Inverse problems in Cardiovascular Mathematics: toward patient-specific data assimilation and optimization

Blood flow through a cylindrical blood vessel can be mathematically described using the principles of laminar flow, a regime in which fluid moves smoothly in parallel layers. In this model, the velocity of the blood is not uniform across the cross-section of the vessel; rather, it varies with the radial distance from the center. The maximum velocity occurs along the central axis, decreasing progressively toward the vessel walls, where it reaches zero due to viscous drag.Approximating Blood...

Physiological pharmacokinetic models, often called flow-limited or perfusion models, typically assume a swift drug distribution between tissue and venous blood, creating a rapid drug equilibrium. This premise is based on the idea that drug diffusion is extremely fast, and the cell membrane presents no barrier to drug permeation. In this scenario, where no drug binding occurs, the drug concentration in the tissue equals that of the venous blood leaving the tissue. This greatly simplifies the...

Cardiac output (CO) is an integral aspect of human physiology, reflecting the heart's efficiency and responsiveness to the body's needs. It represents the volume of blood that the left or right ventricle ejects into the aorta or pulmonary trunk each minute. The CO is calculated by multiplying the heart rate (HR)—the number of heartbeats per minute—by the stroke volume (SV)—the amount of blood pumped out with each heartbeat.
In an average resting adult male, the typical cardiac output averages...

The heart's primary function is to pump blood throughout the body, maintaining a balance between blood sent out (cardiac output) and blood returning (venous return). If this balance is disrupted, it can result in congestive heart failure (CHF), a severe condition where the heart becomes an inefficient pump, leading to inadequate blood circulation.
CHF can occur due to the failure of either side of the heart. Left-side failure leads to pulmonary congestion—the right side continues to send blood...

Cardiac output (CO), the amount of blood the heart pumps per minute, is a parameter in cardiovascular physiology determined by stroke volume and heart rate. Stroke volume, the amount of blood pushed from one of the ventricles per heartbeat, is influenced by preload, afterload, and contractility.
Preload
Preload refers to the initial elongation of the cardiac myocytes before contraction and is related to the volume of blood filling the heart at the end of diastole, or end-diastolic volume. The...

The fundamental mathematical principles, such as calculus and graphs, play crucial roles in analyzing drug movement and determining pharmacokinetic parameters. Differential calculus examines rates of change and helps to determine the dissolution rate of drugs in biofluids, as well as how drug concentrations change over time. For instance, it can help calculate the rate of elimination of a drug from the body based on its concentration-time profile.
On the other hand, integral calculus focuses on...