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Hemodynamic monitoring plays a critical role in peri-operative care, where evaluating cardiovascular function is essential for guiding fluid therapy, detecting circulatory instability, and preventing organ failure. In addition to traditional vital signs, advanced hemodynamic parameters such as cardiac output are particularly valuable in high-risk surgical patients and those in intensive care. Yet, current advanced hemodynamic monitoring techniques are either highly invasive (pulmonary artery catheterization), intermittent (echocardiography) or prone to inaccuracies during hemodynamic fluctuations (pulse contour analysis). These limitations highlight the urgent need for safer, more accurate, and operator-independent alternatives. Ultrasound emerges as a promising solution for non-invasive hemodynamic monitoring as it is safe, portable, cost-effective, and widely available in clinical settings. The common carotid artery (CCA) is particularly suitable for flow monitoring due to its superficial location, consistent geometry, and correlation with central hemodynamic parameters. Multiple studies have demonstrated that CCA-derived metrics, such as corrected carotid flow time and peak systolic velocity, are predictive of fluid responsiveness and cardiac output, making the CCA a valuable target for continuous monitoring. Traditionally, the velocity waveform in the CCA is estimated using Doppler ultrasound in the longitudinal view, where the probe is parallel to the vessel. While this configuration allows for Doppler angle correction by assuming flow is parallel to the vessel wall, it is highly sensitive to probe positioning and motion, making it unsuitable for continuous monitoring. Moreover, velocity estimation in longitudinal Doppler typically uses a single sample volume at the vessel center, requiring assumptions about the velocity profile to estimate average flow that introduce inaccuracies. This thesis proposes and validates a cross-sectional Doppler ultrasound approach. By imaging the vessel in a rotated and tilted view, the full velocity profile and vessel diameter can be captured simultaneously. Under the assumption of a vascular cylindrical geometry, the intersection of the ultrasound plane with the vessel then forms an ellipse, from which the Doppler angle can be estimated geometrically. This configuration is more robust to motion, making it ideal for continuous monitoring applications. Initial in-vitro and in-vivo studies using a commercial ultrasound machine confirmed that cross-sectional Doppler yields accurate flow estimates and is less operator-dependent than conventional methods. A key requirement for this approach is an accurate vessel segmentation. While convolutional neural networks (CNNs) have shown promise in ultrasound image segmentation, their performance relies on large and labeled datasets that are challenging and time-consuming to acquire and manually annotate. To address this, an unsupervised domain adaptation framework was developed, where a CNN was first trained on simulated data and then adapted to in-vivo ultrasound images using model-based constraints derived from the expected elliptical vessel shape. This approach eliminates the need for manual annotations and achieved a median Dice similarity coefficient of 0.951 on a challenging in-vivo dataset, outperforming a domain-adversarial network and an active contour segmentation algorithm. To enable fully automatic operation, an adaptive multi-gate Doppler framework was developed. This system automatically adjusts transmit parameters based on real-time vessel segmentation. The framework was implemented on an ultrasound research platform and achieved high accuracy in both simulation (mean error 0.8%) and phantom experiments (mean error 1.6%). In addition, in-vivo feasibility was demonstrated, capturing diverse velocity profiles throughout the cardiac cycle. The validity of the cross-sectional approach was further investigated through a simulation study. Using a geometric model and computational fluid dynamics (CFD) simulations on ten 3D CCA geometries, this study quantified the impact of vessel curvature and non-circularity on flow estimation. It was found that vessel roundness significantly affects Doppler angle correction, while curvature has a lesser effect. Among three velocity estimation techniques evaluated (single-gate, multi-gate, and plane wave Doppler), multi-gate Doppler achieved the lowest median flow estimation error and was identified as a computationally efficient and accurate method for clinical use. Together, these contributions establish a motion-robust and operator-independent solution for continuous ultrasound-based hemodynamic monitoring, paving the way for wearable systems capable of real-time cardiac-output assessment.