hence the averagerelease rate for a child is 80mL/min.

2.7. Numerical Methods and Algorithms. The simulations andmodeling presented in this paper have been done usingMatlab. We used the trapezoidal integration method with atime step of 0.01 s for numerical integration. The numericaldifferentiation for solving ordinary differential equationspresenting the HLHS model has been done using explicit

Table 2: Circulation parameters.

Parameters ValuesPulmonary circulation

PA𝑅 2mmHg⋅s⋅mL−1

𝐿 0.00412mmHg⋅s2⋅mL−1

𝐶 0.27410mL/mmHgPAB

𝑅 0.41688mmHg⋅s⋅mL−1

𝐶 0.04078mL/mmHgPVB

𝐿 0.00218mmHg⋅s2⋅mL−1

𝐶 0.88750mL/mmHgSystemic circulation

DA𝑅 0.19883mmHg⋅s⋅mL−1

𝐿 0.00287mmHg⋅s2⋅mL−1

𝐶 0.06118mL/mmHgSAB

𝐿 0.00535mmHg⋅s2⋅mL−1

𝐶 0.44296mL/mmHgSVB

𝐶 0.31030mL/mmHgSLV

𝐶 4.07890mL/mmHgPatent ductus arteriosus

𝑅 1mmHg⋅s⋅mL−1

Table 3: Blood gas diffusion parameters.

Parameters ValuesOxygen

𝐷𝐿O2

180mL/minΓ 0.00003 (mL⋅mL−1mmHg−1)𝐵 0.2213 (mL⋅mL−1mmHg−1)𝑁 2.7𝑃avO2

100mmHgCarbon dioxide

𝐷𝐿CO2

450mL/min𝑉cap 3.5mL𝑃avCO2

41mmHg

fourth-order Runge-Kutta method [27]. In our simulationswe have set the time step and time span to 0.01 s and 10 s,respectively. We have then ignored the first few cycles todrop the transients and to achieve steady-state solution. Theinitial condition has been set to 0.1 of the respective unitsfor the pressures and flows. The computer used to run thesimulations has a 32GB of RAM and an Intel Xeon X5680

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.50

Time (s)

PRAPPAB

Figure 5: Pressures in right atrium, pulmonary artery, and pul-monary venous bed. As it is expected in the case of HLHS babies,since blood is pumped from right ventricle to the whole body,pressures in pulmonary arterial bed for the HLHS babies are higherthan those for normal babies.

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.50

Time (s)

PDAPPA

Figure 6: Pressures in right ventricle, descending aorta, andpulmonary artery. The heart rate is assumed to be 160 bpm basedon the average HLHS patient data.

processor. The computation time to run the simulations for10 s was 6.19 s.

3. Results

Numerical results were obtained using the parameter valuesas discussed above. Typical important pressure curves forvarious points of the body are shown in Figures 5 and 6.The pressures depicted by the model in Figures 5 and 6

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5−50

Time (s)

Figure 7: Flow rates in aorta and pulmonary artery. The plots areshown for 𝑅ASD and 𝑅PDA equal to 1mmHg⋅s⋅mL−1. The neonatalsituation shown in this figure is dangerous, because most of the flowgoes to the pulmonary artery and there is not enough flow going tothe vital organs.

10 15 20 25 30 35 40 45 50 55 600

Volume (cm3)

Figure 8: Pressure-volume loop. 𝑅ASD and 𝑅PDA are set to0.1mmHg⋅s⋅mL−1 and HR is 160 bpm.

are consistent with the atrial, ventricular, and pulmonarypressures of the HLHS heart. The pressure values of thedifferent sections of the heart through the two contractioncycles are consistent with a heart that has a blockage/defectof the left ventricle as in patients with hypoplastic leftheart syndrome, where the right ventricle is responsible forcirculating blood to the lungs as well as throughout the body.

The variations of flow rate in the aorta and the pulmonaryartery are shown in Figure 7. Another very important curveis the 𝑃-𝑉 loop for the right ventricle, which is shown inFigure 8. There are no discontinuities or nonsmoothness

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.50

Time (s)

Figure 9: Flow rate in PDA for 𝑅PDA is equal to 10mmHg⋅s⋅mL−1.

in the figure since all the relationships that are used forcalculating the pressures and the volumes are exponential.Note that the heart rate in this paper is assumed to be 160 bpmwhich is calculated from the mean values of patient data.Theend-diastolic and systolic volumes for the right ventricle are49.9 and 18.2 cm3, respectively, with an ejection fraction of63.5%.

Figure 9 displays the PDA flow rate. Increasing theresistance decreases the flow rate as might be expected.

3.1. Parametric Analysis. Weperformed a parametric analysisto investigate the effect of some clinically important param-eters on the model performance and also to provide a wayto compare different stages in the HLHS baby’s life. Whena baby is born, the PDA is fully open; hence 𝑅PDA is low.When the baby takes his first breath, pulmonary resistancefalls sharply in the first few hours but still remains at a highvalue for the first few days of life as it slowly drops down tonormal. To model this phenomenon, we assume 𝑅PDA = 0.1

and 𝑅PA = 2mmHg⋅s⋅mL−1; the resulting pressure curves forthe different parts are as shown in Figure 10.

As can be seen in Figure 10, aortic pressures are in thenormal range of 75–115mmHg and the pulmonary pressuresare also normal. But as the baby ages by 5–10 days, the PDAbegins to close and hence 𝑅PDA increases.

As the pulmonary resistance further drops to 𝑅PA =

0.2mmHg⋅s⋅mL−1, less flow crosses PDA to aorta. In thissituation, since the pulmonary pressure is higher than theaortic pressure, there would be more systemic flow thanpulmonary flow. The pressure curves for this stage areshown in Figure 11. In this situation, the condition becomesextremely dangerous. All the flow goes to the lungs and thereis a reverse pressure gradient across the PDA such that thepressure in the PA is less than the pressure in the aortic archand there is no flow going to the vital organs. Pressure buildsin the RA because of the extra small pulmonary flow anddeath ensues.

Norwood procedure is performed to create a neoaortafrom the common pulmonary artery as the cardiac outflow

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.50

Time (s)

PAOPPA

Figure 10: Pulmonary artery and aorta pressure just after birth inHLHS baby with 𝑅PDA = 0.1 and 𝑅PA = 2mmHg⋅s⋅mL−1.

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.50

Time (s)

PPAPAO

Figure 11: Pulmonary artery and aorta and right ventricle pressurebefore Norwood circulation procedure.

track. A synthetic shunt of constant diameter is placed tostabilize the pulmonary blood flow and the PDA is legated.

The constant diameter of the shunt limits the amount offlow to the low resistance PA, thereby stabilizing 𝑄𝑝/𝑄𝑠, amodel which has been presented in [20].

Three parameters,𝑅ASD, 𝑅PA, and 𝑅PDA, were chosen andwere varied over a range of physiological resistance, and theresulting changes in the hemodynamic parameters were ana-lyzed. Varying ASD resistance (𝑅ASD) corresponds to varyingmagnitudes of the septal defect. Varying the pulmonaryartery resistance (𝑅PA) corresponds to the increasing age ofthe baby. Finally, the last parameter varied is the resistance

0 1 2 3 4 5 6 7 8 9 101

RPDA (mmHg/(mL/s))RASD = 0.01RASD = 4

RASD = 10

Figure 12: Systemic oxygen delivery. Oxygen delivery is calculatedby multiplication of system flow and arterial oxygen concentration.The results are consistent with what is presented in [20].

within the PDA (𝑅PDA). This parameter can be controlled byan infusion of prostaglandin to prevent PDA closure.

Two parameters are varied in each plot to study the effectsof the parametric variation on systemic hemodynamics.The graphs of important hemodynamic parameters such ascardiac output, pulmonary to systemic flow ratio, and oxygenand carbon dioxide partial pressures are presented (Figures12–15). Systemic oxygen delivery as a function of increasing𝑅PDA is shown in Figure 12.We can observe from this plot thatoxygen delivery will decrease with increase of both 𝑅ASD and𝑅PDA.Although this is a fairly obvious fact, the interesting find-ing is that in the case of very high 𝑅ASD the oxygen deliveryis not monotonically decreasing, and it has a maximum forsome intermediate values of 𝑅PDA. We can infer that in thecase of neonates with very high 𝑅ASD an increase in 𝑅PDA tosome extent could be helpful.

As can be observed from Figures 13(a) and 13(b), thepulmonary-to-systemic flow ratio (𝑄𝑝/𝑄𝑠) changes with thechange in the model parameters. As 𝑅ASD increases, this ratiodecreases because the pulmonary flow keeps on decreasing;this is because, at steady state, all the pulmonary flow passesthrough the ASD, and an increase of 𝑅ASD leads to a decreasein the flow. As 𝑅PDA increases, the systemic flow decreasessince all the systemic flow passes through ductus arteriosuswhich leads to an increase in the 𝑄𝑝/𝑄𝑠 ratio.

A 𝑄𝑝/𝑄𝑠 ratio around one is optimal for the body [28]and hence a suitable area for the flow can be selected fromthese graphs accordingly. Considering the parameters thathave to be changed, such a point can be reached.Thedirectionof blood flow depends upon the resistance, and hence, in thecase of HLHS, blood follows the path of least resistance. Inhigh 𝑅PA resistances in constant 𝑅PDA, the ratio decreasesbecause, due to high pulmonary resistance, the amountof pulmonary flow decreases and systemic flow increases

0 1 2 3 4 5 6 7 8 9 101

RPDA (mmHg/(mL/s))

RPA = 0.1RPA = 0.5

RPA = 1RPA = 2

0 1 2 3 4 5 6 7 8 9 100

RPDA (mmHg/(mL/s))

RASD = 0.1

RASD = 1

RASD = 2

RASD = 10

Figure 13: Variation of 𝑄𝑝/𝑄𝑠 by changing the most important model parameters. Values around one are normal.

0 1 2 3 4 5 6 7 8 9 1079

RPA = 2RPA = 0.1RPA = 0.5

RPDA (mmHg/(mL/s))

0 1 2 3 4 5 6 7 8 9 1030

RASD = 0.1RASD = 1

RASD = 10

RPDA (mmHg/(mL/s))

Figure 14: Variation of arterial oxygen saturation by changing the most important model parameters.

leading to a decrease in the 𝑄𝑝/𝑄𝑠 ratio. Plots show thatin some combination of PA, ASD, and PDA resistances theoptimal 𝑄𝑝/𝑄𝑠 can never be achieved. For example, if 𝑅PA =

0.1mmHg⋅s⋅mL−1, the optimal 𝑄𝑝/𝑄𝑠 ratio is unreachable.As shown in Figures 14(a) and 14(b), as 𝑅PDA increases,

the arterial O2saturation also increases and becomes nearly

constant at high values of the resistance; however, it decreasesmonotonously with 𝑅ASD. The decrease with 𝑅ASD becomesmore profound at lower values of 𝑅PDA. This trend is alsoobserved with 𝑅PA. For constant 𝑅PA, systemic arterial O

concentration increases with increasing 𝑅PDA.

As𝑅PDA increases, cardiac output (sumof pulmonary andsystemic flows) decreases slowly as the systemic flow keepson decreasing and this counters the effect of the increasingpulmonary flow as shown in Figures 15(a) and 15(b). As 𝑅ASDincreases, cardiac output decreases significantly due to a rapidreduction in the pulmonary flow. Hence, it can be deducedthat the sensitivity of the pulmonary flow to𝑅ASD is very high.A similar trend can be observed with 𝑅PA playing the samerole as that of 𝑅ASD.

Figure 16 shows a plot of 𝑃CO2

(partial pressure of carbondioxide) for constant 𝑅PA = 0.1mmHg⋅s⋅mL−1 and varying

0 1 2 3 4 5 6 7 8 9 1025

RPDA (mmHg/(mL/s))

RPA = 0.1RPA = 2

RPA = 8

0 1 2 3 4 5 6 7 8 9 1020

RPDA (mmHg/(mL/s))

RASD = 0.1RASD = 1

RASD = 10

Figure 15: Variation of cardiac output by changing the 𝑅ASD, 𝑅PA, and 𝑅PDA.

0 1 2 3 4 5 6 7 8 9 1010

RPDA (mmHg/(mL/s))

RASD = 0.1RASD = 1

RASD = 4RASD = 10

Figure 16: Variation of 𝑃CO2 for constant 𝑅PA = 0.1mmHg⋅s⋅mL−1and changing 𝑅PDA and 𝑅ASD. Previous finding underlined theimportance of 𝑃CO2 as a risk factor for the occurrence of PVL.

values of𝑅PDA and𝑅ASD.We focus on𝑃CO2

since our previousstudies based on computational intelligence (CI) techniqueshave shown that 43mmHg𝑃CO

is an important threshold forpredicting PVL. By inclusion of𝑃CO

in ourmodel we attemptto find a reason for this interesting but somewhat inexplicableresult of the CI technique. Our results show that all values of𝑃CO2

are less than 43mmHg for 𝑅PA = 0.1mmHg⋅s⋅mL−1 and𝑅ASD ≥ 4mmHg⋅s⋅mL−1. Considering the fact that 𝑃CO

tosome extent is measureable and controllable post- and pre-operatively, this finding could well lead to a valuable insight.

4. Conclusion

The goal of this paper has been to develop a lumpedparameter model of HLHS, a fairly common congenital heartdisease. The need for this physical modeling comes from ourultimate goal to predict and prevent one of the consequencesof this abnormal circulation, namely, a form of brain injurycalled periventricular leukomalacia (PVL).

The exact causes of PVL are not well understood and inthis paper our focus has been on understanding how somecritical parameters such as flow resistances at the atrial septaldefect or patent ductus arteriosus might alter systemic flowand systemic oxygen delivery, which could be reasonablyexpected to have a causal relationship to PVL occurrence.

In this paper, we have compared the results of HLHScirculation (preoperative condition) to Norwood circulation(postoperative condition). Results show the manner andextent of changes in the pressure and blood flow in differentparts of the body. In reporting the modeling results, we havemostly focused on the pulmonary to systemic flow ratio(𝑄𝑝/𝑄𝑠), O

2delivery, and O

2saturation because we believe

that these parameters are important risk parameters of thePVL occurrence. By use of the model, the following pointswere demonstrated or confirmed.

Different sizes of the PDA and the ASD and differentresistances of the PDA and the ASD will result in changingthe regulation of pulmonary and systemic flow and hencethese resistances play a critical part in our goal to predict theoccurrence of the PVL.

The direction of blood flow depends upon the PA resis-tance and that of the PDA resistance. Increase in the PDAresistance leads to blood flowing where the resistance isthe least. This results in an increased 𝑄𝑝/𝑄𝑠 ratio. Withan increase in the PDA resistance a greater portion of thecardiac output goes into the lungs.This will lead to a systemic

hypoperfusion and poor O2delivery. This result is consistent

with the results presented in [29].By comparing the slope in cardiac output and pulmonary

to systemic flow variation plots, one could observe that therates of change (slopes) are higher for variation of PA resist-ance in comparisonwith the variation of ASD resistance.Thismeans that the PA resistance is more influential on the COand pulmonary to systemic flow ratio than the ASD resist-ance. This suggests that the PA resistance is more directlyinvolved in the determination of the 𝑄𝑝/𝑄𝑠 ratio and thecardiac output.

The optimal O2delivery is achieved when balanced

pulmonary and systemic perfusion is established, namely,when 𝑄𝑝/𝑄𝑠 is around 1. Our results show that, for somevalues of 𝑅PA and 𝑅ASD, the optimal 𝑄𝑝/𝑄𝑠 can never beachieved. Since insufficient O

2delivery is considered to be a

potential cause for PVL, it seems reasonable to infer that thesevalues of 𝑅ASD and 𝑅PA lead to high risks of PVL.

It is known that the correlation between venous O2

saturation and O2delivery is better than the correlation

between arterial O2saturation and O

2delivery. However,

since measurement of the mixed venous saturation in clinicalpractice remains difficult, we just considered the case ofarterial oxygen saturation [30] in this study.

The developed model also has some other interestingapplications. For example, with some minor modificationsto cover the partly underdeveloped left heart, the proposedmodel could form a good foundation for the analysis ofnewborns with underdeveloped left ventricles and modelbased design of left ventricular assist devices.

Given that the exact causes of PVL are still not clearlyunderstood, the main goal of this paper has been to developa sound mathematical model to investigate HLHS, a severecongenital deformation which has the highest correlationwith the occurrence of PVL as suggested by preliminarypostoperative data collected from Children’s Hospital ofPhiladelphia (CHOP). Also, the paper aims to provide atool to better understand the main factors that drive theHLHS physiology by a suitable variation of physiologicalparameters.

Our previous computational intelligence based findings[5, 7, 10] suggest some important factors in the occurrenceof PVL; however, since the exact physiological reasons andcauses of those findings are not clear, the current model hasbeen developed to discover cases that will lead to the PVLsituation and to analyze them. We hope that the developedmodel could open the door for further investigation of theHLHS syndrome and also to its connection with PVL. Thepostoperative results are consistent with what is reported in[20] which makes this model potentially valuable for clinicaluse.

Some of the limitations of the current work are as follows.

(i) The effect of respiration and regulatory mechanismssuch as HR baroreflex and total peripheral resistancebaroreflex on the cardiopulmonary performance isnot considered in building the model.

(ii) The values of the components of the analog electricalcircuit are estimated and there is still no direct,

reliable, and safe way to measure them with highconfidence.

(iii) The results of themodel should be validated clinically.A parameter adaptation algorithm for patient specificmodeling should be developed and implemented onthe patient specific data. The correlation between thePVL occurrence and different values for ASD, PA, andPDA resistances should be calculated for the aim ofPVL occurrence prediction.

To improve the model further and to add more verisimil-itude with reality, it is important to include other factors intothe system such as autoregulation and to investigate its effects.Our current work is continuing on this important problemand will focus on addressing some of the above limitations.

Finally we would like to comment about the generalutility of the procedures developed in this paper. Withall the obvious limitations, the method of constructing amathematical model pursued in this paper is still appealingbecause it permits the change of parameters very easily tounderstand their influence on the usual key clinical outputssuch as 𝑄𝑝/𝑄𝑠. Such models exist for other conditionsbut none exist for PVL based on preoperative conditions.By use of a parametric analysis on computer models westart to acquire the very useful ability to predict pre- andpostoperative conditions for specific patients and also todevelop patient specific models to investigate physiologicalstates of individual patients. That said, we would like toreiterate the limitations of the current model and hope todevelop and inspire development of improved models withappropriate corroboration.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The research reported in this paper is supported by a grantfrom National Institute of Health (no. 1 R01 NS 72338 01A1).

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