Fluid Dynamics of Heart Development

Abstract

The morphology, muscle mechanics, fluid dynamics, conduction properties, and molecular biology of the developing embryonic heart have received much attention in recent years due to the importance of both fluid and elastic forces in shaping the heart as well as the striking relationship between the heart’s evolution and development. Although few studies have directly addressed the connection between fluid dynamics and heart development, a number of studies suggest that fluids may play a key role in morphogenic signaling. For example, fluid shear stress may trigger biochemical cascades within the endothelial cells of the developing heart that regulate chamber and valve morphogenesis. Myocardial activity generates forces on the intracardiac blood, creating pressure gradients across the cardiac wall. These pressures may also serve as epigenetic signals. In this article, the fluid dynamics of the early stages of heart development is reviewed. The relevant work in cardiac morphology, muscle mechanics, regulatory networks, and electrophysiology is also reviewed in the context of intracardial fluid dynamics.

Appendix

Navier–Stokes Equations

In this section, the governing equations of incompressible, constant viscosity fluid flow is presented. Consider a fluid of constant density ρ (mass per unit volume) and dynamic viscosity μ (indicative of the effects of frictional forces/mixing in a fluid), flowing through a system with velocity components u, v, and w in x, y, and z coordinates, respectively. Let p represent the local fluid pressure. Equation 5 represents the conservation of mass, which requires that the mass flux (mass per unit time) of fluid entering a system must be equal to the mass flux of fluid leaving the system,

$$ {\frac{\partial u}{\partial x}} + {\frac{\partial v}{\partial y}} + {\frac{\partial w}{\partial z}} = 0 $$

(5)

Equations 6–8 represent the conservation of linear momentum in each coordinate direction, and this principle requires that the inertial force must be equal and opposite to the sum of the pressure force, viscous force, and body force (due to gravity in most cases). The inertial forces on the left hand side of the Eqs. 6–8 arise due to the fluid flow and include both unsteady (or transient) and convective (flow velocity and velocity gradient dependent) contributions.

$$ {\frac{\partial u}{\partial t}} + u{\frac{\partial u}{\partial x}} + v{\frac{\partial u}{\partial y}} + w{\frac{\partial u}{\partial z}} = - {\frac{ 1}{\rho }}\,{\frac{\partial p}{\partial x}} + \nu \left( {{\frac{{\partial^{ 2} u}}{{\partial x^{ 2} }}} + {\frac{{\partial^{ 2} u}}{{\partial y^{ 2} }}} + {\frac{{\partial^{ 2} u}}{{\partial z^{ 2} }}}} \right) + f_{{{\text{B,}}x}} $$

(6)

$$ {\frac{\partial v}{\partial t}} + u{\frac{\partial v}{\partial x}} + v{\frac{\partial v}{\partial y}} + w{\frac{\partial v}{\partial z}} = - {\frac{ 1}{\rho }}\,{\frac{\partial p}{\partial y}} + \nu \left( {{\frac{{\partial^{ 2} v}}{{\partial x^{ 2} }}} + {\frac{{\partial^{ 2} v}}{{\partial y^{ 2} }}} + {\frac{{\partial^{ 2} v}}{{\partial z^{ 2} }}}} \right) + f_{{{\text{B,}}y}} $$

(7)

$$ {\frac{\partial w}{\partial t}} + u{\frac{\partial w}{\partial x}} + v{\frac{\partial w}{\partial y}} + w{\frac{\partial w}{\partial z}} = - {\frac{ 1}{\rho }}\,{\frac{\partial p}{\partial z}} + \nu \left( {{\frac{{\partial^{ 2} w}}{{\partial x^{ 2} }}} + {\frac{{\partial^{ 2} w}}{{\partial y^{ 2} }}} + {\frac{{\partial^{ 2} w}}{{\partial z^{ 2} }}}} \right){ + }f_{{{\text{B,}}z}} $$

(8)

Note that ν is the kinematic viscosity of the fluid, which is the ratio of the coefficient of dynamic viscosity to the density of the fluid (ν = μ/ρ), while f _B indicates the body force acting on the fluid flow. The collective set of governing conservation Eqs. 1–4 are also known as the Navier–Stokes equations. The nonlinear mathematical nature of these partial differential equations renders it difficult to solve, and only a few exact analytical solutions for specific problems are known. Further details on these equations may be obtained in the book by Schlichting and Gersten [88], for example.

Fluid Dynamic Scaling

To obtain a physical perspective, it is a useful exercise to non-dimensionalize the terms in the above governing conservation equations using equivalent scaling characteristics as shown below,

$$ x^{\prime } = \frac{x}{L} , \;y^{\prime } = \frac{y}{L} , \;z^{\prime } = \frac{z}{L} $$

(9)

$$ u^{\prime } = \frac{u}{U} , \;v^{\prime } = \frac{v}{U} , \;w^{\prime } = \frac{w}{U} $$

(10)

$$ t^{\prime } = \omega \, t ,\; p^{\prime } = {\frac{p}{{\rho U^{2} }}} $$

(11)

where L, U, and 1/ ω are characteristic flow length, velocity, and time scales respectively, As an example, in the case of blood flow through the arteries of the adult human circulatory system, the diameter of the vessel, the velocity along the centerline of the artery (after some distance devoid of any entrance effects), and the pumping rate of the heart are typically chosen to be the characteristic length, velocity, and time scales respectively. The application of these terms to Eqs. 1–4 results in the following set of equations after neglecting the body force contribution (as its importance in cardiovascular flows is insignificant),

$$ {\frac{{\partial u^{\prime } }}{{\partial x^{\prime } }}} + {\frac{{\partial v^{\prime } }}{{\partial y^{\prime } }}} + {\frac{{\partial w^{\prime } }}{{\partial z^{\prime } }}} = 0 $$

(12)

$$ \left( {{\frac{\omega L}{U}}} \right){\frac{{\partial u^{\prime } }}{{\partial t^{\prime } }}} + u^{\prime } {\frac{{\partial u^{\prime } }}{{\partial x^{\prime } }}} + v^{\prime } {\frac{{\partial u^{\prime } }}{{\partial y^{\prime } }}} + w^{\prime } {\frac{{\partial u^{\prime } }}{{\partial z^{\prime } }}} = - {\frac{{\partial p^{\prime } }}{{\partial x^{\prime } }}} + \left( {{\frac{UL}{\nu }}} \right)\left( {{\frac{{\partial^{ 2} u^{\prime } }}{{\partial x^{\prime 2} }}} + {\frac{{\partial^{ 2} u^{\prime } }}{{\partial y^{\prime 2} }}} + {\frac{{\partial^{ 2} u^{\prime } }}{{\partial z^{\prime 2} }}}} \right) $$

(13)

$$ \left( {{\frac{\omega L}{U}}} \right){\frac{{\partial v^{\prime } }}{{\partial t^{\prime } }}} + u^{\prime } {\frac{{\partial v^{\prime } }}{{\partial x^{\prime } }}} + v^{\prime } {\frac{{\partial v^{\prime } }}{{\partial y^{\prime } }}} + w^{\prime } {\frac{{\partial v^{\prime } }}{{\partial z^{\prime } }}} = - {\frac{{\partial p^{\prime } }}{{\partial y^{\prime } }}} + \left( {{\frac{UL}{\nu }}} \right)\left( {{\frac{{\partial^{ 2} v^{\prime } }}{{\partial x^{\prime 2} }}} + {\frac{{\partial^{ 2} v^{\prime } }}{{\partial y^{\prime 2} }}} + {\frac{{\partial^{ 2} v^{\prime } }}{{\partial z^{\prime 2} }}}} \right) $$

(14)

$$ \left( {{\frac{\omega L}{U}}} \right){\frac{{\partial w^{\prime } }}{{\partial t^{\prime } }}} + u^{\prime } {\frac{{\partial w^{\prime } }}{{\partial x^{\prime } }}} + v^{\prime } {\frac{{\partial w^{\prime } }}{{\partial y^{\prime } }}} + w^{\prime } {\frac{{\partial w^{\prime } }}{{\partial z^{\prime } }}} = - {\frac{{\partial p^{\prime } }}{{\partial z^{\prime } }}} + \left( {{\frac{UL}{\nu }}} \right)\left( {{\frac{{\partial^{ 2} w^{\prime } }}{{\partial x^{\prime 2} }}} + {\frac{{\partial^{ 2} w^{\prime } }}{{\partial y^{\prime 2} }}} + {\frac{{\partial^{ 2} w^{\prime } }}{{\partial z^{\prime 2} }}}} \right) $$

(15)

In the context of vertebrate embryonic heart development, it is useful to examine the limit of low Reynolds and Womersley numbers in the vector form of the Navier–Stokes equations 5–8 as given below:

$$ {\frac{{\partial \vec{u}}}{\partial t}} + \nabla \cdot \left( {\rho \vec{u}\vec{u}} \right) = - {\frac{ 1}{\rho }}\nabla p + \nu \nabla^{ 2} \vec{u} - g $$

(16)

As the Re is sufficiently small, the inertial terms in the momentum equations as well as the gravitational force can be ignored, and for small values of Wo the transient term can be neglected, resulting in the Stokes equations below:

$$ \nabla p = \mu \nabla^{ 2} \vec{u},\;\nabla \cdot \vec{u} = 0 $$

(17)

In this limit of very low Re and Wo, the flow is entirely driven through a balance between the pressure gradient and viscous diffusion.

Shear Stress on Blood Vessels

To analyze the flow within a blood vessel, it is useful to start with a simplified model of internal flow through a pipe. Consider the steady, incompressible, two-dimensional (radial r and axial x, see definitions in Fig. 16), incompressible, internal flow of a fluid of density ρ and uniform dynamic viscosity μ through a cylinder of radius R. The flow velocity is assumed to have no swirling component (u _θ = 0), and the flow is considered to be axisymmetric about the central axis of the pipe, such that $\partial /\partial \theta = 0$.

$$ {\frac{{\partial u_{r} }}{\partial x}} + \frac{1}{r}{\frac{\partial }{\partial r}}\left( {ru_{r} } \right) = 0 $$

(18)

$$ u_{x} {\frac{{\partial u_{x} }}{\partial x}} \, + \, u_{r} {\frac{{\partial u_{x} }}{\partial r}} = - {\frac{ 1}{\rho }}\,{\frac{\partial p}{\partial x}} \, + \, \nu \left( {{\frac{{\partial^{ 2} u_{x} }}{{\partial x^{2} }}} \, + \, {\frac{{\partial^{ 2} u_{x} }}{{\partial r^{2} }}} + \frac{1}{r}{\frac{{\partial u_{x} }}{\partial r}}} \right) $$

(19)

$$ u_{x} {\frac{{\partial u_{r} }}{\partial x}} + u_{r} {\frac{{\partial u_{r} }}{\partial r}} = - {\frac{ 1}{\rho }}\,{\frac{\partial p}{\partial r}} + \nu \left( {{\frac{{\partial^{ 2} u_{r} }}{{\partial x^{2} }}} + {\frac{{\partial^{ 2} u_{r} }}{{\partial r^{2} }}} + \frac{1}{r}{\frac{{\partial u_{r} }}{\partial r}} - {\frac{{u_{r} }}{{r^{2} }}}} \right) $$

(20)

Fig. 16

Internal flow of a fluid through a circular cylinder of radius R (also known as Hagen–Poiseuille flow). Under the simplifications of two-dimensional, time-invariant, axisymmetric (no fluid rotation, u _θ = 0), incompressible flow of a fluid with uniform density and viscosity, the axial velocity u _x remains unchanged along the longitudinal direction x after a certain length (typically 2–4 multiples of the radius) from the entrance, and this condition is also known as a fully developed flow. However, the radial variation of the axial velocity u _x (r) is parabolic about the centerline axis as shown. The shear stress imposed by the flow τ in the radial direction varies linearly from the centerline, and the maximum value τ _w occurs at the walls. Note that δ indicates the thickness of the boundary layer, which is the region near the solid boundaries where the deceleration of the flow on account of fluid viscosity is non-negligible. The coordinate system used for the analysis of this problem (r radial, x axial, θ rotational) is also shown

This problem can be solved analytically by considering a few simplifying assumptions. Typically, in the case of microcirculation, the flow reaches a fully developed state at a short distance from its entrance (small multiple of the vessel diameter) such that there is no variation in the flow velocity along the primary axial direction thereafter ($\partial /\partial x = 0$). This reduces the above equation set 18–20 to the following:

$$ \frac{1}{r}{\frac{\partial }{\partial r}}\left( {ru_{r} } \right) = 0 $$

(21)

$$ u_{r} {\frac{{\partial u_{x} }}{\partial r}} = - {\frac{ 1}{\rho }}\,{\frac{\partial p}{\partial x}} + \nu \left( {{\frac{{\partial^{ 2} u_{x} }}{{\partial r^{2} }}} + \frac{1}{r}{\frac{{\partial u_{x} }}{\partial r}}} \right) $$

(22)

$$ u_{r} {\frac{{\partial u_{r} }}{\partial r}} = - {\frac{ 1}{\rho }}\,{\frac{\partial p}{\partial r}} + \nu \left( {{\frac{{\partial^{ 2} u_{r} }}{{\partial r^{2} }}} + \frac{1}{r}{\frac{{\partial u_{r} }}{\partial r}} - {\frac{{u_{r} }}{{r^{2} }}}} \right) $$

(23)

The above equations of mass and momentum are subject to the following conditions at specific boundaries in the problem domain:

$$ u_{r} \left( {r = 0} \right) = 0 $$

(24)

$$ u_{r} \left( {r = R} \right) = 0 $$

(25)

$$ u_{x} \left( {r = R} \right) = 0 $$

(26)

$$ \left. {{\frac{{\partial u_{x} }}{\partial r}}} \right|_{{r = 0}} = 0 $$

(27)

The boundary conditions 24 and 27 are obtained by considering symmetry about the centerline. The boundary condition 25 ensures that there is no normal flow through the vessel wall, and condition 26 means that the layer of fluid that is in contact with the vessel wall remains at rest (“no slip” of fluid on the solid surface). From the continuity equation 21, $ru_{r}$ must be constant. Applying the boundary conditions 24 and 25, it can be seen that there is no radial flow throughout the vessel, i.e., u _r  = 0 everywhere. This simplifies the r-momentum equation 23 to the form given below:

$$ {\frac{1}{\rho }}\,{\frac{\partial p}{\partial r}} = 0 $$

(28)

As the flow is incompressible, this means that the dynamic pressure is invariant in the radial direction, and is only a function of the axial location, i.e., p = p(x). The axial momentum conservation equation 22 now becomes

$$ {\frac{1}{\rho }}\,{\frac{{{\text{d}}p}}{{{\text{d}}x}}} = \nu \left( {{\frac{{\partial^{ 2} u_{x} }}{{\partial r^{2} }}} + \frac{1}{r}{\frac{{\partial u_{x} }}{\partial r}}} \right) $$

(29)

which can be written as,

$$ {\frac{r}{\mu }}\,{\frac{{{\text{d}}p}}{{{\text{d}}x}}} = {\frac{\partial }{\partial r}}\left( {r{\frac{{\partial u_{x} }}{\partial r}}} \right) $$

(30)

Integrating both sides of the above equation in terms of r, we obtain

$$ {\frac{{r^{2} }}{2\mu }}\,{\frac{{{\text{d}}p}}{{{\text{d}}x}}} = r{\frac{{\partial u_{x} }}{\partial r}} + A $$

(31)

where A is the constant of integration, the value of which is determined by applying 27 to the above equation. The resultant equation can be integrated once again in terms of r to solve for the axial velocity profile u _x (r),

$$ {\frac{{r^{2} }}{4\mu }}\,{\frac{{{\text{d}}p}}{{{\text{d}}x}}} = u_{x} + B $$

(32)

The constant of integration is determined by using boundary condition 26. The solution for the axial velocity is thus given by

$$ u_{x} \left( r \right) = - \left( {{\frac{{R^{2} }}{4\mu }}\,{\frac{{{\text{d}}p}}{{{\text{d}}x}}}} \right) \, \left( {1 - {\frac{{r^{2} }}{{R^{2} }}}} \right) = u_{\max } \left( {1 - {\frac{{r^{2} }}{{R^{2} }}}} \right) $$

(33)

The internal flow through a cylindrical vessel under the previously stated assumptions has a parabolic velocity profile with the peak located along the centerline, the magnitude of which depends on the pressure gradient at the particular axial location of interest, the dynamic viscosity of the fluid, and the vessel radius. The pressure gradient is referred to be adverse when dp/dx > 0 resulting in a decelerating flow, and is favorable when dp/dx < 0 and the flow accelerates.

The viscous fluid flow exerts a tangential shear stress, which can be determined as the gradient of the axial velocity as given below:

$$ \tau_{xr} = \tau_{rx} = \mu {\frac{{\partial u_{x} }}{\partial r}} = - {\frac{{2\mu u_{\max } r}}{{R^{2} }}} = {\frac{r}{2\mu }}\,{\frac{{{\text{d}}p}}{{{\text{d}}x}}} $$

(34)

Of special importance in developmental physiology is the shear stress imposed by blood flow on the walls of blood vessels, which is given by,

$$ \tau_{w} = \left. {\tau_{xr} } \right|_{{r{ = }R}} = - {\frac{{2\mu u_{\max } }}{R}} = {\frac{R}{2\mu }}\,{\frac{{{\text{d}}p}}{{{\text{d}}x}}} $$

(35)

The mean flow velocity through the vessel can be calculated by integrating the axial velocity profile over the cross section,

$$ \bar{u}_{x} = {\frac{1}{{\pi R^{2} }}}\int\limits_{0}^{2\pi } {\int\limits_{0}^{R} {u_{x} \left( r \right)r{\text{d}}r{\text{d}}\theta } } = {\frac{{u_{\max } }}{2}} $$

(36)

The shear stress can be redefined in terms of the volumetric flow rate Q based on mean flow velocity as

$$ \tau_{w} = - {\frac{{4\mu \bar{u}_{x} }}{R}} = - {\frac{4\mu Q}{{\pi R^{3} }}} = - {\frac{32\mu Q}{{\pi D^{3} }}} $$

(37)

Blood Rheology

One way to model the non-Newtonian properties of the blood is to consider it as a generalized Newtonian fluid, where the shear stress is a function of the shear rate at the particular time, and the fluid dynamics do not depend upon the history of deformation. This approach has been used previously to model the blood as a Cross fluid [9] and as a power law fluid [45, 114]. In both cases, the constitutive equations are the same as the traditional incompressible Navier–Stokes equations with the exception that the viscosity is no longer constant and depends upon the shear stress and/or the shear rate. For a power law fluid, the shear stress and effective viscosities are given by the equations:

$$ \tau = K\left( {{\frac{\partial u}{\partial y}}} \right)^{n} $$

(38)

$$ \mu_{\text{eff}} = K\left( {{\frac{\partial u}{\partial y}}} \right)^{n - 1} $$

(39)

where K is the flow consistency index, $\partial u/\partial y$ is the velocity gradient perpendicular to the plane of shear, n is the flow behavior index, and μ _eff is the effective viscosity. Note that for the Newtonian case n = 1. The disadvantage of this model is that it is only appropriate for shear rates over the range for which it was fitted. Notice that the effective viscosity goes to infinity as the shear rate approaches zero for shear thinning fluids (n < 1). A more reasonable generalized Newtonian model might be the Cross model. In this case, the effective viscosity is a function of the shear rate and is given by the equation:

$$ \mu_{eff} \left( {\dot{\gamma }} \right) = {\frac{{\mu_{0} }}{{1 + \left( {{\frac{{\mu_{ 0} \dot{\gamma }}}{{\tau^{*} }}}} \right)^{1 - n} }}} $$

(40)

where μ ₀, τ*, and n are experimentally determined coefficients. The shear rate, $\dot{\gamma }$, is set to the gradient of the velocity of the fluid. In this model, the fluid behaves as a Newtonian fluid at low shear rates (μ ₀ $\dot{\gamma }$ ≪ τ*) and as a power law fluid at high shear rates (μ ₀ $\dot{\gamma }$ ≫ τ*).

For modeling purposes, μ _eff is usually fit in the biologically relevant range of the non-Newtonian characteristics of the blood. These models are typically used for intermediate sized blood vessels (diameter > 22 μm) where blood is treated as a homogenous fluid. If the cell diameter is comparable to the vessel diameter (or on the same order of magnitude), this continuum approximation is not appropriate. A Newtonian fluid approximation for blood viscosity is acceptable typically for larger vessels where the diameter of the vessel (typically >0.5 mm) is well above the diameter of the red blood cells (roughly 8 μm). Skalak and Özkaya [94] present a detailed review of blood rheology, and may be referred to for further information.