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Variation in the Result of the Mitral
Valve Area Calculated through Gorlin's
Formula in Relation to Changes
in Cardiovascular Parameters
Bustamante, John; Vallverdú, Montserrat;
Borrás, Xavier
Cardiovascular Dynamic Group of Pontificia
Bolivariana University and
Cardiovascular Santa Maria Clinic, Colombia Unit of Hemodynamic of
Hospital de la Santa Creu i Sant Pau, Barcelona, Espaņa
Centre de Recerca en Enginyeria Biomedica, Barcelona, Espaņa
SUMMARY
Introduction and Objectives: Determination
of the mitral valvular area using an indirect method, like the Gorlin's formula
(MVA_{G}) or expression that relates the area of the valve to the fall
of transmitral pressure and the flow by the valvular orifice, can be see affected
by certain cardiovascular variables. These alterations become evident by means
of the technique of modeling and simulation.
Methods: With the implementation of a cardiovascular
model in a computer, in which the dynamics of the transmitral flow is modeled,
the isolated influence of diverse parameters and variables are evaluated upon
the result of the MVA_{G}. In order to accomplish this aim, were considered: sanguineous
density, length of valvular tunnel, ventricular compliance, cardiac frequency
and cardiac output.
Results: By means of the sensitivity analysis
(S_{K}) it was found that the sanguineous density (S_{K} 26,2)
is the variable that influences in greater proportion the determination of the
MVA_{G}, follow in its order the cardiac output (S_{K} 11,62), length of
valvular tunnel (S_{K} 11,1), cardiac frequency (S_{K} 4,5)
and compliance (S_{K} 0,35).
Conclusion: It was deduced from the simulation
that the calculation of the MVA_{G} becomes vague in certain hemodynamic situations,
which would be consider for a guessed right calculation.
INTRODUCTION
The valuation of severity of the mitral valve estenosis, as
well as of the success achieved after a therapeutic procedure, is possible by
means of the calculation of the valvular area. This calculation is usually carried
out throughout indirect methods, which have become more necessary due to the
development of techniques of mitral commissurotomy by percutaneous route. One
of the methods includes data collected by means of cardiac catheterization,
and consists of the use of an arithmetical expression that relates the area
of the valve to the transmitral differential pressure and the flow through the
valvular orifice; expression that is known like Gorlin's Formula (1).
The Gorlin's Formula has become a standard method of hemodynamic laboratories to estimate the area of valve orifices in the presence of estenosis; eventhough, it appears to be still many doubts about the precision of the formula in certain clinical conditions, in spite of multiple readjustments that have been made to the values of the mathematical constants that try for to correct the calculated area, considering the losses to the convective component of the flow, as well as the contraction of the sanguineous flow when coming out of the valve.
OBJECTIVES
The current study pretend to evaluate, by means of the simulation
by computer, the isolated influence of diverse parameters and hemodynamic variables
on the result of the Mitral Valve Area Calculated Through Gorlin's Formula (MVA_{G}).
For this, a mathematical model of the Cardiovascular System (CS) must be developed
and implemented in a digital computer. The model must possibility to calculate
the instantaneous values of all the variables considered in the system; this
way, it is possible to analyze, in multiple situations, the collected data when
calculating the MVA_{G} with relation to the wellknown area in the model, real
area (MVA).
MATERIALS AND METHODS
Development of the Model
A model of the CS, as well as of the control of this system
by means of the Autonomic Nervous System (ANS), has been processed; in which
the dynamics of the mitral valve is modeled of detailed way. With it, we looks
for making experiences through the model, since many of the characteristics
of the real system are difficult to evaluate in alive person, due to the complexity
that in the experimental methods imply to analyze in an isolated form the answer
of the system to variations of one or more parameters, or by the danger that
any form would suppose to the subject in study.
It was not considered necessary to carry out simulations of all the vascular segments, due to the specific interest in the transmitral segment, and for that reason the general model could be simplified, becoming a sum of segments within some compartments; so that, although individually not seen each one of these segments, the compartment if it is a suitable representation of the assembly of blood vessel that conform it. The processing of the model was done conserving those compartments related to the variables of defined entrance and exit, and that concerns directly or indirectly to the mitral valve system.
With these considerations, the sanguineous flow through all the cardiovascular network has been described using nine compartments: 1) right auricle RA, 2) right ventricle RV, 3) left auricle LA, 4) left ventricle LV, 5) aorta and other arteries of systemic circulation ACM, 6) cava vein and other veins of systemic circulation VCM, 7) pulmonary artery and other arteries of pulmonary circulation ACP, 8) pulmonary capillary CAP, and 9) veins of pulmonary circulation VCP. The basic equations that describe the hemodynamic component in the nine considered compartments are based on the relation of the variables pressure P(t), volume V(t), sanguineous flow Q(t), and elastancia E, which determine the conditions of the system and characterize each compartimental structure.
The CS is modeled by means of differential equations, starting off from the equations of NavierStokes, and incorporating nonlinear functions (2). The nonlinear elements were considered due to the presence of the venous valves and cardiac valves, as well as to the controllers of the ANS. The heart is modeled both considering the systolic and the diastolic function. The duration of the systolic and diastolic of each cardiac camera is described by means of linear functions of the total cardiac period TH, based on the work of Leaning (3) and the study of validation made with the developed model in this work. The concept of variable elastance in the time E(t) is used to model the contraction and relaxation of the cardiac cavities [auricles E_{A}(t) and ventricles E_{V}(t)], and in this form its action of pumping was modeled.
The blood has a nonlinear behavior, because it is considered a nonNewtonian fluid; nevertheless, the reologic behavior of the blood is practically Newtonian in the physiological conditions of the SC, if phenomena of flow in the microcirculation are not considered, as it is the case. Another consideration to take into account is that the blood behaves like an incompressible fluid, because of the low pressures that it supports in the system, so that the sanguineous volumes corresponding to each modeled compartment are constant, although their distribution and filling are variable in the time by the pulsability of the flow. The sanguineous density is considered for the model and, therefore, the effects of inertia of the blood are considered too.
The
ANS generates regulation signals to the heart and blood vessels, according to
the requirements of the CS, maintaining a suitable sanguineous flow. This system
of control is compound of a measurement element, a unit of processing of the
information and a mechanism to introduce the necessary changes in the parameters
of the hemodynamic system. A representation in block diagram of the content
of the model is in figure
1, where it include the CS and the ANS of control, as well as the
performances of each controller appear on this hemodynamic system.
In this work, the sanguineous density, the length of the valvular tunnel, the
ventricular compliance, the pressure of the left auricle and left ventricle,
the cardiac frequency, the transmitral flow and speed of the flow, were considered
like the entrance variables of the model. The MVA_{G}
was considered like the exit variable of the model.
Figure 1. Diagram of blocks with which it is represented the cardiovascular system and the autonomic nervous system of control. The thoracic and abdominal regions are put under variations of external pressure. 
Implementation of the Model
To simulate the CS the developed mathematical model is implemented
using the Advanced Continuous Simulation Language ACSL platform (4).
A simulation language of general purpose, designed for the study of continuous
systems of concentrated parameters, both linear and nonlinear, described by
means of differentials equations. For it, a computer Sun SP20 was used, under
UNIX atmosphere. The equations are solved within intervals of 3 ms, using the
method of RungeKutta of fourth order.
Validation of the Model
The validation of the model was carried out comparing the
results obtained by means of the simulation to the ones obtained in patients
of the program of percutaneous mitral valvulotomy of the Santa Creu i Sant Pau
Hospital of Barcelona. Where selected those patients who were in sinusal rate,
and those in which the pressures, flows and speeds transmitral registries could
be obtained in an optimal way. For this, the obtained registries of four affected
patients of important mitral valvular estenosis were used, and later they were
practiced valvulotomy by percutaneous via.
With techniques of parametric estimation one determined the values of the unknown parameters of the model, with the purpose of fitting the behavior to each particular patient. During the validation of the model all the values of the output variables had an inferior error to ±10% with respect to the values of the real system obtained by cateterism. These results show that the model was able to reproduce suitably the dynamic behavior of the mitral valvular system, important for the predictive study.
RESULTS
The analysis of Gorlin's Formula behavior was made. The MVA_{G}
is calculated considering different values from hemodynamic parameters in the
model, and conserving the same MVA in 1 cm^{2}.
Ideally, the value calculated by means of the formula would not have to vary
when changing the value of the used parameters, therefore the result would have
to be a horizontal straight line with each point in 1 cm^{2}.
Nevertheless, the calculated area varies in function of these parameters, figure
2.
Figure 2. Relation of valvular area imposed in the model and valvular area calculated with the Gorlin's Formula, when considering different values from hemodynamic parameters in the model. 
With the purpose of evaluating the interference of the parameters related to the determination of the MVA_{G}, an analysis of sensitivity S_{K} was made. This study was carried out considering the nominal values of each of the parameters, and allowed to know the influence of these on variable MVA_{G}. Table 1 shows the results of relative sensitivity S_{K}, obtained when varying each one of the considered parameters.
Table 1. Relative sensitivity analysis of the mitral valvular area. 
From this study of sensitivity, it can be deduced that the sanguineous density is the parameter that influences in greater proportion the determination of MVA_{G}, with a S_{K} of 26.2. It is followed in its order by the cardiac output (S_{K} 11,62), length of valvular tunnel (S_{K} 11,1), cardiac frequency (S_{K} 4,5), and compliance of the ventricle (S_{K} 0,35). It must consider, nevertheless, that while the density usually does not vary more than a 15%, there are other parameters that can vary even in a 100% or more, and like this, it could influence more on the calculation of the area.
CONCLUSIONS
The work has been oriented toward modeling and simulation
of the SC, establishing an analysis of the transmitral flow. The validity of
the model has been demonstrated comparing the results of the simulation, with
the ones obtained in the hemodynamic laboratory in real cases, with excellent
reproductibility.
The evaluation of the results obtained with the percutaneous mitral valvulotomy procedure, demands an objective and quantitative analysis, unfortunately, the evaluation based on Gorlin's Formula, becomes vague in certain situations which become evident by means of the simulation. With the model developed in this work, it has been possible to determine the participation of different cardiovascular parameters considered key in the determination of the mitral valvular area by means of hemodynamic data collected by cateterism, according to what is expressed by Gorlin's Formula.
Extension of the Project: If in the particular case, it were possible to predict with the model the inaccuracy of the data collected through Gorlin's Formula, it would be clear their utility in evaluating new constants that allow to approximate the results obtained by means of the formula with the real values.
REFERENCES
1. Gorlin R, Gorlin SG. Hydraulic formula for calculation of the area of the stenotic mitral valve, other cardiac valves, and central circulatory shunts. Am Heart J 1951; 41:129.
2. Bustamante J. Análisis del flujo transmitral mediante el modelado y la simulación por computador. 1º Edición. Barcelona: universitat Autónoma de Barcelona, 1995, pp. 210.
3. Leaning MS, Pullen HE, Carson ER, Finkelstein L. Modeling a complex biological system: The human cardiovascular system. Trans Inst Meas Control 1983;5:7186.
4. ACSL Library. Advanced continuous simulation language. New York: Mitchell & Gauthier Associates (MGA) inc., 1993.
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