Adaptive Remodeling of Achilles Tendon: A Multi-scale Computational Model
[['Tendons are dense fibrous tissues that transfer tensile forces from muscles to bones .', 'During normal daily activity , human Achilles tendon experiences high intensity cyclic loads , up to 4–8 times the body weight [1–3] .', 'Achilles tendon stores potential strain energy as it is stretched , which is then recovered later in the gait cycle [4 , 5] .', 'This strain energy cycling reduces muscular work and improves the economy of locomotion .', 'Furthermore , the uncoupling of tendon and muscle lengths , due to the elastic deformation of Achilles tendon , enable the muscle fibers to operate at more favorable lengths and velocities , thus improving locomotion economy even further [6] .', 'It is apparent that there are variable tendon geometries and properties , and that tendon tissue has the capacity to adapt to its mechanical environment [7–10] , but how ?', 'Here we develop a biologically plausible computational model of tendon adaptation .', 'The basis of our model is a collagen fiber damage and repair models , which in turn are based on known biological processes .', 'When an Achilles tendon model is incorporated into a simplified model of the Achilles-soleus unit and allowed to adapt over time with usage , we observe the capacity of model tendon to remodel towards a stable equilibrium tendon geometry , which can coincide with minimum metabolic cost of the model musculotendon unit operation .', 'We begin by introducing the key biological processes incorporated into the model .', 'Tendon extracellular matrix ( ECM ) is primarily composed of Type I collagen ( up to 86% dry mass ) .', 'Collagen fibers , run mainly along the axial length of the Achilles tendon , arranged in a hierarchical structure [11 , 12] .', 'At the smallest scale , tropocollagen molecules self-assemble into microscopically visible strands of collagen fibrils [13] .', 'Despite the small diameter of collagen fibrils , typically 100–150 nm in human adult Achilles tendon [11 , 14 , 15] , their total lengths are believed to be much longer , and potentially extend continuously from muscle fibers to bone [16] .', 'A bundle of collagen fibrils form primary collagen fibers [11 , 17] that are then hierarchically aggregated to form primary , secondary and tertiary fascicles across the whole tendon [11 , 18–20] .', 'Tendon ECM is maintained by resident cells known as tenocytes , which mediate the synthesis and degradation of the ECM components [21] .', 'There is compelling evidence for continuous tendon remodeling by the tenocytes [22–24] .', 'Normally , primary collagen fibers are enclosed by a confluent sheet of tenocytes [25] .', 'This implies that collagen fibril adaptation must normally occur via processes acting at a distance from the tenocyte itself .', 'The tenocytes synthesize proteases and new collagen molecules that then self-assemble to repair fibrillar damage .', 'We envisage that these physiologic processes are consistent with homeostatic and adaptive processes within tendon [26] , and it is these processes that are the focus of our proposed tendon adaptation model .', 'Typically , the stress-strain response of a whole tendon is composed of four regions:', '( i ) extension without significant force up to the tendon’s slack length ,', '( ii ) a toe region ,', '( iii ) a linear elastic region , and', '( iv ) a failure region [27] .', 'The slack length is defined as the tendon length at which the tendon first experiences load [28] .', 'Most of the tendon crimp is removed during extension up to the slack length [29] .', 'The toe region corresponds with further sequential straightening of crimped collagen fibers [29] .', "Experimental observations report a unique crimp pattern to each individual collagen fascicle [29–31] indicating that each fascicle's stretched length is also unique .", 'Based on the premise of distribution of collagen fiber lengths , it is possible to reproduce tendon’s non-linear force-extension behavior [32–34] , Fig 1 .', 'Cyclic loading of Achilles tendon during habitual activities , such as walking or running , damages the tendon and initiates repair processes to maintain tendon homeostasis .', 'It is observed that cyclic loading of tendons gradually induces ‘micro-damage‘ or ‘sub-failure injuries’ at the collagen molecular and fibrillar levels , which are collectively referred to as ‘mechanical fatigue damage’ [35–41] .', 'The fatigue response of human Achilles tendon to cyclic mechanical loading is clearly evidenced in the experimental studies of Wren et al . [38] .', 'At the microscale , mechanical fatigue damage to collagen fibrils may present itself in ‘focal’ or ‘generalized’ modes .', 'Generalized damage is evidenced by repeating patterns of kinks and distortions along a number of fibrils [42] , whereas focal damage is evidenced by fracture of collagen fibrils [43] .', 'One mode of damage may dominate the other depending on the prevalence and type of cross-links within collagen fibrils [44] .', 'More cross-linking between tropocollagen molecules results in stiffer tendons , such as Achilles and patellar tendons , and so favors the focal fatigue damage mode [44] .', 'For simplicity , in this paper we have focused on the focal mode of damage only .', 'Nevertheless , the damage model employed here may be modified to include other modes of collagen fiber damage as required .', 'In addition to mechanical damage , tenocyte-mediated proteolytic collagen degradation also occurs , and is an essential component of tendon homeostatic processes [14] as it facilitates tissue remodeling [23 , 42] .', 'In normal tendon , proteolytic damage is usually meditated primarily by members of the matrix metalloproteinases ( MMPs ) family of proteases [45 , 46] .', 'Intriguingly , mechanical tensile strain of collagen fibrils has been shown to reduce , and even completely prevent , proteolytic damage of collagen fibrils by collagenase MMPs at physiologically relevant strains [14 , 22 , 47 , 48] .', 'From the above it is clear that while the basic mechanical and physiological aspects of tendon tissue and adaptation have been studied , how these processes are integrated to produce physiologically relevant outcomes during habitual loading is not completely understood .', 'Here we hypothesize that through utilization of the abovementioned biological processes , Achilles tendon is capable of adaptation by remodeling its geometry , to reach a dynamic stable equilibrium that represents tendon homeostasis .', 'In the following , we demonstrate that for a musculotendon unit with constant muscle fiber length , our proposed tendon model is able to remodel its geometry to reach a stable homeostatic equilibrium , and that this equilibrium state can coincide with minimizing the total metabolic cost of the model musculotendon unit .'], ['In the following we first develop and then test a discretized model of Achilles tendon adaptation .', 'The model tendon is based upon damage and repair processes that take place at the level of primary collagen fibers .', 'The mechanical load experienced by the tendon is based upon a simplified Achilles-soleus model and experimental measurements made during normal human gait mechanics .', 'Over time , repeated cycles of damage and repair of the collagen fibers gradually remodel the whole Achilles tendon .', 'We note here that the timescale for the simulated sequential damage and repair processes are not critical to our proposed adaptation model , but for convenience and definiteness we have assumed a daily cycle of damage and repair .', 'This timescale is likely to accord with at least some of the important cyclic homeostatic processes taking place within tendon [49 , 50] .', 'To frame our model of tendon adaptation in accordance with the known abovementioned tendon physiological process , the following computational sub-models are developed:', '( i ) a discretized fiber model of tendon mechanics , Fig 2', '( a ) , from which the force-extension behavior of the whole tendon can be estimated , Fig 2', '( b ) .', '( ii ) a simplified model of the musculotendon ( Achilles-soleus ) unit , Fig 2', '( c ) , from which the metabolic cost of the ( effective ) soleus muscle mechanical work at the ankle joint is estimated , Fig 2', '( d ) .', '( iii ) a model for calculating the intensity of daily load as a function of metabolic cost , Fig 2', '( e ) ,', '( iv ) models for mechanical damage and repair to individual collagen fibers , Fig 2', '( f ) , and', '( v ) models for proteolytic damage and repair to individual collagen fibers , Fig 2', '( g ) .', 'A single sequential pass through these sub-models , results in a remodeled tendon geometry , with a modified force-extension response , Fig 2', '( h ) .', 'In other words , the model tendon adapts by repeatedly cycling through this algorithm .', 'We now describe each of these sub-models in detail .', 'In principle , either the collagen fibrils or the primary collagen fibers may be the anatomical units regarded as the load carrying ‘string elements’ in our model .', 'However to render the computations more tractable , here we have chosen to represent the primary collagen fibers as the discretized string elements .', 'For the rest of this paper , collagen fibers in our model refer to primary collagen fibers .', 'Human Achilles tendon cross-sectional area ( CSA ) at the mid-section is reported to range from under 50 mm2 [51 , 52] to values higher than 80 mm2 [38 , 53] .', 'Assuming an Achilles tendon to have a CSA of 60 mm2 and assuming a circular cross-section for collagen fibers with an average fiber diameter of 28 μm [54] , the total number of primary collagen fibers ( Ntotal ) in human Achilles tendon is estimated to be:', 'Ntotal=ATAF≈\xa0100 , 000 ( fibers )', '( 1 )', 'where AT and AF denote whole tendon cross-sectional area and average collagen fiber cross-sectional area In the present study , we explicitly model 100 , 000 collagen fibers that are of non-uniform length , but otherwise similar .', 'We note here that it may be appropriate to have fibers with variable stiffness along their length , as it has been reported that the ‘free’ Achilles tendon is more compliant than the aponeurosis [21 , 55 , 56] .', 'Nevertheless , for simplicity we adopt a uniform stiffness along the fiber length .', 'And for simplicity we assume a Gaussian ( normal ) distribution for the initial profile of fiber lengths , and show that it can approximate the non-linear stress-strain curve of a tendon , Fig 1 . Starting mean fiber length is chosen to be in the range 250 to 280 mm to correspond with reported total anatomic length of human Achilles tendon , including the aponeurosis [57–60] .', 'Choosing LT as the mean length of the tendon and Li the slack length of the ith fiber , then the linear extension of the ith fiber ( ΔLi ) as the whole tendon undergoes linear extension ΔLT is calculated from:', 'ΔLi=\u2009{ Li− ( LT+ΔLT ) if\xa0LT+ΔLT>Li0if\xa0LT+ΔLT≤Li', '( 2 )', 'For convenience , we take mean of the slack lengths of all fibers to represent the mean tendon length ( LT≈L¯i ) .', 'While the measured estimates vary , the commonly reported Young’s modulus of the Achilles tendon ( ET ) is about 1 GPa [61 , 62] .', 'Assuming homogenous material properties , and uniform cross-sectional area ( AF ) for all collagen fibers , the stiffness of the ith fiber ( ki ) is then estimated by:', 'ki=AF⋅ETLi', '( 3 )', 'Using Eqs ( 2 ) and ( 3 ) , fiber force ( Fi ) and whole tendon force ( FT ) at a given tendon extension can then be calculated by:', 'Fi=ki⋅ΔLi', '( 4 )', 'FT=Σi=1NtotalFi', '( 5 ) Repeated cyclic loading of tendon during daily activity damages collagen fibers [38 , 63] .', 'Our estimates for the likelihood of mechanical fatigue damage of primary collagen fibers is based on the empirical fatigue damage data for the whole human Achilles tendon obtained by Wren et al [38] .', 'The implicit assumption we employ to use this data is that the whole Achilles tendon fatigue behavior is also representative of the individual primary collagen fibers fatigue behavior making up the human Achilles tendon .', 'The assumption that each part of the Achilles tendon is similar to all others is likely to be a reasonable assumption for healthy tendon , but we note this is less likely to be a reasonable assumption for diseased tendon .', 'The average age of the 25 human subjects from whom the Achilles tendon samples were obtained by Wren et al ( 2003 ) was 75 ( ±12 ) years [38] .', 'Therefore , in order to represent the in vivo damage in young adults , we chose to rescale the ultimate tensile stress value of 70 MPa reported by Wren et al ( 2003 ) [38] to 100 MPa [64–66] , while leaving the slope of the fatigue curve unchanged .', 'Fig 3 shows the normalized fatigue curve for collagen fibers employed in our model .', 'This rescaled human Achilles tendon fatigue curve is probably more representative of younger adults , though other scalings may be deemed appropriate depending on data and the intended purpose of the model .', 'However provided that reasonable values are chosen , the actual values for scaling are not critical , and do not substantially change the findings reported here ( see later sensitivity analysis ) .', 'From Fig 3 , the number of loading cycles to tendon failure ( nfail ) at a given peak fiber stress ( σmax ) are calculated using:', 'σmax=a−b⋅log10 ( nfail )', '( 6 )', '∴nfail=10a−σmaxb', '( 7 )', 'where a corresponds to the ultimate tensile stress at one cycle , in this case 100 ( MPa ) , and b is the slope of the logarithmic fatigue curve in Fig 3 , in this case 8 . 25 ( MPa/log ( n ) ) .', 'Peak tensile stress of the ith fiber ( σmaxi ) in a tendon undergo ) ing linear extension ΔLT is calculated by:', 'σmaxi=ET⋅ΔLiLi', '( 8 )', 'where fiber extension ΔLi is calculated from Eq 2 . It is clear that typical daily activities lead to peak stress levels that rarely ( if ever ) result in complete failure of a normal tendon .', 'Consequently we need to devise a ‘cumulative damage function’ to estimate the amount of damage arising from daily activity .', 'For our ‘string’ tendon model , cumulative tendon damage is assumed to be directly proportional to the fraction of broken fibers .', 'The fraction of broken fibers as a result of daily activity can be estimated from a failure ( or reliability ) function for individual collagen fibers .', 'However due to the lack of experimental data on failure functions ( Pfailmech ) or reliability functions ( R=1−Pfailmech ) for tendon , we employ a commonly adopted ‘exponential failure function’ [67] to describe focal damage failure of individual collagen fibers within the Achilles tendon .', 'Therefore the probability of mechanical failure of an individual fiber experiencing peak fiber stress σmax and n load cycles is estimated by:', 'Pfailmech=−κ+κ⋅eλ\xa0nnfail', '( 9 )', 'The fitting constants κ and λ in ( Eq 9 ) are defined such that Pfailmech=0 at = 0 , Pfailmech=0 . 1 at n = nfail/2 and Pfailmech=1 at n = nfail .', 'These fitting constants are chosen based on reported typical cyclic fatigue test on human Achilles tendon reported in Fig 2', '( b ) and 2', '( c ) of Wren et al ( 2003 ) [38] .', 'Fitting Eq 9 to this figure suggests reasonable parameter values are , κ = 0 . 0125 and λ = 4 . 395 .', 'A typical cumulative damage probability curve is shown in Fig 3 . However clearly these fitting constants can be adjusted to fit experimental results as required , while the influence of these parameters on our model outputs are quantitated in a later sensitivity analysis .', 'In our model for an Achilles tendon with normal physiology , if a fiber mechanically fails , it is always repaired ( which may not happen in a diseased tendon ) .', 'A repaired fiber may ( probabilistically ) be repaired either shorter or longer , however , we bias the repair of mechanically damaged fibers towards lengthening ( Fig 4 ) .', 'A probabilistic interpretation of fiber repair as used in our model is depicted in Fig 4', '( b ) , which shows the probability distribution of relative length changes to a fiber following its repair .', 'We suggest that this repaired length change , depicted Fig 4', '( a ) , is consistent with the following conceptual model of the repair process following mechanical damage .', 'First , the two ends of the broken fiber are enzymatically debrided by proteases to obtain a suitable undamaged surface from which a new portion of collagen fiber can be constructed .', 'A new portion of collagen fiber is then created by polymerization of tropocollagen molecules [68 , 69] .', 'While the section of fiber debrided may be longer than the newly formed portion of collagen fiber , leading to further fiber shortening , on average the broken ends are more likely to lie somewhat apart , so the new portion of fiber bridges both this gap and any fiber debridement , and so the repaired mechanically damaged fiber is on average longer .', 'The gap between the broken ends arises at least partly because mechanically damaged fibers are on average shorter than remaining nearby fibers , but it seems plausible that the gap between broken fiber ends may also be partly promoted by other events , such as the elastic recoil of the fractured ends of a failed fiber or subsequent cyclic friction forces between fibers .', 'The gap between the broken ends is observable in SEM images of damaged collagenous matrices [43 , 70 , 71] .', 'A schematic depiction of this is shown in Fig 4', '( a ) .', 'It has been noted previously by Provenzano et al [43] , that if the gaps between the fractured ends are filled by newly polymerized collagen fiber , then the repaired fibers lengths are increased .', 'Proteases remove damaged or unwanted ECM as part of normal tissue turnover and collagen fiber homeostasis [14 , 23 , 41] , but the rate of collagen degradation is modified by collagen strain .', 'For example , Wyatt el al .', '[72] reports an almost complete cessation in collagenase degradation rate when rat tail fascicles were strained 4–5% [72] .', 'However , Flynn et al . performed similar tests on single collagen fibrils , thereby avoiding rotational deformations of collagen fibers during extension ( rotational deformations are often observed in collagen networks made up of fibrils with a variety of fibril orientations ) [73] .', 'For the experimental tests reported by Flynn et al ( 2013 ) , which probably most closely approximate the ( linear ) fibril structure observed in Achilles tendon , collagenase degradation of fibrils is prevented at strains larger than about 1 . 5% .', 'These test results suggest that in terms of our ‘string’ model of tendon , for a given tendon strain , relatively long collagen fibers are less stretched along their length , which render them more susceptible to being degraded by active proteases [22 , 41 , 48 , 74–76] .', 'Based on the experimental results of Flynn et al [47] , we employ an exponentially decreasing probability of fiber cleavage with increasing strain ( Fig 5 ) , viz:', 'Pfailproteo=e−ϕ⋅εmax', '( 10 )', 'where εmax is the fiber peak strain during a gait cycle and ϕ is a fitting constant .', 'To accord with the observations of Flynn et al . [73] , ϕ is calculated to be 300 .', 'This selection of ϕ results in almost no proteolytic damage in fibers experiencing peak strains εmax ≥ 1 . 5% .', 'Clearly the sensitivity to proteolytic damage can be altered by varying the constant .', 'It is equally clear that Eq 10 is a very crude modeling representation of an actual proteolytic process occurring within tendon , as binding of MMPs to collagen fibers , the movement of MMPs along collagen fibers , the surface state of the collagen fibers , and the history of cyclic strain experienced by the collagen fibers , are all time dependent , and so the level of protection afforded to collagen fibers must also be time dependent .', 'A more sophisticated proteolytic damage model would include such time dependencies , but unfortunately to date there is little data available in the literature to suggest more precise functional relationships .', 'When proteolytic degradation of the collagen fiber is complete , once again new collagen molecules polymerize to create newly formed collagen fiber , which bridges the gap between the degraded ends ( see schematic Fig 4', '( c ) ) .', 'For each fiber that is proteolytically degraded , the fiber repair model determines a probabilistic repair length .', 'As in the case of mechanical damage , the repair length is found by sampling a triangular probability distribution .', 'However , this time the triangular probability distribution has a greater tendency to shorten the original fiber , see Fig 4', '( d ) .', 'In other words , the fiber section removed proteolytically is on average longer than that filled by newly formed fiber section , and so the repaired fiber is shortened .', 'We note that the mechanism for fiber shortening is not known with certainty .', 'However , it is possible that passive mechanical forces may contribute to fiber shortening ( e . g . cyclic frictional forces , or compressive residual stresses in long fibers may relax ) , or active forces generated by cells may contribute to fiber shortening ( e . g . a number of experiments have reported that tenocyte generated contractile forces can actively shorten collagen fibers [41 , 74 , 75 , 77] .', 'To provide an appropriate context to test adaptation of Achilles tendon to its loading environment , we set our new tendon model within a standard three-component Hill-type model .', 'Representing the musculotendon unit , the Hill-type model is composed of a contractile element and two elastic elements , Fig 2', '( c ) .', 'The contractile element and the parallel elastic element simulate the integrative behavior of the human soleus through gait cycles [78 , 79] .', 'The soleus is the muscle of choice for examining our tendon model for the following reasons:', '( i ) among the plantarflexor muscles it is the largest muscle , and from several modeling studies it has become clear that the soleus is the primary muscle responsible for producing ankle power and work in both walking and running [80–82] .', 'Furthermore , modeling results has found the soleus to be among the most important producers of mechanical work during walking and running across all lower limb muscles [83 , 84] .', '( ii ) The soleus only crosses the ankle , unlike the gastrocnemius muscles that cross both the ankle and knee joints ( knee flexor ) .', 'This simplifies the modeling of muscle force using our hill-type model , eliminating potentially complicating factors .', 'Joint torque sharing between the soleus and the other synergist muscles is simplified by initially attributing the torque produced by the soleus to the relative physiological cross sectional area of the soleus and the other ankle plantarflexors combined .', 'Soleus force is subsequently computed from a joint angle-specific soleus moment arm [59] .', 'The activation required to produce this force is modelled incorporating muscle force-length-velocity constraints .', 'Muscle fiber lengths and velocities are influenced in our calculation by ankle joint angle , muscle pennation angle ( we assumed a constant volume muscle model ) as well as tendon stretch .', 'The series elastic element in the musculotendon unit shown in Fig 2', '( c ) represents the Achilles tendon , with its mechanical properties obtained from the discretized tendon model described above .', 'As we allow tendon remodeling over time , mechanical properties of this series elastic element , representing the Achilles tendon , also changes over time .', 'For simplicity the contractile element and its parallel elastic element are taken to have constant properties in all our simulations , though in reality these may also adapt over time [85 , 86] .', 'This simplification is to focus our attention on the adaptation process of the Achilles tendon alone , and to exclude muscle adaptation ( which has been investigated elsewhere [85 , 86] ) .', 'Allowing the muscle to adapt simultaneously with the tendon could potentially complicate our understanding of tendon adaptation , and possibly obscure tendon responses that are of interest here .', 'But clearly muscle does adapt too , and inclusion of such mechanisms is an obvious extension for developing a more realistic future model .', 'Fig 6 illustrates the algorithm for updating the tendon properties in the musculotendon model as a result of activity .', 'The musculotendon model uses an inverse dynamics approach with ankle torque and kinematics as inputs to calculate the required muscle force .', 'The ankle torque and kinematics were experimentally obtained from motion tracking and force measurements from an adult subject during walking [87] .', 'The algorithmic steps in Fig 6 include: ankle torque , Fig 6', '( a ) , and musculoskeletal dimensions ( moment arms ) and ankle angles , Fig 6', '( b ) are employed to first calculate tendon force .', 'The tendon model response to force is then used to calculate tendon extension , Fig 6', '( c ) .', 'Tendon extension and ankle angles , Fig 6', '( b ) and 6', '( c ) , are then used to calculate muscle fiber pennation angle , muscle fiber operating length and muscle force ( FM ) , Fig 6', '( d ) , which balances the calculated tendon force .', 'Muscle activation ( Mact ) to generate the required muscle force is estimated based on muscle length , Fig 6', '( e ) [79] , and muscle velocity , Fig 6', '( f ) [79 , 88] .', 'The expression used to calculate muscle activation , based on formulations by Buchanan et al . [79] , is:', 'Mact=FMFL⋅FV⋅FM , max', '( 11 )', 'Mact ( muscle activation ) is a dimensionless number between zero and one that represents the fraction of maximal muscle activation .', 'FM is the muscle force , Fig 6', '( d ) , while FL is the muscle force at current length as a fraction of the maximum isometric force ( FM , max ) , Fig 6', '( e ) , and FV is the muscle force at the current contraction velocity as a fraction of the maximum isometric force ( FM , max ) , Fig 6', '( f ) .', 'The muscle activation Mact from Eq ( 10 ) is then used to calculate the metabolic cost rate ( power ) , and integration of this quantity with respect to time gives the total metabolic cost during a gait cycle , as described in [78] .', 'The metabolic cost rate ( Q˙ ) during a gait cycle , Fig 6', '( g ) , is expressed as the sum of four terms: activation heat rate ( h˙act ) , maintenance heat rate ( h˙m ) , shortening/lengthening heat rate ( h˙sl ) and the mechanical work rate performed by the muscle ( w˙M ) [78] , viz:', 'Q˙=h˙act+h˙m+h˙sl+w˙M', '( 12 )', 'Now that we have defined our musculotendon unit model , we need to recognize that the musculotendon unit does not operate in isolation from the whole organism , but is in fact part of the whole organism .', 'This relationship between the musculotendon unit and whole organism puts constraints on the muscultotendon units operation , which help guide the tendon to an in vivo equilibrium state .', 'Potential factors at the whole organism level affecting the musculotendon operation in the adult include sensory feedback signaling ( including pain ) , neural muscle activation patterning , higher order cognitive inputs ( e . g . willpower ) and oxygen and metabolic energy availability from the organism to make sure there is matching of supply-demand functions over the whole musculotendon unit .', 'The equilibrium state achieved by tendon operating within the musculotendon unit , operating within the whole organism , depends on these interactions/constraints , which may be formulated mathematically as optimization of a multi-objective function .', 'Clearly this multi-objective function can vary over time with changes in environmental and sensory inputs , nutritional status and determination of the individual , and it is these changes that usually drive tendon adaptation in vivo .', 'But for our modeling purposes , how can we simply and reasonably take into account this substantial in vivo complexity ?', 'At equilibrium it is likely that for everyday repetitive activities such as walking , an important contributor to the multi-objective function is musculotendon unit economy .', 'For habitual repetitive activities such as walking , energy minimization is regarded by some as a key optimization criterion dictating locomotor behavior [89 , 90] .', 'There is much evidence pointing to movement patterns that minimize energy expenditure , from the selection of preferred walking speeds in humans and other species [91 , 92] to preferred stride frequencies [93 , 94] and preferred gaits [95 , 96] .', 'Consequently , it is likely that Achilles-soleus unit economy is close to being maximized when the tendon geometry has reached its geometrical equilibrium state , and that Achilles-soleus unit economy will ‘fall’ on either side of the tendon’s geometrical equilibrium state .', 'Making these assumptions , the simplest way for us to approximate the change in walking economy with tendon geometry in our model is to reflect the effect of current metabolic cost on the musculotendon unit load intensity .', 'This can be implemented most simply by scaling the ankle torque while keeping both gait pattern and number of load cycles per day constant .', 'Taking this approach effectively acts as a constraint on the musculotendon unit operation , preventing the total metabolic cost from becoming physiologically unrealistic at some tendon geometries .', 'Therefore ankle torque ( τa ) can be calculated via:', 'τa=β⋅τa , m', '( 13 )', 'where τa , m is the lab-measured ankle torque and β is an activity scale factor calculated by:', 'β=QminQ', '( 14 )', 'where Q refers to the total metabolic cost of muscle activation and Qmin refers to the minimum total metabolic cost of muscle activation .', 'Fig 6', '( h ) , shows the relationship between the activity scale factor ( 0 < β ≤ 1 ) and the metabolic cost for a range of tendon lengths and constant fiber length dispersion .', 'The calculated activity scale factor β is used in the next cycle to determine subsequent ankle torques , from which flows tendon forces , muscle forces and metabolic cost , which are calculated via cycling through the algorithm for muscultotendon unit operation depicted in Fig 6 .', 'Clearly a more realistic model would take into account changes in the multi-objective function governing musculotendon unit interactions/constraints with the whole body in a much more sophisticated way , and also involve changes in both gait patterns and number of load cycles .', 'Relaxing the assumptions made here represent an interesting direction for future research .', 'We chose a single cycle of remodeling , shown in Fig 2 , to represent a 24-hour period .', 'To correspond the tendon activity level with this time-frame , we subjected all our tendon models to a total of n = 5 , 000 load cycles per day , which approximates the number of gait cycles of active adults [97–99] .', 'To reduce possible time discretization errors , the tendon model is subjected to loading cycles in three equally spaced blocks of simulated activity cycles during a day .', 'At the end of an activity block , the tendon’s fatigue damage is assessed as outlined above , resulting in loss of some intact fibers .', 'With an updated metabolic cost , tendon peak force for the next loading block is updated wherein tendon continues undergoing mechanical loading .', 'At the end of the third loading block , proteolytic damage and finally repair of the mechanically and proteolytically damaged fibers take place , as previously explained , resulting in a new fiber length distribution ( and restored fiber number ) .', 'This daily cycle is repeated to simulate tendon remodeling over weeks or months , and adaptation of tendon properties can be tracked over time .', 'A summary of parameter symbols and values used in the model are shown in Table 1 below .'], ['We first consider fiber length distribution of the model tendon , as fiber dispersion strongly influences its mechanical properties .', 'A model tendon with reduced fiber length dispersion ( i . e . reduced standard deviation ) experiences more rapid fiber recruitment and force development with strain , while a tendon with a larger fiber length dispersion ( i . e . larger standard deviation ) exhibits slower fiber recruitment and force development with strain ( see Fig 7 ) .', 'Strain energy is the potential energy stored by elastic materials as they undergo deformation .', 'For elastic materials such as tendon , it is quantified as the dot product of force and dis
['Introduction', 'Methods', 'Results', 'Discussion']
['While it is known that musculotendon units adapt to their load environments , there is only a limited understanding of tendon adaptation in vivo .', 'Here we develop a computational model of tendon remodeling based on the premise that mechanical damage and tenocyte-mediated tendon damage and repair processes modify the distribution of its collagen fiber lengths .', 'We explain how these processes enable the tendon to geometrically adapt to its load conditions .', 'Based on known biological processes , mechanical and strain-dependent proteolytic fiber damage are incorporated into our tendon model .', 'Using a stochastic model of fiber repair , it is assumed that mechanically damaged fibers are repaired longer , whereas proteolytically damaged fibers are repaired shorter , relative to their pre-damage length .', 'To study adaptation of tendon properties to applied load , our model musculotendon unit is a simplified three-component Hill-type model of the human Achilles-soleus unit .', 'Our model results demonstrate that the geometric equilibrium state of the Achilles tendon can coincide with minimization of the total metabolic cost of muscle activation .', 'The proposed tendon model independently predicts rates of collagen fiber turnover that are in general agreement with in vivo experimental measurements .', 'While the computational model here only represents a first step in a new approach to understanding the complex process of tendon remodeling in vivo , given these findings , it appears likely that the proposed framework may itself provide a useful theoretical foundation for developing valuable qualitative and quantitative insights into tendon physiology and pathology .']
['It is now widely acknowledged that tendon plays a vital role in locomotion , while experiments have revealed that tendon is much more metabolically active than previously believed .', 'There have been increasing numbers of papers describing the responses of tenocytes to mechanical loading and speculation about the origins of tendinopathy , but to date there is currently no basic theoretical framework describing how tendon maintains tissue homeostasis consistent with the experimental data , or indeed how tendon adapts to its environmental load conditions .', 'Based on established biological principles of tendon damage and repair , for the first time we develop a dynamic model of tendon homeostasis that is capable of adaptation .', 'We show that for a model soleus musculotendon unit with muscle fiber length kept constant , our model tendon is ‘capable’ of dynamically adjusting itself to find a stable equilibrium tendon geometry , which coincides with minimum metabolic cost of muscle activation .', 'This new theoretical framework for tendon homeostasis and adaptation offers the possibility of refocusing research in basic and clinical science .']
['material fatigue', 'medicine and health sciences', 'legs', 'classical mechanics', 'limbs (anatomy)', 'biomechanics', 'collagens', 'physiological processes', 'homeostasis', 'materials science', 'damage mechanics', 'materials physics', 'musculoskeletal mechanics', 'muscle physiology', 'musculoskeletal system', 'proteins', 'connective tissue', 'biological tissue', 'torque', 'physics', 'biochemistry', 'anatomy', 'tendons', 'physiology', 'biology and life sciences', 'physical sciences', 'ankles', 'motion']