Vaccination can be performed on everyone in the world, throughout our lives. These vaccinations are often used as a protective measure against such diseases like Hepatitis B for newborns or to fight off infections as adults aged 65 and older. Sometimes an injection such as platelet-rich plasma (PRP) is needed to recover from acute or tendon injuries, an anesthetic solution is required before surgery, or steroid or stem cell injection needs to be accurately located. For these and numerous other injections, the medication’s accurate delivery optimizes the efficacy of its application. Physicians who perform these injections often use needles, cannulas, and other sharp objects, which can be invasive to the human body. The injected needle needs to travel through the body, passing nerves, blood vessels, and other internal body organs to reach the minute target area without causing a wound to the patient. In the current image-guided medication process, the practitioner first identifies the region of interest using an ultrasound probe [1]. When the desired anatomy is in view, the practitioner estimates the needle trajectory and insertion point. The inability to perfectly identify the needle tip makes it dangerous to advance the needle, leading to life-threatening seizures [2]. Therefore, the placement of this sharp object must be done with high precision. For this reason, navigated procedures are required to avoid unintended injury. Currently, many of these procedures require more experienced practitioners with the fine motor skill to navigate the needle inside the body safely [2]. Another disadvantage of this current system is that it requires two practitioners to perform the whole procedure: one performing the procedure and assisting with the ultrasound adjustments and medication injection. The requirements of the second practitioner for this task increase cost and limit efficiency. So, making ultrasound system with needle guidance easier to perform medicines in a cost-effective way [3]. Ultrasound with needle guidance is depicted in recent days for medical imagining years due to its lack of harmful effects on the human body. But the detailed analysis of the needle guidance system is still in a pristine state [4]. In this study, we tried to analyze this system mathematically and extract ultrasound waves’ behavior from transducers in different mediums and obstacles. Generally, an array of a transducer is used over the skin to generate an ultrasound wave into the body, and it collects the reflected signals. 2D images showing human organs or body parts are developed with the help of the reflected signals. The traditional ultrasound methods of using 2D imaging to find a 3D route lack precision. Even a skilled physician with a high degree of proficiencies, understanding, and synchronization from the clinician, the 2D guidance does not accurately show the precise location in all three dimensions, which is the end goal of the needle placement. To lessen this impediment, an ultrasound flashlight (USFL) consisting of a needle inserted in a target location of a living body to create an ultrasound image was invented (US Patent: 10,172,588 B1) [4]. Based on this idea, this paper proposes an experimental setup which can identify the presence and nature of obstacles between two transducers and can show varying grayscale image in real-time.

Imaging using piezoelectric crystals in a needle has the restriction of needle size that sets the limit of the needle’s crystal size. The crystal size limit then sets the frequency used in the imaging. The needle size also establishes the maximum array size. With these parameters for imaging, the focus of this research was to establish the mathematical foundation for imaging and the importance of near field imaging. In this paper, only 1D ultrasound images from a single transducer are demonstrated. Two needles, each consisting of one piezoelectric material and a delivery opening, are used as a transmitter and a receiver. Various metal objects of different sizes are used as obstacles to represent organs in the human body. Grayscale color mapping is used to describe the concentration or the obstacles present between the transducers.

An eSNR pulse-echo system was used that uniquely incorporates the effects of shift-varying impulse responses, spatiotemporal coding, and various beam-formers. Closed-form expressions were found to predict the measured eSNR gains using SMF and STF beamformers, and the gains are significant for an SMF in the near field [5] [6] [7]. In the case of an echo-signal model , a linear equation is considered to define pulse-echo RF signals g ( t ) in terms of the spatial distribution of an object function f ( x , t ′ ) with the spatiotemporal impulse response h ( t ) = h ( x , t ) [3] [4] [5] and it is written as,

g ( t ) = s ( t ) + e ( t ) (1)

where, s ( t ) = ∮ d x h ( x , t ) f ( x , t ′ ) .

Here, s ( t ) defines the RF signals, and e ( t ) is the noise.

Equation (1) can be transformed from continuous to discrete by sampling the signal in discrete times,

g [ l , m ] = s [ l , m ] + e [ l , m ] (2)

Here, l is described as 0 ≤ l ≤ L − 1 , and L is the range of sampling with interval T for one A-line in the frame, and m is described as 0 ≤ m ≤ M − 1 , and M defines the number of sequential A lines in the recorded frame. For a stationary scatter, f ( x , t ′ ) = f ( x ) and t ′ = ( l + m L ) T is the acquisition time of the echo signal. The echo signal to noise (eSNR) is another important term for echo-signal to measure the signal strength over noise [8] [9] [10] within the acquisition time. The eSNR for beamformed RF ultrasound data may be written as,

eSNR = E { ∑ l , m [ E { g [ l , m ] } e | f ] 2 } f / { ∑ ​   E { e 2 [ l , m ] } } e (3)

The notation E ( g ) denotes ensemble average of the echo frame g over the noise process e for a fixed realization of stochastic object f . For random scattering media, Equation (3) becomes,

eSNR = σ f 2 σ e 2 ( E h ) (4)

Ignoring the physical order of the signal formation process, the order of the sum and integral can be interchanged to give,

eSNR = σ f 2 σ e 2 ∮ d x ϕ h ( x , 0 ) (5)

The product of temporal correlation and volume is just the average pulse energy. The shift-invariant resultis,

eSNR r = σ f 2 σ e 2 ϕ h ( x , 0 ) (6)

A is the reflective amplitude of a single point scatterer, located at x for all t'. Combining the equations for eSNR_r and eSNR_p yields this relation,

eSNR p ( x ) = A 2 σ e 2 ϕ ′ h (7)

After applying conditions, shift-invariant systems for a single point scatterer yield,

eSNR r ( x ) = σ f 2 A 2 ∮ d x ϕ ′ h eSNR p ( x ) (8)

“Spatial filtering” [8] is a term often applied to the processing of RF echo data g to facilitate beamforming in the process of forming image f ^ . Spatial filtering effects are observed in the image. We reserve the term “temporal filtering” to refer to compression of coded pulses, and “spatiotemporal filtering” (STF) as a combination of these two methods [9].

The spatial matched filter (SMF) is beamforming filterable to concentrate more of the pulse energy than dynamic receive, focusing on providing a greater eSNR_p [2]. For a point target and matched-filter processing,

∑ l , m eSNR p ( m ) ( i ) ( x , [ l , m ] ) (9)

The eSNR r ( m ) ( x ) is the eSNR from random scattering media and matched-filter processing:

eSNR r ( m ) = σ f 2 A 2 ∮ d x eSNR p ( m ) ( x ) (10)

The corresponding eSNR gain for closed region Ω is,

γ r ( m ) Ω = ∮ d x eSNR p ( m ) ( x ) ∮ d x eSNR p ( x ) ≥ min γ p ( m ) ( x ) (11)

In the above case, the SMF is applied after the RF signals are passed through a beamformer, meaning the echo-signals are filtered after receive-channel summation. Alternatively, an SMF can be applied to signals from each receiver channel before they are added. The latter approach, called ESMF, generates higher lateral resolution and lower sidelobe amplitudes. The disadvantage is that the computational load goes up. The eSNR for the ESMF beamformer is

eSNR e p ( m ) ( x ) = A 2 N σ e 2 ∑ l , m Q 1 2 ( x , [ l , m ] ) Q 1 ( x , [ 0 , 0 ] ) (12)

The eSNR gain from temporal filtering of coded pulses is,

γ p ( t ) ( x ) = eSNR p ( t ) ( x ) eSNR p ( x ) (13)

temporal coded excitation enhancement γ r ( t ) and SMF enhancement γ r ( m ) . If decoding is successful, the eSNR is increased by K. If it is incomplete, range sidelobes are generated that distort h ( x , t ) / ϕ h . Because of this, the expression that follows is an approximation. It is assumed that the pulse compression is complete. The combined gain from spatiotemporal filtering is

γ r ( m , t ) = γ r ( m , t ) × γ r ( m , t ) = K ∮ d x e S N R p ( m ) ( x ) ∮ d x e S N R p ( x ) (14)

eSNR varies significantly throughout the field for instruments with shift-varying impulse responses. Unfocused near-field regions of a fixed-focus aperture system contain greater eSNRs compared to the focal region and far-field. The greatest gain from the SMF occurs in the far-field. Pulse compression of beamformed signals produces geometric errors in the near field more than in the far-field, thus generating range lobes.

In this experiment, two piezo materials are placed in the air medium, and data was taken from varying distances between two piezo materials. A sinusoidal signal with 10V peak-peak and 5 MHz is applied to the transmitter. Different obstacles were placed in between two piezo, and the detailed specification of obstacles is shown in Table 1. By varying distances and mediums, different obstacles were placed in between the transmitter and receiver, and the amplitude of the received signal was recorded as shown in Table 2 and Table 3, respectively.

Two piezoneedles are placed face to face at a distance (d), as shown in Figure 1. A sinusoidal signal is applied to a piezoelectric material as a transmitter (T) and received by another piezo as a receiver (R). The experimental laboratory setup for this work is shown in Figure 2. The received signal is then analyzed by MATLAB using an interface, and a decision is made about the medium through an image output. In our experiment, we have used three different mediums such as air, gelatin, and water, by changing the distance from 1 cm to 6 cm. We labeled the three mediums with three different colors. Black; means highly dense (gelatin), grey; means medium density (water), and white; means a less dense medium (air). Anything between less and highly dense is expressed as a grey. Decision-making about the medium will be discussed later.