Biventricular Interaction During Acute Left Ventricular Ischemia in Mice: A Combined In-Vivo and In-Silico Approach

Computational models provide an efficient paradigm for integrating and linking multiple spatial and temporal scales. However, these models are difficult to parameterize and match to experimental data. Recent advances in both data collection and model analyses have helped overcome this limitation. Here, we combine a multiscale, biventricular interaction model with mouse data before and after left ventricular (LV) ischemia. Sensitivity analyses are used to identify the most influential parameters on pressure and volume predictions. The subset of influential model parameters are calibrated to biventricular pressure–volume loop data (n = 3) at baseline. Each mouse underwent left anterior descending coronary artery ligation, during which changes in fractional shortening and RV pressure–volume dynamics were recorded. Using the calibrated model, we simulate acute LV ischemia and contrast outputs at baseline and in simulated ischemia. Our baseline simulations align with the LV and RV data, and our predictions during ischemia complement recorded RV data and prior studies on LV function during myocardial infarction. We show that a model with both biventricular mechanical interaction and systems-level cardiovascular dynamics can quantitatively reproduce in-vivo data and qualitatively match prior findings from animal studies on LV ischemia. Supplementary Information The online version contains supplementary material available at 10.1007/s10439-023-03293-z.


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Blood volume and pressure estimates
Nominal parameters are generated using a combination of pressure and volume data or literature.Total blood volume in the mouse is determined by (S1) as postulated by Riches et al. 7 .Bodyweight for mice 1, 2, and 3 were 29, 32.

Cardiac Equations
The sarcomere model is embedded within a cardiac tissue model of atrial dynamics and biventricular inte 4 ).Changes in blood volume ( l) cause distension in the cardiac chambers, giving rise to the myocardial strain Here, (cm 2 ) is the current mid-wall area of the chamber, (cm 2 ) is the reference mid-wall area, and (dimensionless) is a curvature variable related to the ratio of wall volume, (cm 3 ), and radius of mid-wall curvature (cm -1 ) 4 .Once has been calculated and the corresponding is obtained from the sarcomere model, the mid-wall tension can be calculated as A balance in axial and radial tensions, and is enforced across the septal wall (S4) providing two differential algebraic equations 6 .The cavity tensions are used to calculate the cavity pressures.
The mid-wall volume, (cm 3 ), mid-wall curvature, (cm -1 ), and mid-wall crosssectional area, (cm 2 ), are the driving variables for cardiac chamber dynamics.In the atria, these are described by Since the LV, RV, and S are mechanically coupled, a separate formulation for , , and is required (based on the TriSeg model 4 ).Utilizing the common radius of midwall junction point and denoting the maximal axial distance from each chamber wall surface to the origin as 4 , we get for the LV, RV, and S. We can also relate to the blood volume in the chamber which is updated at each time point.
The atrial transmural pressure is determined from the wall tension and mid-wall curvature (S13) The ventricular mid-wall tension is broken up into the axial ( ) and radial ( ) tensions based on the geometry of the spherical chambers and the angle of the sphere opening 4 , giving (S14) These tensions must be balanced across the LV, RV, and septal wall, and serve as the algebraic constraints for the system as described above.The axial tensions in the ventricular cavities are then used to calculate the transmural pressure (S15) In total, the cardiac chambers and TriSeg model contribute two algebraic constraints ( and being zero), five wall volume parameters ( ), and five reference area parameters ( ).

Heteroskedastic Errors and Asymptotic Uncertainty Quantification
We assume that the measurement errors in our data are independent and stem from a zero mean Gaussian distribution, i.e., , for each sequential data point and measurement source .Our measurement sources include RV and LV pressure and volumes, as well as systemic arterial pressure.Parameter inference is performed on the natural-log scaled parameters using the negative loglikelihood, equivalent to minimizing where is the measured pressure and volume data, is the corresponding simulations of the pressure and volumes.Under the assumption of independent errors, the covariance matrix, , is block diagonal where the subscripts RVP, RVV, LVP, LVV, and SAP represent RV pressure, RV volume, LV pressure, LV volume, and systemic arterial pressure, respectively, and is the identity matrix.Since the parameters, , and covariance matrix, , are coupled, we pursue a two-step updating scheme as follows 1 : 1. Using the initial guess for the natural-log scaled parameters, and .
This is equivalent to solving the ordinary least squares problem, which provides the current optimal value of the scaled parameter .
2. Calculate the residual vector using the OLS estimate from step 1.

Set the error variance estimators as (S18)
where denotes the measurement source and is the number of samples associated with that measurement.In our work, for each sample.
4. Set using the results from equation (S17) in the block diagonal of (S18).
5. Minimize equation (S18) using from the previous iteration and the updated covariance.
6. Repeat steps 2-5 until the scaled parameter vector has converged.We use the cutoff to stop the algorithm.
Asymptotic uncertainty quantification is carried out using the final values of and .
As described in detail elsewhere 1,8 , the statistical properties of the parameter estimate can be approximated using asymptotic theory, which gives rise to (S19) where the parameter variance-covariance estimator is approximated by The term approximates the matrix, which we calculate using sensitivity of the model outputs at the optimal parameter estimate in combination with the error covariance 1,8 .Using this approximation, the parameter confidence intervals can be constructed using a t-statistic and the parameter estimator variance-covariance matrix, giving The error variances for the three animals can be found in Table S1.The pairwise correlation between parameters at the inferred value is defined by and shown below in figure S1.
S21) where and are the total number of data points and the number of parameters, respectively.The confidence and prediction intervals for the model output can be derived from the expected variance of the model response (S22) and the variance of a new, predicted value, ,
Cardiac output (CO) was calculated as the product of stroke volume and heart rate.The assumed nominal pressures in the other cardiovascular compartment are calculated as a function of the LV and RV pressure data, as detailed in Table1.The unstressed volumes (i.e., the blood not ejected during a cardiac cycle) and stressed volumes in each compartment are based on previous work 2 and provided in Table2.The compartment pressures and stressed volumes are used to construct nominal estimates of vascular resistance and compliance, shown in Table3.
determined from left (LV) and right ventricle (RV) volume measurements, set to be the max of the two indices.

Table S2 .
Formulas for nominal pressure values (presented in mmHg for convenience).Data values are provided in brackets for mouse 1, 2, and 3, respectively.