# Real-Time HIFU Treatment Monitoring Using Pulse Inversion Ultrasonic Imaging
## 2 Materials and Methods  
### Proposed High-Intensity Focused Ultrasound (HIFU) Interference Elimination and Its Implementation  
Figure 1 shows the conceptual diagram of the proposed algorithm for HIFU interference elimination in which two steps of transmission and reception are performed to obtain one HIFU interference-free scanline. In the first step, the HIFU signal and the ultrasound imaging signal with \(0^{\circ}\) phase are simultaneously transmitted and the imaging transducer with the impulse response of \(h_{XDCR}(t)\) receives [17]:  
\[ \begin{array}{ll}r_{+}(t)&=\left\{\left[h_{f}(t)+h_{odd}(t)+h_{even}(t) \right]+\left[u_{f}(t)+u_{odd}(t)+u_{even}(t)\right]\right\}*h_{XDCR}(t)\\ &\approx\left[h^{\prime}{}_{f}(t)+h^{\prime}{}_{odd}(t)+h^{\prime}{}_{even}(t )\right]+u^{\prime}{}_{f}(t)\end{array}  \tag{1}\]where \(h_{f}(t)\), \(h_{add}(t)\), and \(h_{even}(t)\) are the fundamental, odd, and even harmonic components of the HIFU signal, \(u(t)\), \(u_{odd}(t)\), and \(u_{even}(t)\) are those of the ultrasound imaging signal. These signals are the ultrasound waves that are reflected from the tissues and reach the transducer. The asterisk in Equation (1) indicates the convolution operation. Since the full bandwidth of the imaging transducer is used to transmit ultrasound for imaging, only the fundamental component of the imaging signal \(u/(t)\) remains in the received signal. Note that only the fundamental and harmonic components of the HIFU signal of which frequency is in the passband of the imaging transducer remain in the received signal; the components play a role of HIFU interference represented by \(h/(t)\), \(h_{odd}(t)\), and \(h_{even}(t)\) in Equation (1). The apostrophe in the functions implies the convolutional result between the received ultrasound and the impulse response of the transducer. After \(T_{PRI}\) indicating the interval between the ultrasound transmissions with inverted phases, the HIFU signal and the ultrasound imaging signal with 180\({}^{\circ}\) phase are simultaneously transmitted and subsequently received. This step can be expressed as:  
\[r_{-}(t-T_{PRI})\approx\left[h^{\prime}_{f}(t-T_{PRI})+h^{\prime}_{odd}(t-T_{ PRI})+h^{\prime}_{even}(t-T_{PRI})\right]+u^{\prime}_{f}(t-T_{PRI}). \tag{2}\]  
Since the ultrasound imaging signal has a phase of 180\({}^{\circ}\) in the second step of transmission and reception, the fundamental and odd harmonic components of the imaging signal have a negative sign [19]. To obtain one scanline signal, the subtraction of \(r_{+}(t)\) and \(r_{-}(t-T_{PRI})\) is conducted as follows:  
\[SL(t)=r_{+}(t)-r_{-}(t-T_{PRI})=\Big{[}h^{\prime}_{f}(t)-h^{\prime}_{f}(t-T_{ PRI})\Big{]}+\Big{[}u^{\prime}_{f}(t)-u^{\prime}_{f}(t-T_{PRI})\Big{]}. \tag{3}\]  
Under the assumption that the target lesion is stationary, i.e.,  
\[h^{\prime}_{f}(t)=h^{\prime}_{f}(t-T_{PRI})\text{ and }u^{\prime}_{f}(t)-u^{ \prime}_{f}(t-T_{PRI}), \tag{4}\]  
Equation (3) becomes:  
\[SL(t)=2u^{\prime}_{f}(t), \tag{5}\]  
which is a HIFU interference-free scanline signal. In the proposed method, the maximum pulse length of HIFU is less than \(T_{PRI}\) because the HIFU and ultrasound imaging signal should be simultaneously transmitted. This means that the pulse repetition of HIFU should be equal to \(T_{PRI}\). Note that the frame rate of monitoring ultrasound imaging decreases at a given number of scanlines as the maximum pulse length of HIFU increases. Equation (5) is an ideal case because in practice the target lesion moves between the successive transmission and reception events, i.e., \(T_{PRI}\) due to patient motion and  
Figure 1: Conceptual diagram of the proposed method for high-intensity focused ultrasound (HIFU) interference elimination. breathing during HIFU surgery. This hampers the complete elimination of the HIFU interference signal. The amount of residual HIFU interference after the subtraction of \(r_{*}(t)\) and \(r_{-}(t-T_{PRI})\) depends on how fast the target lesion is moving; their relationship will be demonstrated below.