A Prediction of Blood Flow through a Bypass Graft Using Statistical Methods

Introduction

Coronary revascularization belongs among the most frequent cardiosurgery operations in the world as well as in our republic (in the year 2002, 10797 cardiosurgery operations were performed of which 7051 were CABG surgeries) [1]. Perioperative and longterm survival depend on the patency of the graft used and the anastomotic quality. Many authors were concerned with finding characteristics which determine graft patency and anastomotic quality from both a short-term and a long-term point of view. As it turns out, haemodynamical characteristics help to verify anastomotic quality as well as longterm graft patency [2], [3], [4], [5], [6]. Louagie indicates resistance as a dominant characteristic for longterm graft patency. Resistance can be calculated as blood pressure divided by blood flow through a vessel [7]. Although this formula is only a simplification of the reality (blood flow is not a steady flow and blood is not a Newtonian fluid) [8], it is reasonable to think about blood pressure and blood flow as characteristics which determine resistance. Hata [9] confirmed on a set of patients with low free blood flow through an arterial bypass that a left mammary artery (LIMA) is able to adapt its diameter in relation to storage demands of a target coronary bed. It is also known that a flow through a bypass is affected by the competitive flow of a native coronary bed. Furthermore, flow through a bypass is speculated to affect changes of graft patency in relation to a competitive flow and the sensitivity of different kind of bypass grafts (LIMA, RIMA, SVG) [10], [11], [12]. During myocardial revascularization (CABG on a heart bypass machine), various measurements of blood flow through the bypass are taken: free-flow (the bypass is not anastomosed on a target coronary bed yet), before removal of the clamp from the aorta (a coronary bed is stored only by bypass – there is no competitive flow) and after removal of the cross clamp from the aorta (with a competitive flow). Sometimes, a surgeon takes two measurements before removal of the clamp from the aorta: one before removal of clamps from other bypasses and one after removal of these clamps (blood flow through a bypass also depends on collaterals). A flowmeter (produced by the Norwegian company Medi-Stim) is used for the measurement of blood flow during aortocoronary bypass surgery at the Faculty Hospital in Olomouc. This machine works by the so called Transit-Time method. The principle is based on the fact that the time required for the ultrasound to pass through blood is slightly longer when it is passing upstream than downstream [13]. A surgeon can observe perioperatively with the aid of this machine the blood flow curve, the blood flow value, the mean flow, the Pulsatility Index [PI=(max. flow – min. flow)/mean flow] and other characteristics. It is possible to connect to this machine two flow probes, two pressure inputs and two auxiliary devices, for example ECG input. The flowmeter can calculate many characteristics from recorded data, for example the Fast Fourier Analysis for a saved curve (of blood flow, pressure etc.). In a small group of patients (35), mean arterial pressure and blood flow through a bypass (a left mammary artery grafted to the left anterior descending artery – LIMA-LAD) were measured during CABG surgery at the time when the cross clamp is applied to the aorta (blood is flowing to a coronary bed only through a bypass) and also at the time when the clamp is removed (a competitive flow). The aim of this article is to find a statistical model for the prediction of blood flow through a bypass after removal of the cross clamp from the aorta based on the blood flow value with the clamp still in the place on the aorta. When this prediction is good, it would be able to decrease a number of measurements with keeping whole information about an object.

1. Input data and model

1.1 Input data

Data obtained by measurement at the Clinic of Cardiosurgery, Faculty Hospital in Olomouc are organized in a vector of input data ζ = (x ₁,y₁,...,x_n,y_n, ξ₁ ,η₁,... ξ_n, η_n) ' (' signs vector transposition), where

x₁, x₂,..., x_n is blood flow through a bypass LIMA-LAD I,
y₁, y₂,..., y_n is mean arterial blood pressure LIMA-LAD I,
ξ₁, ξ₂,..., ξ_n is blood flow through a bypass LIMA-LAD III,
η₁, η₂,..., η_n is mean arterial blood pressure LIMA-LAD III.

Table 1. Input data.

1.2 Statistical model

The model of the relationship between blood flow and pressure at the time when the clamp is applied to the aorta (LIMA-LAD I) and at the time when the clamp is removed (LIMA-LAD III) can be assumed in a few forms. The simplest possible form is the classical linear regression model.

where x_i and y_i are values of the patient’s blood flow and blood pressure at the time when the cross clamp is applied to the aorta and ν_i,1 and ν_i,2 are predicted values of these parameters after removal of the cross clamp. There are random errors only on the left side in ν_i,1 and ν_i,2 values in this model. Parameters α₁, α₂, β₁₁ , β₁₂, β₂₁, β₂₂ are estimated by the least square method.

where E(ζ) satisfies the conditions:

Note: E(ζ) and Var (ζ) denote the mean value and the variance of the random vector ξ.

2. Prediction of blood flow and pressure

2.1 The classical regression model

Estimations of parameters α₁, α₂, β ₁₁, β₁₂, β₂₁, β₂₂ , which we obtain for our data set with usage of function lm(y~x)(available in software R), are shown in the table 2.

Table 2. Estimated parameters in the classical regression model.

(4)

Table 3. The residuals in the classical regression model.

Fig. 1. Estimated and measured values of blood pressure and blood flow through a bypass after removal of the cross clamp from the aorta in the classical regression model.

2.2 The “Crystal ball” model

We establish predicted values for patient number i as follows:

(5)

where x_i and y_i are measured values of blood flow and blood pressure at the time when the cross clamp is in the place on the aorta and and are predicted values of these parameters after removal of the cross clamp.

An algorithm of the calculation of estimations of parameters β _i, i=1,2,...,6, which is iterative, is presented in the appendix A.1. Now we are testing the efficiency of the suggested algorithm on a data set which was used for the estimate of parameters β_i. We compare estimated values with directly measured values. The points for the initial iteration lie on a border of the data field (i=14, 17, 18). The numbers mean order numbers of observations (patients).

Values of parameters β_i obtained in six iterative steps for the points of initial iteration mentioned above and our data set are shown in the table 4.

Table 4. Estimated parameters in the “Crystal ball” model.

For our data set we obtain following variance matrix and estimation of σ ² (the algorithm is presented in the appendix A.1):

For numerical expression of accuracy of estimates obtained in our model, we show values of the following residuals in the table 5:

(6)

Table 5. The residuals in the “Crystal ball” model.

In the figure 2, there are measured (black) and estimated (bold red) values of blood flow and blood pressure after removal of the cross clamp from the aorta.

Fig. 2. Predicted and measured values of blood pressure and blood flow through a bypass after removal of the cross clamp in the “Crystal ball” model.

Already from the picture and also from the values of residuals we can see that the “Crystal ball” model, which contains more parameters and this fact leads to better approximation of the measured values by our estimates, describes reality better than the classical linear regression model.

2.3 Location of outlier points in the “Crystal ball” model

In this section, we try to locate outlier observations in the “Crystal ball” model. Thereafter, we will exclude the outliers from the data set used in calculation of parameters and study changes in the model.

We find the indexes for which the expressions (35) are greater than or equal to 1.96 (or approximately 2). The points of these indexes are our suspect outliers.

For our data, set we have found as outliers the points with indexes 2, 9, 13, 14, 25, as you can see in the table 6.

Table 6. Suspect outliers in the “Crystal ball” model.

After exclusion of our suspect points from calculation of parameters in the “Crystal ball” model, we obtain the results shown in the table 7.

Table 7. Estimated parameters after exclusion of the suspect outliers.

Fig. 3. Predicted and measured values of blood pressure and blood flow through a bypass after removal of the cross clamp from the aorta after exclusion of suspect outliers.

Already in the figure 3, where estimates of blood flow are marked in bold red and blood pressure and in black, we can see that the residuals got lower after the exclusion of suspect outliers. Also, the table of residuals confirms this statement.

Table 8. Residuals after exclusion of the suspect outliers.

2.4 Location of leverage points in the “Crystal ball” model

We want to find the maximum of absolute values of derivation of the terms (38) and (39), i.e. maximum of:

(7)

For our data set without the exclusion of suspected outliers, we obtain by calculation as a leverage point for estimates of increments δμ, δν and also for parameters β the point with index 2. After the exclusion of suspected outliers, we obtain as leverage point for estimates of increments δμ and δν the point with index 17 and for estimate of increments of parameters β the point with index 30.

Conclusions

If we use the classical linear model for estimate of blood pressure and blood flow after removal of the cross clamp from the aorta during CABG surgery, we obtain less accurate estimates than when we use the “Crystal ball” model. Moreover, when we exclude suspected outliers our estimates are even more accurate. This statement leads to the hypothesis that patients are divided into several groups and with our algorithm we can calculate certain values of parameters for one group and generally different values for other groups. However, inclusion of patients in the wrong group may be manifested as outliers. To confirm this hypothesis we must complete a deeper analysis of a larger group of patients and include in the surveyed parameters those which can affect blood flow through a bypass before and after removal of the cross clamp from the aorta in the way that they define the above mentioned groups. We can use for example the percentage of stenosis in the target coronary bed, the ejection fraction, FFT ratio, the type of bypass used (LIMA, RIMA, SVG), time of dilatation, the bypass as pedicle/scelet and others [2], [3], [12].

In this orientation period of research a very simplified structure of covariance matrix was used. In further research the covariation matrix, which respects differences in dispersions of the measure of blood flow and the measure of blood pressure must be studied.

Appendix The “Crystal ball” model

The model of the relationship between blood flow and pressure at the time when the clamp is applied to the aorta (LIMA-LAD I) and at the time when the clamp is removed (LIMA-LAD III) we assume in this form (for notation see section 1.1):

(8)

where E(ζ) satisfies the conditions:

(9)

Note: E(ζ) and Var(ζ) denote the mean value and the variance of the random vector ζ.

A.1 Estimate of parameters of the model

After linearization we get the model of a direct incomplete measurement of a vector parameter:

with constraints

(11)

where

(12)

The matrix B₁ of the type 2nx4n is (a vertical line separates first 2n columns):

(13)

The matrix B₂ of the type 2nx6 is:

(14)

Estimations of parameters δμ_i,j, δν_i,j , i=1,2,...,n, j=1,2 and δβ_i, i=1,2,...,6 are obtained by minimization of function:

(15)

with satisfaction of the constraints (11).

After tedious calculation we get the result in the form:

(16)

(17)

where

(18)

This result is considered as the result of the first iteration step, i.e.:

(19)

Following vectors are used in the next iteration step

The resulting estimate can be written as follows:

(20)

where , , i=1,2,...,n, j=1,2 and , i=1,2,...,6 are estimates of parameters ν and β from the previous iteration.

The vector Z^(k) in the k-th iteration step is:

(21)

In the iterative process we correct the estimates of the parameters ν with these relationships:

(22)

For the initial (zero) iteration we choose

(23)

and parameters ν_i,j⁽⁰⁾, i=1,2,...,n, j=1,2 and β_i⁽⁰⁾, i=1,2,...,6 we calculate from equations for three points (in the xy plane), which lay at borders of points field, i.e. for certain coordinates:

For these points we solve the system of equations:

(24)

Parameters β_i⁽⁰⁾, i=1,2,...,6 are the solution of this system and they are used for the calculation of the initial iteration of parameters ν_i,j⁽⁰⁾, i=1,2,...,n, j=1,2 in following equations:

(25)

where parameters μ_i,j⁽⁰⁾, i=1,2,...,n, j=1,2 are set up in this way:

(26)

The iterative calculation is finished when estimates of the increments δμ _i,j, δν_i,j, i=1,2,...,n, j=1,2 and δβ _i, i=1,2,...,6 can be omitted, i.e. on the following condition:

(27)

where ε is a preset constant.

Estimates of parameters β_i obtained in six iterative steps (ε=0,1) and our data set are shown in the table 4.

An estimator of a covariance matrix for these parameters is calculated by following relationship:

(28)

where

(29)

whereas

64 = number of measurements(140)+number of constraints(70)–number of parameters(140+6)

and

(30)

For our data set we obtain following numerical results:

A.2 Location of outlier points in the “Crystal ball” model

When the iteration is stopped (k-th step), for our vector of residuals Z ^(k) from the equation (21), the following relationship is certainly valid :

(31)

(32)

Thereafter, when we denote

(33)

we can write

(34)

For every i=1,2,...,2n we calculate these expressions:

(35)

A.3 Location of leverage points in the “Crystal ball” model

When the iteration process is stopped (k-th step) we use equations for increments

(36)

(37)

which we can write as follows:

(38)

(39)

Bibliography