## 2 Materials and Methods  
### Measurement Principle  
Figure 2 shows the device we designed and built, featuring a grab handle and a fixture containing US transducers.  
Two array transducers with four elements each gently press against the arm laterally and medially, respectively. Both transducer arrays on each side of the arm are used for acquiring pulse-echo reflection signals and receiving transmission signals. To increase the chances of measuring the potentially existing growth plate, the system moves all transducers in distal and proximal directions during data acquisition while the subject is holding a grab handle tightly with her hand. The use of US array transducers eliminates the need to reposition the transducer as the skin's elasticity permits epidermal movements of up to 15 mm. The grab handle restrains the subject's arm movements, preventing any involuntary actions, while the transducers are moving along the wrist and arm. It also serves as a reference point for the arm, reducing the variation in measurement location. The linear orientation of the transducer elements enables measurement locations of subjects with different arm lengths to be taken into account. We designed the system for forearm sizes ranging from 31 to 55 mm at the epiphyseal plate of the radius bone, which was suitable in most cases. Our system (see Figure 3a) does not perform 2D US imaging and acquires only raw A-scans. We use rectangular tristate burst excitation signals with a length of 3 or 5 periods at center frequency for each measurement path. In each measurement modality, 25 A-scans containing 2048 values each are sampled as a motorized sled moves the transducers along the forearm (see Figure 3b). The lateral resolution of subsequent ultrasound measurements is at least 0.75 mm in the middle of the motion area and slightly  
Figure 2: The portable integrated ultrasound-based measurement device tested and evaluated.  
smaller at the sides due to acceleration and deceleration of the motor. This lateral resolution is smaller than the lateral extent of the ultrasound beam resolution of a single array element. Figure 4 shows the simulated acoustic pressure distribution of such a single array element. For this simulation, we assume an average speed of sound of 2500 m/s for the measurement through the wrist consisting of tissue (average speed of sound of 1540 m/s) and bone (average speed of sound of 3900 m/s for girls around maturity [38]) in non-equal parts. The processing and analysis presented later do not rely on any assumptions regarding the average speed of sound through the tissue and bone.  
Figure 5 depicts a sample reflection (left) and transmission (right) signal. The signals include clipped digitization of the transmitted burst pattern in the first approximately 5 \(\upmu\)s. Depending on the measurement mode the reflected ultrasound wave is directly  
Figure 4: The simulated acoustic pressure distribution of a single array element for a speed of sound of approximately 2500 m/s in bone.  
Figure 3: The compact, stackable ultrasound electronic module used in the device (**a**); the inbuilt ultrasound array (**b**).  
measured again at the same transducer (left), or the transmitted ultrasound wave is detected after traveling through the arm or wrist (right). While the reflected ultrasound signal is much stronger being measured after traveling only through skin and tissue layers, the transmitted signal amplitude is much lower, caused by the attenuation of the bone. The reflected signal is also detected over a longer period of time caused by multiple reflections between the transducer and bone. The transmission signal is only a superposition of the direct propagation through bone, the growth plate (if present), and the ultrasound wave traveling around the bone. In all cases, we analyze all significant parts of the signal not only containing the first arriving signal (FAS) but also multi-path propagations and multi-reflections (highlighted with orange boxes in Figure 5).