Download Simulation of Biomechanical and Chemical Transport in

Simulation of Biomechanical and Chemical Transport in
Human Tissue
Doruk Seren, Metin Ozen Ph.D.
OZEN ENGINEERING, INC.
Abstract:
This article presents the formulation of a novel method for applying the FEA code ANSYS to large-scale
study of drug distribution and chemical transport in the human body. The methods described here can be
used in combination with other simulation methodologies developed for studying mechanical responses of
biological tissue. Various single-tissue or single-system models have been executed, as well as more
traditional analytical treatments or physiologically based pharmacokinetic (PBPK) simulations, but to date,
large-scale numerical models are not in use. In order to provide a complete picture of varying drug
transport in different tissue on a macro scale, a more sophisticated modeling effort is necessary. This paper
incorporates finite element theory and various clinical studies to propose an approach for expanding the
scope and scale of delivery mechanisms, and convection-diffusion transport analyses.
Introduction
Understanding and visualization of drug distribution and transport effects in the human body as a whole is
important to medical device and drug manufacturers. This is particularly true in the case of innovative
delivery methods, such as stent-based or diffusive-patch delivery.
In 2004, Teague and Feezor [1] described the problems inherent in creation of complex finite element
models of biological tissue, even in terms of creating a useable geometry file. They described some
available methods for coping with this problem, but focused mostly on difficulties in solution of
mechanical problems in a single tissue.
In 2001, Hwang (et al.) [2] developed a numerical model to accompany laboratory testing of bovine aorta,
and were able to capture anisotropic diffusive properties in the arterial wall, as well as variation of transport
properties from one drug to another, in their model. This, again, focused on effects in one tissue.
While extensive research has focused on single-tissue [2, 5, 6, 7, 8] or single-organ [4] scale computational
modeling, these are very limited in scope. More precisely-targeted therapies, such as Trans Dermal Patch
(TDP) or patch-delivered muscle pain drugs [9] for example, may require a more 'macro' view of the
human body to ensure that the desired effect is only realized at the desired site.
ANSYS finite element software and its proprietary finite-volume code, CFX, have been used to model
haemodynamics and pulmonary delivery problems, and some biomechanics or tissue mechanics problems.
However, large scale applications have not been sufficiently pursued.
The work presented here explores and demonstrates how finite element methodology can be used for large
scale convective-diffusive transport simulations.
Case Studies
Several test cases were performed in order to compare results of this method to intuitive predictions and
previous models. These include:
•
a whole-body heat conduction problem
•
a simple arterial wall diffusion problem, as if by stent delivery
•
a two-part stent problem:
•
deployment (stent and wall modeled)
•
•
diffusion of the drug coating on the stent through the arterial tissues
Integration of systemic circuit and 'bulk' soft tissue model, omitting other structure for the time
being
Finite Element Model Formulation
ANSYS Element SOLID 70 is a heat transfer element that includes mass transport (diffusion) and nonlinear
fluid flow (convection) as optional settings. These are built into the SOLID 70 thermal elements purely out
of convenience, as the equations describing these effects are remarkably similar. However, as the bulk of
transport in human tissue is attributed to convection-diffusion effects, the inclusion of these three groups of
equations in the same element formulation is also convenient for the solution of transport problems in the
human body.
Special element settings (KEYOPT (8) > 0) includes mass transport effects (ie, Fickian diffusion) and even
checks Peclet (Pe) number (a ratio of convection effects to diffusion effects) automatically to ensure that
convective forces are relatively small and the solution will therefore be physically valid. This is useful in
the solution of the majority of flow-through-tissue problems.
Normally, ANSYS SOLID 70 thermal elements solve the Laplace equation:
∂ 2T ∂ 2T ∂ 2T
+
+
=0
∂x 2 ∂y 2 ∂z 2
With ∇T , temperature gradient, described by:
q heat x = kA
∂T
∂T
∂T
, q heat y = kA
, q heat z = kA
∂z
∂x
∂y
However, when KEYOPT (9)>0, ANSYS SOLID 70 solves the Laplace equation instead with ∇C ,
concentration gradient for Fickian Diffusion, as shown below:
q molarx = D
∂C
∂C
∂C
, q molary = D
, q molarz = D
∂x
∂y
∂z
To produce a dimensionally correct flow rate, the normal specific heat and density material property inputs
are instead set to arbitrary values of unity, allowing diffusion coefficient to replace thermal conductivity as
the fundamental material property. So, dimensionless concentration units are exchanged for temperature in
the applied boundary conditions, and diffusivity in area per unit time replaces thermal conductivity.
ANSYS also calculates a Peclet number and checks that it is less than one, as Pe > 1would represent
relatively weak diffusion as compared with convection. This ensures that the solution will be physically
valid.
The same thermal element also allows activation of a nonlinear fluid flow option which is analogous to
convective transport. Activation of KEYOPT (7) allows solution of pressure-driven, convection-dominated,
high-Pe transport. This is done by a similar variable analogy:
q permeatingfluid =
kA
µ
∇P
Temperature gradient and its accompanying proportional tensors from the Fourier heat equation can thus be
replaced by an appropriately grouped set of tensors and pressure gradients here, just as they were above by
concentration gradient and diffusivity.
This means that in certain tissues such as the Vitreous Humor [10], where convection plays a much greater
role (up to 40% of total mass transport in this case), a submodel can be used to simulate the sum of
convective and diffusive forces by solving in two steps.
The problem of modeling the circulatory system can be addressed as well. ANSYS includes FLUID 116
elements; one-dimensional (beam-like) 'coupled' physics elements that solve pressure and temperature
equations simultaneously, representing 'pipe'-like simple flows that may also be conducting heat. The
temperature solution can be adapted to chemical transport problems by the same analogy discussed above,
and the pressure solution will allow appropriate, pulsed-pressure boundary conditions for mass transfer
from tissue to blood and vice versa.
FLUID 116 elements solve the following:
N is the number of flow channels, C is specific heat, Kt is thermal conductivity and Kp is pressure
conductivity; In converting to dimensionless concentration units as above, specific heat must be eliminated
from the calculations by similar means, and Qg must be set to zero (internal heat generation). By these
means, FLUID 116 elements can complement the ability of SOLID 70 elements to form soft-tissue models.
The system may be modeled such that the pressure drop across capillary beds is taken into full account.
This is done by modifying N, the number of flow channels, and programmed radius of said channels. By
reducing diameter significantly and increasing the number of flow channels by one to two orders of
magnitude, different capillary layers may be modeled. This allows a clear distinction between injection on
the high-pressure or low-pressure side of the systemic circuit. This is realistic as capillaries, the smallest
vessels in the circulatory system, are often in the range of one red-blood cell diameter themselves, by
contrast to the aorta, the largest individual vessel; these are commonly in the range of ~2.5 cm in diameter.
More than realistic, however, it is also beneficial in forcing the pressure drop necessary to distinguish
effects in the low-pressure side (as a point of drug entry for example) from the high-pressure side.
The pulsing pressure load applied by the heart can also be modeled. By setting time steps to roughly half a
minute, a 'do' loop can be used to vary pressure between 120 and 80 mmHg with each time step by a simple
on / off algorithm: (time step number) % 2, returns 0 for an even number or 1 for an odd number time step;
this results in a 'heart rate' of about 60 beats per minute, within the range of normal resting heart rates.
Results
Bulk Heat Transfer
The first and simplest verification problem was a test of SOLID 70 thermal elements over a transient load
case, as the non-angular surfaces (which necessitate a tetrahedral mesh) are known to sometimes cause
significant errors in transient thermal analyses. These errors are often manifested by results that fall outside
the boundaries defined in the problem; As this was not observed in here, the mesh seems viable. In fact, the
inclusion of all three sets of equations as options for the SOLID70 element type was especially useful here,
because the body model did not have to be re-meshed after checking results. Option settings were changed
were merely changed to mass transport option for the circulatory system simulation to follow.
Figure 1. Simple, single-material body mesh. Curvature of surfaces necessitates
tetrahedral elements, initially creating concern for stability in transient case
Figure 2. Accurate results in a bounded heat-transfer problem, using the above mesh
Simplified Artery Wall Transport
Figure 3. ¾ symmetry artery wall showing orthotropic diffusivity
Another simple verification case was a ¾ symmetry model of an artery wall, with drug 'sources' applied as
if by stent coating. Results of this simple case indicate that SOLID 70 elements can effectively capture the
heterogeneity of final drug concentration due to anisotropic transport properties (see fig. 3) as observed in
smooth muscle tissue, like artery walls; As seen below, concentration contours begin with a nonuniform
distribution on the inner surface of the artery but are seen to merge and become a uniform radial gradient,
due to the larger D θ and DZ as compared with DR.
Body with Circulatory System Model
The FLUID 116 elements explained above were added to the existing bulk soft tissue model. As these are
line elements, this required the addition of small volumes within the existing solids. In this way, the lines to
be meshed were part of the boundary of the solid that makes up the soft tissue model, ensuring
‘temperature’ communication between the two element types. A further advantage of this approach is that it
does not require complex operations like multiple steps, contact elements, or convection surface elements.
As seen in the iso-surface contour plots (Fig. 4, 5) below, the addition of the circulatory system has
numerous advantages over the extremely coarse, single-material model used in the initial heat conduction
problem. Concentration gradients can now be seen moving to and around the heart. As the model increases
in complexity, capillary beds will be included, further taking advantage of the flexibility of this element
type, and providing an accurate picture of redistribution by the body’s active transport mechanism.
Figure 4. Mesh including FLUID 116 line elements and initial results. In early load step the
concentration contours show effect of circulatory system especially at heart
Figure 5. Closer image, later load time. Shows concentration gradients dissipating through
tissues near point of delivery but gathered at the ‘heart’, to be re-distributed by the
bloodstream
Stent Case Verification
One other verification case that represents the flexibility of these tools with regard to biomedical /
biotechnology problems is the following- A simplified stent case, where first, ANSYS is used to solve the
structural problem of the expansion of the stent into the artery wall by angioplasty balloon, and then the
subsequent drug transport problem. This stent case represents a 'reality check', demonstrating that the
element formulation presented here can solve a more traditional problem than the proposed large-scale
approach.
A stent is expanded into the artery wall by a uniform fixed displacement, which is reasonably accurate
given that angioplasty balloons are 'inflated' with saline water solution, and have fairly rigid walls. This
displacement forces the stent into the artery walls and stretches the artery walls somewhat. At this point, the
structural model has been solved; Many FEA tools also have the capability to display additional
information like exact penetration depth at points of contact and contact pressure. As an example of the
utility of this information, significant contact pressure may warrant modifications to the problem, such as
creation of a localized increased convection condition due to the pressure on the drug coating layer from
the stent strut, or modification of transport properties of the artery wall in regions of high distention.
Here, the above tissue modeling method is applied. The structural elements used to model the contact
problem of stent expansion are converted to thermal elements, with the appropriate settings discussed
above. ANSYS contact elements are also reset from structural contact to thermal contact equations.
Figure 6. mesh
Figure 7. deformed artery wall after expansion
Figure 8. Stent penetration
Figure 9. Concentration gradients when run to steady-state. Poor stent design does not
penetrate enough to yield good distribution
Discussion
While this model is an effective proof of the idea that the problem may become manageable, the model
requires a great deal of refinement- addition of layers of tissue, modeling of components through which
diffusion is significantly reduced (bones, cartilage, selective or semi-permeable membranes), additions of
capillary bed models, and other, more complicated tissues such as electrophysiological pathways.
Beyond mere geometric refinements, for accurate modeling of soft tissue transport, laboratory testing to
determine diffusivity of particular tissues will be necessary. Of course, diffusivity is not a material property
but a relationship- a given tissue will yield different diffusivities for different compounds. However, broad
guidelines could be developed to use as a range, as diffusion coefficients in a given medium often vary in
an almost-linear fashion with molecular weight- the smaller the molecule, the more easily it is conducted.
That said, the post-processing capabilities in ANSYS can be used to make results very easy to interpret and
discuss, especially if one were to complicate the problem with time-dependency, such as, other drug inputs
delivered by the balloon catheter, then removed after expansion, for example. As research continues, the
spectrum of problems and accuracy of this method will improve.
Conclusions
A novel method for applying the FEA code ANSYS to large-scale study of drug distribution and chemical
transport in the human body is presented in this paper. Positive initial results suggest that in the future,
incorporation of diffusivity relationships from simple laboratory testing can allow characterization of new
drugs, delivered by new techniques. This will benefit new device, drug and delivery designs in the same
way advanced simulations have done in many other industries. More sophistication and advanced features
will be incorporated into the current model in the near future.
References
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