Bolus characteristics based on Magnetic Resonance Angiography
- Zhijun Cai^{1},
- Alan Stolpen^{2},
- Melhem J Sharafuddin^{3},
- Robert McCabe^{1},
- Henri Bai^{4},
- Tom Potts^{1},
- Michael Vannier^{5},
- Debiao Li^{6},
- Xiaoming Bi^{6},
- James Bennett^{2},
- Jafar Golzarian^{2},
- Shiliang Sun^{2},
- Ge Wang^{2} and
- Er-Wei Bai^{1}Email author
https://doi.org/10.1186/1475-925X-5-53
© Cai et al; licensee BioMed Central Ltd. 2006
Received: 10 August 2006
Accepted: 17 October 2006
Published: 17 October 2006
Abstract
Background
A detailed contrast bolus propagation model is essential for optimizing bolus-chasing Computed Tomography Angiography (CTA). Bolus characteristics were studied using bolus-timing datasets from Magnetic Resonance Angiography (MRA) for adaptive controller design and validation.
Methods
MRA bolus-timing datasets of the aorta in thirty patients were analyzed by a program developed with MATLAB. Bolus characteristics, such as peak position, dispersion and bolus velocity, were studied. The bolus profile was fit to a convolution function, which would serve as a mathematical model of bolus propagation in future controller design.
Results
The maximum speed of the bolus in the aorta ranged from 5–13 cm/s and the dwell time ranged from 7–13 seconds. Bolus characteristics were well described by the proposed propagation model, which included the exact functional relationships between the parameters and aortic location.
Conclusion
The convolution function describes bolus dynamics reasonably well and could be used to implement the adaptive controller design.
Background
To diagnose vascular disease, high quality imaging is extremely important. Many imaging studies utilized an intravascular injection of contrast material that must be delivered to the field-of-view (FOV). For a typical four channel CT scanner, the FOV is the imaging aperture, which has a width of about 10 mm [1]. Estimating the peak density of the contrast bolus during the entire period of data acquisition is highly desirable for planning image acquisition, processing, and display. A detailed knowledge of bolus characteristics helps to optimize vascular imaging and thereby improve depiction of vessels, lesions, and tumors [2–5].
Currently, Magnetic Resonance Imaging (MRI), digital subtraction fluoroscopy and CT are three imaging modalities used for angiography. One determinant of image quality is the accurate spatial and temporal synchronization of the peak bolus density with the imaging acquisition window. For CT, this involves translating the scanner table appropriately after intravascular injection [6, 7]. MRA and Digital Subtraction Angiography (DSA) have a relatively large FOV, which makes the synchronization easier. However, compared to CTA, MRA has inferior spatial resolution. DSA has the best spatial resolution, but it is invasive, and 2D projection images may limit its diagnostic utility. Although many researchers have investigated bolus chasing in DSA [8] and MRA [9, 10], there has been insufficient work devoted to synchronizing contrast material delivery and propagation with the CT imaging aperture. Our group has previously proposed the adaptive bolus chasing control scheme to address this issue [11].
The goal of this paper is to investigate arterial bolus characteristics (focusing on the aorta) in order to develop an adaptive control mechanism for bolus-chasing CTA. Properties of the contrast bolus, such as bolus velocity, peak position, and dispersion, were investigated. Using sequential images from MRA contrast bolus timing datasets, temporal density curves and spatial density curves were extracted. These curves were displayed as a bolus Distance-Time-Density profile and used to formulate a mathematical model.
As it travels through the artery, the bolus fills the vessel for some length. Therefore, a complete description of bolus dynamics requires a large field of view. CT images a cross-section of the body, usually in the transverse plane, and thus provides information about the bolus density in only a single slice at any given time. CT is incapable of interrogating a bolus along the length of an artery that travels orthogonal to the imaging plane. Because they offer a large FOV, both MRA and DSA could potentially provide this information. Of these, MRA is our first choice due to similarities with CTA regarding the contrast injection site, and the likelihood that both have similar bolus dynamics. In addition, MRA does not expose human subjects to ionizing radiation or nephrotoxic contrast agents.
Quantitative analysis of MRA data is difficult because of the complex relationship between MR signal intensity and contrast material concentration or density. However, the analysis can be simplified by using relative (as opposed to absolute) contrast bolus density, wherein a linear relationship between signal intensity and contrast density is assumed. Other assumptions used in our work are: 1) the MRA imaging plane is assumed to pass through the middle of the aorta, thereby neglecting effects due to vessel curvature; 2) the effects of blood flow-related enhancement on MR signal intensity are not considered. Using the assumptions above, the pixel value (after subtraction of background) in the MRA image corresponds to the relative bolus density. Through this paper, "density" refers to the relative bolus density.
In this paper, the procedure for analyzing the MRA images will be described first, followed by an analysis of the bolus characteristics. A complete description of the bolus dynamics is formulated by fitting the data to a set of standard mathematical functions; the results are then used to facilitate the development of an adaptive controller design.
Methods
Data acquisition and extraction
MRA datasets were collected from two sources: 1) Routine clinical MRA studies with bolus timing runs performed on patients at the University of Iowa Hospital and Clinics (UIHC), and 2) studies performed on volunteers at Northwestern University (NU). UIHC studies were performed on a 1.5 Tesla GE CV/i scanner using a 2D gradient echo sequence (TR = 5.7 ms, TE = 1.5 ms, 40 deg flip angle), 5 mm slice thickness, 38 cm FOV, 32 kHz bandwidth, and a 256 × 224 matrix. For these studies, a 3 cc contrast bolus (1.5 mmole gadodiamide) was injected into a peripheral vein followed by a 25 cc saline flush, both at a rate of 3 cc/sec. Northwestern University studies were performed on a 3 Tesla SIEMENS MR scanner using a 2D gradient echo sequence (TR = 4.2 ms, TE = 1.11 ms, 8 deg flip angle), 11 mm slice thickness, 40 cm FOV, and a 580 Hz/pixel bandwidth. A 2 cc contrast bolus was injected into a peripheral vein followed by a 10 cc saline flush, both at a rate 2 of cc/sec. At both sites, images were acquired in a sagittal or oblique sagittal plane through a long segment of the thoracoabdominal aorta. The header information in the Digital Imaging and Communications in Medicine (DICOM) format provided information about pixel spacing. All datasets were comprised of a series of image frames at one second time intervals. Patient information was also collected from DICOM, including gender and age.
From the large pool of bolus timing studies, we selected those demonstrating a clear, continuous longitudinal segment of the aorta. The bolus analysis algorithm was developed using MATLAB (ver. 7.1. [R14], SP3) with Image Processing Toolbox (ver. 5.1).
Data analysis
- 2)
Each frame was filtered with a 9 × 9 average (low pass) filter to reduce the noise;
- 3)
Five to ten consecutive frames were averaged starting with the 3^{rd} frame and ending before the upslope of the contrast bolus. This average was used as the background and subtracted from each frame (Figure 1c);
- 4)
The ROI from each frame was masked, that is, all the pixels outside the ROI was set to zero, while pixels inside the ROI untouched. (Figure 1d);
- 5)
All pixel values in each row were summed and divided by the number of pixels in that row of the ROI. The result, for each frame, was a curve showing bolus average density of that cross section versus longitudinal displacement;
- 6)
Steps 3–6, above, were repeated for each frame.
Data fitting
Our purpose was to study bolus characteristics by establishing a bolus propagation model that can be used to develop bolus-chasing CTA. Once an actual bolus 3D profile was obtained, the next step was to fit it into a mathematical model. Others have previously used a variety of functions to fit the bolus profile in order to find a bolus time-density curve at a specified location (i.e., temporal curve). For example, in [13], A gamma variate curve fitting $C=k{(t-{t}_{0})}^{a}{e}^{\frac{(t-{t}_{0})}{b}}$ was suggested to obtain an aortic time-attenuation curve, which could be used to correct the recirculation effect. In the above formula, C and t represent density and time, respectively, whereas a, b, t _{0} are fitting parameters. It has been reported that the gamma variate model provides a poor fit to the time density curve when the contrast bolus is small and rapid [14]. In [15], an empirical formula $C={C}_{p}{e}^{-k{\mathrm{log}}_{e}^{2}\frac{(t-{t}_{a})}{({t}_{p}-{t}_{a})}}$ was proposed, where C, C _{ p }and t represent concentration (density), maximum concentration, and time, respectively, whereas k, t _{ a }, t _{ p }are fitting parameters. This function was shown to be incapable of fitting the obtained 3D bolus profile by several trials. In [16], a lagged normal density function (i.e., the convolution of a Gaussian and an exponential function) was used to fit the bolus density at the femoral and dorsalis pedis arteries. Recently, [8] used an updated lagged normal density function by including the effects of the injection pattern. Using this approach, the bolus temporal profile was converted into a spatial profile, which was used to resist the noise and feature extraction errors in bolus tracking for X-Ray Peripheral Angiography. The fitting functions used in the above work focused on bolus attenuation at a specified location, but paid little attention to bolus dispersion in the artery. However, to design an adaptive controller for bolus chasing CTA, a whole picture of bolus density, including its attenuation along the distance and time axis, is crucial. To that end, we need to extend the above work to the 3D fitting.
In this work, we initially examined all three bolus fitting functions mentioned above, and found that the data were best-fitted by the lagged normal function. Therefore, the lagged normal density model
$b=C\times \frac{1}{\sqrt{2\pi}\sigma}{e}^{-\frac{{(t-{t}_{c})}^{2}}{2{\sigma}^{2}}}\otimes \frac{1}{\tau}{e}^{-\frac{t}{\tau}}\left(1\right)$
was adopted to fit our datasets. In (1), t is time, and C, σ, t _{ c }and τ are four parameters, which are functions of the position z. After integrating Equation (1), we obtained
$b(t,z)=\frac{C}{2\tau}\times {e}^{-\frac{1}{\tau}+\left(\frac{{t}_{c}}{\tau}+\frac{{\sigma}^{2}}{2{\tau}^{2}}\right)}\left[erf\left(\frac{1}{\sqrt{2}\sigma}\left(t-{t}_{c}-\frac{{\sigma}^{2}}{\tau}\right)\right)-erf\left(\frac{1}{\sqrt{2}\sigma}\left(-{t}_{c}-\frac{{\sigma}^{2}}{\tau}\right)\right)\right],\left(2\right)$
- 1)
The main part of the bolus was selected. Avoiding the lowest signal intensity portions of the bolus helps to make the fitting problem numerically well-conditioned.
- 2)
At a given position z _{ k }, the parameters σ, t _{ c }, C and τ are fixed because they are functions of distance. The temporal curve at z _{ k }was found by adjusting σ, t _{ c }, C and τ to minimize the sum of the squares of the errors; that is, ${\left[C,\sigma ,{t}_{c},\tau \right]}_{k}=\underset{\left[C,\sigma ,{t}_{c},\tau \right]}{\mathrm{arg}\mathrm{min}}\left\{{\displaystyle \sum _{t}{\left[b(t,{z}_{k})-B(t,{z}_{k})\right]}^{2}}\right\}$ where B(t,z _{ k }) is actual density value at time t and position z _{ k }.
- 3)
Steps 1 and 2 were repeated at every temporal position;
- 4)
The parameters σ, t _{ c }, C and τ were expressed as a function of the position z.
Summary of 30 MRA bolus timing datasets: selected features of interest and fitting errors.
Source | Patient number | Patient Information | Bolus Velocity (cm/sec) | Travel Time (sec) | Travel Length (cm) | Relative Fitting error |
---|---|---|---|---|---|---|
UIHC 'GE' MR machine | 1 | M/50 | 9 | 10 | 25 | 0.122 |
2 | F/63 | 8 | 10 | 27 | 0.170 | |
3 | M/15 | 9 | 10 | 28 | 0.056 | |
4 | M/54 | 10 | 11 | 28 | 0.153 | |
5 | F/21 | 13 | 7 | 23 | 0.039 | |
6 | F/41 | 11 | 8 | 29 | 0.102 | |
7 | M/71 | 6 | 7 | 16 | 0.046 | |
8 | M/64 | 9 | 7 | 26 | 0.038 | |
9 | F/49 | 5 | 10 | 14 | 0.051 | |
10 | F/66 | 12 | 11 | 26 | 0.065 | |
11 | M/51 | 7 | 9 | 19 | 0.107 | |
12 | M/50 | 10 | 10 | 21 | 0.089 | |
13 | M/23 | 6 | 8 | 23 | 0.093 | |
14 | F/55 | 6 | 11 | 25 | 0.098 | |
15 | M/45 | 6 | 8 | 23 | 0.129 | |
16 | F/22 | 13 | 6 | 19 | 0.180 | |
17 | F/60 | 8 | 7 | 21 | 0.064 | |
18 | M/58 | 10 | 11 | 22 | 0.135 | |
19 | F/25 | 10 | 11 | 24 | 0.087 | |
20 | M/48 | 11 | 8 | 23 | 0.049 | |
21 | M/26 | 6 | 12 | 26 | 0.063 | |
22 | M/25 | 8 | 10 | 22 | 0.057 | |
NU 'Siemens ' MR machine | 23 | M/34 | 13 | 8 | 23 | 0.005 |
24 | M/19 | 13 | 13 | 16 | 0.003 | |
25 | F/40 | 8 | 8 | 21 | 0.008 | |
26 | M/36 | 9 | 9 | 20 | 0.006 | |
27 | M/57 | 10 | 12 | 21 | 0.003 | |
28 | F/37 | 8 | 8 | 19 | 0.005 | |
29 | M/34 | 9 | 10 | 11 | 0.024 | |
30 | F/27 | 6 | 9 | 10 | 0.015 |
Summary of bolus model fitting parameters.
patient number | t _{ c } | σ | τ | C | ||||
---|---|---|---|---|---|---|---|---|
a _{0} | a _{1} | C _{0} | C _{1} | C _{2} | ||||
UIHC 'GE' MR machine | 1 | 15 | 0.0076 | 1.80 | 4.89 | 4.2 | 0.044 | -1.2e-4 |
2 | 17 | 0.017 | 7.75 | 5.25 | 4.4 | -0.0032 | -7e-5 | |
3 | 17 | -0.0029 | 2.34 | 8.10 | 5.4 | 0.068 | -5.2e-4 | |
4 | 13 | 0.019 | 2.04 | 5.30 | 2.7 | 0.014 | -4e-5 | |
5 | 13 | 0.0063 | 2.06 | 5.17 | 4.7 | 0.026 | -1.7e-4 | |
6 | 13 | 0.00047 | 1.67 | 6.37 | 5.1 | -0.012 | -4.3e-5 | |
7 | 22 | 0.044 | 2.81 | 19.70 | 5.5 | 0.25 | -3e-3 | |
8 | 18 | 0.0089 | 2.06 | 6.79 | 5.8 | -0.11 | -2.4e-5 | |
9 | 14 | 0.029 | 2.63 | 7.72 | 2.6 | 0.056 | -5.6e-4 | |
10 | 15 | 0.015 | 2.36 | 4.75 | 8.5 | -0.052 | 3.5e-4 | |
11 | 16 | 0.0092 | 1.87 | 8.27 | 5.7 | -0.021 | 4.4e-5 | |
12 | 15 | 0.0085 | 1.77 | 7.03 | 6.3 | 0.034 | -4e-4 | |
13 | 16 | 0.001 | 2.05 | 4.81 | 5 | 0.016 | -3.6e-5 | |
14 | 18 | 0.0082 | 2.48 | 18.20 | 16 | -0.12 | 7.3e-4 | |
15 | 14 | 0.0096 | 2.02 | 4.98 | 6.2 | 0.012 | -8.9e-5 | |
16 | 10 | 0.004 | 1.35 | 3.85 | 6.7 | -0.051 | 1.8e-4 | |
17 | 14 | 0.0085 | 1.85 | 4.98 | 4.3 | 0.11 | -6.5e-5 | |
18 | 12 | 0.0076 | 1.60 | 5.41 | 4.7 | -0.021 | 1.4e-5 | |
19 | 15 | -0.0034 | 1.77 | 5.42 | 7.2 | -0.0014 | 4.7e-6 | |
20 | 13 | 0.0054 | 1.62 | 6.10 | 3.5 | 0.013 | -9e-5 | |
21 | 15 | 0.013 | 2.25 | 12.13 | 8.1 | -0.067 | 3.9e-4 | |
22 | 15 | 0.0062 | 2.24 | 9.16 | 14 | -0.034 | -1.3e-4 | |
NU 'Siemens ' MR machine | 23 | 16 | 0.0049 | 1.79 | 4.08 | 0.88 | 0.00074 | -1.7e-5 |
24 | 23 | 0.0089 | 2.99 | 7.42 | 1.4 | -0.0087 | 7.4e-6 | |
25 | 24 | 0.0067 | 2.06 | 6.62 | 1.4 | -0.0032 | -1.4e-5 | |
26 | 25 | 0.0061 | 2.08 | 6.52 | 1.3 | -0.0035 | -1.5e-5 | |
27 | 28 | 0.0086 | 2.90 | 19.07 | 1.4 | 0.12 | -1.3e-4 | |
28 | 15 | 0.0027 | 1.86 | 4.09 | 0.81 | -0.0034 | 5.7e-6 | |
29 | 19 | 0.0078 | 2.62 | 3.48 | 1.5 | 0.036 | -4.9e-4 | |
30 | 20 | 0.0043 | 2.63 | 4.25 | 0.66 | 0.019 | -5e-4 |
Results
The datasets from 30 patients were studied. Datasets were analyzed using the algorithm described above. Table 1 summarizes the bolus characteristics, where "Bolus Velocity", "Travel Time" and "Travel Length" were used to describe bolus characteristics, while "Relative Fitting error" shows the accuracy of the fitting. The "Relative Fitting error" is given by average fitting error square for each point divided by the maximum signal intensity of the dataset. From Table 1, we can see that most of the relative error is smaller than 0.1 (22 out of 30), which shows the great fitting ability of the lagged normal density model. In Table 1, the fitting error for UIHC datasets is bigger than that for NU datasets, this may be due to the different MR machine technical parameters and it is beyond the scope of this paper. Because of the fast movement of contrast bolus in the aorta and poor temporal resolution of MRI, we did not extract bolus peak trajectory. Furthermore, in this paper, all of the datasets used a small amount of contrast material (2–3 cc) to determine the bolus arrival time in clinical MRA studies (see section 2.1). In clinical parlance, these datasets are referred to as test bolus or bolus timing runs. For these reasons, discussing bolus maximum velocity here may not be a good practical fit. In this study, "Bolus Velocity" for thirty datasets ranged from 5 to 13 cm/sec with average about 9 cm/sec. Compared to the reported blood average velocity in the aorta, which is about 27 cm/sec [17], the obtained velocity is relatively low. Considering the higher density of the contrast material and its dispersion, this is reasonable. Besides, the current clinical CT scanning protocols do not require CT table to be moved at a speed higher than 6 cm/sec.
Discussion
Analysis of MRA bolus timing data provides information about contrast bolus dynamics. The 3D bolus profile is a useful tool for simulating bolus-chasing CTA. The physicians always want the best diagnosis image, which requires the highest SNR (signal to noise ratio) at each vascular position for CT scanning. To that end, scanning each vascular position with highest density bolus inside is highly desired. However, during the CT scanning, it is very unlikely to know the maximum density time at each vascular position due to narrow imaging window of CT machine and complicated dynamics of contrast bolus. Therefore, an adaptive controller is urgently needed to predict the bolus maximum density time for the next scanned position. The controller requires the information of the bolus peak density in order to function correctly. On the other hand, the bolus fitting function (i.e., the propagation model) can provide the necessary information and be used to facilitate the controller design. For example, identification of σ, t _{ c }, C and τ in Equation (1) enables us to have an optimal trajectory of CT table movement. In fact, the more information about the propagating bolus, the better the controller design would be. Our ultimate goal is to design an adaptive controller, which is robust to track the bolus peak density of all patients with all diseases on CT machine. Currently, the CT machine operators moves CT table at a pre-set constant velocity, which does not consider the complicated bolus dynamics, the patient characteristics, and the disease effects. It is unlikely to have a good tracking result unless overdose is used. The adaptive controller will benefit both patient and physician in 1) better diagnosis CT image, 2) reduction of dose, 3) reduction of radiation exposure. In our latest work, we have designed an adaptive optimal controller, which has been proved to have an ability to track the bolus peak position very well and maximize the signal intensity at each scanned position.
Our analysis has several limitations related to the MR data. 1) Only a portion of the aorta was evaluated. For some CTA studies, the entire arterial tree from the chest to the toes will be scanned, but the bolus dynamics vary with arterial location. It is expected to have the MR sequence images of the entire arterial tree. 2) MR images demonstrate a complex and non-linear relationship between signal intensity and contrast bolus concentration. Thus, our analysis, which assumes a linear relationship, represents an approximation. 3) The temporal resolution of the MR data is less than that of CTA. The MR frame rate of one per second may be inadequate to capture some key features of the contrast bolus.
In conclusion, we have not only developed a tool for analyzing the characteristics of the MRA contrast bolus using a widely available software MATLAB and the Image Processing Toolbox, but found a mathematical model that fits the bolus well. Our results provided realistic simulations for bolus-chasing CTA and critical information for the design of an adaptive controller. Future work will focus on controller design and validation by simulation and experiment.
Declarations
Acknowledgements
This work was supported in part by NIH/NIBIB EB004287.
Authors’ Affiliations
References
- Fleischmann D, Hallett RL, Rubin GD: CT angiography of peripheral arterial disease. J Vasc Interv Radiol 2006,17(1):3–26.View ArticleGoogle Scholar
- Bae KT, Heiken JP, Brink JA: Aortic and hepatic contrast medium enhancement at CT: Part I. Prediction with a computer model. Radiology 1998, 207: 647–655.View ArticleGoogle Scholar
- Jakobs TF, Bernd J: MDCT-Imaging of Peripheral Arterial Disease. Seminars in Ultrasound CT and MRI 2004, 25: 145–155. 10.1016/j.sult.2004.02.003View ArticleGoogle Scholar
- Schweiger GD, Chang PC, Brown BP: Optimizing contrast enhancement during helical CT of the liver: A comparison of two bolus tracking techniques. AJR 1998, 171: 1551–1558.View ArticleGoogle Scholar
- Silverman PM, Roberts SC, Ducic I, Tefft MC, Olson MC, Cooper C, Zeman RK: Assessment of a technology that permits individualized scan delays on helical hepatic CT: A technique to improve efficiency in use of contrast material. AJR 1996, 167: 79–84.View ArticleGoogle Scholar
- Kopka L, Rodenwaldt J, Fischer U, Mueller DW, Oestmann JW, Grabbe E: Dual-phase helical CT of the liver: Effects of bolus tracking and different volumes of contrast material. Radiology 1996, 201: 321–326.View ArticleGoogle Scholar
- Sheafor DH, Keogan MT, DeLong DM, Nelson RC: Dynamic helical CT of the abdomen: Prospective comparison of pre- and postprandial contrast enhancement. Radiology 1998, 206: 359–363.View ArticleGoogle Scholar
- Wu Z, Qian J: Real-time tracking of contrast bolus propagation in X-ray peripheral angiography. IEEE Workshop on Biomedical Image Analysis 1998, 164–172.Google Scholar
- Ho VB, Choyke PL, Foo TK, Hood MN, Miller DL, Czum JM, Aisen AM: Automated Bolus Chase Peripheral MR Angiography: Initial Practical Experiences and Future Directions of This Work-In-Progress. Journal of Magnetic Resonance Imaging 1999, 10: 376–388. 10.1002/(SICI)1522-2586(199909)10:3<376::AID-JMRI20>3.0.CO;2-UView ArticleGoogle Scholar
- Kruger DG, Riederer SJ, Polzin JA, Madhuranthakam AJ, Hu HH, Glockner JF: Dual-Velocity Continuously Moving Table Acquisition for Contrast-Enhanced Peripheral Magnetic resonance Angiography. Magnetic Resonance in Medicine 2005, 53: 110–117. 10.1002/mrm.20324View ArticleGoogle Scholar
- Bennett JR, Bai EW, Halloran JI, Wang G: A Preliminary Study on Adaptive Field-of-View Tracking in Peripheral Digital Subtraction Angiograph. J of X-ray Science and Technology 2003, 11: 149–159.Google Scholar
- Shpilfoygel SD, Close RA, Valentino DJ, Duckwiler GR: X-Ray videodensitometric methods for blood flow and velocity measurement: A critical review of literature. Med Phys 2000,27(9):2008–2023. 10.1118/1.1288669View ArticleGoogle Scholar
- Blomley MLK, Dawson P: Bolus dynamics: Theoretical and experimental aspects. The British Journal of Radiology 1997, 70: 351–359.View ArticleGoogle Scholar
- Cunningham I, Hobbs B, Fenster A: A new system for quantitative arterial imaging and blood flow measurements. Invest Radiol 1986, 21: 465–471. 10.1097/00004424-198606000-00004View ArticleGoogle Scholar
- Stow RW, Hetzel PS: An Empirical Formula for Indicator-Dilution Curves as Obtained in Human Beings. J Appl Physiol 1954, 7: 161–167.Google Scholar
- Bassingthwaighte JB, Ackerman FH, Wood EH: Applications of the lagged normal density curve as a model for arterial dilution curves. Circulation Research 1966,18(4):398–415.View ArticleGoogle Scholar
- Zierler RE, Zierler BK: Duplex sonography of the lower extremity arteries. Semin Ultrasound CT MR 1997,18(1):39–56. 10.1016/S0887-2171(97)90037-8View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.