^{1,a)}

### Abstract

A fluorophore-quencher (F-Q) labeled microbubble system is proposed as a sensor for measuring externally applied static pressure distribution in a tumor. To quantify the sensitivity of such an F-Q bubble system to the externally applied pressure, a model describing bubble response to the static pressure was derived. Additionally, a model connecting the fluorescence lifetime and bubble radius was developed for the basic F-Q bubble system. The sensitivity is quantified based on these models given typical parameters. Results show that it is possible to resolve as low as pressure variation when both the F-Q bubble system and the measurement system are optimized. Strategies for optimizing an F-Q bubble system are discussed.

I. INTRODUCTION

II. THEORETICAL MODELS: THE RESPONSE OF A MICROBUBBLE SYSTEM TO STATIC PRESSURE PERTURBATION

II.A. Review of microbubbles as pressure sensors

II.B. Response of microbubble radius to static pressure perturbation

II.C. Response of fluorescence lifetime to variation in microbubble radius

II.D. Phase delay and accuracy of phase measurement

II.E. Phase delay when considering bubble size distribution

II.F. Summary of the models

III. RESULTS AND DISCUSSION

III.A. Bubble response to externally applied pressure

III.B. Sensitivity of fluorescence lifetime and phase delay to external pressure

III.C. Effect of bubble size distribution

III.D. Measurement accuracy of phase

III.E. Challenges in manufacturing F-Q microbubbles

IV. SUMMARY AND CONCLUSION

### Key Topics

- Bubble dynamics
- 56.0
- Fluid bubbles
- 48.0
- Mercury (element)
- 29.0
- Cancer
- 24.0
- Fluorescence
- 24.0

## Figures

A perfluorocarbon-filled microbubble showing the inward and outward pressure components. The pressure caused by the elasticity of the shell is not shown because it is considered to be zero for a lipid shell in this study. Revised from Ref. 37. (b) Gas diffusion through the bubble shell and the liquid medium. is the gas concentration at the gas-shell boundary and is the gas concentration at the shell-liquid boundary (both are within the shell). represents the gas concentration at the shell-liquid boundary within the liquid and is the gas concentration at infinity in the liquid.

A perfluorocarbon-filled microbubble showing the inward and outward pressure components. The pressure caused by the elasticity of the shell is not shown because it is considered to be zero for a lipid shell in this study. Revised from Ref. 37. (b) Gas diffusion through the bubble shell and the liquid medium. is the gas concentration at the gas-shell boundary and is the gas concentration at the shell-liquid boundary (both are within the shell). represents the gas concentration at the shell-liquid boundary within the liquid and is the gas concentration at infinity in the liquid.

A flowchart describing the relationships among the models.

A flowchart describing the relationships among the models.

Normalized bubble radius as a function of time . The external pressure is applied at after the bubble is injected into the liquid medium and released at (upward and downward arrows). In (a) and , 0.4, and 0.2. In (b), and , 0.5, and 0. The multiple lines for each case represent the externally applied pressures from with increment.

Normalized bubble radius as a function of time . The external pressure is applied at after the bubble is injected into the liquid medium and released at (upward and downward arrows). In (a) and , 0.4, and 0.2. In (b), and , 0.5, and 0. The multiple lines for each case represent the externally applied pressures from with increment.

Relationship between the externally applied pressure and the relative change in bubble radius. (a) Effect of XF and on the relative change in the bubble radius. (b) Effect of surface tension and initial bubble radius on the relative change in the bubble radius.

Relationship between the externally applied pressure and the relative change in bubble radius. (a) Effect of XF and on the relative change in the bubble radius. (b) Effect of surface tension and initial bubble radius on the relative change in the bubble radius.

A typical relationship between the normalized fluorophore lifetime and the F-Q distance when [Eq. (14)]. The dashed lines indicate the lifetime ratio equals 0.5 when .

A typical relationship between the normalized fluorophore lifetime and the F-Q distance when [Eq. (14)]. The dashed lines indicate the lifetime ratio equals 0.5 when .

Phase difference between the phases when and as a function of the externally applied pressure when the surface tensions are 0.02, 0.04, 0.05, and .

Phase difference between the phases when and as a function of the externally applied pressure when the surface tensions are 0.02, 0.04, 0.05, and .

Phase sensitivity (deg/mm Hg) as a two-dimentional function of the initial bubble radius and the surface tension of the bubble: (a) , (b) , and (c) . The initial F-Q distance is .

Phase sensitivity (deg/mm Hg) as a two-dimentional function of the initial bubble radius and the surface tension of the bubble: (a) , (b) , and (c) . The initial F-Q distance is .

Phase difference between the phases when and as a function of the initial F-Q distance when the surface tensions are 0.02, 0.03, and .

Phase difference between the phases when and as a function of the initial F-Q distance when the surface tensions are 0.02, 0.03, and .

Phase sensitivity (deg/mm Hg) as a two-dimentional function of the initial bubble radius and the initial F-Q distance: (a) and , (b) and , (c) and , and (d) and .

Phase sensitivity (deg/mm Hg) as a two-dimentional function of the initial bubble radius and the initial F-Q distance: (a) and , (b) and , (c) and , and (d) and .

Effect of the STD of the bubble size distribution on the phase sensitivity to the external pressure (deg/mm Hg).

Effect of the STD of the bubble size distribution on the phase sensitivity to the external pressure (deg/mm Hg).

(a) Measurement error of phase as a function of signal to noise ratio and (b) the required SNR as a function of the number of averaging at specific phase resolutions.

(a) Measurement error of phase as a function of signal to noise ratio and (b) the required SNR as a function of the number of averaging at specific phase resolutions.

## Tables

Parameters for the microbubble, fluorophore, and modulation frequency (Refs. 36, 43, and 52). Typical parameters for microbubble were taken from Ref. 36, 43, and 52 and the parameters of fluorohores were chosen based on fluorescein. The modulation frequency is a typical value for detecting flurescence lifetime of the order of nanoseconds (Ref. 45 and 46).

Parameters for the microbubble, fluorophore, and modulation frequency (Refs. 36, 43, and 52). Typical parameters for microbubble were taken from Ref. 36, 43, and 52 and the parameters of fluorohores were chosen based on fluorescein. The modulation frequency is a typical value for detecting flurescence lifetime of the order of nanoseconds (Ref. 45 and 46).

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