GPU-accelerated Bloch Simulations for bSSFP

The signal evolution of IR-bSSFP sequence has been well described in previous studies^{9 (link), 10 (link)}. We chose to use IR-bSSFP because of its simplicity in implementation, and its enhanced contrast by mixing the T1 and T2 relaxations in the signal evolution. We used GPU-based parallel computing to simulate bSSFP signal evolution in real-time for training the neural network, as shown in Figure 1. In the simulation algorithm, we used simplifications for the excitation and procession in one TR, and they are given as:
$$\text{Excitation}:R_{ex}\left(\alpha ,\varphi \right)=R_x\left(\varphi \right)R_z\left(\alpha \right)R_x\left(-\varphi \right)\text{and} M_{k,+}=R_{ex}\left(\alpha ,\varphi \right)M_{k,-}$$
where R_x and R_z are the standard SO(3) rotation matrices, α denotes the flip angle, and ϕ is the phase of RF, M_k,- and M_k,+ is the magnetizations before or after kth excitation respectively, and
$$\text{Procession}: M_{k+1,-}=diag\left(\left[E_2, E_2, E_1\right]\right)R_z\left(\phi \right)M_{k,+}+m_0\left[0, 0, 1-E_1\right]$$
where $E_{\mathrm{1,2}}=e^{-TR/T\mathrm{1,2}}$ , m₀ proton density, φ phase accumulation due to the frequency offset during procession period and M_k,+/M_k+1,- the magnetization at the beginning or ending time points of the k^th TR. The TR-by-TR evolution was sequentially computed, while different spins were simulated in parallel on GPU with a batch size of 100. On typical GPU hardware, e.g., NVIDIA TITAN X or 1070/1060 GTX, simulating hundreds of signal evolutions with 600 to 1000 TRs normally could be done within a few seconds.