Ultrasound Array Research Articles

Broadband Ultrasound Detection Using Silicon Micro-Ring Resonators

High sensitivity, broadband bandwidth, small size and scalability to a fine-pitch matrix of ultrasound (US) sensors are essential for high-resolution ultrasonography and photoacoustic tomography. In this work, we demonstrated a method to build micro-ring resonators (MRR) as US sensors on the Silicon-On-Insulator (SOI) platform. To improve the sensitivity, transverse magnetic (TM) mode is utilized to increase the mode interaction with the cladding as well as reduce the scattering loss caused by sidewall roughness. Besides, the SU-8 elastomer is introduced as cladding to enhance US response. The fabricated 10 μm-radius MRR with only one standard lithography process shows a high quality (Q) of 7.4 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> and a steep slope of 351 dB/nm. The US sensing bandwidth reaches 165 MHz at −6 dB with a high sensitivity of 14.5 mPa Hz <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1/2</sup> . The high refractive index of the silicon waveguide contributes to an order of magnitude smaller sensing area than the polymer-based MRR. The proposed sensitive MRRs on the SOI platform makes it very promising for mass manufacturing integrated dense US arrays.

A Coregistered Ultrasound and Photoacoustic Imaging Protocol for the Transvaginal Imaging of Ovarian Lesions.

Ovarian cancer remains the deadliest of all the gynecological malignancies due to the lack of reliable screening tools for early detection and diagnosis. Photoacoustic imaging or tomography (PAT) is an emerging imaging modality that can provide the total hemoglobin concentration (relative scale, rHbT) and blood oxygen saturation (%sO2) of ovarian/adnexal lesions, which are important parameters for cancer diagnosis. Combined with coregistered ultrasound (US), PAT has demonstrated great potential for detecting ovarian cancers and for accurately diagnosing ovarian lesions for effective risk assessment and the reduction of unnecessary surgeries of benign lesions. However, PAT imaging protocols in clinical applications, to our knowledge, largely vary among different studies. Here, we report a transvaginal ovarian cancer imaging protocol that can be beneficial to other clinical studies, especially those using commercial ultrasound arrays for the detection of photoacoustic signals and standard delay-and-sum beamforming algorithms for imaging.

A Coregistered Ultrasound and Photoacoustic Imaging Protocol for the Transvaginal Imaging of Ovarian Lesions

Ovarian cancer remains the deadliest of all the gynecological malignancies due to the lack of reliable screening tools for early detection and diagnosis. Photoacoustic imaging or tomography (PAT) is an emerging imaging modality that can provide the total hemoglobin concentration (relative scale, rHbT) and blood oxygen saturation (%sO2) of ovarian/adnexal lesions, which are important parameters for cancer diagnosis. Combined with coregistered ultrasound (US), PAT has demonstrated great potential for detecting ovarian cancers and for accurately diagnosing ovarian lesions for effective risk assessment and the reduction of unnecessary surgeries of benign lesions. However, PAT imaging protocols in clinical applications, to our knowledge, largely vary among different studies. Here, we report a transvaginal ovarian cancer imaging protocol that can be beneficial to other clinical studies, especially those using commercial ultrasound arrays for the detection of photoacoustic signals and standard delay-and-sum beamforming algorithms for imaging.

Fully electronically steerable high power ultrasound phased arrays for therapy—A review of progress

When combined with imaging-guidance focused ultrasound (FUS) provides means for localized delivery of mechanical energy deep into tissues. This focal energy deposition can modify tissue function via thermal or mechanical interactions with the tissue. Traditionally, ultrasound has been focused using spherically curved phased arrays that allow limited electronic steering around geometric focus of the array. To have a fully electronically steered arrays requires the transducer element size to be &amp;lt; wavelength/2 of the ultrasound wave. One way to accomplish this is to use sparse arrays with small transducer elements, the other is to have fully populated arrays with thousands of elements and required electronic complexity. However, the electrical impedance of small transducer elements is high and their efficiency is low and it has been difficult to manufacture arrays with adequate acoustic power. In this talk we will describe how these challenges can be solved with novel manufacturing methods and how we developed complete image-guided systems for both animal and clinical testing. We will also describe our experience with these systems for both brain and body treatments.

Detecting myocardial perfusion deficit with contrast-enhanced high frame rate ultrasound

Imaging the myocardial perfusion after a heart attack has a high prognostic value, because it can guide the clinicians for better treatment before, during, or after percutaneous coronary intervention (PCI). Contrast-enhanced ultrasound is a promising imaging modality because it can be performed quickly on bedside within or outside the Cathlab. However, the image quality may generally not be good enough for accurate assessment of the myocardial perfusion. We have developed a technique that uses higher-order singular value decomposition of high frame rate echocardiographic data, in combination with contrast-specific pulsing schemes, to detect flow and perfusion. To assess the sensitivity of our technique to detect the local interruption of myocardial perfusion, we performed in vivo measurements on a porcine model. The left anterior descending coronary artery was occluded by using a balloon catheter which mainly interrupted the flow into the apex of the heart. Data were acquired with an ultrasound research scanner and phased array, and processed offline. The results showed that we can localize the blood flow inside the myocardium and detect the interrupt when the vessel is occluded, which shows the first proof of concept.

Super-resolution cellular ultrasound imaging via localization of nanodroplets

Ultrasound localization microscopy (ULM) had made it possible to differentiate microbubble contrast agents that are separated only by a few micrometers and, thus, opening the path for super-resolution imaging. However, due to the innate limits of acoustic waves and microbubble imaging, similar levels of resolution to optical imaging (i.e., submicron resolution) are still very challenging. In this paper, we utilize ULM to achieve cellular imaging by using phase-change perfluorocarbon nanodroplets (PFCnDs) as a contrast agent that were fabricated with spontaneous nucleation method. To identify the point spread function (PSF) of a single nanodroplet, gaussian fit function was applied for reconstruction. Nanodroplets were injected into cells through patch clamping and later captured with two transducers: a single element transducer for focused ultrasound and linear array transducer for imaging. Nonlinear imaging (NLO) was used to maximize the sound to noise ratio (SNR), which enables optimal amplitude. With the PSF of a phase-transitioned nanodroplet known, stochastic activation of multiple nanodroplets within a cell was accumulated to image a full cell morphology. These findings can lead researchers to develop effective ultrasound imaging paradigms that can visualize intracellular level of organs in deep tissue invivo.

Aluminum Nitride Piezoelectric Micromachined Ultrasound Transducer Arrays for Non-Invasive Monitoring of Radial Artery Stiffness

An aluminum nitride (AlN) piezoelectric micromachined ultrasound transducer (PMUT) array was proposed and fabricated for non-invasive radial artery stiffness monitoring, which could be employed in human vascular health monitoring applications. Using surface micromachining techniques, four hexagonal PMUT arrays were fabricated within a chip area of 3 × 3 mm2. The mechanical displacement sensitivity and quality factor of a single PMUT were tested and found to be 24.47 nm/V at 5.94 MHz and 278 (in air), respectively. Underwater pulse-echo tests for the array demonstrated a -3 dB bandwidth of 0.76 MHz at 3.75 MHz and distance detection limit of approximately 25 mm. Using the PMUT array as an ultrasonic probe, the depth and diameter changes over cardiac cycles of the radial artery were measured to be approximately 3.8 mm and 0.23 mm, respectively. Combined with blood pressure calibration, the biomechanical parameters of the radial artery vessel were extracted using a one-dimensional vascular model. The cross-sectional distensibility, compliance, and stiffness index were determined to be 4.03 × 10-3/mmHg, 1.87 × 10-2 mm2/mmHg, and 5.25, respectively, consistent with the newest medical research. The continuous beat-to-beat blood pressure was also estimated using this model. This work demonstrated the potential of miniaturized PMUT devices for human vascular medical ultrasound applications.

An Ultrasonic Adaptive Beamforming Method and Its Application for Trans-Skull Imaging of Certain Types of Head Injuries; Part II: Reception Mode and Adaptive Imaging.

Background theory and a new algorithm for single-point adaptive focusing in transmission mode through ultrasonic barriers via one-dimensional phased arrays were reported in part I. In this paper the algorithm is further extended and implemented into a full adaptive beamforming process, including complete transmission and reception modes. Corrected time delay patterns, adapted to the local acoustical and geometrical properties of the barrier, are calculated and applied in both modes. Further, an adaptive imaging process is also developed that implements the proposed beamforming process for two-dimensional imaging through randomly shaped multilayered phase-aberrating structures. The method is optimized for the case of human skull as the ultrasound barrier and its application for transcranial imaging is discussed. Laboratory results of adaptive imaging through realistic skull-mimicking phantoms are presented. The algorithms are implemented on a 64-channel ultrasound open-source phased array platform controlling a standard 128-element biomedical phased array. Irregularly shaped reflectors with characteristic dimensions of the order of ∼0.5 mm to ∼4.5 mm were used as targets behind the skull phantoms in our experiments. Minimum and maximum distortional target displacements of 2.2 mm and 25.3 mm (in 12 cm depth) were observed in sonograms when uncompensated time delays were used. By contrast, the positioning errors ranged from 0.0 to 0.9 mm when our algorithm was employed. The adaptive imaging results demonstrate strong potential of the proposed technique for diagnostic imaging of acoustically reflective head injuries directly through intact human skull.

Image-Guided Measurement of Radiation Force Induced by Focused Ultrasound Beams.

The radiation force balance (RFB) is a widely used method for measuring acoustic power output of ultrasonic transducers. The reflecting cone target is attractive due to its simplicity and long-term stability, at a reasonable cost. However, accurate measurements using this method depend on the alignment between the ultrasound beam and cone axes, especially for highly focused beams utilized in therapeutic applications. With the advent of dual-mode ultrasound arrays (DMUAs) for imaging and therapy, image-guided measurements of acoustic output using the RFB method can be used to improve measurement accuracy. In this article, we describe an image-guided RFB measurement of focused DMUA beams using a widely used commercial instrument. DMUA imaging is used to optimize the alignment between the acoustic beam and reflecting cone axes. In addition to image-guided alignment, DMUA echo data is used to track the displacement of the cone, which provides an auxiliary measurement of acoustic power. Experimental results using a DMUA prototype with [Formula: see text] shows that 1-2 mm of misalignment can result in 5%-14% error in the measured acoustic power. In addition to the use of B-mode image guidance for improving measurement accuracy, we present preliminary results demonstrating the benefit of displacement tracking using real-time DMUA imaging during the application of (sub)therapeutic focused beams. Displacement tracking provides a direct measurement of the radiation force with high sensitivity and follows the expected dependence on changes in amplitude and duty cycle (DC) of the focused ultrasound (FUS) beam. This could lead to simpler, more reliable methods for measuring acoustic power based on the radiation force principle. Combined with appropriate computational modeling, the direct measurement of acoustic radiation force could lead to reliable dosimetry in situ in emerging applications such as transcranial FUS (tFUS) therapies.

Integration of microlenses on surface-micromachined optical ultrasound transducer array to improve detection sensitivity for parallel data readout.

This Letter reports the integration of microlenses (MLs) on a surface-micromachined optical ultrasound transducer (SMOUT) array to enable parallel ultrasound data readout from a multiplicity of elements. The MLs are fabricated by photoresist patterning and reflow, and their focal lengths are optimized with parametric studies. Experiments are conducted to characterize the acoustic responsivity and its uniformity of the SMOUT-ML elements under different conditions. The temporal stability of SMOUT-ML elements immersed in water is assessed by monitoring their acoustic response continuously for 1 week. Parallel ultrasound signal readout is simulated with a small group of SMOUT-ML elements. Experimental results show that high acoustic sensitivity and excellent long-term stability can be achieved by the ML-integrated SMOUT array, which could provide a promising approach for enabling parallel ultrasound data acquisition for improving the imaging speed of 3D acoustic tomography.

Super-resolution ultrasonic Lamb wave imaging based on sign coherence factor and total focusing method

Improving the resolution of ultrasound array imaging systems for multiple defects in a measured medium is a popular research topic in nondestructive testing technology. However, because of the diffraction of acoustic waves, the resolution of ultrasound array imaging systems must follow Rayleigh's criterion and hence is limited, and defects with a spacing smaller than the resolution limit cannot be identified using conventional ultrasound imaging methods. Ultrasonic Lamb waves have dispersion and mode conversion characteristics, and the scattering acoustic field after the interaction between Lamb waves and defects is usually complex, making it even more difficult to improve the imaging resolution. In this paper, ultrasonic Lamb wave imaging is investigated with the sign coherence factor-total focusing method (SCF-TFM) when the spacing of the detected defects is smaller than the resolution limit, and the imaging resolution limit of SCF-TFM is analyzed. Experiments on a single blind-hole defect show that the lateral resolution of the SCF-TFM imaging is nearly twice as high as that of the total focusing method (TFM), with a higher signal-to-noise ratio. For experiments on multiple blind-hole defects, the resolution of the SCF-TFM improves by 71.32%, which is higher than that of the TFM when the spacing of the detected defects is larger than the resolution limit. The SCF-TFM enables super-resolution imaging and achieves a higher resolution limit than time-domain topological energy when the spacing of the detected defects is smaller than the resolution limit.

Reconstruction of thermoacoustic emission sources induced by proton irradiation using numerical time reversal

Objective. Mapping of dose delivery in proton beam therapy can potentially be performed by analyzing thermoacoustic emissions measured by ultrasound arrays. Here, a method is derived and demonstrated for spatial mapping of thermoacoustic sources using numerical time reversal, simulating re-transmission of measured emissions into the medium. Approach. Spatial distributions of thermoacoustic emission sources are shown to be approximated by the analytic-signal form of the time-reversed acoustic field, evaluated at the time of the initial proton pulse. Given calibration of the array sensitivity and knowledge of tissue properties, this approach approximately reconstructs the acoustic source amplitude, equal to the product of the time derivative of the radiation dose rate, mass density, and Grüneisen parameter. This approach was implemented using two models for acoustic fields of the array elements, one modeling elements as line sources and the other as rectangular radiators. Thermoacoustic source reconstructions employed previously reported measurements of emissions from proton energy deposition in tissue-mimicking phantoms. For a phantom incorporating a bone layer, reconstructions accounted for the higher sound speed in bone. Dependence of reconstruction quality on array aperture size and signal-to-noise ratio was consistent with previous acoustic simulation studies. Main results. Thermoacoustic source distributions were successfully reconstructed from acoustic emissions measured by a linear ultrasound array. Spatial resolution of reconstructions was significantly improved in the azimuthal (array) direction by incorporation of array element diffraction. Source localization agreed well with Monte Carlo simulations of energy deposition, and was improved by incorporating effects of inhomogeneous sound speed. Significance. The presented numerical time reversal approach reconstructs thermoacoustic sources from proton beam radiation, based on straightforward processing of acoustic emissions measured by ultrasound arrays. This approach may be useful for ranging and dosimetry of clinical proton beams, if acoustic emissions of sufficient amplitude and bandwidth can be generated by therapeutic proton sources.

High-fidelity deep functional photoacoustic tomography enhanced by virtual point sources

Photoacoustic tomography (PAT), a hybrid imaging modality that acoustically detects the optical absorption contrast, is a promising technology for imaging hemodynamic functions in deep tissues far beyond the traditional optical microscopy. However, the most clinically compatible PAT often suffers from the poor image fidelity, mostly due to the limited detection view of the linear ultrasound transducer array. PAT can be improved by employing highly-absorbing contrast agents such as droplets and nanoparticles, which, however, have low clinical translation potential due to safety concerns and regulatory hurdles imposed by these agents. In this work, we have developed a new methodology that can fundamentally improve PAT’s image fidelity without hampering any of its functional capability or clinical translation potential. By using clinically-approved microbubbles as virtual point sources that strongly and isotropically scatter the local pressure waves generated by surrounding hemoglobin, we can overcome the limited-detection-view problem and achieve high-fidelity functional PAT in deep tissues, a technology referred to as virtual-point-source PAT (VPS-PAT). We have thoroughly investigated the working principle of VPS-PAT by numerical simulations and in vitro phantom experiments, clearly showing the signal origin of VPSs and the resultant superior image fidelity over traditional PAT. We have also demonstrated in vivo applications of VPT-PAT for functional small-animal studies with physiological challenges. We expect that VPS-PAT can find broad applications in biomedical research and accelerated translation to clinical impact.

3-D Protoacoustic Imaging Through a Planar Ultrasound Array: A Simulation Workflow.

Bragg peak range uncertainties are a persistent constraint in proton therapy. Pulsed proton beams generate protoacoustic emissions proportional to absorbed proton energy, thereby encoding dosimetry information in a detectable acoustic wave. Here, we seek to derive and model 3D protoacoustic imaging with an ultrasound array and examine the frequency characteristics of protoacoustic emissions. A formalism is presented through which protoacoustic signals can be characterized considering transducer bandwidth as well as pulse duration of the incident beam. We have also collected an experimental proton beam intensity signal from a Mevion S250 clinical machine to analyze our formalism. We also show that proton-acoustic image reconstruction is possible even when the noise amplitude is larger than the signal amplitude on individual transducers. We find that a 4μ s Gaussian proton pulse can generate a signal in the range of MHz as long as the spatial heating function has sufficiently high temperature gradients.

Particle Size Measurement of Liquid–Solid Two-Phase Media Using Multifrequency Ultrasound Array

Particle Size Measurement of Liquid–Solid Two-Phase Media Using Multifrequency Ultrasound Array

Estimating Thrombus Elasticity by Shear Wave Elastography to Evaluate Ultrasound Thrombolysis for Thrombus With Different Stiffness.

There is uncertainty about deep vein thrombosis standard treatment as thrombus stiffness alters each case. Here, we investigated thrombus' stiffness of different compositions and ages using shear wave elastography (SWE). We then studied the effectiveness of ultrasound-thrombolysis on different thrombus compositions. Shear waves generated through mechanical shaker and traveled along thrombus of different hematocrit (HCT) levels, whereas 18-MHz ultrasound array used to detect wave propagation. Thrombus' stiffness was identified by the shear wave speed (SWS). In thrombolysis, a 3.2 MHz focused transducer was applied to different thrombus compositions using different powers. The thrombolysis rate was defined as the percentage of weight loss. The estimated average SWS of 20%, 40%, and 60% HCT thrombus were 0.75 m/s, 0.44 m/s, and 0.32 m/s, respectively. For Thrombolysis, the percentage weight loss at 8 MPa Negative pressure for the same HCT groups were 23.1%, 35.29%, and 39.66% respectively. SWS is inversely related to HCT level and positively related to thrombus age. High HCT thrombus had higher weight loss compared to low HCT. However, the difference between 20% and 40% HCT was more significant than between 40% and 60% HCT in both studies. Our results suggest that thrombus with higher SWS require more power to achieve the same thrombolysis rate as thrombus with lower SWS. Characterizing thrombus elastic property undergoing thrombolysis enables evaluation of ultrasound efficacy for fractionating thrombus and reveals the appropriate ultrasound parameters selection to achieve a certain thrombolysis rate in the case of a specific thrombus stiffness.

Estimation of the Thickness Profile of a Human Skull Phantom by Ultrasound Methods Using a Two-Dimensional Array

The paper presents the results of evaluating the thickness profile of a skull phantom using a two-dimensional ultrasound array consisting of piezoelectric elements with a center frequency of 2.1 MHz. Two pulse-echo ultrasound methods were used in the experiment: the A-mode elementwise measurements and scanning with a focused probing beam created by the entire array using delay-and-sum (DAS) beamforming. The obtained thickness profiles are compared with the reference thickness profile obtained using X-ray computed tomography. It was shown that ultrasound DAS beamforming with a focused probing beam makes it technically possible to estimate the thickness profile of the skull phantom.

A Rapid Prototyping Method for Sub-MHz Single-Element Piezoelectric Transducers by Using 3D-Printed Components.

We present a rapid prototyping method for sub-megahertz single-element piezoelectric transducers by using 3D-printed components. In most of the early research phases of applying new sonication ideas, the prototyping quickness is prioritized over the final packaging quality, since the quickness of preliminary demonstration is crucial for promptly determining specific aims and feasible research approaches. We aim to develop a rapid prototyping method for functional ultrasonic transducers to overcome the current long lead time (>a few weeks). Here, we used 3D-printed external housing parts considering a single matching layer and either air backing or epoxy-composite backing (acoustic impedance > 5 MRayl). By molding a single matching layer on the top surface of a piezoceramic in a 3D-printed housing, an entire packaging time was significantly reduced (<26 h) compared to the conventional methods with grinding, stacking, and bonding. We demonstrated this prototyping method for 590-kHz single-element, rectangular-aperture transducers for moderate pressure amplitudes (mechanical index > 1) at focus with temporal pulse controllability (maximum amplitude by <5-cycle burst). We adopted an air-backing design (Type A) for efficient pressure outputs, and bandwidth improvement was tested by a tungsten-composite-backing (Type B) design. The acoustic characterization results showed that the type A prototype provided 3.3 kPa/Vpp far-field transmitting sensitivity with 25.3% fractional bandwidth whereas the type B transducer showed 2.1 kPa/Vpp transmitting sensitivity with 43.3% fractional bandwidth. As this method provided discernable quickness and cost efficiency, this detailed rapid prototyping guideline can be useful for early-phase sonication projects, such as multi-element therapeutic ultrasound array and micro/nanomedicine testing benchtop device prototyping.

An endoluminal cylindrical sectored-ring ultrasound phased-array applicator for minimally-invasive therapeutic ultrasound.

The size of catheter-based ultrasound devices for delivering ultrasound energy to deep-seated tumors is constrained by the access pathway which limits their therapeutic capabilities. To devise and investigate a deployable applicator suitable for minimally-invasive delivery of therapeutic ultrasound, consisting of a 2D cylindrical sectored-ring ultrasound phased array, integrated within an expandable paraboloid-shaped balloon-based reflector. The balloon can be collapsed for compact delivery and expanded close to the target position to mimic a larger-diameter concentric-ring sector-vortex array for enhanced dynamic control of focal depth and volume. Acoustic and biothermal simulations were employed in 3D generalized homogeneous and patient-specific heterogeneous models, for three-phased array transducers with 32, 64, and 128 elements, composed of sectored 4, 8, and 16 tubular ring transducers, respectively. The applicator performance was characterized as a function of array configuration, focal depth, phasing modes, and balloon reflector geometry. A 16-element proof-of-concept phased array applicator assembly, consisting of four tubular transducers each divided into four sectors, was fabricated, and characterized with hydrophone measurements along and across the axis, and ablations in ex vivo tissue. Simulation results indicated that transducer arrays (1.5 MHz, 9mm OD × 20mm long), balloon sizes (41-50 mm expanded diameter, 20-60 mm focal depth), phasing mode (0-4) and sonication duration (30s) can produce spatially localized acoustic intensity focal patterns (focal length: 3-22 mm, focal width: 0.7-8.7 mm) and ablative thermal lesions (width: 2.7-16 mm, length: 6-46 mm) in pancreatic tissue across a 10-90 mm focal depth range. Patient-specific studies indicated that 0.1, 0.46, and 1.2cm3 volume of tumor can be ablated in the body of the pancreas for 120s sonications using a single axial focus (Mode 0), or four, and eight simultaneous foci in a toroidal pattern (Mode 2 and 4, respectively). Hydrophone measurements demonstrated good agreement with simulation. Experiments in which chicken meat was thermally ablated indicated that volumetric ablation can be produced using single or multiple foci. The results of this study demonstrated the feasibility of a novel compact ultrasound applicator design capable of focusing, deep penetration, electronic steering, and volumetric thermal ablation. The proposed applicator can be used for compact endoluminal or laparoscopic delivery of localized ultrasound energy to deep-seated targets.

Optimized auto-focusing method for 3D ultrasound imaging in NDT

In this work, we propose an optimized auto-focusing algorithm to be used with ultrasound matrix arrays for three-dimensional imaging in the presence of interfaces. Its main application is for non-destructive-testing (NDT), where usually two mediums with different propagation velocities are present. The refraction at the interface between those mediums complicates the calculation and the application of the focusing delays, which requires iterative processes that are usually the main bottleneck for high-speed inspections.We tackle this problem by generalizing the virtual array concept previously proposed for linear arrays and plane images to the case of matrix-arrays and three-dimensional (3D) beamforming. Furthermore, we propose a two-dimensional iterative algorithm for exact time-of-flight calculation to the two foci per scan line required by the virtual array. The mathematical formulation for the 3D case is obtained and the time-delay errors are analyzed by simulation. Finally, image quality with the proposed method is compared with that obtained with conventional Total Focusing Method (TFM) and exact delay calculation. The principal conclusion is that the virtual array approach generates images with equivalent quality to those obtained by exact calculation, while it reduces more than two orders of magnitude the computation load.