The Power‐law Mathematical Model for Blood Damage Prediction: Analytical Developments and Physical Inconsistencies

Abstract

Blood trauma caused by medical devices is a major concern. Complications following the implantation/application of devices such as prosthetic heart valves, cannulae, blood pumps, tubing, and throttles lead to sublethal and lethal damage to platelets and erythrocytes. This damage is provided by the alterations in fluid dynamics, providing a mechanical load on the blood corpuscle's membrane by means of the shear stress. An appropriate quantification of the shear-induced hemolysis of artificial organs is thought to be useful in the design and development of such devices in order to minimize device-induced blood trauma. To date, a power-law mathematical relationship using the time of exposure of a blood corpuscle to a certain mechanical load and the shear stress itself (derived under the peculiar condition of uniform shear stress) has served as a basic model for the estimation of the damage to blood, investigated by means of numerical and/or experimental fluid dynamical techniques. The aim of the present article is to highlight the effect of a time-varying mechanical loading acting on blood cells based on the usual power-law model; furthermore, the effect of the loading history of a blood particle is discussed, showing how the past history of the shear acting on a blood corpuscle is not taken into account, as researchers have done until now. The need for a reassessment of the power-law model for potential blood trauma assessment is discussed by using a mathematical formulation based on the hypotheses of the existence of damage accumulation for blood with respect to time and with respect to shear stress, to be applied in complex flow fields such as the ones established in the presence of artificial organs.

Biological effects of dynamic shear stress in cardiovascular pathologies and devices

“…Platelet activity decreased with increasing T2 or 'relaxation time' between the peak shear stress. The experimental results showing platelet damage and recovery under shear stress obtained herein were in excellent agreement with mathematical damage accumulation models proposed by Grigioni et al [72,73], originally developed for RBC hemolysis and adapted for platelets in the study described previously [65].…”

“…The mechanics and kinetics of recovery of platelets after acute shear-stress exposure have been recently experimentally investigated in the HSD as discussed previously [65]. These results show remarkable agreement with numerical predictions from a cumulative damage accumulation model (i.e., cumulative effect of previous activated state, shear loading history and exposure time) originally proposed to investigate RBC hemolysis [72,73]. More importantly, the study demonstrates the utility of dynamic viscometers in emulating shear-loading histories (and determination of their effects on platelets) typically found in arterial circulation and recirculation devices.…”

Platelet Activation Due to Hemodynamic Shear Stresses: Damage Accumulation Model and Comparison to In Vitro Measurements

“…However, because of the complexity and the variety of phenomena involved in blood damage, a universal approach to the problem is incongruous. 8 A successful approach in which an analytic formulation was based on empirically calibrated parameters has been previously demonstrated to predict hemolysis in prosthetic heart valves. 9 There are many established methods for the in vitro investigation of platelet activation.…”

Hemolysis Prediction by using a Power-Law Mathematical Model in an Aortic Arterial Section

“…The term into the square brackets in (11) represents the whole mechanical dose acting on the erythrocyte when it moves along the pathline from the initial observation time until the th instant (conventionally is assumed 0 = 0), while ( ( ) ⁄ Δ ) is the mechanical dose sustained by a RBC in the th interval, (from instant −1 until instant ). Equation (11) states that the th experienced dose (the term inside the square brackets) is a weighted summation on the observation time intervals Δ coming before time ( = 1, … , ), the th weight is the value of shear stress ( ) sustained during the th time interval elevated to / value.…”

“…Hemolysis prediction with mathematical models must fulfill 3 conditions proposed in [3]:  Condition 1: It must not have a reduction of damage by a decreasing shear stress [11].…”