SOFT TISSUE MECHANICS DAVID SHREIBER BME MEASUREMENT AND ANALYSIS LABORATORY 125:315

WHY STUDY SOFT TISSUE MECHANICS? Elastic Cartilage Blood Vessel Tendon MANY BIOLOGICAL TISSUES HAVE EVOLVED TO PERFORM SPECIFIC MECHANICAL FUNCTIONS. SOMETIMES, THESE TISSUES FAIL (PHYSICALLY AND/OR FUNCTIONALLY).

WHY IS IT IMPORTANT TO UNDERSTAND THE MECHANICAL FUNCTION OF TISSUES? Obviously, prior to implantation, a replacement must be able to meet the structural requirements necessary to perform its mechanical function. –Make sure bioartificial ligament is strong enough to support loads In many, if not all, cases you must be careful not to “exceed” the requirements. –In typical engineering mechanics, we include “safety factors” that overestimate the structural integrity of the device, typically by at least a factor of –This may not be appropriate for bioengineering bioartificial tissues. –What might happen if we implanted a ligament that was 3X as stiff as a normal ligament? –Can you think of other examples where the “correct” mechanical properties (not too weak/compliant, not too strong/stiff) are necessary to ensure long term function?

IN VIVO FLOW PREDICTIONS What happens to blood flow if a vessel’s mechanical properties change, say from intimal hyperplasia?

MECHANOTRANSDUCTION MECHANOSTRUCUTRAL PROPERTIES CELLULAR PHYSIOLOGY You must consider the cellular response to abnormal mechanical conditions, not just what you might think would be the “worst case scenario.”

CRITICAL QUESTIONS What kinds of loads will the tissue experience in vivo? What dictates the mechanical response of the tissue? What happens if the tissue/cells experience abnormal loading conditions? CRITICAL SCIENCE Biological: Kinesiology, cell biology Engineering: Statics, strength of materials, dynamics

WHAT MAKES BIOLOGICAL MATERIALS DIFFERENT THAN “TRADITIONAL” ENGINEERING MATERIALS?

HOW DO WE DESCRIBE THE MECHANICAL PROPERTIES OF A “TYPICAL” MATERIAL?

Generally, for most materials you have encountered as engineers, stress is linearly proportional to strain. Furthermore, the time scale of loading doesn’t matter….you can bend aluminum quickly or slowly and the stiffness will be the same.

TYPICAL BIOLOGICAL RESPONSE MOST BIOLOGICAL TISSUES DEMONSTRATE A NON-LINEAR STRESS-STRAIN CURVE. MOST OF THESE CURVES SHOW “STRAIN-STIFFENING” – IE THE MATERIAL IS STIFFER AT HIGHER STRAINS THAN AT LOWER STRAINS, ALTHOUGH SOME SHOW THE OPPOSITE (“STRAIN-SOFTENING”). WHY IS THIS SO?

NON-LINEAR ELASTICITY REASON #1: A tissue is not a continuum, but rather a composite of many fibers. If you look at the structural heirarchy of the tendon, for instance, it consists of collagen fibers, bundled into microfibrils, bundled into subfibrils, bundled into fibrils, into fascicles, and finally into a tendon. These subunits may have different “spring constants” and may experience loads differently because of their orientation and their crimp.

FIBERS ARE STRONG IN TENSION Schematic of stress-strain relationship and fiber orientation in skin. As a tissue is stretched, the fibers orient in the direction of stretch, and they become “straight”. Think of a fiber as a string. If a string is wavy (crimped), it is very easy to pull it. The string has very little resistance to tension until it is perfectly straight. What if you had a bundle of strings, and different ones became “straight” at different strains? As you increase strain, more and more strings straighten, and your effective stiffness increases!

CRIMP AND ELASTICITY If each fiber is linearly elastic (ie Stress = strain x stiffness), then when all of the fibers are straight, the effective stiffness of the tissue should remain constant, and the slope of the stress-strain curve should be linear. Roughly linear

MULTIPLE COMPONENTS REASON #2: Biological tissues are composites of different kinds of structural- supporting fibers. The most common ones are collagen and elastin. They have different mechanical properties. Elastin is generally believed to contribute to the low-strain behavior of tissues, and collagen to the high- strain behavior.

TIME-DEPENDENT BEHAVIOR Strain rate 1 Strain rate 2 Strain rate 3 Strain rate 1 > strain rate 2 > strain rate 3

WHY DOES THIS HAPPEN? Tissues are not only composites of “elastic” structures (if we assume that collagen and elastin are, indeed, elastic), but also of highly charged molecules called proteoglycans that attract water and maintain tissue hydration. The fluid component of tissues adds complexity to the mechanical response.

HOW CAN YOU DESCRIBE THE MECHANICAL PROPERTIES OF BIOLOGICAL TISSUES?

CONSTITUTIVE RELATIONSHIPS In mechanics, a “law” that describes the fundamental response of a class of materials. Each material in a class may have unique properties, but they all follow the same law. Example: Stress = Young’s Modulus x strain This is a constitutive law that describes homogeneous, isotropic, linear elastic materials. Different materials materials of this type have different values for E, but they all follow the same constitutive law. So what are good constitutive laws for biological tissues?

ELASTICITY A material is elastic if you can recover all of the energy you put into the material. In other words, if you stretch an elastic material, it will return to its underformed length upon removal of the load.

LINEAR ELASTIC VS. HYPERELASTIC Both of these materials are (potentially) elastic…however, one is linearly elastic (red line), where stress is proportional to strain, and the other is non-linearly elastic, and more specifically hyperelastic (black line). To quantify a linearly elastic material is easy….calculate the slope of the line, and this = the Elastic Modulus. To quantify a non-linearly elastic material is not so easy… One way is to develop a more complicated constitutive law: Another is to caclulate the slope of the “linear” region, and approximate the strain at which this region begins. Neo-hookean Stress- Strain law

TIME-DEPENDENT MECHANICAL BEHAVIOR As discussed previously, all biological tissues demonstrate time-dependent mechanical behavior This allows the tissue to optimally respond to the dynamics of mechanical loading What if your cartilage was just like an elastic piece of rubber?

VISCOELASTICITY Viscoelasticity describes the mechanical response of tissues that both store and dissipate energy. Think of these materials as part viscous liquid and part elastic solid. –The viscous part dissipates energy, while the elastic part stores it.

ELASTIC MODEL “Elastic” Spring The behavior of viscoelastic materials in uni-axial stress closely resembles that of models built from discrete elastic and viscous elements.    = extension (strain) Typically with springs, we relate force, F to extension,  through a “spring constant”, k. We can also think of the spring representative of a material (like a bar).    = extension (strain) Normally, F=k  But in terms of material constants,  =E  Where E is stiffness or Young’s modulus.

VISCOUS MODEL “Fluid” Element =“Viscous” Dashpot (or damper) The behavior of viscoelastic materials in uni-axial stress closely resembles that of models built from discrete elastic and viscous elements.    = extension (strain) Viscous elements respond to the rate of loading: The stronger the force, the faster the piston will move; or The faster the extension, the more force (or stress) is generated. Recall fluid shear stress:

LUMPED PARAMETER MODELS We now have descriptions of elastic and viscous elements Behavior of simple, linear, viscoelastic materials can be defined by combining these elements in different configurations.

STANDARD VISCOELASTIC TESTS Creep: Stress Relaxation: Stress Deformation Load Time Deformation

MAXWELL FLUID 1 2 A MAXWELL FLUID IS A SPRING AND DASHPOT IN SERIES

MAXWELL FLUID….WHAT DOES THE RESPONSE LOOK LIKE? Think about it intuitively….what happens at t=0 when we “instantaneously” apply a stress,  0 ? 12

STANDARD VISCOELASTIC TESTS Creep: Stress Relaxation: Stress Deformation Load Time Deformation

MAXWELL FLUID….WHAT DOES THE RESPONSE LOOK LIKE? Stress = constantStrain = constant

KELVIN SOLID 1 2 A KELVIN SOLID IS A SPRING AND DASHPOT IN SERIES

KELVIN SOLID…..WHAT DOES THE RESPONSE LOOK LIKE? 1 2 Subject this to same load history as before…

STANDARD VISCOELASTIC TESTS Creep: Stress Relaxation: Stress Deformation Load Time Deformation

KELVIN SOLID…..WHAT DOES THE RESPONSE LOOK LIKE?

STANDARD LINEAR SOLID WHAT DOES THE RESPONSE LOOK LIKE?

STANDARD VISCOELASTIC TESTS Creep: Stress Relaxation: Stress Deformation Load Time Deformation

STANDARD LINEAR SOLID …..WHAT DOES THE RESPONSE LOOK LIKE?

THIS WEEK’S LAB Testing of samples in uniaxial tension Compare mechanical properties of 2 kinds of “material” –Stretch 2 types of samples (rubber and chicken skin) at different rates Make sure to take careful measurements of the dimensions of the samples! (Why??) Plot Stress vs. Strain for different materials Quantify modulus (moduli?) and compare

WHAT REALLY HAPPENS 1.You use the Instron to apply a displacement to one end of the sample. The computer records how far the grip (and thus one end of the sample) has moved. 2.The other end of the sample is fixed (doesn’t move)and is attached in series to a force transducer that converts force to voltage that can be measured with an analog-to-digital converted (somewhere in the instrumentation). The force is also recorded by the computer. 3.You know how much the sample has deformed (the displacement). If you measure the gage length of the sample, then you can calculate the engineering strain: 4.You know how much force is produced in the material to resist that deformation. If you know the cross-sectional area of the sample, you can calculate the engineering stress:

HOW TO QUANTIFY STRESS-STRAIN PLOTS –If linear elastic, the slope –If non-linear elastic….depends on your ambition –Basically you need a consistent method of evaluating mechanical properties to compare one case to the next.

STRESS-STRAIN PLOTS We are running load-deflection experiments, not instantaneous creep or stress-relaxation tests (although you can run these if you like and there is extra time! Could be educational (but who would want that). Slow displacement Fast displacement

SAMPLE DIMENSIONS Gage Length, L 0 Length Width Area = length x width Sample viewed from side

BONUS EXPTS WITH THIS WEEK’S LAB If time allows….or you are interested…. Additional experiments: 1)Crosslink the chicken skin with a formaldehyde solution or UV light and examine mechanical response 2)Compare the effect of orientation on the stress-strain plot in rubber and chicken samples: 1 2 Vs. 1 2