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Introduction

The primary cause of pressure ulcers is some form of prolonged mechani- cal load, although environmental and intrinsic factors have an influence on the development. Without the mechanical loading no wound will develop.

This means that studying the response of soft biological tissues to mechan- ical loading is a critical issue and this may be achieved by the use of theo- retical models. There is a surprising dearth of published studies on pres- sure ulcers backed by some form of theoretical analysis. Speculative and intuitive statements on the role of mechanical loads and the influences of shear, pressure and traction on tissue behaviour and on the assumed occlu- sion of blood vessels are abundant, however, on occasion supported by ex- perimental evidence. One can only guess why this is the case. One reason might be the paucity of bioengineers involved in pressure ulcer research, although it is also true that the level of sophistication of theoretical models has long been far fromrealistic. However, it is believed that the current prospects are very good, and this is an opportune moment to examine the potential of available theoretical, numerical models as a tool in pressure ulcer research. The objective of this chapter is to show where mechanical modelling can be important or even indispensable. This will include a de- scription of how models were used in the past, what can be expected in the near future and which are the bottlenecks in this research.

Basically there can be two reasons for the development of theoretical models. The first is to use models as a design tool for patient support sur- faces in the most general sense, ranging from beds and wheelchairs or wheelchair cushions to prostheses for patients with an amputation. This approach will obviate the need for extensive prototyping and may lead to computer-aided optimization of patient support surfaces. The first section is devoted to modelling that is used for this purpose.

The second reason to use theoretical models is to understand the very complex phenomena in soft tissues such as skin, fat and muscle under pro- longed mechanical load. In studies on the aetiology of pressure ulcers these models are indispensable. The second section is devoted to these models.

Perspectives of Numerical Modelling in Pressure Ulcer Research

Cees Oomens

10

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Models for the Design of Supporting Surfaces

Most of the theoretical models that were developed as design tools for sup- porting surfaces were devoted to prosthetic sockets for upper- and lower- limb amputees. This work was reviewed recently by Zhang et al. [1] and is discussed in Chap. 9 by Sanders. The present section is limited to support- ing surfaces for seated persons in wheelchairs.

Chow and Odell [2] published one of the first finite-element studies with particular reference to this research area. To evaluate the quality of cush- ions, these authors postulated that it was necessary to predict the complete state of stress and strain inside the soft body tissues of a sitting person. In other words, knowledge of the pressure distribution at the interface was in- sufficient. Their motivation for this approach was based on the observation that even very high hydrostatic pressures are not damaging to the soft tis- sues, suggesting that tissue deformation is the critical factor. That is why a finite-element analysis seemed appropriate. Based on measured pressure distributions, Chow and Odell [2] assumed that an axi-symmetric model was sufficient and studied the stress/strain state under different loading conditions with a range of support cushions. The contact properties were modelled by applying a modified cosine pressure distribution at the inter- face. Their most important conclusion was: ªThe distortion of tissues is of- ten more severe at internal locations than on the surface of the buttocksº.

Todd and Thacker [3] presented a three-dimensional computer model of the human buttocks to understand the risk of developing a pressure ulcer while sitting in a wheelchair. Their objective was to show that a finite-element model could be developed that gave results consistent with experimental data and to relate buttock/cushion interface pressures to internal pressures at the ischium. A finite-element model was developed, based on MR images with a 1-mm resolution. Subjects were placed in supine position in the tube of a whole-body MR system. From transverse cross sections a 3-D image was re- constructed. Bone, soft tissue and cushion properties were modelled. A linear elastic, isotropic model with small deformations was used. For the seated per- sons the same model was used, but the stiffness values for the tissues were increased to simulate characteristics in the seated position. The load was based on the individual bodyweight of the test persons, with half bodyweight transferred in the seated position. The results were based on displacements, principal stresses and von Mises stresses. The latter parameter appeared to provide a good representation of the areas corresponding to the highest val- ues for tissue deformation. Indeed, von Mises stresses were found near bony prominences, which may explain why in many cases severe ulcers are in- itiated within the muscle tissues.

Dabnichki et al. [4] attempted to demonstrate that a finite-element model of a buttock can be a valuable tool for evaluation of support surfaces. They were the first group to perform a geometrically nonlinear analysis and em- ploy contact elements to obtain an improved description of the mechanical state at the interface between buttock and cushion. Although some critical re-

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marks can be made with respect to the material laws used for both the soft tissues and the cushion [5], the use of the large deformation theory and con- tact elements can be considered a considerable step forward in this type of modelling. They used a geometrical model of the buttock comparable to the Chow and Odell model. Frictionless contact and contact with friction were simulated. In the former case, the results suggested a minimal (negative) principal stress near the bony prominence. The highest shear strain was found in the middle of the tissue, although a local maximum was found ad- jacent to the bony prominence. The case with friction showed only minimal changes near to the bone, but higher compressive stresses were estimated near to the buttock/tissue interface. A softer cushion reduces peak stresses and leads to a change of location for the maximum shear stress.

A logical extension of this model was performed by the present author [6] by employing improved material laws for the soft tissues composite, with distinct properties for skin, fat and muscle layers. For the reference model material properties were used which were obtained from animal ex- periments. To account for the uncertainty in these data a sensitivity analy- sis was performed with respect to fat properties and the thickness of indi- vidual layers. In addition, variations with cushion properties and different frictional properties of the interface were performed. The major conclu- sions from previous model studies [4] were confirmed by this study. A re- markable result was that a reduction of the stiffness properties of the cush- ion led to very high changes in the maximum shear strain in the fat layer.

By contrast, the same reduction in cushion stiffness only marginally influ- enced the high local maximum shear strain in the muscle near the bony prominence. As expected, a softer cushion led to a more equal pressure distribution at the interface with lower peak stresses than with a stiffer cushion. This is an important result, because it confirmed that even though the interface pressure was lower and the shear strains in the fat were lower, the shear strains in the muscle were still at a high risk level. These findings have significant consequences for the design of supporting surfaces.

How realistic is it to assume that mechanical models can really play a role in the future design of supporting surfaces? Finite-element models should satisfy specific demands:

z They should be realistic, meaning that the geometry, loading history and other boundary conditions have to be realistic.

z Sufficiently accurate material laws have to be available for the differ- ent tissues involved, preferably with information on individual tissue tolerance.

z As an interactive design tool, the model has to be developed in a lim- ited amount of time.

z Simulations with the model have to be fast, especially when the model is used in an optimization loop.

At the present time it is possible to develop models with a fairly realistic ge- ometry using loading conditions close to reality. Figure 10.1 shows a solid model of a part of a skeleton and a finite-element model of the buttock region a Models for the Design of Supporting Surfaces z 151

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and upper limbs of a seated person [7]. This model is based on data of a seated cadaver that was digitized for a large European project (HUMOS I) on human modelling. Starting from this database the meshing of this region took 3 months with a highly sophisticated mesh generator. One single anal- ysis with an explicit code on a medium sized computer (Pentium III) takes approximately 3 days. However, there are reports that with a semi-two-di- mensional approach real-time finite-element method comes into reach [8].

Compared with the time it takes to develop mannequins for experimental testing and the costs associated with such experiments, theoretical models of this kind will certainly be competitive. Although the CPU time for the anal- ysis is high for optimization purposes, developments in computer hardware and especially in parallel processing will reduce this problemconsiderably.

Models to Be Used as a Tool to Understand Aetiology

The alternative approach is to utilize theoretical models to provide an im- proved understanding of the aetiology of pressure ulcers. This section will include a description of relevant studies on this subject, as well as a com- ment on future developments.

Sacks et al. [9] provided a simple, but elegant, attempt to understand the inverse relationship between external pressure levels and the risk of de- veloping an ulcer by using a dimensional analysis. The authors assumed that a definable allowable pressure exists that will initiate an ulcer. More- over, this pressure was assumed to depend on physical properties of the tissue (tissue density q, tissue elastic modulus E), time t and the local blood flow Q before the load is applied. Thus a mathematical relationship was formulated, such that:

psˆ f …q; Q; E; t† …10:1†

By using a simple dimension analysis the assumption led to a relation- ship between the allowable pressure and time:

Fig. 10.1.Mesh used for comfort analysis of human sittingon a cushion [7]

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psˆ a ‡ Bt 4=3 …10:2†

Apart fromthe constants A and B, which, of course, vary for different types of experiments, the exponent of ±4/3 of the time in Eq. 10.2 fitted re- markably well to experimental data. An important remark in the paper is the rationale that the formation of pressure ulcers primarily depends on tissue properties, irrespective of the type of loading. These properties change with age and pathology and can make individuals more or less sus- ceptible to pressure ulcers.

In a comparable dimension analysis Sacks [10] demonstrates that ex- periments on skin blood flow changes can best be presented in dimension- less formof blood flow decrease against tissue strain and not as a relation of blood flow versus pressure. The first presentation allows a comparison between similar groups of patients, while the second method depends on too many geometric variables.

Mak et al. [11] and Zhang et al. [12] explored an idea that had been pub- lished earlier by Reddy [13], namely that interstitial fluid flow may play a part in pressure ulcer development. By means of a biphasic poro-elastic analysis these later authors studied the behaviour of skin as a function of time under a prolonged mechanical load. What is clear from the theory is that the inter- stitial fluid takes up most of the loading just after the external pressure is ap- plied, but with time this fluid diminishes and the load is increasingly carried by the solid matrix. The authors tried to compare their model results with the experimental data from Reswick and Rogers [14]. Although a little premature in view of more recent evidence suggesting the important roles of fixed charge density and charge of ions, causing osmotic swelling effects, this study made clear that the mechanical state of the soft tissues changes with time, even with a constant external load. Keeping in mind that the duration of the load is a risk factor, this must be an important observation.

Fromthe above it should be appreciated that the local tissue deformation is a better predictor of damage leading to a pressure ulcer than the external load or pressure. This becomes more evident when discussing the assumed greater risk of shear than of compression. Shear is supposed to be a higher risk because vessels can be occluded more easily by shear than by compres- sion. However, this discussion was somewhat confused by abuse of terminol- ogy. In 1994 Zhang et al. [15] attempted to clarify this confusion with a com- bined theoretical and experimental analysis. These authors were careful to distinguish clearly between the external load that was applied (which can be a shear force, a compressive force or a combination of both) and the re- sulting internal stress and strain within the tissue. The best way to illustrate this is by looking at uniaxial compression and simple shear. Consider the uniaxial compression, for which Fig. 10.2a and Fig. 10.2b represent the same object with the same deformation. In the centre point of the specimen the strain tensor of this mechanical system is given by:

e ˆ e11~e1~e1‡ e22~e2~e2‡ e33~e3~e3with e11 ˆ e33 …10:3†

a Models to Be Used as a Tool to Understand Aetiology z 153

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If this equation is written in matrix form, with respect to a vector basis f~e1;~e2;~e3g this leads to the following strain matrix:

e ˆ e11 0 0 0 e22 0 0 0 e11

0

@

1

A …10:4†

If the object is rotated by 45o the matrix representation of the same strain tensor with respect to basis f~e1;~e2;~e3g is:

e ˆ1 2

e11‡ e22 e22 e11 0 e22 e11 e11 e22 0

0 0 2e11

0

@

1

A …10:5†

This means that if a small specimen is cut from the object, like the cen- tre square in Fig. 10.2a, it stretches in the same way as the macroscopic specimen. However, if a centre square is cut from the object oriented in a way like the square in Fig. 10.2b it will stretch and shear! Thus, although many might consider that the macroscopic deformation was compressive, it is in fact a combination of compression and shear in nature. For each ma- terial point we can find principal strain directions that, if used as a basis, would lead to a matrix like that in Eq. 10.4. We can also find a direction for which the maximum shear strain in a material is reached. In the above example this is an angle of 458 with respect to basis f~e1;~e2;~e3g. The princi- pal strains and the maximum shear strain are invariants and are mechani- cal properties that may be suitable to use in damage criteria (depends on the kind of material we look at).

Zhang et al. (1994) examined the maximum shear strain inside the tissue for different external loading conditions, namely, compression, shear, and a combination of the two modes. Their findings may be summarized as:

1. The maximum compressive stress (strain) inside the tissue is deter- mined by the resultant of shear and compression.

2. When an external shear force is applied, the location within the tis- sue with the highest shear strain will shift in the direction of the load if compared to the location for an external compressive load.

Fig. 10.2a, b.Schematic drawingof uniaxial compression test. Solid line, undeformed configuration; dotted line, deformed

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3. The external shear force mainly influences the stress/strain state near the surface, while the compressive load mainly influences this state in the deeper layers.

The aforementioned studies, representing global analyses using fairly sim- ple theoretical models, elucidate some of the confusing discussions on pressure ulcers in the literature. However, to understand the conditions within the tissue more complicated models need to be adopted. This inevi- tably necessitates the use of micro- or mesoscopic models or a combina- tion of models in a multi-scale approach. One such model, developed at Eindhoven University of Technology in the Netherlands, will be described in the following section.

Multi-scale Modelling of Tissues

The model is based on a number of experimental observations, which will be discussed first. Bouten et al. [17] and Breuls et al. [16] compressed tis- sue-engineered constructs with well-controlled mechanical loads and re- lated local cell deformation to cell death. From these studies it has become clear that muscle cells are susceptible for deformation, even if transport processes through the construct are not disturbed. It may be that the de- formation changes the transport properties of the cell membrane, even- tually leading to damage, but it is not caused by a reduced supply of oxy- gen and nutrients fromthe environment. Fromthis they concluded that a damage threshold for cells can be defined in relation to deformation of the cells (see Fig. 10.3).

In an alternative approach Bosboomet al. [18, 19] performed animal studies involving the loading of the tibialis anterior muscle of rats with an indenter for 2 h. After 24 h the animals were sacrificed and, by means of histological techniques, damaged muscle cells were localized and quanti- fied. This study was subsequently repeated using T2-weighted MRI (see

a Multi-scale Modellingof Tissues z 155

Fig. 10.3.Percentage dead cells as a function of time for different straining regimes (reprinted with permission from [21])

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Chaps. 12 and 18). The large variation in damage distributions between samples precluded the adoption of a direct relationship between a defor- mation index and the resulting damage. To find this relationship it is prob- ably necessary to follow damage development as a function of time, which is possible with MRI. Nonetheless, these studies did indicate that in the initial phases of damage healthy cells were present adjacent to dead cells. It may be postulated that the most vulnerable cells first lose viability, although it may be argued that due to the heterogeneous structure of the tissue, some cells have to carry a much higher load than others.

These two ideas, namely that muscle cells can only withstand a deforma- tion for a limited amount of time and that the inhomogeneous structure of the tissue plays an important role in damage development, were explored the- oretically by Breuls et al. [20], using a multi-level finite-element approach. A global macroscopic model of, for example, a skeletal muscle is created by dis- cretizing the geometry in a finite-element mesh. The microstructure at a par- ticular point of the tissue is given by a representative volume element (RVE).

In the multi-level finite-element approach, microstructural finite-element models, representing the RVEs, are assigned to each integration point of the macroscopic elements. By using periodic boundary conditions and stress/strain averaging, the global and the microscopic mesh are coupled.

In this way it is possible to determine the actual load on individual cells as a result of a typical microstructure and, at the same time, determine how ma- terial properties change on a macro scale when cells are dying.

Breuls assumed a damage evolution based on the assumption that if the strain energy density (SED) of a cell exceeds a damage threshold, governed by the adaptive capacity of the cell, a, damage starts to accumulate in the

Fig. 10.4. a Macroscopic mesh, representingmuscle and skin layers that are compressed against a bony prominence by an external pressure P. The nine circles in one particular macro- scopic element indicate the location of the macroscopic integration points to which the micro- structural models are assigned. b Microstructural mesh representinga simplified geometry of randomly dispersed cells embedded in extracellular matrix (ECM) material (reprinted with per- mission from [21])

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cell. This adaptive capacity, or tissue tolerance, may change with age or disease. When the accumulated damage becomes higher than a threshold value Tcell, the cell will die. It is evident that in a heterogeneous micro- structure not all cells ªfeelº the same SED and, hence, evolution of damage will be different between cells. By fitting the damage evolution of RVEs on the experimental data from cell culture experiments, Breuls et al. were able to determine the parameters and Tcell for skeletal muscle cells.

This theory was subsequently applied to a macroscopic muscle and skin layer over a bony prominence, as depicted schematically in Fig. 10.4. The tissue was loaded with a uniformly distributed load at the skin surface.

This approach resulted in the observations that damaged areas became evi- dent within the tissue at different points in time.

By performing a number of analyses of this type, a pressure/time risk curve could be produced, as illustrated in Fig. 10.5. Such a curve indicates the importance of cell tolerance, which itself will be influenced by pathol- ogy and age, the microstructure of the skeletal muscle and, often ignored, the location in the tissue.

Future Developments

It is difficult to predict the future, but a few trends are visible, which prob- ably will influence developments in the immediate future.

In material science and mechanics of polymers and composites, re- searchers have acknowledged in the last two or three decades that to really

a Future Developments z 157

Fig. 10.5.Damage evolution in a particular microstructure at four different external pressures P (4 kPa, 4.4 kPa, 5 kPa and 6 kPa). The vertical axis of each subplot is the external pressure P, while the horizontal axis denotes the time of compression. The bar on the right represents the percentage of damaged cells in the microstructure (reprinted with permission from [21])

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understand and predict the behaviour of materials it is necessary to exam- ine details at the micro-structural level. They realized that the heterogene- ity of materials has a profound influence on the way damage initiates and evolves. By understanding the microstructural behaviour it became possi- ble to predict material behaviour and to design new materials on demand.

It was also clear that cooperation between different disciplines, including chemistry, mechanical engineering and physics, was necessary to reach this goal. This has led to new theoretical developments, such as non-local con- tinuum mechanics, continuum damage mechanics and multi-level finite- element method, etc.

In biomechanics a comparable development has taken place. By means of studies on cultured cells and by tissue engineering it has become possi- ble to study metabolic processes and damage and adaptation in very well controlled microscopic environments. All kinds of vital imaging techniques are becoming available. At the level of animal experiments and human studies, imaging techniques are also improving, leading to detailed micro- scopic information that is obtained in an in-vivo set-up. The inhomoge- neous structure of all biological materials was cited as a major complica- tion in the past, but was often ignored in the analysis. Currently, the inho- mogeneous structure is recognized as an essential feature that has to be in- cluded in studies attempting to achieve understanding of damage and adaptation processes in tissues. The aforementioned developments make it possible to adopt such an approach.

From a theoretical point of view phenomenological models at a macro- scopic level are no longer sufficient. It is becoming increasingly important to develop models at a microscopic level, including models on the lymph- and microcirculation, detailed models on transport processes of nutrients, small molecules like oxygen and CO2, models on cell growth and cell death, consumption and synthesis. These will be indispensable to predict the time and location of cell death. Nonetheless it is essential to connect this micro- structural research with the macroscopic world of the patient with pressure ulcers. Multi-level type approaches can be a valuable tool for this.

References

1. Zhang M, Mak AFT, Roberts VC (1998) Finite element modelling of a residual lower-limb in a prosthetic socket: a survey of the development in the first decade. Med Eng Phys 20:360±373

2. Chow WW, Odell EI (1978) Deformations and stresses in soft body tissues of a sitting person J Biomech Eng 100:79±87

3. Todd BA, Thacker JG (1994) Three-dimensional computer model of the hu- man buttocks, in vivo. J Rehab Res Dev 31(2):111±119

4. Dabnichki PA, Crocombe AD, Hughes SC (1994) Deformation and stress ana- lysis of supported buttock contact, Part H: J Eng Med 208:9±17

5. Barbenel JC, Lee VSP (1994) Discussion on paper of Dabnichki et al, Part H.

J Eng Med 208:263±266

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6. Oomens CWJ, Bressers OFJT, Bosboom EMH, Bouten CVC, Bader DL (2003) Can loaded interface characteristics influence strain distributions in muscle adjacent to bony prominences? Comp Meth Biomech Biomed Eng 6(3):171±

7. Verver MM, Hoof J van, Oomens CWJ, Wismans JSHM, Baaijens FPT (2004)180 A finite element model of the human buttocks for prediction. Comp Meth Biomech Biomed Eng 7:193±203

8. Linder-Ganz E, Gefen A (2004) Mechanical compression-induced pressure sores in rat hindlimb: muscle stiffness, histology, and computational models.

J Appl Physiol 96:2034±2049

9. Sacks AH, O'Neill H, Perkash MD (1985) Skin blood flow changes and tissue deformations produced by cylindrical indentors. J Rehab Res Dev 22(3):1±6 10. Sacks AH (1989) Theoretical prediction of a time-at-pressure curve for avoid-

ing pressure sores. J Rehab Res Dev 26(3):27±34

11. Mak AFT, Huang L, Wang Q (1994) A biphasic analysis of the flow dependent subcutaneous tissue pressure and compaction due to epidermal loadings: is- sues in pressure sores. J Biomech Eng 116:421±429

12. Zhang JD, Mak AFT, Huang LD (1997) A large deformation biomechanical model for pressure ulcers. J Biomech Eng 119:406±119

13. Reddy NP (1981) Interstitial fluid flow as a factor in decubitus ulcer forma- tion. J Biomech 14:879±881

14. Reswick J, Rogers J (1976) Experience at Rancho Los Amigos Hospital with devices and techniques to prevent pressure sores. In: Kenedi RM, Cowden JM, Scales J (eds) Bed sore biomechanics. Macmillan, London, pp 301±310 15. Zhang M, Lord M, Turner-Smith AR, Roberts VC (1994) Development of a

nonlinear finite element modeling of the below-knee prosthetic socket inter- face. Med Eng Phys 17:559±566

16. Breuls RGM, Bouten CVC, Oomens CWJ, Bader DL, Baaijens FPT (2003) Compression induced cell damage in engineered muscle tissue. Ann Biomed Eng 31(11):1357±1365

17. Bouten CVC, Knight MM, Lee DA, Bader DL (2001) Compressive deformation and damage of muscle cell sub-populations in a model system. Ann Biomed Eng 29(2):153±163

18. BosboomEMH, Hesselink M., Oomens CWJ, Bouten CVC, Drost MR, Baai- jens FPT (2001) Passive transverse mechanical properties of skeletal muscle under in-vivo compression. J Biomech 34 (10):1365±1368

19. BosboomEMH, Bouten CVC, Oomens CWJ, Straaten HWM van, Baaijens FPT, Kuipers H (2001) Quantification and localisation of damage in rat mus- cles after controlled loading; a new approach to study the aetiology of pres- sure sores. Med Eng Phys 23:195±200

20. Breuls RG, Sengers BG, Oomens CWJ, Bouten CVC, Baaijens FPT (2002) A multi-level finite element model to study muscle damage in tissue engineered skeletal muscle. J Biomech Eng 124:198±207

21. Breuls RGM, Oomens CWJ, Bouten CVC, Bader DL, Baaijens FPT (2003) A theoretical analysis of damage evolution in skeletal muscle tissue with refer- ence to pressure ulcer development. J Biomech Eng 125:902±909

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