Model Study of the Pressure Build-Up during Subcutaneous Injection

In this study we estimate the subcutaneous tissue counter pressure during drug infusion from a series of injections of insulin in type 2 diabetic patients using a non-invasive method. We construct a model for the pressure evolution in subcutaneous tissue based on mass continuity and the flow laws of a porous medium. For equivalent injection forces we measure the change in the infusion rate between injections in air at atmospheric pressure and in tissue. From a best fit with our model, we then determine the flow permeability as well as the bulk modulus of the tissue, estimated to be of the order 10−11–10−10 m2 and 105 Pa, respectively. The permeability is in good agreement with reported values for adipose porcine tissue. We suggest our model as a general way to estimate the pressure build-up in tissue during subcutaneous injection.


Supporting Information: Total pressure drop along the needle Entrance Flow in the Needle
We shall now consider the flow in the needle. The Reynolds number for the characteristic flow rates achieved during injection can be expressed in terms of the inner needle diameter D, the flow rate Q and the kinematic viscosity ν, For a flow rate of approximately 200µL/s and a needle diameter of 135µm, we obtain Re ∼ 1900. Even though this value is below the critical value for onset of turbulent flow in a pipe, we see that the inertial forces are not negligible. The flow in the entrance region of the pipe therefore differs from the laminar Poiseuille flow. Due to the friction with the wall, boundary layers with increasing widths downstream will be formed at small distances from the needle inlet. In the boundary layers, the fluid particles experience a reduction of their velocity and owing to mass conservation the velocity of the particles near the axis must increase. In the entrance region the velocity field will vary not only in the radial direction but also in the axial direction 1 . However, sufficiently far from the inlet where the width of the boundary layer is approximately equal to the needle radius, the viscous forces will dominate and the velocity profile will become asymptotically parabolic as in the Poiseuille flow. The length of the entrance region when the boundary layers merge into each other is called the entry length, and we shall denote it by L e . Since the inertial forces are not negligible in the entrance region a higher pressure gradient is needed to establish a given flow rate than in the case of a Poiseulle flow.
The development of flow patterns in the inlet of a 2D channel with parallel walls was studied in [1] and [2]. Details on the flow in the entrance of a pipe can be found in [3].
We shall here follow the calculations of [3] in order to calculate the pressure drop along the needle.

Governing equations
For this axially symmetrical flow the mass conservation equation, the radial and longitudinal Navier-Stokes equations read respectively, and The boundary conditions are, We shall now simplify the Navier-Stokes equations given the relatively high Reynolds numbers estimated for the injection problem. The longitudinal velocity in the bulk of the flow is U o . The boundary layer thickness, δ, is proportional to the square root of the kinematic viscosity of the fluid and it can be rigorously proven that near the inlet (see [1]) in which L denote a characteristic length of the variation of the axial velocity along the z coordinate, and δ(z) denote the boundary layer thickness at position z. The variation of the longitudinal velocity v z takes place over distances equal to the entry length, in which δ(L e ) = R leading to the estimate 2 With the above estimate we conclude that From the continuity equation we can see that v r ∼ Uo Re . Similarly, making the estimation for the pressure variation in the entrance region from Eq. (4) we have whereas from Eq. (3) we have for the transverse variation Thus, we can consider that for high Reynolds number the pressure in a cross-section of the pipe is practically constant when compared with the z-variation. Then, for the typical Reynolds numbers of the injection we can describe the flow inside the needle by the following systems of symplified Navier-Stokes equations, known as Prandtl's boundary layer equations The solution of this system of equations was solved in [3]. There, the axial velocity was determined as in which I n is the hyperbolic Bessel function of n'th order and β = β(z) is a function of the z coordinate which can be determined as [3].
The functions f (β) and g(β) are determined by [3] f The entry length is defined as the point where the central velocity has reached 99 of its terminal value. From the expression for the axial velocity the value of β at the entry length is β o ≈ 0.69. Thus, from Eq. (16) the ratio between the entry length and the pipe diameter is determined by Evaluating the latter equation in r = 0, leads to where we have used the fact that the pressure gradient is constant along the cross-section of the pipe and that both v r and ∂ r v z are identically zero at the pipe axis. In order to calculate the pressure drop form the inlet to the entrance length L e let us integrate Eq. (20) in the interval z ∈ [0; L e ]. Integration yields v 2 Using the boundary conditions given by Eq. (5) and (6) we obtain The first term in the right hand side of the Eq. (22) is essentially the pressure drop due to the acceleration of the fluid in the entrance region outside the boundary layer, while the second one accounts for the pressure losses owing to friction. In order to calculate the integral in Eq. (22) is more convenient to perform the integration changing, with the help of the definition in Eq. (16), the variable z to β.
Pressure losses in the entrance region can be calculated accordingly, Let us calculate now the pressure drop from the entry length to the outlet of the needle. In this region we have a fully developed parabolic profile, and the pressure difference can be determined by Since we are in the region of the fully developed flow the integrand can be calculated formally taking the limit 1 r ∂ r (r∂ r v z ) We thus obtain The total pressure losses in the needle can be obtained summing the pressure differences From Eqs. (28) and (24) we can derive the expression for the total pressure drop in the needle as ∆P = 128ηL πD 4 Q(t) + 16.16ρ π 2 D 4 Q 2 (t).