Maternal Blood-Flow Remodeling During Pregnancy

Why Blood-Flow Remodeling Matters

A maternal PBPK model needs more than a gestational cardiac-output value. It also needs a defensible account of how that output is distributed among the modeled tissues. If cardiac output increases while an organ receives a smaller fraction of it, that organ’s absolute blood flow may rise, remain similar, or fall. Therefore, multiplying every baseline organ flow by the same cardiac-output ratio is not automatically a valid pregnancy model.

The introductory material on gestational trajectories for maternal PBPK parameters establishes the value of indexing physiological inputs to gestational age. Here the focus is narrower: updating maternal organ blood flows, maintaining coherent flow bookkeeping, and distinguishing measured gestational changes from modeling assumptions. Pregnancy-related flow values vary across individuals, measurement methods, and gestational stages; no single set of regional changes should be treated as universal.

The relationship \(Q_{i,g}=CO_g f_{i,g}\) separates two distinct physiological inputs: the total volume of blood pumped per unit time and the share allocated to a region. It is the absolute flow \(Q_{i,g}\), not the fraction alone, that is used as a perfusion input for a tissue compartment. A fraction can decline while absolute flow rises if cardiac output rises sufficiently.

Define the Flow Bookkeeping Before Updating Inputs

The flow-balance equation applies to the set of flows that leave the modeled arterial supply as mutually exclusive branches. The set must be defined consistently. For example, if the model represents the uteroplacental circulation as a separate maternal flow, that flow must be included in the allocation and the remaining organ flows must be defined so they do not include it again. A model that does not represent this circulation separately may account for its effects differently, but should document that choice.

Similarly, a flow entering an organ through an internal pathway is not an additional independent share of cardiac output. In a conventional liver representation, hepatic arterial inflow and portal venous inflow have different origins: portal venous blood is returning from splanchnic tissues. Adding both the portal return and all the upstream splanchnic flows as independent systemic arterial branches would double-count blood. The model’s pathway structure, rather than a list of organ names alone, determines which flows belong in the cardiac-output sum.

Flows should also use consistent units and time bases, such as liters per minute throughout. A flow total that exceeds cardiac output is not fixed by changing a tissue volume or a concentration input. It signals either inconsistent inputs, mismatched definitions, or double-counting.

Use Gestational Evidence Without Inventing Precision

When data are available, represent the evidence at the level it supports. A study may report an absolute flow, a regional fraction, a change from a reference condition, or a measurement at only one gestational stage. These are not interchangeable. An absolute flow measurement can inform \(Q_{i,g}\) directly; a fraction must be combined with a compatible \(CO_g\); and a reported relative change requires a clearly defined baseline before it can be converted to an absolute flow.

Many PBPK applications lack direct measurements for every organ at every gestational age. A model may then use a literature-supported trend for some regions and an explicit assumption for others. For example, it might allow cardiac output to vary with gestational stage while holding a particular organ’s fraction constant. That choice implies the organ’s absolute flow scales with cardiac output; it is not evidence that the organ’s measured perfusion fraction is constant in pregnancy.

A useful record for each input therefore includes the quantity being modeled (absolute flow or fraction), the reference condition, gestational stage, units, data source, and whether the value is measured, derived, or assumed. Where evidence supports only a broad trend, reporting a highly precise gestational curve can imply more knowledge than the data provide. A piecewise or simpler trajectory may be preferable to an elaborate curve whose parameters cannot be supported.

Deriving a Constrained Reconciliation

When regional absolute-flow estimates \(m_i\) and cardiac output do not balance exactly, one principled option is to reconcile them using their measurement uncertainties. Suppose the estimates have independent errors with standard deviations \(s_i\), and impose the flow constraint \(\sum_i q_i=CO_g\). A weighted least-squares reconciliation chooses flows \(q_i\) that minimize the sum of squared standardized adjustments:

To derive the solution when the nonnegativity constraints are inactive, form the Lagrangian \(L=\sum_i(q_i-m_i)^2/s_i^2+\lambda(\sum_iq_i-CO_g)\). Differentiating with respect to \(q_i\) and setting the result to zero gives \(2(q_i-m_i)/s_i^2+\lambda=0\), so each adjustment is proportional to its variance. Summing the resulting flows and applying the constraint yields:

Thus, if the raw estimates fall short of cardiac output, the discrepancy is allocated more to estimates with larger uncertainty; if they exceed it, those estimates are reduced more. This result assumes the stated error model and that the adjusted flows remain nonnegative. If a result would be negative, a nonnegative constrained optimization is required. Reconciliation makes a coherent set of inputs; it does not resolve biased data or justify treating an uncertain estimate as a precise measurement.

Worked Example: A Measured Fraction Can Counter Simple Scaling

Suppose a hypothetical nonpregnant model has \(CO_0=5.2\) L/min and a kidney flow fraction of \(0.20\). At a specified pregnancy stage, suppose \(CO_g=6.2\) L/min and an available regional estimate supports a kidney fraction of \(0.17\). Compare the fraction-based flow with the result of scaling baseline kidney flow by cardiac output alone.

The baseline kidney flow is:

Using the pregnancy-stage fraction gives:

By contrast, scaling kidney flow in direct proportion to cardiac output while holding its fraction at \(0.20\) gives:

The two estimates differ by \(1.240-1.054=0.186\) L/min. The difference arises from the assumptions: one uses a pregnancy-stage fraction of \(0.17\), whereas the other assumes the baseline fraction remains \(0.20\). Neither calculation establishes which flow is physiologically correct; that depends on the quality and relevance of the evidence supporting the fraction.

Diagnostics, Alternatives, and Limitations

After updating inputs, check that the modeled parallel flows sum to the corresponding \(CO_g\) at each gestational stage. Also inspect each regional fraction and absolute flow separately. A smooth cardiac-output trajectory can coexist with an implausible discontinuity in one organ’s fraction, and conversely a smooth fraction can produce a substantial absolute-flow increase as cardiac output changes.

A flow-balance failure may indicate a unit conversion error, mixing of gestational stages, incomplete branch coverage, or double-counting of a pathway. A flow that becomes negative is physiologically impossible and usually signals an invalid adjustment or parameterization. Large changes caused only by normalization are also a warning: normalization preserves the relative weights but can conceal that their original scale or interpretation was unsuitable.

There are several alternatives, each answering a different evidence problem:

Pregnancy changes can be heterogeneous, and a measured value may represent a population average, a particular gestational window, or a method-specific estimate. A PBPK model that enforces flow balance remains an approximation of physiology. Sensitivity analysis can show whether predictions depend strongly on uncertain regional flows, but it cannot determine which assumed trajectory is true. Keep uncertainty and provenance visible, and do not infer a pregnancy-specific flow change merely because the model requires a complete set of inputs.

Common Mistake

Incorrect reasoning: “Cardiac output rises by 25%, so every organ flow rises by 25%.” This follows only if every organ’s fraction of cardiac output remains constant. It is not a consequence of the cardiac-output increase itself.

Correction: Specify the gestational cardiac output and the regional fraction separately, then calculate \(Q_{i,g}=CO_g f_{i,g}\). If the fraction is unknown and held constant, label that as an assumption. Check the complete modeled flow allocation and avoid counting internal return pathways as new independent cardiac-output branches.