This study was conducted using the HeartCycle v1.0.0 multimodal dataset, which is publicly available through PhysioNet (https://physionet.org/content/heartcycle/1.0.0/)^[26]. The dataset comprises synchronously acquired signals from 17 healthy subjects, including ECG, ICG, PCG, and ultrasound-derived reference information, as illustrated in Figure 1. These data provide a reliable basis for cardiac cycle-level event localization and hemodynamic parameter evaluation. Unlike studies relying solely on ICG, the HeartCycle dataset offers concurrent observations of electrical activity, mechanical vibration, and impedance variation, making it particularly well suited for multimodal temporal alignment and fusion modeling. All model training, validation, and testing in this study were performed exclusively on this dataset, without the inclusion of external data sources.

To preserve physiologically relevant components while suppressing noise, modality-specific preprocessing was performed. For ECG, the conventional rule-based analysis first used a 50 Hz notch filter to suppress power-line interference (Q = 30), followed by a fourth-order Butterworth 0.5-40 Hz band-pass filter to remove baseline wander and high-frequency electromyographic noise. In the deep learning model, ECG was resampled and standardized beat by beat as the temporal reference for electrical activation. For ICG, the conventional morphological analysis first applied a fourth-order Butterworth 0.7-20 Hz band-pass filter to the impedance signal Z, and then used a Savitzky-Golay filter to compute the first derivative dZ/dt^[27]. The S-G differentiation window length was 11 samples (approximately 55 ms at 200 Hz), the polynomial order was 3, and the derivative step was delta = 1/200 s. A fourth-order Butterworth 25 Hz low-pass filter was subsequently applied to ICG to suppress residual high-frequency noise. In the deep learning model, the ICG channel was represented by the first gradient of the resampled impedance signal and standardized by z-score normalization within each beat window. For PCG, the conventional ECG + PCG-assisted pipeline used a fourth-order Butterworth 20-200 Hz band-pass filter and smoothed the downsampled amplitude envelope with a 15 Hz low-pass filter. In the deep learning model, PCG was resampled to 4,000 Hz when necessary, filtered using a fourth-order Butterworth 20-800 Hz band-pass filter, transformed into an analytic-signal amplitude envelope by the Hilbert transform^[28,29], and resampled to 200 Hz for strict synchronization with ECG and ICG. After single-modality preprocessing, beat-level alignment and segmentation were performed using the ECG R peak as the temporal anchor. The main deep learning model used a fixed input window from 0.20 s before to 0.80 s after the R peak, with a unified sampling rate of 200 Hz, corresponding to 200 time points per beat, as shown in Figure 2. To further reduce the impact of amplitude differences among subjects on subsequent morphological analysis and model training, all segmented heartbeat segments underwent amplitude normalization or z-score standardization. After these steps, a multimodal beat-level input with strict temporal alignment, controlled noise levels, and clear physiological significance was obtained, laying a unified data foundation for subsequent ICG waveform morphological analysis and event localization based on multimodal constraints.

The reference time points for AVO and aortic valve closure (AVC) were obtained directly from the event annotations publicly released with the HeartCycle dataset rather than manually relabeled in this study. Specifically, the AVO and AVC event coordinates provided in the HeartCycle measurement files were used as the common reference standard for the onset and termination of ventricular ejection. The algorithmic B and X points reported in this study therefore denote feature locations predicted on the ICG waveform by rule-based or learning-based methods, whereas their supervised training and evaluation were based on the dataset-provided AVO and AVC reference labels.

Because the present work relies on public dataset annotations, potential label uncertainty may arise from ultrasound-derived event identification, multimodal synchronization, finite sampling resolution, beat matching, and small timing differences between waveform-derived ICG landmarks and true valvular mechanical events. Such uncertainty can affect feature-point localization and propagate into LVET estimation through the B-X interval. It may also influence downstream SV and CO estimates because these parameters are derived from LVET, heart rate, and ICG morphological descriptors. For this reason, the present study emphasizes parameter-level evaluation in addition to point-localization performance, and reports error distributions, confidence intervals, and paired statistical tests to characterize the impact of label and localization uncertainty on final hemodynamic estimates.

To systematically characterize morphological variation in ICG waveforms across cardiac cycles, unsupervised clustering and morphological classification analyses were performed on the valid preprocessed ICG beat segments. Given that physiological signals commonly exhibit local temporal stretching, phase shifts, and timing misalignment, dynamic time warping (DTW) distance was adopted to quantify waveform similarity and to construct the pairwise distance matrix. Compared with conventional point-to-point Euclidean distance, DTW is better suited for evaluating the overall similarity of ICG waveforms in the presence of temporal fluctuation^[30].

Based on the resulting distance matrix, K-medoids clustering was applied to the ICG segments to identify latent waveform morphological subtypes^[31]. To examine the distribution of waveform patterns under different levels of clustering granularity, the number of clusters, K, was set to 2, 3, 4, and 5, and the class composition and waveform differences under each clustering scheme were compared. It should be emphasized that the purpose of this analysis was not to establish a fixed clinical taxonomy, but rather to determine, from a data-driven perspective, whether single-modality ICG waveforms exhibit substantial morphological heterogeneity and feature overlap. This analysis also provides a basis for understanding the limitations of conventional rule-based methods in subsequent investigations.

To support both the discovery of ICG waveform morphologies and the evaluation of classification performance, this study conducts unsupervised morphological clustering and supervised classification at two separate levels. First, DTW-K-Medoids clustering is applied to all 1,908 valid ICG beat segments from 17 subjects to characterize the overall morphological heterogeneity of the ICG waveforms in the HeartCycle cohort, identify representative waveforms, and examine the class composition under different numbers of clusters. This step describes the morphological patterns present in the study cohort and defines candidate morphology classes for the subsequent classification tasks. On this basis, the supervised classification experiments adopt a fixed subject-level split to evaluate model generalization to unseen subjects. No subject overlap exists among the training, validation, and test sets, ensuring that multiple beat segments from the same subject do not appear in different data subsets. For each K value, the clustering prototypes, data standardization parameters, and class weights are determined exclusively from the training set. The validation and test sets are transformed, assigned labels, and evaluated using only the parameters obtained during training.

The same fixed subject-level data split is used for all classification tasks with K = 2, K = 3, K = 4, and K = 5 to ensure comparability across different numbers of clusters and model architectures. The training set contains 1,145 beat segments from 11 subjects, the validation set contains 381 beat segments from 3 subjects, and the test set contains 382 beat segments from 3 subjects. The three subsets are completely disjoint at the subject level, and all valid beat segments from each subject are assigned to only one subset. The fixed split comprising 11 training subjects, 3 validation subjects, and 3 test subjects is used exclusively for the ICG morphology classification experiments and is not applied to the subsequent LVET, SV, and CO estimation experiments. Morphology classification and hemodynamic parameter estimation use independent data construction and model training pipelines. Therefore, the subject composition and number of beat segments should be interpreted separately for the two tasks. To prevent information leakage from the test set into model training, all input standardization parameters are estimated using only the training set. Specifically, the mean μ_train(t) and standard deviation σ_train(t) are calculated at each time point t from the training data, and the same training-derived statistics are then used to standardize the training, validation, and test sets. For each K value, the K-Medoids clustering prototypes are fitted using only the training beat segments. Morphology labels for the validation and test segments are determined according to their DTW distances to the training-derived prototypes, with each segment assigned to the nearest prototype. Class weights are also calculated exclusively from the class frequencies in the training set. Let n_c denote the number of training samples in class c, and N_train denote the total number of training samples. The initial weight for class c is defined as N_train/n_c, after which all class weights are divided by their mean so that the normalized weights have an average value of 1. These weights are used only in the weighted cross-entropy loss during training, while the class frequencies in the validation and test sets are not involved in their calculation. The validation set is used only for model selection and for saving the model parameters associated with the best validation performance. The test set is evaluated only once after model training and parameter selection are complete.

To further justify the selection of the number of clusters, the clustering results obtained with K = 2, 3, 4, and 5 were evaluated in terms of validity and stability. Specifically, for each value of K, K-Medoids clustering was repeated using five different random seeds, and several clustering validity metrics were calculated, including the Silhouette score, Calinski-Harabasz index, Davies-Bouldin index, inter-/intra-cluster distance ratio, and minimum cluster proportion. To assess the sensitivity of the clustering results to random initialization, the adjusted Rand index (ARI) and normalized mutual information (NMI) were further used to evaluate label consistency across different random seeds. For each value of K, the five repeated clustering runs generated a total of 10 pairwise comparisons, which were used to quantify the consistency and stability of the cluster assignments under different random seeds.

Based on the clustering labels, we further developed an ICG morphology classification model, MS-GBiLSTM-TA, to evaluate the learnability of waveform features under different clustering granularities, as illustrated in Figure 3. The model takes a single ICG segment as input and employs a multi-branch convolutional architecture to extract local morphological features at different scales. A bidirectional long short-term memory network is then used to capture contextual dependencies along the temporal dimension. Subsequently, a temporal attention mechanism is introduced to adaptively weight informative time intervals, thereby enhancing discriminative capability for complex waveform patterns. Because the sample distribution was imbalanced across different clustering schemes, a weighted training strategy was adopted to mitigate the influence of class imbalance on classification performance. By comparing classification results across different clustering granularities, the inter-class separability of ICG waveforms and the clarity of their morphological boundaries could be further assessed.

This study adopts a single cardiac cycle (beat) as the fundamental unit of analysis. This design is motivated by the intrinsic periodicity of key ICG waveform components, including the systolic upstroke, incisura, and diastolic decay. In both clinical and engineering practice, hemodynamic parameters - such as LVET, SV, and CO - are typically reported on a beat-to-beat basis or through sliding-window averaging. Accordingly, defining model inputs, performing feature-point localization, and deriving parameters at the beat level not only aligns with real-world application requirements but also mitigates the influence of inter-cycle drift on model performance and estimation stability.

For the b-th cardiac cycle, the synchronized multimodal input segment is defined as

(1) $$ \begin{equation} \begin{aligned} \mathbf{X}^{(b)}=\left[\mathbf{x}_{\mathrm{ICG}}^{(b)}, \mathbf{x}_{\mathrm{ECG}}^{(b)}, \mathbf{x}_{\mathrm{PCG}}^{(b)}\right], \end{aligned} \end{equation} $$

where $$\mathbf{x}_{\mathrm{ICG}}^{(b)}$$, $$\mathbf{x}_{\mathrm{ECG}}^{(b)}$$ and $$\mathbf{x}_{\mathrm{PCG}}^{(b)}$$ denote the time-domain segments of the ICG, ECG, and PCG signals, respectively, within the b-th cardiac cycle. To ensure comparability across subjects and recordings, all modalities were cropped into fixed-length beat windows, with the ECG R peak serving as the temporal anchor for alignment. This alignment strategy preserves a consistent phase reference across beats, thereby reducing the influence of temporal offsets on model learning and preventing alignment errors from being misinterpreted as physiological variation. Within each input window, the first derivative of the ICG signal was used as the primary representation to emphasize dynamic morphological changes associated with mechanical ejection events; the ECG signal provided temporal reference for cardiac electrical activity, whereas the PCG signal supplied complementary acoustic information on valvular mechanical events.

This study compares four feature-point localization strategies, each formulated in a unified manner as a mapping from multimodal inputs to the temporal positions of key feature points. For the b-th cardiac cycle, the m-th localization strategy can be expressed as

(2) $$ \begin{equation} \begin{aligned} \hat{\mathbf{p}}_{m}^{(b)}=f_{m}\left(\mathbf{x}^{(b)}\right), \hat{\mathbf{p}}_{m}^{(b)}=\left[\hat{t}_{B, m}^{(b)}, \hat{t}_{X, m}^{(b)}\right], \end{aligned} \end{equation} $$

where f_m(·) denotes the m-th feature-point localization strategy, and $$ \hat{t}_{B, m}^{(b)}$$ and $$\hat{t}_{X, m}^{(b)}$$ represent the predicted temporal positions of the B point and X point, respectively, in the b-th cardiac cycle. In general, the B point corresponds to an event temporally adjacent to AVO and is used to characterize the onset of ventricular ejection, whereas the X point corresponds to an event adjacent to AVC and is used to characterize the termination of ejection. Together, these two points define the key systolic interval most closely associated with the ejection process and directly determine the accuracy and stability of ICG-derived parameters, including LVET, SV, and CO. Accordingly, in this study, localization errors of the B and X points are regarded as one of the major sources of downstream error in hemodynamic parameter estimation.

To avoid conceptual ambiguity, it should be clarified that the B and X points referred to in this study primarily represent the feature-point locations identified on the ICG waveform by the proposed model or rule-based method. Both supervised training and performance evaluation used the publicly available AVO and AVC event timestamps as reference labels in the HeartCycle dataset. Therefore, the algorithm-derived B and X points were not treated as independently annotated ground-truth labels. Instead, the AVO and AVC reference labels were used as unified reference standards for the onset and termination of ventricular ejection, respectively, against which ICG feature-point localization and LVET estimation were evaluated. It should be noted that there may be systematic deviations between the B/X annotation and the actual valve event. For example, changes in waveform morphology may lead to different algorithms having inconsistent definitions of point B. Accordingly, point-wise localization error alone is insufficient to fully capture the practical value of a given method. A more meaningful evaluation is to examine whether such localization errors materially affect the final hemodynamic estimates, which is precisely why this study places greater emphasis on parameter-level performance. After obtaining $$ \hat{t}_{B, m}^{(b)}$$ and $$\hat{t}_{X, m}^{(b)}$$, a unified deterministic post-processing procedure was applied to map the localized feature points to beat-to-beat hemodynamic parameters. First, the left ventricular ejection time is defined as

(3) $$ \begin{equation} \begin{aligned} \hat{\mathrm{LVET}_{m}^{(b)}}=\hat{t}_{X, m}^{(b)}-\hat{t}_{B, m}^{(b)} . \end{aligned} \end{equation} $$

Subsequently, $${\mathrm{\hat{LVET}}_{m}^{(b)}}$$ is combined with the corresponding ICG-derived morphological descriptor $$\phi_{\mathrm{ICG}}^{(b)}$$ and fed into a predefined parametric function g(·) to yield the beat-to-beat estimate of SV:

(4) $$ \begin{equation} \begin{aligned} \hat{\mathrm{SV}}_{m}^{(b)}=g\left(\hat{\mathrm{LVET}}_{m}^{(b)}, \phi_{\mathrm{ICG}}^{(b)}\right). \end{aligned} \end{equation} $$

Finally, CO is computed from the corresponding heart rate HR^(b) for that beat:

(5) $$ \begin{equation} \begin{aligned} \hat{\mathrm{CO}}_{m}^{(b)}=\frac{\hat{\mathrm{SV}}_{m}^{(b)} \times \mathrm{HR}^{(b)}}{1000} . \end{aligned} \end{equation} $$

Specifically, the ICG morphological descriptor for each cardiac beat was defined as the absolute peak amplitude of the ICG signal at the systolic C point. Its detailed definition and parameter-calibration procedure are provided in Section “Parameter derivation and evaluation metrics”. To ensure a fair comparison across methods, all experiments employed the same parameter derivation function, g(·), thereby minimizing the influence of subsequent parameter-conversion procedures. Accordingly, performance differences between methods were intended to primarily reflect their ability to localize ICG fiducial points.

This study reformulates the ICG fiducial point localization problem as a deep learning-based sequence modeling task. The proposed framework automatically learns the temporal patterns associated with AVO and closure events from multimodal signals, including ECG, ICG, and PCG, in a data-driven manner, thereby improving the stability of fiducial point localization under complex waveform conditions. The overall workflow is illustrated in Figure 5. The primary advantage of the deep learning approach lies in its ability to extract high-level representations directly from raw signals and to integrate complementary information across modalities within an end-to-end architecture, thereby reducing reliance on handcrafted rules and prior assumptions inherent to conventional methods. In this section, the proposed model is described in detail, including its input representation, network architecture, training strategy, and inference procedure.

For each cardiac cycle, the ECG R peak was used as the temporal anchor, and multimodal signal segments from 0.20 s before to 0.80 s after the R peak were extracted to construct a fixed 1.00 s beat-level analysis window. This window covers the main physiological process from electrical activation to the end of mechanical systole and includes the typical temporal range of the B and X points. To ensure temporal consistency across modalities and reduce computational complexity, all signals were resampled to 200 Hz; therefore, each beat-level input sample contained 200 time points. The ECG signal was z-score standardized beat by beat after resampling and used as the electrical timing reference. The ICG signal was represented by its first derivative, computed as the temporal gradient of the resampled impedance signal and then z-score standardized within each beat, to emphasize rapid hemodynamic waveform changes. For PCG, when the original sampling rate was high, the signal was first resampled to 4,000 Hz, then filtered using a fourth-order Butterworth 20-800 Hz band-pass filter, transformed into an analytic-signal amplitude envelope using the Hilbert transform, resampled to 200 Hz, and standardized within each beat. Compared with the raw high-frequency PCG waveform, the envelope representation preserves key valvular acoustic events such as S1 and S2 while reducing training instability caused by high-frequency oscillations.

To fairly evaluate the contribution of PCG, two input configurations were evaluated: a trimodal setting (ECG + ICG + PCG) and a bimodal setting (ECG + ICG, with the PCG channel set to zero). The network architecture was kept identical between settings so that performance differences could be attributed to information content rather than model capacity. The proposed deep learning model consists of four core components: modality-specific encoders, a multimodal fusion and contextual modeling module, an event-query localization head, and an auxiliary regression constraint module. The model was designed to capture both local waveform morphology and global temporal dependencies, while introducing physiological consistency constraints into the localization task. To avoid mixing the different statistical properties of the modalities too early, an independent one-dimensional convolutional encoder was assigned to each modality. Each encoder maps a single-channel sequence to a high-dimensional feature representation:

(6) $$ \begin{equation} \begin{aligned} \mathrm{F}_{m}=E_{m}\left(\mathrm{x}_{m}\right) \in \mathbb{R}^{B \times C \times T} \end{aligned} \end{equation} $$

Each encoder consists of an initial stem convolutional layer with kernel size 7, followed by stacked residual dilated convolutional blocks. The residual blocks use convolutions with kernel size 5 and a dilation-rate sequence of [1, 2, 4, 8, 16, 8, 4, 2], forming a multi-scale receptive field that captures millisecond-level local waveform details and broader morphological context over hundreds of milliseconds. The three modality-specific feature representations are then concatenated along the channel dimension and projected into a shared latent space with hidden dimension d = 64 through a convolutional layer, yielding the fused temporal representation:

(7) $$ \begin{equation} \begin{aligned} \mathrm{U}=\operatorname{Convld}_{1 \times 1}\left(\left[\mathrm{~F}_{\mathrm{ECG}} ; \mathrm{F}_{\mathrm{ICG}} ; \mathrm{F}_{\mathrm{PCG}}\right]\right) \in \mathbb{R}^{B \times d \times T} \end{aligned} \end{equation} $$

The fused representation is transposed into a temporal-token sequence and combined with sinusoidal positional encoding to inject absolute timing information. The resulting sequence is passed to a module composed of two Transformer encoder layers, whose self-attention mechanism captures long-range dependencies across the entire cardiac window:

(8) $$ \begin{equation} \begin{aligned} \mathrm{H}=\operatorname{Transformer}(\mathrm{S}+\mathrm{P}) \in \mathbb{R}^{B \times T \times d} \end{aligned} \end{equation} $$

In the event localization head, an event-query retrieval mechanism was adopted to jointly exploit the complementary temporal information provided by ECG, ICG, and PCG signals. Specifically, the model defines two learnable event query vectors, $$\mathrm{q}_{1}, \mathrm{q}_{2} \in \mathbb{R}^{d}$$, corresponding to the AVO and AVC events, respectively. The hidden dimension of the fused temporal representation was set to d = 64 , and the Transformer output sequence is denoted as $$\mathrm{H} \in \mathbb{R}^{B \times T \times 64}$$, where B is the batch size and T is the number of time steps. The two event query vectors are stacked row-wise to form $$ \mathrm{E} \in \mathbb{R}^{2 \times 64}$$. The linear projection matrices W_q and W_k are both implemented using bias-free fully connected layers, i.e., Linear(64,64), such that $$ \mathrm{W}_{q}, \mathrm{W}_{k} \in \mathbb{R}^{64 \times 64}$$. For the b-th sample, the event queries and temporal context are projected as $$\mathrm{Q}^{(b)}=\mathrm{EW}_{q}^{\mathrm{T}} \in \mathbb{R}^{2 \times 64}$$ and $$\mathrm{K}^{(b)}=\mathrm{H}^{(b)} \mathrm{W}_{k}^{\mathrm{T}} \in \mathbb{R}^{T \times 64}$$, respectively. Equivalently, the query vector for the e-th event type and the key vector at the t-th time step can be expressed as $$\tilde{\mathrm{q}}_{e}=\mathrm{W}_{q} \mathrm{q}_{e}$$ and $$\mathrm{k}_{t}^{(b)}=\mathrm{W}_{k}\mathrm{h}_{t}^{(b)}$$. The logits for event e at time point t are then computed using scaled dot-product attention:

(9) $$ \begin{equation} \begin{aligned} Z_{e}(t)=\frac{\tilde{\mathrm{q}}_{e}^{T} \mathrm{k}_{t}}{\sqrt{d}} \cdot \exp (s)+b_{e}, e \in\{\mathrm{AVO}, \mathrm{AVC}\} \end{aligned} \end{equation} $$

where s is a learnable scaling parameter and b_e is the event-specific bias term. This yields a logit heatmap $$Z \in \mathbb{R}^{B \times 2 \times T}$$. This design is intended to allow the model to learn adaptive event-related representations and perform retrieval-based matching on the fused temporal representations, thereby providing an adaptive event-localization strategy for addressing inter-individual variability and noise interference.

In addition to the primary localization task, the model incorporates an auxiliary regression branch to impose physiological consistency constraints. First, an additive temporal attention module is applied to H obtain a global context vector, $$c \in \mathbb{R}^{B \times d}$$. Next, the event logits are transformed by a softmax operation into temporal attention weights w_e(t), which are then used to compute event-specific context vectors, $$c_{e} \in \mathbb{R}^{B \times d}$$, through weighted summation over H. The concatenated representation [c, c_AVO, c_AVC, HR] is subsequently fed into a two-layer multilayer perceptron (MLP) to predict CO and an auxiliary estimate of Δ_LVET. This regression design, combining global context and event-specific context, allows the regression and localization tasks to share a common temporal representation while explicitly focusing on event-relevant segments, thereby providing an additional physiological plausibility constraint for event localization. Model training is performed using a multi-task joint optimization strategy, in which the overall loss consists of both a heatmap-based localization loss and a physiological parameter regression loss. For the localization task, after mapping the reference event times of AVO and AVC, denoted by c, to the corresponding window indices, one-dimensional Gaussian soft labels were constructed as the supervision targets:

(10) $$ \begin{equation} \begin{aligned} y(t)=\exp \left(-\frac{(t-c)^{2}}{2 \sigma^{2}}\right) \end{aligned} \end{equation} $$

where σ corresponds to a temporal tolerance of approximately 18 ms (converted into sample points according to the sampling rate), which helps mitigate annotation jitter and provides more stable gradients during training. The heatmap localization loss is implemented using focal binary cross-entropy with logits to address the extreme class imbalance along the temporal axis between positive samples (event locations) and negative samples (non-event time points).

For the regression task, predictions of CO and Δ_LVET were optimized using the Smooth L1 (Huber) loss, with missing labels masked during training. To further enforce internal physiological consistency, the model derived the predicted SV from the predicted mean $$\bar{CO}$$ and the known heart rate HR according to $$\bar{SV}$$ = $$\bar{CO}$$·1000/HR, and the discrepancy between $$\bar{SV}$$ and the ground-truth SV was incorporated as an additional consistency regularization term. The total loss was defined as a weighted sum of all components, with fixed weights throughout the experiments. The total loss consisted of the event heatmap localization loss, CO regression loss, SV consistency constraint loss, and auxiliary LVET regression loss, with their respective weights fixed at 1.00, 0.20, 0.05, and 0.20. A focal binary cross-entropy loss with logits was used for AVO and AVC event heatmap localization, with the focusing parameter γ set to 2.0 and the class-balancing parameter α set to 0.25. Smooth L1, also known as Huber loss, was used for all regression terms related to CO, SV, and LVET. All deep learning models were trained using the AdamW optimizer with an initial learning rate of 3.0 × 10^-4, a weight decay coefficient of 1.0 × 10^-2, a batch size of 32, a fixed random seed of 42, and a maximum of 50 epochs. To improve training stability, gradient clipping is applied after each backward pass, with the maximum gradient norm fixed at 1.0.

During inference, the heart rate used to calculate the RR interval is constrained to 30-220 beats/min. The AVO heatmap decoding window extends from 15 ms after the R peak to min(250 ms, 0.55RR) after the R peak. The lower bound of the AVC decoding window is set to 80 ms after AVO, and the upper bound is set to min(750 ms, 0.85RR) after AVO. Heatmap logits outside these intervals are masked, and the event positions from the heatmap branch are calculated within the constrained intervals using soft-argmax with a temperature parameter of τ = 0.35. The LVET predicted by the auxiliary regression branch is constrained to 120-650 ms and is used to construct a second AVC candidate, defined as the AVO position plus the auxiliary LVET. This candidate is further constrained to the same physiological AVC interval used by the heatmap branch. The AVC heatmap confidence is defined as the maximum probability obtained after temperature-based normalization with τ = 0.35 within the constrained AVC interval. The final AVC position is obtained by fusion according to AVC_final = (1 - w)AVC_heatmap + wAVC_aux. When the heatmap confidence is below 0.18, the fusion weight w for the auxiliary LVET pathway is set to 0.45. When the confidence is at least 0.18, w is set to 0.20. The corresponding weights of the heatmap branch are therefore 0.55 and 0.80, respectively. These search ranges, thresholds, and fusion weights remain fixed across all training and test folds, and neither the validation set nor the test set is used to determine these values.

The primary hemodynamic parameter estimation experiment uses a fixed subject-disjoint data split. The training set includes CH08 and CH09, with 152 candidate beat segments in total, comprising 52 segments from CH08 and 100 from CH09. The validation set includes CH10 with 59 candidate beat segments, while the test set includes CH07 with 70 candidate beat segments. After filtering for complete reference labels, 61 valid test beats with reference values for LVET, SV, and CO are retained for unified beat-level performance evaluation in the main experiment. No subject overlap exists among the training, validation, and test sets. These 61 beats are used only to report beat-level performance on the fixed test set and are not treated as independent samples in the subsequent paired statistical analysis involving 17 subjects. Subject-level inference is conducted separately using independent test results obtained when each subject is completely held out in the 17-fold leave-one-subject-out (LOSO) evaluation.

To evaluate computational efficiency, the forward inference time and parameter number of the Proposed model are measured on a Windows 10 workstation. The evaluation environment includes an Intel Core i7-14650HX CPU, an NVIDIA GeForce RTX 4060 Laptop GPU, Python 3.11.14, and PyTorch 2.5.1. The Proposed model contains 395,270 parameters, and the checkpoint file size is approximately 1.65 MB. The model input is a beat-level multimodal signal segment with a length of 1 s, consisting of 200 temporal points and three channels. On the CPU, the average forward inference time for a single heartbeat window is 23.72 ± 3.69 ms with a batch size of 1, corresponding to 1.62 ms/beat with a batch size of 64. On the GPU, the average inference time is 6.24 ± 1.13 ms for batch size 1 and 0.15 ms/beat for batch size 64. These results indicate that the proposed model has a low computational burden and can support beat-level offline analysis and near-real-time processing.

To further compare the performance of different learning-based models, three baseline models, including CNN-only, BiLSTM, and ICG-only, are established using the same beat-level data partitioning, input modalities, preprocessing procedures, and evaluation metrics as the main experiment. The CNN-only baseline employs conventional one-dimensional residual convolution for feature extraction and heatmap localization without incorporating the event-token attention mechanism. The BiLSTM baseline uses bidirectional LSTM networks to model temporal dependencies and estimates LVET only from AVO/AVC heatmap localization results without an additional direct LVET regression head. Both learning-based baselines use ECG, ICG (dZ/dt), and PCG envelope as multimodal inputs and follow the same training epochs, validation-based model selection strategy, and test evaluation procedure. In addition, an ICG-only baseline model is introduced to investigate the limitations of single-channel ICG-based localization and parameter estimation. This model maintains the same data partitioning, R-peak alignment windows, preprocessing pipeline, training epochs, validation selection strategy, and test evaluation procedure as the main experiment. However, it only uses single-channel ICG signals as input and does not incorporate ECG waveform information, PCG envelope features, or the event-token attention mechanism. The ICG-only model directly predicts AVO/AVC heatmaps from the single-channel ICG representation, and LVET is calculated from the detected fiducial points. SV and CO are subsequently obtained using the same output conversion procedure as other learning-based models.

To ensure fair comparison among localization strategies, the same hemodynamic parameter derivation pipeline was used for all methods. Regardless of whether the method was rule-based (ECG-only or ECG + PCG-assisted) or deep learning-based (DL w/o PCG or DL + PCG), each method first produce the B-point and X-point locations for every valid cardiac cycle, and LVET, SV, and CO were then calculated using the same deterministic post-processing procedure. The auxiliary regression branch in the deep learning model was used only to impose physiological consistency during training and to assist decoding for low-confidence samples during inference; it was not used directly as the source of the final reported SV and CO results.

For the i-th valid cardiac cycle, if the B and X points output by the model are first represented as sample indices, they are converted to the physical time axis according to the sampling rate:

(11) $$ \begin{equation} \begin{aligned} \hat{t}_{B}^{(b)}=\frac{\hat{n}_{B}^{(b)}}{f_{s}},~\hat{t}_{X}^{(b)}=\frac{\hat{n}_{X}^{(b)}}{f_{s}} . \end{aligned} \end{equation} $$

On this basis, left ventricular ejection time is defined as:

(12) $$ \begin{equation} \begin{aligned} \mathrm{\hat{LVET}}^{(b)}=\hat{t}_{X}^{(b)}-\hat{t}_{B}^{(b)} . \end{aligned} \end{equation} $$

After obtaining $$\mathrm{\hat{LVET}}^{(b)}$$, SV and CO were calculated using a unified deterministic parameter derivation function. For the b-th valid cardiac cycle, the C-point time t_C^(b) was first determined from the filtered and polarity-corrected ICG waveform according to the systolic dominant-peak localization rule shared by all methods, as described in Section ECG-only method. The C point was identified within a predefined systolic search window following the R peak, with the specific search range consistent with that used in the conventional localization procedure of this study.

The absolute peak amplitude of ICG at the C point was then defined as:

(13) $$ \begin{equation} \begin{aligned} A_{C}^{(b)}=\left|\frac{d Z}{d t}\left(t_{C}^{(b)}\right)\right| \end{aligned} \end{equation} $$

Here, A_C^(b) denotes the only ICG morphological descriptor used in this study for SV derivation. No waveform integral, area-based measure, second-derivative feature, template similarity measure, or other manually engineered ICG morphological indices were used. The $$\mathrm{\hat{LVET}}^{(b)}$$ determined by the B and X points was incorporated as a temporal feature together with A_C^(b) for subsequent parameter calculation.

As the HeartCycle dataset does not provide the complete subject-specific parameters required by conventional Kubicek or Sramek-Bernstein equations, including thoracic length, electrode spacing, baseline impedance, and blood resistivity, fixed constants reported in the literature were not directly adopted. Instead, the scaling coefficient K_SV was robustly calibrated once within each training split using the VE reference sequence provided by HeartCycle. Specifically, the VE time series was first linearly interpolated at the timing of the current R peaks to obtain beat-wise reference SV values:

(14) $$ \begin{equation} \begin{aligned} S V_{\text {ref }}^{(b)}=\operatorname{Interp}_{\text {lin }}\left(V E, t_{R}^{(b)}\right) \end{aligned} \end{equation} $$

The reference ejection time was calculated from the public AVO and AVC reference events:

(15) $$ \begin{equation} \begin{aligned} L V E T_{\mathrm{ref}}^{(b)}=t_{\mathrm{AVC}, \mathrm{ref}}^{(b)}-t_{\mathrm{AVO}, \mathrm{ref}}^{(b)} \end{aligned} \end{equation} $$

In the valid training-beat set $$\mathcal{D}_{\text {train }}$$, the beat-wise proportional estimate was defined as:

(16) $$ \begin{equation} \begin{aligned} r^{(b)}=\frac{S V_{\text {ref }}^{(b)}}{A_{C}^{(b)} \cdot L V E T_{\text {ref }}^{(b)}} \end{aligned} \end{equation} $$

To reduce the influence of aberrant cardiac beats, outliers were first removed within the training set based on the median absolute deviation of logr^(b), yielding the valid set $$\mathcal{D}_{\text {train }}^{*}$$. The following quantity was then calculated:

(17) $$ \begin{equation} \begin{aligned} K_{\mathrm{SV}}=\operatorname{median}_{b \in \mathcal{D}_{\text {train }}^{*}}\left(r^{(b)}\right) \end{aligned} \end{equation} $$

Accordingly, the predefined beat-level SV function g(∙) used in this study was:

(18) $$ \begin{equation} \begin{aligned} \hat{S V}^{(b)}=g\left(\hat{L V E T}^{(b)}, \phi_{\text {ICG }}^{(b)}\right)=K_{\mathrm{SV}} \cdot A_{c}^{(b)} \cdot \hat{L V E T}^{(b)}, \phi_{\mathrm{ICG}}^{(b)}=\left\{A_{C}^{(b)}\right\} \end{aligned} \end{equation} $$

Heart rate was obtained from adjacent R-R intervals:

(19) $$ \begin{equation} \begin{aligned} HR^{(b)}=\frac{60}{R R^{(b)}} \end{aligned} \end{equation} $$

Finally, CO was calculated as:

(20) $$ \begin{equation} \begin{aligned} \hat{C O}^{(b)}=\frac{\hat{S V}^{(b)} \cdot H R^{(b)}}{1000} \end{aligned} \end{equation} $$

Here, $$\hat{SV}^{(b)}$$ is expressed in mL and HR^(b) in beats/min; accordingly, $$\hat{CO}^{(b)}$$ is expressed in L/min. The ICG signal used to calculate A_C^(b) retained its physical amplitude information and was not normalized on a beat-by-beat basis. Amplitude normalization was applied only to the deep learning model inputs and was not involved in the deterministic derivation of SV or CO. The reference values for SV and CO were obtained from the VE and DC parameter sequences provided by HeartCycle, respectively, and were linearly interpolated at the timing of the current R peaks. In this study, VE and DC were used only as device-derived reference values for model training and performance evaluation, rather than being considered independent ultrasound-based or invasive gold standards.

Let $$\hat{y}^{(b)}$$ and y^(b) denote the predicted value and reference value, respectively, of parameter y for the b-th valid cardiac cycle. Given N_y valid samples, the mean absolute percentage error (MAPE) of parameter y is defined as:

(21) $$ \begin{equation} \begin{aligned} \mathrm{MAPE}(y)=\frac{1}{N_{y}} \sum_{b=1}^{N_{y}}\left|\frac{\hat{y}^{(b)}-y^{(b)}}{y^{(b)}}\right| \times 100 \% \end{aligned} \end{equation} $$

Based on the above definitions, this study calculates the MAPE values for LVET, SV, and CO and evaluates ECG-only, ECG + PCG-assisted, DL w/o PCG, and DL + PCG using a unified parameter derivation procedure and consistent evaluation metric definitions. Because the number of valid beats available for parameter evaluation after filtering for complete reference labels differs across methods, cross-category comparisons between the conventional rule-based and deep learning methods are presented primarily as descriptive analyses. Method comparisons and statistical inference based on the same set of valid beats are conducted separately in the corresponding controlled experiments. MAPE was selected as the primary metric because the magnitudes of SV and CO can differ substantially among subjects; percentage error reduces the influence of scale differences on the overall statistics and better reflects the relative impact of feature-point localization errors on final functional parameter outputs.

To quantify the intrinsic complexity of ICG waveform morphology, unsupervised clustering was first performed on 1,908 valid ICG beat segments extracted from the HeartCycle dataset. The clustering results provide an intuitive representation of the morphological diversity of ICG waveforms as well as the distributional differences across clusters. In the waveform visualizations, solid lines represent the mean morphology of each cluster, while the shaded regions indicate the mean ± one standard deviation (±SD), thereby illustrating intra-class variability and the extent of feature overlap between different clusters. The corresponding visualization results are shown in Figure 6.

The statistical results are summarized in Figure 7. A total of 1,908 valid ICG segments from the HeartCycle dataset were included in the clustering analysis. For K = 2, the two clusters contained 625 and 1,283 segments, respectively. For K = 3, the cluster sizes were 818, 484, and 606 segments. For K = 4, the distribution further split into 713, 670, 430, and 95 segments. For K = 5, the cluster sizes were 297, 340, 526, 612, and 133 segments. Across both coarse- and fine-grained clustering settings, pronounced class imbalance was consistently observed. These results indicate substantial morphological heterogeneity of ICG waveforms across cardiac cycles, accompanied by significant inter-class overlap. As clustering granularity increases, both boundary ambiguity between clusters and imbalance in class distribution become more evident. These findings suggest that conventional feature-point localization methods based on single-modality local morphological rules are unlikely to generalize robustly across diverse waveform patterns.

To further justify the choice of cluster number, the validity and stability of K = 2-5 clustering solutions were evaluated using internal validity metrics and repeated random seeds. K = 2 achieved the best global geometric separation according to the Silhouette, Calinski-Harabasz, and Davies-Bouldin indices, suggesting a coarse two-pattern structure in the ICG waveforms. However, K = 3 showed the highest cross-seed stability (ARI, 0.777 ± 0.145; NMI, 0.796 ± 0.116) while maintaining an interpretable minimum cluster proportion of 24.916% ± 0.368% without extremely small clusters. In contrast, K = 4 and K = 5 showed smaller minimum-cluster proportions and a higher risk of over-partitioning. Therefore, K = 3 was considered more suitable for subsequent morphology-stratified analysis when geometric separation, stability, and cluster-size interpretability were considered jointly. The quantitative results are summarized in Table 1 and Figure 8.

To further validate the morphological diversity and learnability of ICG waveforms, this study performs morphology classification on the preprocessed ICG beat segments using the clustering-derived labels. For each K value, the test set contains n = 382 beat segments. In the classification tasks with K = 2, K = 3, K = 4, and K = 5, the complete MS-GBiLSTM-TA model achieves Accuracy values of 95.03%, 91.36%, 89.79%, and 76.96%, respectively. The corresponding Macro-F1 scores are 0.9432, 0.9134, 0.8507, and 0.7686, while the Weighted-F1 scores are 0.9501, 0.9135, 0.8982, and 0.7714, respectively. Detailed class-level Precision, Recall, and F1-score results are presented in Tables 2-5. As the number of clusters increases, the classification task becomes more difficult and model performance declines, indicating that finer-grained partitioning of ICG waveform morphologies leads to greater feature overlap and less distinct boundaries between classes.

Figure 9 presents the corresponding confusion matrices. As the number of classes increases, the number of off-diagonal elements also increases, indicating more pronounced class confusion in fine-grained ICG morphology recognition. Further analysis of the class-level results reveals several difficult classes in the multiclass tasks. For example, in the four-class classification task, Class 3 achieves a Recall of 0.6842, a Precision of 0.6500, and an F1-score of 0.6667, indicating that this class remains difficult to identify accurately. This finding suggests that some ICG waveform subtypes exhibit high local morphological similarity to other classes, making it difficult for the model to learn stable and discriminative feature representations. The differences between the macro-averaged and weighted-averaged metrics further reflect the influence of class imbalance on performance evaluation. When minority classes are difficult to identify, the macro-averaged metrics decrease more markedly, indicating that the model does not perform uniformly across classes. Overall, the morphology classification results provide data-driven support for the preceding clustering analysis, showing that ICG waveforms exhibit substantial morphological diversity and class overlap under real-beat conditions. These findings indicate that rigidly matching specific mechanical events using only local morphological features from single-modal ICG can lead to temporal ambiguity and localization drift in complex waveforms. This observation provides further motivation for incorporating ECG and PCG into the subsequent multimodal fiducial point localization framework.

The structural ablation experiments evaluate the contributions of multiscale convolution, bidirectional temporal modeling, and temporal attention across different ICG morphology classification tasks. All models are compared using the same training, validation, and test splits, and the test set for each K value contains n = 382 beat segments. The structural ablation results of the MS-GBiLSTM-TA classification model are presented in Table 6. In the binary classification task with K = 2, the CNN-only model achieves the highest Accuracy, Macro-F1, and Weighted-F1. For K = 3, the complete MS-GBiLSTM-TA model performs best across all three metrics. For K = 4, the complete model achieves the highest Accuracy and Weighted-F1, whereas the BiLSTM-only model obtains a higher Macro-F1. For K = 5, the CNN + BiLSTM model achieves the highest Accuracy and Weighted-F1, while the complete model yields the highest Macro-F1.

The preceding clustering and classification analyses demonstrate that ICG waveforms exhibit substantial morphological variability and feature overlap. This implies that relying solely on local morphological characteristics from single-modality ICG to match specific mechanical events is prone to temporal ambiguity and localization drift, which can further propagate as cumulative errors in beat-to-beat hemodynamic parameter estimation. Therefore, single-modality local morphological analysis still faces challenges in achieving stable event localization under complex waveform conditions. Motivated by these findings, we further compared the performance of conventional rule-based methods and multimodal deep learning approaches in both key feature-point localization and downstream hemodynamic parameter estimation.

The comparative methods fall into two categories. The conventional rule-based approaches include the ECG-only method, which relies exclusively on ECG for temporal anchoring, and the ECG + PCG-assisted method, which further incorporates soft constraints derived from PCG heart sound events. The deep learning approaches include deep learning (DL) w/o PCG, which excludes PCG input, and DL + PCG, which explicitly integrates PCG into the multimodal framework. Error statistics were computed on the basis of the relative error for each valid beat and then aggregated as the MAPE for comparison. For overall reference, Table 7 summarizes the mean MAPE (%) of all methods for LVET, SV, and CO.

To address statistical uncertainty, this study retains the beat-level mean MAPE reported in the original manuscript as the primary result and additionally reports the mean ± SD, median [IQR], and 95% CI of the error distributions for each method, as shown in Table 8. The beat-level MAPE statistics in Table 8 describe the per-beat error performance and dispersion of each method within its corresponding evaluation cohort, whereas statistical inference for between-method comparisons is conducted at the subject level.

Because each subject can contribute multiple beats, individual beats are not treated as independent observations in subject-level statistical inference. The paired analysis reported in Table 9 is based on the results of 17-fold LOSO cross-validation. In each fold, one subject is completely held out as the independent test subject, one subject is used for model selection, and the remaining 15 subjects are used for model training. CH01 through CH17 serve once each as the independent test subject, and the statistical result for each subject is obtained exclusively from the fold in which that subject is completely held out. DL + PCG and DL w/o PCG use identical training, validation, and test subject assignments in every fold and are evaluated on the common test beats with complete reference values. The MAPE values for LVET, SV, and CO are first aggregated within each test subject, after which paired tests are conducted using the aggregated results from the 17 independent test subjects. For both method comparisons, ΔMAPE is calculated within each subject as the MAPE of the method without PCG minus that of the corresponding PCG-assisted method. Specifically, ΔMAPE is calculated as ECG-only minus ECG + PCG-assisted for the conventional rule-based comparison and as DL w/o PCG minus DL + PCG for the deep learning comparison. The ΔMAPE reported in the table is the mean of the 17 subject-specific differences; therefore, a positive value indicates that incorporating PCG reduces the estimation error. Both the paired t-test and the Wilcoxon signed-rank test use the aggregated results from the 17 subjects as paired observations. All 17 paired observations are derived from independent test results obtained when the corresponding subject is excluded from both model training and model selection, with the statistical results presented in Table 9. Because the conventional rule-based methods do not involve model training, their paired comparison is likewise aggregated at the subject level across all 17 subjects and is calculated using only the common beats for which both methods yield valid results and complete reference labels.

Based on the independent test results from the 17-fold LOSO evaluation reported in Table 9, DL + PCG shows overall reductions in subject-level MAPE for SV and CO compared with DL w/o PCG, with both the paired t-test and the Wilcoxon signed-rank test indicating statistically significant differences. The ΔMAPE for LVET is also positive, although the difference does not reach statistical significance. For the conventional rule-based methods, ECG + PCG-assisted methods yield positive ΔMAPE values for LVET, SV, and CO compared with ECG-only, but none of the paired tests for these three metrics reach statistical significance. These results indicate that, under subject-independent LOSO evaluation, the mechanical-event timing information provided by PCG offers a clearer complementary contribution to SV and CO estimation within the deep learning framework, whereas the improvements observed in the conventional rule-based methods are primarily numerical trends.

To further address subject independence and model generalization, LOSO cross-validation was supplemented in this study. In each fold, one subject was completely held out as the independent test set, one subject was used as the validation set, and the remaining 15 subjects were used for model training, thereby avoiding cardiac beat segments from the same subject appearing simultaneously in the training and test sets. By iteratively treating each subject as the independent test subject, this setting provides a supplementary assessment of model generalization to unseen subjects. Unlike the fixed subject-level split used in the main experiment, LOSO cross-validation covers all 17 subjects, with each subject serving as the test subject once.

In 17-fold LOSO validation, the DL + PCG model was evaluated for beat-level LVET, SV, and CO estimation under subject-independent testing in Table 15. When summarized at the subject level, the MAPE values for LVET, SV, and CO were 21.39 ± 13.87%, 27.42 ± 21.79%, and 26.69 ± 22.39%, respectively, with 95% confidence intervals of 14.26-28.52%, 16.22-38.63%, and and15.18-38.20%. These results indicate that the multimodal framework is feasible under strict subject-independent evaluation, while the observed inter-subject variability indicates that larger samples and more diverse populations are needed to improve and validate cross-subject generalization.

To justify the fixed input window, the temporal distributions of AVO, AVC, and conventional ICG B/X landmarks relative to the R peak were analyzed in Table 16 and Figure 19. Across 129 HDF5 record files, 1,036 AVO and 1,069 AVC events were matched from HeartCycle, and the conventional morphology-based pipeline yielded 50 B points and 50 X points. AVC, B, and X all fell within the fixed window from 0.20 s before to 0.80 s after the R peak, with 100% coverage. AVO coverage was 99.52%, and its 99% distribution range was approximately 5.30-781.44 ms, still within the upper boundary of the window. Only five AVO events were outside the window, possibly due to boundary beats, event-matching errors, or rare extreme timing. These results support the use of a fixed 1.00 s R-peak-aligned window for beat-level modeling, while also indicating that atypical timing patterns may require further consideration in broader populations.

From the perspective of model generalization, the advantage of the fixed-window strategy is that it provides a unified input length and a clear R-peak-aligned reference, facilitating fusion of ECG, ICG, and PCG signals on the same time axis. At the same time, the 0.20 s window before the R peak preserves local baseline and early morphological information, whereas the 0.80 s window after the R peak covers major events such as AVO, AVC, the B point, and the X point. It should be noted that this strategy also has potential limitations. Under extreme heart rates, R-peak misdetection, event-label shifts, or severe signal drift, a small number of events may approach the window boundary or even fall outside the fixed window. In addition, because the fixed window is aligned only to the R peak and no time-scale normalization is performed according to the R-R interval, the model retains true event-delay differences across beats. However, this may also increase the difficulty of model learning under marked heart-rate variability. Therefore, the window setting adopted in this study is reasonable under the current HeartCycle data distribution, but future external validation should further assess its applicability in broader heart-rate ranges and pathological populations.

This study establishes a comprehensive framework for ICG fiducial point localization and hemodynamic parameter estimation, covering waveform morphology analysis and multimodal deep fusion modeling. Clustering and classification analyses based on the HeartCycle dataset demonstrate pronounced morphological diversity, class overlap, and ambiguous boundaries among ICG waveforms under real beat conditions, suggesting that relying solely on local morphological rules from single-modality signals for B/X point localization remains limited under complex waveform conditions. Based on these findings, this study proposes an end-to-end deep learning model integrating ECG, ICG, and PCG and compares it with conventional rule-based methods and learning-based baseline models under a unified parameter derivation and evaluation framework. Experimental results demonstrate that the proposed multimodal framework can achieve beat-level LVET, SV, and CO estimation under the current evaluation conditions, with lower errors particularly observed in SV and CO estimation. The ablation experiments and baseline comparisons are consistent with the interpretation that PCG-based mechanical event context, multimodal temporal information, and event-related modeling may jointly contribute to improved parameter estimation stability, highlighting the potential of multimodal information fusion for ICG fiducial point localization and hemodynamic parameter estimation. Further LOSO validation demonstrates the feasibility of the framework for beat-level LVET, SV, and CO estimation under subject-independent testing conditions, providing experimental support for future studies involving larger cohorts, multicenter datasets, and subject-adaptive modeling. Overall, the current results primarily support the capability of the proposed method for beat-to-beat hemodynamic parameter estimation under the evaluated conditions and provide a methodological foundation for continuous noninvasive hemodynamic trend assessment. Reliable continuous trend monitoring and within-subject tracking of hemodynamic changes require further establishment through longitudinal data, dynamic physiological scenarios, and prospective studies. Future work will incorporate independent external datasets, multicenter validation, controlled hemodynamic state variation experiments, and fine-grained structural ablation analyses to systematically evaluate the model’s external generalization capability, within-subject change tracking ability, component contributions, and clinical applicability.