All methods were performed in accordance with the relevant guidelines and regulations as stipulated by University of Manchester. This was a retrospective pilot study based on data from St. Mary’s Hospital, Manchester, UK, collected with the approval of NHS Research Ethics Committees (RECs). Informed consent was received from all participants in this study.A computational model of the maternal circulation and wave reflectionThis study utilizes models by Carson et al.24,26,27 comprising larger arteries, veins, vascular beds (including organs and capillary systems), and a heart model. Larger vasculatures are modelled for wave propagation using one-dimensional flow models, while electric circuit models describe the vascular beds. Figure 1 demonstrates how the maternal arterial system, utero-ovarian vasculature, and non-invasive measurements are employed to create personalised models.Fig. 1Maternal arterial network model (left). The pressure upstream of the uterine artery (P1, blue dots) are calculated using global maternal measurements of heart and vasculature together with the uterine Doppler waveform (S – peak systolic velocity, D – end-diastolic velocity) resulting in the prediction of downstream pressure (P2, yellow dots); P1 – pressure at start of uterine artery, P2 – pressure at end of uterine artery.Parameters as SBP, DBP, PWV, CO, HR and blood flow in the uterine artery (measured using standard methods such as blood pressure monitors, Doppler scan etc.) are used to converge the computational model (Fig. 1).The resulting personalised models form a “digital twin” of each patient that characterises the patient’s cardiovascular physiology. It can provide predictions of pressures/flow waveforms throughout the larger vasculature, but also in the arcuate/spiral arteries in the utero-ovarian vasculature (Fig. 1).Patient characteristicsThis was a case control pilot study amongst a high risk cohort (see Table 1). To determine an appropriate sample size, a one-tailed t-test with a typical power of 80% was employed. Given the integration of various measurements into the model, the effect size was anticipated to be large (equal to 1), whilst alpha was chosen to be 0.1. From this a target sample size of 10 for each of the groups was estimated, which led to the final study groups of 9 and 12 for whom the required combination of measurements were taken.Women were selected from a cohort attending high-risk clinics, where they were referred due to risk factors for developing pre-eclampsia and fetal growth restriction (FGR)28. Participants were randomly chosen from the clinic database, ensuring a complete dataset of measurements (required for running the computational model, see Table 1) was available. The two groups outcomes were defined as following: NPE group contains women with no medical conditions or late onset pre-eclampsia or late onset FGR while PE contains women that were diagnosed with early onset pre-eclampsia or early onset FGR. Diagnosis of pre-eclampsia was made in accordance with ISSHP guidelines29.Table 1 The measurements were obtained during routine visits. SD – standard deviation; FDIU – fetal death in utero, NPE – no early onset pre-eclampsia group, PE – early onset pre-eclampsia group* (*= with 2 early onset FGR), GA – gestational age.Defining potential classifiersThe new biomarkers proposed for the assessment of pre-eclampsia were derived using Buckingham Pi theorem and can be found in Eqs. 1–6 (\({\pi _1}\), \({\pi _2}\), \({\pi _3}\), \({\pi _4}\), \({\pi _5}\) and \({\pi _6}\)). The parameters that were used to compose the dimensionless terms are uterine vessel resistance \({R_{ut}}\), Stroke volume SV, Cardiac output CO, Systemic compliance \({C_{syst}}\), Peripheral resistance \({R_{periph}}\), Pulse Wave Velocity PWV, Systolic Blood Pressure \({P_{syst}}\), Pulse pressure \(\Delta {P_{pulse}}\), Aortic area A.$${\pi _1}=\frac{{{R_{ut}}}}{{{R_{periph}}}}$$
(1)
$${\pi _2}=\frac{{S{V^2}}}{{{A^3}}}$$
(2)
$${\pi _3}=\frac{{CO\,{R_{periph}}}}{{{P_{syst}}}}$$
(3)
$${\pi _4}=\frac{{{C_{syst}}\,{P_{syst}}}}{{{A^{\frac{3}{2}}}}}$$
(4)
$${\pi _5}=\frac{{{R_{periph}}\,A\,PWV}}{{{P_{syst}}}}$$
(5)
$${\pi _6}=\frac{{\Delta {P_{pulse}}}}{{{P_{syst}}}}$$
(6)
Two further dimensionless terms are proposed based on the pressure and velocity in the utero-ovarian vessels such as ascending uterine artery, arcuate artery and radial/spiral arteries (the radial and spiral arteries were modelled together so when referring to radial arteries or subscript rad, spiral arteries were also included). The formulation of pressure index for the utero-umbilical system was introduced by Adamson et al.30. An alternative metric proposed clinically is the pulmonary artery pulsatility index (PAPi) which was defined as the pulmonary artery pulse pressure (PAP) divided by the right atrial pressure (RAP)31. Here we propose pressure pulsatility index, PPI, as:$$PPI=\frac{{{P_{max}} – {P_{min}}}}{{{P_{mean}}}}$$
(7)
where \({P_{min}}\) is the minimum pressure in the selected utero-ovarian vessel and \({P_{max}}\) is the maximum pressure during the cardiac cycle. PPI relates pulsatility to pressure and should not be confused with the PI which captures the pulsatility of the velocity.RI has the generic form shown in Eq. 8. The RI calculated from the Doppler scan will be noted as RI and the RI calculated from the computational model will be noted with the appropriate subscript.$$RI=\frac{{{V_{max}} – {V_{min}}}}{{{V_{max}}}}$$
(8)
Classification analysisThe terms stated above were used in a binary classification problem (PE and NPE). The analysis was performed using the Classification Learner App in MATLAB R2021b. To analyse the data both Logistic Regression (supervised learning) and k-means clustering (unsupervised learning) were used. The supervised learning training sample size was 19 and testing sample size was 2, where a two-fold cross-validation was used during training. This process was repeated 5 times to reduce bias. The purpose of the supervised classification is to assess the features’ classification performance while the unsupervised machine learning focuses on the features’ ability to classify the two groups in a bias-free manner.The dimensionless terms, PPI, RI, \({\pi _1}\), \({\pi _2}\), \({\pi _3}\), \({\pi _4}\), \({\pi _5}\), \({\pi _6}\), \({\pi _7}\) and the clinical parameters PI, RI, DBP and SBP were selected as classifiers. The metrics used to assess the classification were: A – accuracy (%), 95% CI – confidence interval (%), SE – sensitivity (%), and SP – specificity (%). Accuracy is defined as the number of correct predictions divided by the total number of predictions.
Development of non-invasive biomarkers for pre-eclampsia through data-driven cardiovascular network models
