Poisson Statistical Model of Ultrasound Super-Resolution Imaging Acquisition Time

Abstract

A number of acoustic super-resolution techniques have recently been developed to visualize microvascular structure and flow beyond the diffraction limit. A crucial aspect of all ultrasound (US) super-resolution (SR) methods using single microbubble localization is time-efficient detection of individual bubble signals. Due to the need for bubbles to circulate through the vasculature during acquisition, slow flows associated with the microcirculation limit the minimum acquisition time needed to obtain adequate spatial information. Here, a model is developed to investigate the combined effects of imaging parameters, bubble signal density, and vascular flow on SR image acquisition time. We find that the estimated minimum time needed for SR increases for slower blood velocities and greater resolution improvement. To improve SR from a resolution of λ/10 to λ/20 while imaging the microvasculature structure modeled here, the estimated minimum acquisition time increases by a factor of 14. The maximum useful imaging frame rate to provide new spatial information in each image is set by the bubble velocity at low blood flows (<150 mm/s for a depth of 5 cm) and by the acoustic wave velocity at higher bubble velocities. Furthermore, the image acquisition procedure, transmit frequency, localization precision, and desired super-resolved image contrast together determine the optimal acquisition time achievable for fixed flow velocity. Exploring the effects of both system parameters and details of the target vasculature can allow a better choice of acquisition settings and provide improved understanding of the completeness of SR information.

I. Introduction

Imaging of the microvasculature is crucial for the early detection and intervention of diseases such as cancer [1], [2], ischemia [3], and peripheral arterial disease [4]–[6].

Ultrasound (US) super-resolution (SR) techniques have recently been developed to visualize microvascular structure and flow beyond the diffraction limit using microbubble contrast agents [7]–[18]. A crucial aspect of all methods based on single-bubble localization is the detection of spatially isolated microbubble signals. Obtaining these isolated signals may be achieved in a number of ways, including: suitable microbubble concentrations with contrast imaging modes and background subtraction techniques [7], [8], linear imaging techniques with high-pass spatiotemporal filtering [11], [12], differential imaging [13], differential imaging with spatiotemporal nonlocal means filtering [14], and background subtraction methods [9], [10], [15]. Furthermore, the recent use of nanodroplets for SR has the benefit of providing sparsely activated microbubbles without the requirement for sufficient blood flow [19], [20].

Essential to the clinical translation of SR techniques is the ability to generate images in clinically viable acquisition times. This not only limits patient–clinician time requirements but also reduces the unwanted effects of motion during long scan times. In all the aforementioned cases, the number of individual microbubble signals detected and localized per frame is restricted by the diffraction limited nature of the acquired data. Since these methods rely upon the combined localization information gathered over a series of frames, minimizing potentially long acquisition times is crucial.

A high localization rate is critical in reducing acquisition times and can be achieved by a combination of factors. First, an increase in frame rate allows more frequent sampling of microbubble flow through the vasculature. Due to the need for bubbles to circulate through the entire vasculature during acquisition, slow flows associated with the microcirculation limit the minimum acquisition time needed to obtain adequate spatial information. This, therefore, presents a limit to the benefit that increased frame rates can provide on the SR acquisition time.

Second, each frame should contain a high number of spatially separable bubble signals. This can be achieved, for example, by increasing the frequency [reducing the point spread function (PSF) size] or by increasing the bubble concentration. The latter, however, does not necessarily improve the localization rate since overlapping or interfering signals should be rejected. If included, subsequent incorrect localizations may be positioned outside of the vessel diameter, and, thus, are likely to degrade the final image. Therefore, the optimization of signal density is crucial for time-efficient SR acquisition.

The following model has been developed based on a simplified Poisson statistical model which aims to investigate the combined effects of acquisition imaging parameters on localization rate, and, ultimately, image acquisition time.

II. Materials and Methods

A. Number of Interrogations Required

The acquisition time required to create a super-resolved image is given by

$$\text{Acquisition}\text{Time}=t_i·N_i$$ (1)

where t_i is the time to acquire a single interrogation and N_i is the total number of interrogations needed to obtain sufficient localizations. An interrogation is defined as the sequence of pulse-echoes required to produce a single 2-D image or 3-D volume.

To estimate the number of interrogations required, the likelihood of imaging spatially isolated microbubbles is required. Here, we image a cubic volume V with sides of length l_x,l_y, l_z, at a maximum depth d = l_y, as shown in Fig. 1. The proportion of the local volume containing blood vessels is the local tissue blood volume fraction, equal to V_ν, ranging from 0% to 100%. The density of bubble signals detected in the acquisition is termed the signal density. Using a contrast agent infusion, the signal density in the blood is assumed to be constant and equal to C_b.

Fig. 1.

Illustration of volume V imaged by the US system. Vascular structures occupy a subvolume, defined by a variable local tissue blood volume fraction V_ν.

The PSF volume, or resolution voxel, V_PSF, shown in Fig. 2, can be approximated by

$$V_{\text{PSF}}={\text{FWHM}}_x{\text{FWHM}}_y{\text{FWHM}}_z$$ (2)

where FWHM_x and FWHM_y are the lateral and axial full-width at half-maximum of the PSF, respectively, and FWHM_z is the elevational resolution, or if acquiring 2-D data, this can be defined as the “slice thickness,” Δz.

Fig. 2.

Imaging volume V can be divided in approximately PSF sized voxels, corresponding to the 3-D diffraction-limited resolution of the imaging system.

This is representative of the original diffraction-limited resolution of the system. Thus, two scatterers within the same resolution voxel cannot be resolved. For simplicity, V_PSF is represented as an isotropic voxel in diagrams, such that FWHM_x = FWHM_y = FWHM_z. In reality, the in-plane components are anisotropic and these often differ greatly to the elevational resolution or the slice thickness.

If independent and discrete events occur with a known average rate, then Poisson statistics can be used to express the probability of a given number, k, of events occurring within a fixed interval of time or space. In this work, an event is defined as the presence of a bubble, and the fixed spatiotemporal interval of observation is a diffraction-limited sized region within the imaging volume [23], [24]. It is assumed that bubbles do not cluster, and therefore bubble events can be defined for a finite set of values of k. Poisson statistics can thus be used to generate an initial distribution of bubbles from an initial expectation value [25]. We can then use this distribution to examine the probability of imaging single bubbles in a resolution voxel. In this case, the probability, P_PSF, of having k bubbles in a sample volume can be given by the following relation:

$$P_{\text{PSF}}\left(k\right)=\frac{{\mu }^k{e}^{-\mu }}{k!}$$ (3)

where μ is the Poisson expectation value, given by the known number of events occurring in one sample volume, V_PSF

$$\mu =C_{\text{tis}}·V_{\text{PSF}}.$$ (4)

Here, C_tis is the concentration of bubbles in the local tissue

$$C_{\text{tis}}=C_b·V_v.$$ (5)

Since there exists a precision associated with the localization of point scatterers using an US imaging system [7], a SR pixel, A_SR, or voxel, V_SR, can be approximated as an area or volume with sides equal in length to the localization precision in the x, y, and z dimensions, denoted by σ_x, σ_y, and σ_z (Fig. 3), given by

$$V_{\text{SR}}={\sigma }_x{\sigma }_y{\sigma }_z.$$ (6)

Fig. 3.

Illustration of the change of resolution represented by a change in resolution voxel size, where the original resolution voxel is the PSF measured from the US imaging system, and the SR voxel is given by the localization precision in the x, y, and z dimensions.

The localization precision determines the FWHM of the Gaussian localization profile plotted for each localization in the final SR rendering as performed in our previous work in 2-D [8].

The concentration of single bubble events, C_SB, is the concentration of bubbles in the tissue multiplied by the probability that no bubble falls into the same resolution cell

$$C_{\text{SB}}=C_{\text{tis}}{e}^{-C_{\text{tis}}·V_{\text{PSF}}}.$$ (7)

For each interrogation, the average number of single bubble detections in an SR voxel, n_SB, is given by

$$n_{\text{SB}}-C_{\text{SB}}·V_{\text{SR}}.$$ (8)

The average number of localizations in an SR voxel, N_l, after N_i interrogations is

$$N_l=N_i{C}_{\text{SB}}{V}_{\text{SR}}.$$ (9)

This value can be chosen to define the average number of detections per SR voxel to reach a sufficient signal-to-noise ratio (SNR). The number of interrogations required can then be given by

$$\begin{array}{l}{N}_i=\frac{N_l}{C_{\text{SB}}{V}_{\text{SR}}}\hfill \\ N_i=\frac{N_l}{C_{\text{tis}}{V}_{\text{SR}}}{e}^{C_{\text{tis}}·V_{\text{PSF}}}.\hfill \end{array}$$ (10)

B. Imaging Rate

The imaging rate, I, can be considered as the number of interrogations created per second, given by

$$I=\frac{1}{t_i}.$$ (11)

During imaging, microbubbles flow through the vasculature at an average velocity ν_b. For the microbubbles to provide new spatial information in each consecutive interrogation, its movement should exceed the magnitude of the system’s localization precision, σ = (σ_x, σ_y, σ_z). Since this is often not isotropic, the average localization precision over all directions is taken. Thus, the distance moved by a bubble in each interrogation, d_b, should be

$$d_b\ge \overline{\sigma }$$ (12)

thus

$${\upsilon }_b\ge \overline{\sigma }·I$$ (13)

and the flow rate limit on the minimum time between interrogations, t_iFL, can be given by

$$t_{\text{iFL}}\ge \frac{\overline{\sigma }}{{\upsilon }_b}.$$ (14)

This limit ensures acquisition times found using this model consider the challenge associated with slow microvascular flow rates. Nevertheless, imaging rates above this, or conversely, bubbles with velocity below this, will still contribute localizations to the final image, and thus will enhance the final SNR.

The fundamental limit on the imaging rate is determined by the time of flight and image acquisition procedure implemented. Using contrast enhanced US (CEUS) imaging techniques such as pulse inversion (PI) [26] or contrast pulse sequencing (CPS) [27], multiple pulse-echoes, N_pulses, of varying phase or amplitude are used to create each interrogation. Multiple-angle plane wave compounding also requires echoes from several directed wavefronts to generate each image. Thus, the minimum interrogation time, termed the acoustic limit, t_iAL, can be given by

$$t_{\text{iAL}}=\frac{2d{N}_{\text{pulses}}}{c}$$ (15)

where c is the speed of sound in the medium, assumed to be constant, and d is the imaging depth.

C. Acquisition Time

By combining relations from previous equations, an approximate overall relation can be given by

$$\text{Acquisition}\text{Time}=t_i\left(\frac{N_l{e}^{C_{\text{tis}}·{\text{FWHM}}_x{\text{FWHM}}_y{\text{FWHM}}_z}}{C_{\text{tis}}{\sigma }_x{\sigma }_y{\sigma }_z}\right)$$ (16)

or

$$\text{Acquisition}\text{Time}=t_i\left(\frac{N_l{e}^{C_{\text{tis}}.{\text{V}}_{\text{PSF}}}}{C_{\text{tis}}{V}_{\text{SR}}}\right)$$ (17)

where

$$t_i=\{\begin{array}{ll}{t}_{\text{iFL}},\hfill & t_{\text{iFL}}>t_{\text{iAL}}\hfill \\ t_{\text{iAL}}\hfill & t_{\text{iFL}}\le t_{\text{iAL}}\hfill \end{array}.$$

D. Investigations/Specific Models

If not varied, parameters within models are fixed under conditions typical for abdominal imaging with: transmit frequency 4 MHz; depth 5 cm; flow velocity 5 mm/s; N_l = 1; and resolution improvement λ/30, where λ is the transmit wavelength, and localization precision is 10 μm in 2-D, where V_PSF = FWHM_xFWHM_yΔz and A_SR is used in place of V_SR. The FWHM is estimated by λ/2. Acquisitions were also modeled over a range of acquisition parameters, SR localization precision values, and bubble signal densities to explore the impact on acquisition time.

1). Tissue Variations

a). Probability of bubble events (Infusion)

The probability of imaging bubble events across varying tissue types was determined using (3)-(5), where the tissue blood volume fraction ranges between that of cholangiocellular cancer, hepatocellular carcinoma, normal liver parenchyma, and other highly perfused tissue or macrovessels. Models were performed using a constant microbubble infusion, where the microbubble signal density in the blood, C_b, is optimized for imaging normal liver parenchyma, i.e., where C_b · V_ν · V_PSF = 1.

b). Probability of bubble events (Bolus Injection)

The probability of imaging bubble events was determined as described above at various time-points post bolus injection. This was performed using a peak blood microbubble concentration optimized for imaging normal liver parenchyma.

c). Effect of vascular velocity

SR acquisition times were determined using (17), where bubble velocities, ν_b, were varied between the slow flow of the microvasculature, up to fast flow in the aorta, using a constant microbubble infusion optimized for normal liver parenchyma.

d). Effect of transmit frequency

The number of frames required to create an SR image with fixed abdominal parameters was modeled using transmit frequencies between 0.5 and 15 MHz.

e). Combined effects on acquisition time

SR acquisition times were again determined using (17), where the combined effects of transmit frequency, signal density, level of resolution improvement, and vascular velocities were modeled.

III. Results

Fig. 4 shows the probability of imaging bubble events within the diffraction limit across varying tissue blood volume fractions, V_ν. Here, the bubble concentration in the blood is optimized for imaging normal liver parenchyma (orange region) of 26% [28] using a constant infusion. Ranges of tissue blood volumes are also shown for regions of cholangiocellular cancer (pink) and hepatocellular carcinoma (yellow). Blood volumes above 40% (purple) may represent highly perfused tissue or macrovessels. Vessels with a diameter above λ/2 are at 100% blood volume per diffraction limit. For these vessels, the probability of imaging individual bubbles at this concentration is 0.09, while the chance of multiple bubbles occurring within the diffraction limit reaches 0.88.

Fig. 4.

Probability of imaging a single bubble within the diffraction limit across tissues containing varying tissue blood volume fractions (% of total tissue) when bubble concentration in the blood is optimized for imaging the normal liver parenchyma (orange region). Ranges of blood volume fractions are also shown for regions of cholangiocellular cancer (pink), hepatocellular carcinoma (yellow), and for blood volumes above 40% which may represent highly perfused tissue or macrovessels (purple). Vessels with a diameter above λ/2 are assumed to have a vascular volume fraction of 100%.

Fig. 5 shows the concentration of sulfur hexafluoride in the blood following intravenous administration of SonoVue in healthy volunteers as a bolus 0.03-ml/kg dose [Fig. 5(a), blue curve]. This shows the probability of imaging a single microbubble within the diffraction limit for various time points post-injection [Fig. 5(b)] demonstrated by corresponding dashed lines as shown in Fig. 5(a). In this example, peak bubble concentration in the blood is optimized for imaging cancerous tissue, i.e., for V_ν = 13%, between cholangiocellular cancer (pink) and hepatocellular carcinoma (yellow). Ranges of blood volume fractions are shown in Fig. 4. Fig. 5(b) demonstrates the difference in potential detections using a bolus injection in contrast to constant infusion shown in Fig. 4. In this case, the optimum probability of detecting isolated bubbles occurs over a range of tissue types during bolus circulation. The steep concentration increase at inflow means optimal signal density moves from regions of high blood volume to those with lower as the blood concentration peaks. Again, at the latter part of bolus circulation, signal detections in tissues of high blood volume are likely to surpass those in less vascular regions due to the bubble concentration decrease [29].

Fig. 5.

(a) Concentration of sulfur hexafluoride in the blood following intravenous administration of SonoVue in healthy volunteers for 0.03-ml/kg dose (thin blue curve), spline fit (thick blue curve), data taken from [29]. (b) Probability of imaging a single microbubble within the diffraction limit over varying tissue blood volume fractions is shown, where each curve is generated for various time points post-injection. (A) Time points are shown by corresponding dashed lines. Here, peak bubble concentration in the blood is optimized for imaging cancerous tissue, cholangiocellular cancer (pink), and hepatocellular carcinoma (yellow). Ranges of blood volumes are also shown for regions of the normal liver parenchyma (orange region) and for blood volumes above 40% which may represent highly perfused tissue or macrovessels (purple).

The estimated number of frames required to create an SR image decreases with increasing transmit frequency (Fig. 6) for a fixed depth of 5 cm, localization precision, and at optimum bubble concentration. In general, relative improvements in the final SR precision compared with the initial diffraction-limited resolution ultimately determines the number of frames required.

Fig. 6.

Number of frames required to create an SR image for a given localization precision, see the legend. The number of frames decreases with increasing transmit frequency for a fixed depth of 5 cm at optimum bubble concentration. Increasing the SR localization precision will increase the number of localizations required in the final image, and therefore increase the number of frames needed.

At 5-cm depth with 10-μm localization precision, Fig. 7(a) shows that the effective frame rate limits for blood velocities ranging from the microvasculature (<1 mm/s) to the aorta (~45 cm/s) are restricted by the bubble velocity at low blood flows (<150 mm/s), but are only restricted by the time-of-flight at higher flows. Therefore, the estimated minimum time needed for SR increases for slower blood velocities, and for improved resolution [Fig. 7(b)]. In this example, to obtain an SR of λ/10 while imaging the microvasculature, the estimated minimum time is 56 s, while for λ/20 improvement, this increases to 13 min.

Fig. 7.

(a) Frame rate limits are defined by the bubble velocity at low blood flows, while imaging at high blood flows becomes restricted only by the time-of-flight. Localization precision 10 μm. (b) Acquisition time required to create SR images for a target region at 5-cm depth with blood velocities ranging between the microvasculature (yellow), veins and arterioles (red), vena cava (blue), arteries (purple), and aorta (green). The graph demonstrates an increase in acquisition time with improvements in SR, shown as a fraction of the transmit wavelength λ.

Fig. 8 shows the acquisition time needed to produce SR images at a fixed depth of 5 cm with varying bubble signal density and resolution improvement when imaging vasculature of varying flow velocities (b), resolution improvement (c), and frequencies (d) and (e), with fixed resolution improvement and fixed localization precision, respectively. The curve shown in green shows the same conditions and is, therefore, identical throughout Fig. 8(b)–(e) using the typical abdominal imaging parameters. The acquisition time is shown to increase considerably away from the optimum bubble signal density, determined by the transmit frequency. Fig. 8(b) and (c) shows how the minimum possible acquisition time changes with blood flow velocity and resolution improvement, respectively. As the frequency lowers, the optimal signal density range narrows; however, the minimum possible time at the optimum concentration remains the same for fixed resolution improvement [Fig. 8(d)]. In contrast, reducing the frequency increases the acquisition time needed to achieve the same localization precision even at optimal signal density [Fig. 8(e)].

Fig. 8.

(a) Acquisition time needed to produce SR images at a fixed depth of 5 cm with varying bubble signal density and resolution improvement. Graphs show the time required when imaging vasculature of varying (b) flow velocities, (c) resolution improvement, and (d) and (e) frequencies, with a fixed level of resolution improvement (λ/30) and fixed localization precision (10 μm), respectively. Resolution improvement values for (E) are λ/62, λ/31, λ/21, and λ/15 for 2, 4, 6, and 8 MHz, respectively. If not varied, parameters are set with transmit frequency: 4 MHz, depth: 5 cm, flow velocity: 5 mm/s, improvement λ/30, and localization precision: 10 μm. The curve shown in green is constant throughout (a)–(d) and corresponds to the red line in (a). Dashed line in (a) indicates the optimum signal density.

IV. Discussion

The acquisition procedure, transmit frequency, localization precision, bubble concentration, and desired SR image contrast together determine the minimum acquisition time for given flow velocity and tissue blood volume fraction.

This model aims to provide insight into the relationship between imaging parameters and microbubble concentrationon the localization rate and overall acquisition time of US SR imaging. Under the assumptions of a Poisson distribution, this provides estimations of ideal bubble concentrations for minimizing acquisition time.

As expected, acquisition time increases with those parameters which tend to increase the acquisition time for all CEUS imaging techniques, as shown in (15), including: increased imaging depth, increased number of pulses within multipulse CEUS imaging techniques, increased compounding angles, and a decreased speed of sound in the medium of interest. However, this interrogation rate becomes further limited when imaging slow flows of the microvasculature, which impose an effective frame rate limit. The SR component [bracketed term of (17)] demonstrates that a better diffraction limited resolution, as provided by a higher transmit frequency, acts to decrease the acquisition time. This is due to the opportunity to extract a higher number of spatially isolated bubble signals in each frame. The desired number of localizations per SR voxel, N_l, relates to the SNR in the resulting image; hence for a factor of improvement in the SNR, the acquisition time required will increase by the same factor. This relies upon the algorithm’s ability to separate multiple bubble events. In the event that multiple bubbles falling within one voxel are detected by the algorithm as a single detection, an incorrect localization position may be found. If the algorithm is not able to adequately isolate bubble signals, incorrect localizations from overlapping or interfering signals will decrease the SNR in the resulting image. This highlights the importance of accurate signal detection and isolation.

Equation (17) also demonstrates an inverse relationship between imaging time and localization error; this is due to the increased size of the Gaussian localization plots in the final SR image.

The results show that the use of an optimal concentration is crucial in reducing acquisition times, and thus in ensuring clinical feasibility. It was shown that to obtain an SR of λ/10 while imaging the slow-moving microvasculature flow at 5-cm depth, the acquisition time could be as low as 56 s. If SonoVue¹ microbubbles are administered as an intravenous infusion (VueJect¹, Bracco, Milan) at a rate of 5 ml/min, it is estimated that with an average of 300 million microbubbles per ml [31], and an average human blood volume of 4.9 L, 5102 microbubbles will be introduced per milliliter per second. After 30 s, 1.5 × 10⁵ bubbles/ml will have been introduced, compared to an optimum signal density provided by the model of approximately 0.1–1.5 × 10⁵ signals/ml. Accounting for a reduction in the number of bubbles which are detected during imaging due to issues such as dissolution, the proportion of bubbles reaching the target area, and the polydispersed nature of the bubble population, current estimations of signal density appear to be in the practical range for clinical imaging.

Dencks et al. [32] provided an exponential expression for the acquisition time needed for the localization coverage in an image to saturate to a value assumed to be proportional to relative blood volume (rBV). The reliability of rBV estimates from shortened measurement times is then examined experimentally for a specific imaging target and imaging parameters (bolus scans of mouse tumors). Our study aims to develop a generalized model that is able to predict the required SR imaging time from user-input imaging parameters, microbubble concentrations, and target vasculature. The model aims to predict the optimal bubble signal density for specific imaging conditions for SR. With imaging parameters and tissue targets comparable to Dencks et al. [32] (40-MHz transmit frequency, 5-μm SR, 50-Hz frame rate, and bolus injection), the estimated acquisition time to obtain 90% localization coverage using our model was comparable to the 2017 study for tumor imaging (78–139 s depending on the peak blood concentration during bolus injection, and 50–101 s, respectively).

Signal density will vary depending on many details of the imaging acquisition, including transmit frequency, bubble population and behavior, bandwidth, and background noise. The corresponding suitable injection concentration will depend not only on these factors but also on aspects of the practical setup, e.g., the proportion of bubbles reaching the target area, the disease condition, and administration type (bolus/infusion), so could be patient- and disease-dependent.

An increase in temporal resolution using fast plane waves should provide a higher bubble localization rate for a given microbubble concentration (providing bubbles are not destroyed), and moreover, may improve velocity estimations due to more frequent sampling. Since SR imaging relies upon the combined contributions of many localizations over time, for a given microbubble concentration, a greater frame rate could, therefore, result in a decrease in the overall acquisition time. This, however, does not take into account the flow, and therefore this justification is more relevant to situations in which individual targets are activated and deactivated. Nevertheless, the SR technique requires that the microbubbles sample the entire microvascular structure during acquisition to provide full spatial information. This, therefore, places a limit on the minimum imaging time possible for adequate visualization of slow flow in microvessels.

In the case of fast imaging of moving bubbles, two competing factors are at play when thinking about acquisition time; these relate to blood flow velocity and frame rate. First, in order for microbubbles to provide new spatial information in each frame, the bubbles must be moving and their position in each frame should contribute additional spatial information to the final rendering. In this study, additional spatial information is provided when a bubble has moved a distance beyond the localization precision in consecutive frames.

Conversely, the frame rate should be high enough that bubble motion during multipulse acquisition strategies, e.g., PI or CPS, does not drastically affect the result of coherent compounding. Bubble movement between each plane wave transmission may mean plane waves may not be added coherently and could result in artifacts, incomplete suppression of linear targets, or smearing or spreading of nonlinear signals in the direction of motion. It is noted that an axial displacement of approximately half a pulse wavelength during the time required to acquire a frame will lead to destructive interference in the compounding operation, and as a result causes image degradation [31]. Higher phase coherence is required to avoid motion artifacts in the axial direction than the lateral since the spatial frequency in the axial direction is much higher, whereas, in the lateral direction, the PSF acts as a spatial low-pass filter [31], [32]. Smearing due to fast bubble movement in compounded images may cause a reduction in localization accuracy. Nevertheless, this should not cause a problem to the final visualization if the bubble trajectory remains within the lumen of the vessel, i.e., without a sharp change in direction during each pulse sequence.

Results demonstrated that the optimum probability of detecting isolated bubbles occurs over a range of tissue types throughout the bolus circulation. While this means that the signal density varies with time post-injection for each region-of-interest, it provides the opportunity to obtain SR information over a range of tissue types. These results also demonstrate that there will be preferential times post-injection to visualize certain tissue types. For example, optimum SR imaging is likely to occur in areas of high tissue blood volume immediately after contrast arrival. After the fast inflow phase, the peak probability will change to lower tissue blood volumes and will slowly change back to higher blood volumes as the contrast agent dissolves. Even with constant signal density, some regions may not be fully represented in SR images if there are large variations in blood volume fractions.

There are a number of assumptions implemented in this model which could lead to discrepancies between modeled and experimental findings. The model assumes an ideal SR algorithm which is able to correctly detect single bubbles in all cases and reject those from multiple bubbles within one PSF sized volume. A more realistic imaging scenario would incorporate noise, as well as varying bubble signals such as ringing or interference signals created by clouds of bubbles; these would affect the ability of the algorithm to identify and accurately localize bubbles. A condition to account for the uneven spatial distribution of events could also be added in future models [33]. Additionally, in general, V_PSF is not isotropic; the elevational resolution is typically much larger than in-plane resolutions in 2-D, and this can be readily modeled by the method presented here. 3-D results can be easily extrapolated from the model by setting a diffraction-limited resolution and localization precision measure in the elevational direction, FWHM_z and σ_z, respectively.

Acquisition times in the order of 10 s of minutes would be undesirable in a clinical setting due to motion effects and the use of both clinician and patient time. Acquiring data with a 3-D probe is vital for future clinical application. Modeling of US-SR in a 3-D setting is required to explore the parameters required to minimize this time.

The standard clinical dose of SonoVue microbubbles in bolus form has been shown in clinical experiments to be far higher than desired for SR at its peak concentration. Instead, a slow infusion of the same clinical dose is preferred due to its lower and more constant signal density, and longer potential imaging time due to replenishment. Calculation of the required bubble concentration prior to imaging based on the imaging volume is challenging and is likely to be unfeasible in a clinical environment. Sustaining a suitable concentration of microbubble scatterers within the imaging volume may instead require the development of an automatic feedback system that regulates the bubble concentration. By automatically monitoring the signal density per frame during image acquisition according to the optimum predicted by Poisson statistics, the concentration information could be used to drive an infusion pump delivering adjustable microbubble infusion rates. This work could, therefore, form a basis for the development of more complex and realistic models for SR in the future. One possible approach for overcoming this limitation is high-density imaging. By increasing the density of potential localizations per frame, shorter acquisition times could be achieved. High density methods which exist for optical microscopy, such as DAOSTORM, which fits multiple overlapping PSFs in an iterative manner by analyzing pixel clusters in the residual image and obtains localizations by minimizing a least-squares criterion, and compressed sensing STORM (CS-STORM), which imposes sparsity priors on the distribution of signal sources and localizes based on a convex optimization problem, can provide increased recall rates. Various sparsity-based techniques have recently been adopted in the US field [16], [34], [35] to reduce acquisition times.

The number of required frames has been shown to increase for a decrease in the transmit frequency, and for an increase in SR precision. Lower frequencies will therefore increase the acquisition time for any given frame rate.

V. Conclusion

Calculations based on Poisson statistics demonstrate the importance of retaining an appropriate microbubble signal density to maintain viable acquisition times for clinical implementation. Too high and the occurrence of multiple, inseparable signals will limit the number of isolated signals detected; too low and the requirement for a large number of frames will result in long acquisition times. This, along with the acquisition procedure, transmit frequency, localization precision, vascular flow properties, and desired super-resolved image SNR together determine the optimal acquisition time for SR imaging. Exploring the effects of both system parameters and details of the target vasculature can allow a better choice of acquisition settings and provide an improved understanding of the completeness of SR information.

Biographies

Kirsten Christensen-Jeffries received the B.Sc. degree in maths and physics from the University of Warwick, Coventry, U.K., in 2010, and the M.Res. degree in biomedical imaging with Imperial College London, London, U.K., in 2011.

She spent eight months with IXICO, London in 2011 and 2012. She currently holds a post-doctoral research position with the Ultrasound (US) Imaging Group, Kings College London, London. Her research interests include contrast-enhanced US imaging, with a focus on the development of super-resolution (SR) US imaging techniques for visualization of the microvasculature.

Jemma Brown received the M.Phys. degree from the University of Oxford, Oxford, U.K., in 2015. She is currently pursuing the Ph.D. degree in medical imaging with the ESPRC Centre, Kings College London, London, U.K. and Imperial College London, London, with a focus on super-resolution (SR) ultrasound, under the supervision of Dr. Eckersley and Dr. Tang.

Sevan Harput received the B.Sc. degree in micro-electronics engineering and the M.Sc. degree in electronic engineering and computer sciences from Sabanci Üniversitesi, Istanbul, Turkey, in 2005 and 2007, respectively, and the Ph.D. degree from the University of Leeds, Leeds, U.K., in 2013.

He is currently a Research Associate with the Department of Bioengineering, Imperial College London, London, U.K., and a Visiting Scholar with King’s College London, London, and the University of Leeds, Leeds. His research interests include high frame-rate ultrasound (US) imaging, super-resolution (SR) imaging, US contrast agents, signal processing for biomedical imaging, nonlinear acoustics, US sensor modeling, and biomedical device development.

Dr. Harput has been an Editorial and Administrative Assistant with the IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control since 2013.

Ge Zhang received the B.Eng. degree in business finance from London School of Economics, University College London, London, U.K., in 2015, and the M.Sc. degree in biomedical engineering from Imperial College London, London, U.K., in 2016, where he is currently pursuing the Ph.D. degree in medical imaging, with a focus on contrast-enhanced ultrasound and photoacoustic imaging, super-resolution (SR) imaging, and molecular imaging, under the supervision of Prof. M.-X. Tang.

Jiaqi Zhu received the B.Eng. degree in optoelectronic and information engineering from the Nanjing University of Science and Technology, Nanjing, China, in 2015, and the M.Eng. degree in electronic and communications engineering from Nottingham University, Nottingham, U.K., in 2016. She is currently pursuing the Ph.D. degree in bioengineering from Imperial College London, London, U.K., with a focus on lymph node ultrasound imaging under the supervision of Prof. M.-X. Tang.

Meng-Xing Tang (M’05-SM’17) is currently a Professor and Chair of Biomedical Imaging with the Department of Bioengineering, Imperial College London, London, U.K. He leads the Ultrasound (US) Laboratory for Imaging and Sensing (ULIS), Imperial College London. He has authored more than 90 peer-reviewed journal papers. His current research mainly focuses on developing and applying new imaging techniques, particularly of ultrahigh temporal and spatial resolution using US, contrast agents, and computing.

Prof. Tang is an Associate Editor of the IEEE Transactions on Ultra-sonics, Ferroelectrics, and Frequency Control.

Christopher Dunsby received the M.S. degree from the University of Bristol, Bristol, U.K., in 2000, and the Ph.D. degree from Imperial College London, London, U.K., in 2003.

He is currently a Reader with a joint postition between Photonics, Department of Physics, and the Centre for Pathology, Department of Medicine, Imperial College London. His research interests include the application of photonics and ultrafast laser technology to biomedical imaging, multiphoton microscopy, multiparameter fluorescence imaging and fluorescence lifetime imaging, and methods for super-resolution (SR) ultrasound imaging.

Robert J. Eckersley received the B.Sc. degree in physics from the King’s College London, London, U.K., in 1991, and the Ph.D. degree from the Institute of Cancer Research, Royal Marsden Hospital, University of London, London, in 1997.

He was with Hammersmith Hospital NHS Trust, London and was awarded an MRC Research Training Fellowship in 1999. As part of this fellowship, he spent some time at the Sunnybrook Health Sciences Centre, University of Toronto, Toronto, ON, Canada. He subsequently continued as a Post-Doctoral Researcher with the Imaging Sciences Department, Imperial College London, where he became a Nonclinical Lecturer in ultrasound (US) in 2007. In 2012, he joined the Biomedical Engineering Department, King’s College London, as a Senior Lecturer. His research interests range from fundamental studies to clinical applications. Examples include image and signal analysis of US data for functional imaging or tissue characterization, nonlinear imaging for improved detection of microbubbles, and understanding errors and artifacts in US contrast imaging.