Semi-Automatic Algorithms for Estimation and Tracking of AP-Diameter of the IVC in Ultrasound Images

Acutely ill patients presenting with conditions such as sepsis, trauma, and congestive heart failure require judicious resuscitation in order to achieve and maintain optimal circulating blood volume. Increasingly, emergency and critical care physicians are using portable ultrasound to approximate the temporal changes of the anterior–posterior (AP)-diameter of the inferior vena cava (IVC) in order to guide fluid administration or removal. This paper proposes semi-automatic active ellipse and rectangle algorithms capable of improved and quantified measurement of the AP-diameter. The proposed algorithms are compared to manual measurement and a previously published active circle model. Results demonstrate that the rectangle model outperforms both active circle and ellipse irrespective of IVC shape and closely approximates tedious expert assessment.

definition of AP-diameter. Clinicians measures the AP-diameter as the largest vertical diameter in the 48 IVC and obviously, if the CSA of the IVC is deviated from the horizontal angle, the circle diameter 49 slightly defers from the clinically measure AP-diameter. 50 In this paper, we propose two algorithms based on ellipse and rectangle models. The height of a thin 51 rectangle fitted inside the IVC can efficiently model its clinically measured AP-diameter. We also 52 develop another algorithm based on ellipse fitting just for comparison purpose. 53 The remainder of this paper is organized as follows -Section 2 discusses the background and related 54 work. The proposed active rectangle and active ellipse algorithms are presented in Section 3 while 55 experimental results are in Section 4, the results are discussed in Section 5 and the paper is concluded 56 in Section 6. In [23], authors showed that the AP-diameter of the IVC can be accurately modeled with the diameter of a circle fitted in inside the IVC. The active circle algorithm proposed in [23] is based on the following evolution functional: where u and v are the mean of the intensities for the pixels inside and outside the contour, respectively, and I is the intensity of the pixels on the contour. This functional is used to evolve the parameters of the circle, i.e., R as the circle radius and (x c , y c ) as its center coordinates. For this, the circle is sampled at K points with polar angles θ k = 2kπ N , k = 0, 1, ..., K − 1, where the normal vector and the Cartesian coordinates corresponding to the kth sampled point are denoted as and respectively. The evolution functional generates forces along the normal vectors. The forces shift the sampled contour points to new positions governed by where f k is the value of the evolution functional at kth contour point. It is shown that the average evolution functional approaches zero when the contour points are on the IVC boundary. In active circle algorithm, the center of the circle is evolved as where [x c , y c ] and [x c ,ỹ c ] the center coordinates of the circle before and after evolution, respectively, and f k is the evolution force for the kth contour points obtained from 1, and n k is its corresponding normal vector that can be obtained as Eq. (6) indicates that for a given set of forces the circle center is simply shifted by the average of the force vectors f k n k . The center of the circle is evolved asR where R andR is the radius of the circle before and after evolution, respectively. This indicates that 59 for a given set of forces, the circle radius is evolved with the average of the force values f k .  In [23], authors showed that the active circle algorithm estimates the IVC AP-diameter much more accurate than the star-Kalman algorithm in [24] which is based on ellipse fitting. But does this indicate that a circular model can estimate the AP-diameter more accurate than an elliptical model? To answer this question, we develop an active ellipse algorithm based on the same evolutional functional as the one in the active circle algorithm. In general case, during the evolution, the kth contour point is evolved as To fit an ellipse to these K evolved points, we use the following conic equation.
where x and y are the coordinate of the points on the conic, a, b, c, d, and e are the conic parameters. Note that with an ellipse, the values of a and b must be positive. With K points with coordinates [x k ,x k ], the best ellipse is fitted by minimizing the following cost function.
Eq. (11) can be rewritten in matrix form where the vector of conic parameters defined as A = [a, b, c, d, e] T , X is a matrix with [x 2 k ,x k ,ỹ k ,ỹ 2 k ,x k ,ỹ k ] as its kth row, 1 K is K × 1 all-one vector, and superscript T is the transpose operator. After setting the gradient C(A) to zero, the vector A is obtained aŝ The active ellipse algorithm is summarized in table 1. Fig. 1 shows the ellipse evolution versus the number of iterations for a sample IVC frame.
Active ellipse algorithm for estimation and tracking of the IVC AP-diameter Input: An IVC video and parameter α = 10 −4 .
-Read one frame from the input video.
-If it is the first frame, select one point inside the IVC by a mouse click. This point is assumed as the initial value of the ellipse center. The initial ellipse is created as circle centered at this point and initial radius R=8 pixels.
--Compute the forces for the points on each side of the rectangle using (4).
-Repeat the last two steps until a convergence is achieved. We assume the algorithm has reached the convergence when the maximum of the change in the elements of the A is less than 10 −4 , i.e., where max is the element-wise maximum, andÂ n is the vector A estimated at nth iteration.
-Return to the first step for the next frame.

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The intuition to use a rectangular model is that the AP-diameter is clinically defined as the largest vertical diameter of the IVC contour which may practically deviate from the actual diameter of an circle or even an ellipse. The AP-diameter can be modeled as the height of a vertical thin rectangle. As a start point, we assume that the fitted rectangle has a fix width w=3pixels and the only parameters that have to be modified are the center and the height of the rectangle. With the forces defined as (4), either of the upper and lower sides of the rectangle move with the average of the forces applied on that side. Hence, the center of the rectangle is moved as where P l , P r , P u , and P b are the subsets of the contour points on the left, right, upper, and lower sides of the rectangle, respectively, and K l , K r , K u , and K b are the number of points in each of these sets. Similary the height of the rectangle is evolved as Although a thin rectangle model accurately models the clinically measured AP-diameter, it might iterations.
-Repeat the last two steps for N =200 iterations.
-Return to the first step for the next frame.

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The experimental data was collected from eight young healthy male subjects. The study protocol perform less accurate than the case in Fig. 4. This is mainly due to the fact that although the 95 second clip seems to have a better quality than the first one, it has a more fuzzy contour, making the 96 semi-automatic algorithms less accurate than the first clip. Similar to the first clip, we can see that the 97 active rectangle algorithms performs better than the other two methods.     In five out of the eight cases, the active circle algorithm performed better than the active ellipse 126 algorithm. This is mainly due to the fact that a circle has less degree of freedom compared to an ellipse 127 and therefore, circle evolution can be performed more accurate than ellipse evolution. Furthermore, based on weighted variance. International journal of computer theory and engineering 2009, 1, 7.