## Abstract

Collagen is the most abundant protein in mammals, often serving as the main load bearing constituent in soft tissues. Collagen undergoes continuous remodeling processes in which present collagen degrades, and new collagen is formed and deposited. Experiments find that modestly strained fibrillar collagen is often stabilized to enzymatic degradation, a mechanism that is essential in approaching and maintaining a homeostatic balance in the tissue remodeling process for healthy tissue. At larger strains, this decline in the collagen degradation rate may be reversed. This article reviews different modeling approaches that seek to account for the effect of straining on collagen remodeling, both with respect to collagen amount and to resulting mechanical properties. These models differ in the considered length scale starting from the molecular scale up to the larger continuum scale.

## 1 Introduction

Biological tissues consist of cellular and noncellular components. The extracellular matrix (ECM) is comprised of several constituents that provide the physical and biochemical support for the embedded cells regarding growth, migration, differentiation, survival, homeostasis, and morphogenesis [16]. ECM constituents include collagens, proteoglycans/glycosaminoglycans, elastin, fibronectin, laminins, and several other glycoproteins [710]. In the ECM of vertebrates, the primary function of collagen and elastin is to resist tensile loading, while proteoglycans and hydroxyapatite resist compressive loadings [11]. With around 30% of the total protein mass, collagen is the most abundant protein in mammals [12]. Based on their molecular structure and on their assembly, these collagens can be subdivided into 28 groups [1214], although there is no well-defined boundary that distinguishes between collagen and collagen-like protein [15].

This review focuses on collagen in fibrillar form. Collagen of types I and III account for 34–77% of the dry weight of cervical tissue [16,17]. This weight varies through the course of the pregnancy, with a collagen dry weight of 54–77% in nonpregnant woman, a collagen dry weight of around 48.4% at the tenth week of pregnancy, and a collagen dry weight of 23–40% during the third trimester of pregnancy [18]. In human skin, collagen represents around 70–80% of the dry weight, where collagen of type I represents around 80% of the total collagen content and collagen of type III around 15% [19]. Arteries have different layers (intima, media, and adventitia), each with a different arrangement of collagen fibers (see, e.g., the diagrammatic artery model of Fig. 1 of Ref. [20]). The collagen content of arteries exhibits considerable variation. For the combined intima and media of the human aorta, an examination of 85 vessels found that the collagen content increases from 20% at the age of 20 yr to 30.5% at the age of 70 [21]. Another study [22] on atheroma-free human aortas in both females and males with different causes of death found a collagen content of 31.24% in fetal tissue, which decreases to 21.75±4.60% in the 70–79 yr age group. In blood vessel walls, the collagen content depends on the layer, age, disease, and its location in the body [23]. In bone, collagen of type I represents around 95% of the collagen content and, at lower levels, collagen of types III and V has also been found [24]. Collagen types II, IX, and XI function as the core fibrillar network in developing cartilage [25], while collagen of type III is reported to act as a fibril network modifier [26].

It is common for collageneous tissue to exhibit ongoing processes of growth and remodeling. In this context, growth simply describes an increase in mass or size, whereas Humphrey and Rajagopal [27] describe remodeling as “a change in structure (intracellular, cellular, or extracellular) that is achieved by reorganizing existing constituents (e.g., altered orientation or crosslinking) or by synthesizing new constituents that have a different organization; remodeling may or may not alter the mass density, but it does change other material properties such as stiffness or symmetry.”

Distinct molecular mechanisms of mechanosensing and mechanotransduction are involved in the translation of mechanical stimuli into biochemical response by living cells [2831]. These stimuli are capable of inhibiting or facilitating new interactions in the tissue [32], which may result in either a stiffening [3337] or a softening mechanical response [3840] of the tissue. The development, repair, and remodeling of tissue are regulated by a timely breakdown of the particular constituents including collagen in the extracellular matrix, whereas an interruption of this breakdown process may result in various diseases [41]. Experiments suggest that straining of collagen has a strong influence on the enzymatic degradation of collagen. Many articles report that strain reduces the rate of enzymatic collagen degradation [4252]. A few works report the opposite effect, namely, strain that promotes enzymatic collagen degradation [5355]. Some studies have linked the enzymatic collagen degradation behavior to both effects [5659]; relatively small amounts of strain inhibit enzymatic degradation, and after a strain-threshold has been passed, enzymatic degradation increases with any additional increment to the now relatively large value for the strain.

The mechanical modeling of biological materials must contend with challenges due to the complexity of the microstructure, the nutrition transport, diffusion processes, and numerous other factors that result in a change of the material properties. This complexity often results in tissue models that are restricted to capturing a limited number of phenomena within a certain time frame. Discussion of theoretical treatments for growth and remodeling can be found in Refs. [27] and [6065]. Ongoing processes of tissue degradation and resynthesis result in new and different arrangements of the individual constituents. For example, fibrillar components can change their orientation. The articles [6670] account for such reorientation processes in their modeling approaches.

The effect of mechanical stimuli on the tissue remodeling behavior is modeled in Refs. [7183]. Baaijens et al. [84] review the broader modeling of collagen remodeling, in which also some approaches are discussed that study mechanosensitive enzymatic collagen turnover. Chang and Buehler [85] review the impact of mechanical force on the collagen degradation process with a special focus on the biomechanics of collagen molecules. Gaul [86] examines strain-mediated degradation of arterial collagen, and it provides a detailed review on experiments, on imaging techniques for collagen visualization in soft tissues, and on models that explore the relation between mechanical loading and enzymatic collagen remodeling. The comprehensive review by Saini et al. [87] surveys the molecular mechanisms that are implicated in collagen-tension related suppression of enzymatic degradation.

This article seeks to provide a timely survey on the mechanical modeling of strain-mediated enzymatic turnover of fibrillar collagen. The collagen modeling efforts reviewed here are motivated by the results of experiments involving distinct types of tissues, enzymes, and loading scenarios that have served to clarify the relation between strain and degradation. The reviewed models can be categorized in several ways, such as by the length scale on which the remodeling process is simulated (e.g., molecular level, collagen fibril or fiber level, tissue level, etc.) and the degradation effects that are taken into account (e.g., constant versus mechanosensitive collagen deposition, tissue swelling, etc.). While a variety of experiments and general modeling techniques are mentioned in this review, its main focus is to present a comprehensive overview of modeling approaches that may find their application in different continuum mechanics and finite element studies.

Section 2 provides a synopsis of the biological processes that are involved in the remodeling of fibrillar collagen. Section 3 reviews experiments that yield insight into the relation between mechanical stimuli and enzymatic collagen degradation. Section 4 discusses the broader modeling framework of fibrous soft tissue in order to provide a basis for the rest of this paper, which especially focus on the modeling of strain-stabilized enzymatic degradation. The heart of this technical review consists of Secs. 57. Section 5 is devoted to modeling at the molecular length scale, for example, in the framework of molecular dynamics simulations, and it provides an insight into the relation between the collagen trimer arrangements under mechanical loading and the formation of accessible cleavage sites. On the fibril and fiber length scale, different modeling approaches treat the fibrillar collagen as a rod whose radius decay is mediated by its axial stretch; these approaches are reviewed in Sec. 6. Section 7 then reviews higher length scale continuum models that account for development of the fiber volume or density. Section 8 concludes the review with an outlook on the research issues that require additional consideration in broader modeling efforts.

In reviewing and highlighting equations from the literature, alternative notations from the original are sometimes employed in order to avoid duplicate notations in comparison between works.

## 2 Remodeling of Fibrillar Collagen

The word collagen is derived from the Greek words for glue and to produce [88], and it can be roughly translated as glue former [89]. Collagens are structural proteins, which are a part of the ECM of biological tissues. van der Rest and Garrone [90] describe collagen molecules as “structural macromolecules of the extracellular matrix that include in their structure one or several domains that have a characteristic triple-helical conformation.” Kadler et al. [13] point out that there is no common standard definition for collagen; there exist triple-helical proteins that are regarded as collagens and there exist proteins with triple-helical domains that are not regarded as collagens. They refer to collagens as triple-helical proteins with the purpose of assembling and maintaining tissue components [13].

The collagen superfamily contains 28 members [1214], and the different types of collagen are categorized by using Roman numbers. Ricard-Blum [12] and Karsdal [14] organize some of these members into six different groups based on their supramolecular assembly: (i) fibril forming or fibrillar collagens; (ii) fibril-associated collagens with interrupted triple helices that correspond with the surface of fibrils, but do not form fibrils on their own; (iii) network forming collagens (four molecules form tetramers by assembling via their aminoterminal 7S domain, and two molecules form NC1 dimers by assembling via their carboxyterminal NC1 domain); (iv) hexagonal network forming collagen; (v) beaded filaments forming collagens; and (vi) anchoring fibrils. Despite the high structural diversity of the different collagen types, all collagens have right-handed triple helices formed by three α-chains as a common characteristic [91]. These collagen types are summarized in Table 1.

Table 1

Classification in Refs. [12] and [14] of some types of collagen into six categories based on their molecular structure and supramolecular assembly

Collagen categoryExamples of collagen types
Fibril formingI, II, III, V, XI, XXIV, XXVII
Fibril-associated collagens with interrupted triple helicesIX, XII, XIV, XVI, XIX, XX, XXI, XXII
Network formingIV
Hexagonal network formingVIII, X
Anchoring fibrilsVII
Collagen categoryExamples of collagen types
Fibril formingI, II, III, V, XI, XXIV, XXVII
Fibril-associated collagens with interrupted triple helicesIX, XII, XIV, XVI, XIX, XX, XXI, XXII
Network formingIV
Hexagonal network formingVIII, X
Anchoring fibrilsVII

We refer to the book [92] for additional detailed discussions on collagen, in which all collagen types I–XXVIII are discussed chapterwise.

Collagen fulfills a variety of mechanical functions in biological tissue [93]. It bears mechanical loading and is considered to serve as a template in the embryonic formation of different load-supporting tissues [11]. Collagen is involved in wound healing processes [94]; its use in different therapeutic measures is examined in Refs. [95] and [96].

### 2.1 Fibril Forming Collagen.

In biological tissue, those collagen types that form fibrillar structures represent one of the main load bearing constituents of the ECM. Collagen of type I is a key structural component in many tissues [97] and as a result is one of the most intensively studied collagen types. Figure 1 depicts an idealized and simplified representation of the hierarchical organization of collagen fibers. Usually in collagen of type I, the collagen molecules are organized in heterotrimer triple helices consisting of two α-1-chains and one α-2-chain. However, in fetal tissues, fibrosis, and cancer in humans, collagen of type I may also appear in homotrimetric forms with three α-1-chains [98]. These trimers form microfibrils that assemble into larger fibrils. These fibrils in turn form collagen fibers. It has been suggested that the interfibrillar load transition is provided by proteoglycan bridges [101]. The collagen fibril deformation micromechanisms and the impact of crosslinks on their aggregated mechanical properties are explored in Refs. [99] and [100]. For tendon collagen, Fratzl and Weinkamer [102] report a thickness of $≈1.3 nm$ for collagen molecules and a thickness of $50−500 nm$ for collagen fibrils. These form fascicles of a thickness in the range $50−300 μm$, which in turn form the tendon fibers with a thickness of $100−500 μm$. The Young's moduli of the collagen fibrils are much larger than those of elastin fibers (100–1667 times lager [18,103]). In vertebrates, where it presents around 90% of the bone and tendon mass, it is the major collagen in interstitial connective tissue, skin, and ligaments [104]. Different Young's moduli have been reported for type I collagen in various soft tissues for different states of the tissue and under different testing methods: 0.2–7 GPa [105,106] in bovine Achilles tendon; 4.1 GPa in lathyritic rat skin [107]; and 0.1–21 GPa in rat tail tendon [108110].

Fig. 1
Fig. 1

Collagen of type II is typically coassembled with type XI collagen, and it forms the main constituent of cartilage with around 60% of the dry mass and around 95% of the collagens in it [111]. Collagen of type III is secreted by fibroblasts and other mesenchymal cell types, and together with collagen of type I these collagens form the main constituents of the interstitial matrix [112]. Collagen of type V is found to be essential for the fibrillation of collagen of types I and III [113], and therefore it is significant in determining the quality of the fibrillary formation in the tissue. Collagen of type XI maintains both the radius of type II collagen and the spacing between these type II collagens; it is considered to be a nucleator of the fibrillogenesis of collagens of types I and II [114]. Collagen of type XXIV is found in the brain, bone, cornea, muscle, kidney, spleen, liver, lung, testis, and ovary tissue. The function of this type of collagen is as yet not fully resolved [115]. Collagen of type XXVII is reported to be shorter than other fibrillar collagens, which suggests that it may function in a somewhat different manner from that of other collagens; it is indicated that during skeletogenesis this collagen type plays a role during the transition from cartilage to bone [116]. Table 2 provides an overview of the occurrence of the fibrillar collagen types in mammalian tissue.

Table 2

Occurrence of the fibrillar collagen types in mammalian tissue [15,91,97,111116]

Fibrillar collagen typeExamples of occurrence in mammalian tissue
Collagen INoncartilaginous tissues, bone, tendon, skin, ligaments, cornea, vasculature, interstitial connective tissues
Collagen IICartilage, vitreous body, nucleus pulposus, intervertebral disk
Collagen IIIEmbryonic skin, lung vasculature, vessel wall, elastic tissues, reticular fibers of lungs, liver, spleen
Collagen VCornea, embryonic tissues, bone, interstitial matrix, lung, liver, fetal membranes
Collagen XICartilage, testis, trachea, tendons, trabecular bone, skeletal muscle, placenta, lung, neoepithelium of the brain, vitreous body
Collagen XXIVDeveloping cornea and bone, muscle, kidneys, spleen, liver, lung, testis, ovary
Collagen XXVIIEmbryonic cartilage, developing dermis
Fibrillar collagen typeExamples of occurrence in mammalian tissue
Collagen INoncartilaginous tissues, bone, tendon, skin, ligaments, cornea, vasculature, interstitial connective tissues
Collagen IICartilage, vitreous body, nucleus pulposus, intervertebral disk
Collagen IIIEmbryonic skin, lung vasculature, vessel wall, elastic tissues, reticular fibers of lungs, liver, spleen
Collagen VCornea, embryonic tissues, bone, interstitial matrix, lung, liver, fetal membranes
Collagen XICartilage, testis, trachea, tendons, trabecular bone, skeletal muscle, placenta, lung, neoepithelium of the brain, vitreous body
Collagen XXIVDeveloping cornea and bone, muscle, kidneys, spleen, liver, lung, testis, ovary
Collagen XXVIIEmbryonic cartilage, developing dermis

### 2.2 Biological Processes in the Enzymatic Turnover of Fibrillar Collagen.

A schematic illustration of the collagen remodeling process in living tissue is given in Fig. 2. Fibroblast cells secrete ECM components, of which some assemble into the fibrillar collagen. Enzymes have a key role in cleaving the collagen. These processes are mediated by different factors including mechanical loading. Mechanical stimuli regulate the fibroblast cells secretion of ECM components. While this figure depicts the frustrated ability of enzymes to cleave strained collagen, it should also be mentioned that different cases have been observed in which the collagen cleavage rate remains constant or even increases [117].

Fig. 2
Fig. 2

The lifespan of collagen may vary from tissue to tissue. A rapid collagen turnover rate has been observed by Bienkowski et al. [118], who report cases in which 20–40% of the α-chains in newly synthesized collagen were broken down within several minutes after being synthesized. Humphrey [119] reports a collagen half-life of 3–90 days under normal conditions in various soft tissues. The study by Nissen et al. [120] on skin, tail tendon, aorta, and mesenteric artery of 12 week old rats found a collagen half-life of 60–70 days in normotensive animals. In hypertensive rats, the half-lifes of skin and tail tendon collagen remain the same, while the half-lifes of collagen in the aorta and mesenteric artery were decreased to about 17 days. The collagen turnover rate decreases with age, and to give an example, Mays et al. [121] report that at 24 months of age the collagen synthesis rates had decreased by at least tenfold in all tissues compared to the synthesis rates at 1 month of age. In this regard, one must distinguish between in vivo collagen lifespans and a hypothetical chemical lifespan in an in vitro setting. For example, Thorpe et al. [122] report a relatively long lifespan in horse superficial-digital flexor tendon, in which a collagen half-life of 197.54±18.23 yr is estimated.

Within a specific tissue, the lifespan is dependent upon overall age and state of health. Abdul-Hussien et al. [123], for example, report that the growth of abdominal aortic aneurysms and their failure correlates with elevated rates of collagen turnover. The carbon-14 bomb pulse method permits the tracking of the life-long collagen turnover in various tissues. Using this method, Heinemeier et al. [124] found minimal collagen turnover in healthy adult human cartilage. They also found that the occurrence of osteoarthritis does not initiate collagen renewal, a finding that contradicts those of previous works [125,126], which had indicated an increased collagen synthesis in osteoarthritis. In a later publication, Heinemeier et al. [127] apply the carbon-14 bomb pulse method to tendons. They report a very limited collagen turnover in adulthood as well as an increased collagen turnover in tendinopathic tendons. They further summarize that this tendinopathy-related turnover affects just one part of the ECM, whereas other parts containing old collagen tend to remain stable.

In an open dynamic system, homeostasis refers to a steady-state that is regulated by internal processes. This definition is often refined within the context of a specific area of study. In their research on mechanical response to mechanical loading in three-dimensional substrates, Brown et al. [128] characterize tensional homeostasis in dermal fibroblasts as a regulatory mechanism by which fibroblast cells contract the ECM as a means to oppose the influence of mechanical stimuli. Stamenović and Smith [129] describe tensional homeostasis as the ability of a tissue to provide a consistent tension with a low variability near a certain point and across different length scales. In the framework of this review, we simply refer to homeostasis as the macroscopic balance between the dynamic processes of collagen production and degradation on smaller length scales (also see Refs. [130132]). Such a homeostatic balance is necessary in order to maintain the function of the tissue, and it is essential in the wound healing process [133,134]. A disruption of the homeostatic balance may result in different tissue disorders. An example for collagen creation under reduced collagen degradation is myocardial fibrosis, which can be linked to many cardiovascular disorders [135]. Different mechanisms that are involved in disrupting the otherwise regular process of ECM degradation are discussed in Sec. 2.2.2. Eichinger et al. [136] survey recent insights regarding mechanical homeostasis in tissue equivalents (e.g., collagen gels seeded with living cells) with a focus on uniaxial and biaxial experiments.

#### 2.2.1 Collagen Synthesis.

Fibroblast cells are responsible for numerous functions that govern the tissue architecture and the ECM turnover rate [137]: (a) fibroblasts synthesize and secrete different ECM components such as collagens, proteoglycans, tenascin, laminin, and fibronectin; (b) fibroblasts create matrix metalloproteinases (MMPs) that are capable of degrading ECM components such as collagen; and (c) fibroblasts generate tissue inhibitors of metalloproteinases (TIMPs), which suppress the MMPs (see Sec. 2.2.2). An essential task in the connective-tissue remodeling process during both normal and pathological wound healing is the modulation of fibroblastic cells toward the myofibroblastic phenotype [138]. This fibroblast cell activity can be charted in terms of two perpendicular axes: one axis lists the growth factor environments (promigratory versus procontractile), and the other axis lists the cell–matrix interactions (high tension versus low tension) [139,140].

Collagen biosynthesis takes place on a wide range of length and time scales where, in general, it proceeds from very small and very fast intracellular processes to somewhat larger and slower for extracellular processes [141]. Specifically, fibroblasts invoke gene transcription originating within the cell nucleus, the post-translational modifications of procollagen molecules in multiple steps, the secretion of the triple-helical molecules from the cellular into the extracellular space, and culminating in its extracellular processing and modification [91,142144]. Six specific stages are identified in Ref. [145]: (i) In the first stage, ribosomes synthesize procollagen chains. (ii) These procollagen chains are transported into the rough endoplasmic reticulum. After several post-translational steps, three chains are assembled into a triple-helical procollagen molecule. (iii) These procollagen molecules are then packaged into secretory vesicles which are denoted as Golgi to plasma membrane carriers. (iv) The collagen fibrils are assembled and stabilized by the creation of crosslinks. (v) The fibril deposition is then arranged. (vi) In the sixth and final step, the fibrils are deposited into the ECM.

A study [146] on cultured smooth muscle cells from rabbit aortic media suggests the involvement of angiotensin II and transforming growth factor-β in stretch-mediated collagen synthesis. As a result, collagenous tissue may not be free from stresses in the absence of external loading. The natural or undeformed configurations of the soft-tissue constituents may differ, which may result in residual stresses. For example, blood vessels may have a resting length that is 25–30% less than their in situ length [138]. Many articles report a prestretch (or deposition stretch) of collagen fibers in different types of soft tissue, so that the fibers tend to contract the surrounding tissue in the fiber direction. Some examples are listed in Table 3.

Table 3

Examples of prestretched collagen in different tissues

Type of tissueAmount of prestretch
Saccular cerebral aneurysms1.02–1.04 [147]
Human coronary arteries1.039–1.05 [148] (in situ)
Healthy cervical stroma1.03 [149]
Aged human iliac artery1.07 [150] (in situ)
Human femoral artery adventitia1.09 [151] (in situ)
Type of tissueAmount of prestretch
Saccular cerebral aneurysms1.02–1.04 [147]
Human coronary arteries1.039–1.05 [148] (in situ)
Healthy cervical stroma1.03 [149]
Aged human iliac artery1.07 [150] (in situ)
Human femoral artery adventitia1.09 [151] (in situ)

While the previous examples focus on fibers that contract the surrounding tissue, the surrounding tissue may increase its volume and swell, which in turn may stretch the embedded collagen fibers. There are numerous biological processes that involve some form of swelling response, for example, edema [152], mechanical trauma [153], inflammatory responses [154], and the development of cervical tissue during pregnancy [18]. These processes are often mediated by hormonal changes in the tissue (discussed in Sec. 3.4).

In certain situations, collagen may be deposited in a crimped, wavy, or helical arrangement within the soft tissue. Wavy collagen is found, for example, in liver [155], in tendon [156], and in the arterial adventitia [157]. It has been reported that due to the waviness of the collagen fibers in the dermis of balloon-fish (Diodon holocanthus) skin, minimal force is required to achieve an extension of up to 40% of its initial length. Upon sufficient extension, such fibers are recruited so as to bear a portion of the tensile loading. Different collagen crimp patterns emerge, which can be distinguished by the length, angle, and location of the crimp peaks [156,158].

Tissue development, morphogenesis, repair, and remodeling are accompanied by regulated and timely degradation of ECM constituents. A disruption in the regulation of the degradation processes may lead to diseases such as arthritis, nephritis, cancer, encephalomyelitis, chronic ulcers, and fibrosis [41].

The main enzymes that degrade collagen in the ECM are MMPs, which are calcium-dependent zinc-containing endopeptidases. Based on domain organization and substrate preference, these MMPs can be classified into collagenases, gelatinases, stromelysins, matrilysins, membrane-type MMPs (transmembrane enzymes and glycosylphosphatidilyinositol anchored enzymes), and other nonclassified MMPs [159161]. Another classification into five groups according to bioinformatic analysis has been presented by Cerofolini et al. [162] as follows: nonfurin regulated MMPs, MMPs bearing three fibronectin-like inserts on the catalytic domain, MMPs anchored to the cellular membrane by a C-terminal glycosylphosphatidylinositol moiety, MMPs bearing a transmembrane domain, and the group comprising all the other MMPs (see Table 4). Jackson et al. [163] provide a comprehensive list of all MMPs in humans and mice by their trivial names, grouping by domain composition, and also grouping by phylogenetic alignment and chromosomal location. MMPs fulfill a multitude of functions such as tissue repair and healing [164]. Collagenases are key enzymes in the remodeling processes of collagen [165]. TIMPs regulate the MMP activities. A deregulation of the MMPs activities may result in different pathologies, which can be grouped according to the resulting destruction of tissue, fibrosis, and the weakening of the ECM [164]. In humans, four TIMPs (TIMP-1, TIMP-2, TIMP-3, and TIMP-4) are known to regulate the activities of MMPs; the structure, function, and evolution of these TIMPs are discussed in Ref. [166].

Table 4

Classification of the different types of MMPs based on domain organization and substrate preference [159161] and according to bioinformatic analysis [162]

MMP classification based on domain organization and substrate preference [159161]
MMP groupMMP type
CollagenasesMMP-1, MMP-8, MMP-13
GelatinasesMMP-2, MMP-9
StromelysinsMMP-3, MMP-10, MMP-11
MatrilysinsMMP-7, MMP-26
Transmembrane enzymesMMP-14, MMP-15, MMP-16, MMP-24
Glycosylphosphatidilyinositol anchored enzymesMMP-17, MMP-25
Further MMPsMMP-12, MMP-19
MMP classification into five groups according to bioinformatic analysis [162]
MMP groupMMP type
Nonfurin regulated MMPsMMP-1, MMP-3, MMP-7, MMP-8, MMP-10, MMP-12, MMP-13, MMP-20, MMP-27
MMPs bearing three fibronectin-like inserts on the catalytic domainMMP-2, MMP-9
MMPs anchored to the cellular membrane by a C-terminal glycosylphosphatidylinositol moietyMMP-11, MMP-17, MMP-25, MMP-1-like
MMPs bearing a transmembrane domainMMP-14, MMP-15, MMP-16, MMP-24
The group comprising all the other MMPsMMP-19, MMP-21, MMP-23, MMP-26, MMP-28
MMP classification based on domain organization and substrate preference [159161]
MMP groupMMP type
CollagenasesMMP-1, MMP-8, MMP-13
GelatinasesMMP-2, MMP-9
StromelysinsMMP-3, MMP-10, MMP-11
MatrilysinsMMP-7, MMP-26
Transmembrane enzymesMMP-14, MMP-15, MMP-16, MMP-24
Glycosylphosphatidilyinositol anchored enzymesMMP-17, MMP-25
Further MMPsMMP-12, MMP-19
MMP classification into five groups according to bioinformatic analysis [162]
MMP groupMMP type
Nonfurin regulated MMPsMMP-1, MMP-3, MMP-7, MMP-8, MMP-10, MMP-12, MMP-13, MMP-20, MMP-27
MMPs bearing three fibronectin-like inserts on the catalytic domainMMP-2, MMP-9
MMPs anchored to the cellular membrane by a C-terminal glycosylphosphatidylinositol moietyMMP-11, MMP-17, MMP-25, MMP-1-like
MMPs bearing a transmembrane domainMMP-14, MMP-15, MMP-16, MMP-24
The group comprising all the other MMPsMMP-19, MMP-21, MMP-23, MMP-26, MMP-28

MMP-1, MMP-8, MMP-13, and MT1-MMP catalyze the degradation of the fibrillar collagens of types I and III into 1/4 and 3/4 fragments; solubilized monomers of collagens I, II, and III are digested by MMP-2; solubilized monomers of collagens I and III are digested by MMP-9 [167]. Central to these processes are the diffusion of MMP to the susceptible sites [168,169], the vulnerability of cleavage sites in collagen fibrils [170,171], and the different mechanisms for collagen triple helix engagement and distortion [167].

The first collagenases that have been found and studied in detail are those produced by the bacteria Clostridium histolyticum. Even though these micro-organisms are not the only ones to produce collagenase, the terms clostridium collagenase, bacterial collagenase, and microbial collagenase are often used interchangeably [172]. Contrary to MMP-1, MMP-8, and MMP-13 that split collagens at specific sites, bacterial collagenases are described as being more efficient in degrading collagen as they cleave collagen at multiple sites [87,173].

Trypsin is a pancreatic enzyme that is found in the digestive system of different vertebrates [174]. The article by van Deemter et al. [175] examines the role of trypsin in the enzymatic degradation of collagen in the human vitreous body. Ellsmere et al. [53] report that trypsin is capable of cleaving denatured type I and type III collagen. Kirkness et al. [117] report an increased sensitivity of collagen triple helices to trypsin when subject to mechanical loading.

Homotrimers of collagen of type I are reported to be resistant to cleavage by all mammalian collagenases (e.g., MMP-1, MMP-8, and MMP-13). The molecular mechanisms behind this resistance are studied by Han et al. [176], Chang et al. [177], and Teng and Hwang [178] (also see Sec. 5).

Binding of enzymes, unwinding of the helices at the cleavage locations, and sequential hydrolysis of the chains are involved in the cleavage of collagen by MMP [177]. Models have been proposed that invoke collagen cleavage before [179] and after [180] MMP attachment to the cleavage location.

The complex relationship between MMP expression and diseases such as cancer has motivated several clinical investigations [181]. The degradation of collagen by means of enzymes is utilized in several medical applications. Fang et al. [182] describe the role of collagen in tumor progression as a “double-edged sword,” in which collagen has been regarded as a barrier that resists the expansion of tumors, whereas it is now considered as evident that collagen has an active role in the progression of tumors. It does so by creating a local micro-environment that destabilizes the cell polarity and the cell–cell adhesion, while augmenting growth factor signaling [182]. It has been reported that the enzymatic degradation of collagen in tumors with a high collagen content may increase the diffusion of agents that are used to target cancerous cells [183,184].

## 3 Mechanosensitive Collagen Remodeling

Experiments have shown that strain may stabilize collagen to enzymatic degradation. Conversely, other studies report accelerated collagen degradation under mechanical loading. Table 5 provides a concise summary of different models that are addressed in this section.

Table 5

Bacterial collagenase
Collagen moleculesUp to $16 pN$Adhikari et al. [55]
Bovine collagen I$≤50%$ network strainBhole et al. [42]
Recombinant human type I collagen monomers$0.06 pN, 3.6±1.1 pN, 9.4±1.3 pN$Camp et al. [43]
Bovine hearts, intact pericardia$1 g,10 g,60 g$ static loading, $40−80 g$ dynamic loadingEllsmere et al. [53]
Bovine sclera collagen I fibrils$2 pN$/monomer, $24 pN$/monomer (see Fig. 7 in Ref. [47])Flynn et al. [46,47]
Pig heart pericardial tissue$5%→ 40%$↓ ($<20%$ strain) ↑ ($>20%$ strain)Ghazanfari et al. [57]
Bovine tendon, reconstituted$≤7%$ strain↓ ($<4%$ strain) ↑ ($>4%$ strain)Huang and Yannas [56]
Patella-patellar, tendon-tibia(New Zealand White rabbits)$4 %$ elongationNabeshima et al. [48]
Bovine cornea6% strainRobitaille et al. [11]
Bovine cornea15–30 pN/monomerRuberti and Hallab [50]
Mice tail tendon fasciclesStrain up to $∼5−8%$ and larger (in cell-free (i.e., protein synthesis free) tissue)↓ ($<∼5−8%$ strain)Saini et al. [59]
– ($>∼5−8%$ strain)
Rat tail tendon type I collagen1–10% strainWyatt et al. [51]
C57BL/6 mice$20%,60%,80%$ strain (static), 30–50% and 20–60% strain (0.1 and 1 Hz)↓ ($<20%$ strain) ↑ ($>20%$ strain)Yi et al. [58]
Bovine cornea1.1, 2.75, 5.5 pN/monomerZareian et al. [52]
MMP-1
Collagen moleculesUp to $16 pN$Adhikari et al. [54,55]
Rat tail tendon type I collagenInternal strain regulated remodelingDittmore et al. [44]
Swiss-Webster mice33% strain in mouse anulus fibrosus tail intervertebral diskLotz et al. [49]
Mice tail tendon fasciclesStrain up to $∼5−8%$ and larger (in cell-free (i.e., protein synthesis free) tissue)↓ ($<∼5−8%$ strain)Saini et al. [59]
– ($>∼5−8%$ strain)
MMP-8
Bovine collagen I$≤50%$ network strainBhole et al. [42]
Bovine collagen IStretching to over 100%Flynn et al. [45]
MMP-9
Rat tail tendon type I collagenInternal strain regulated remodelingDittmore et al. [44]
Unspecified MMP
Human dentin collagen100,000 cycles, 2 Hz, 49 N (applied to dentin beams)Toledano et al. [185]
Trypsin
Collagen type III$0 pN$/trimer, $9 pN$/trimerKirkness et al. [117]
Bacterial collagenase
Collagen moleculesUp to $16 pN$Adhikari et al. [55]
Bovine collagen I$≤50%$ network strainBhole et al. [42]
Recombinant human type I collagen monomers$0.06 pN, 3.6±1.1 pN, 9.4±1.3 pN$Camp et al. [43]
Bovine hearts, intact pericardia$1 g,10 g,60 g$ static loading, $40−80 g$ dynamic loadingEllsmere et al. [53]
Bovine sclera collagen I fibrils$2 pN$/monomer, $24 pN$/monomer (see Fig. 7 in Ref. [47])Flynn et al. [46,47]
Pig heart pericardial tissue$5%→ 40%$↓ ($<20%$ strain) ↑ ($>20%$ strain)Ghazanfari et al. [57]
Bovine tendon, reconstituted$≤7%$ strain↓ ($<4%$ strain) ↑ ($>4%$ strain)Huang and Yannas [56]
Patella-patellar, tendon-tibia(New Zealand White rabbits)$4 %$ elongationNabeshima et al. [48]
Bovine cornea6% strainRobitaille et al. [11]
Bovine cornea15–30 pN/monomerRuberti and Hallab [50]
Mice tail tendon fasciclesStrain up to $∼5−8%$ and larger (in cell-free (i.e., protein synthesis free) tissue)↓ ($<∼5−8%$ strain)Saini et al. [59]
– ($>∼5−8%$ strain)
Rat tail tendon type I collagen1–10% strainWyatt et al. [51]
C57BL/6 mice$20%,60%,80%$ strain (static), 30–50% and 20–60% strain (0.1 and 1 Hz)↓ ($<20%$ strain) ↑ ($>20%$ strain)Yi et al. [58]
Bovine cornea1.1, 2.75, 5.5 pN/monomerZareian et al. [52]
MMP-1
Collagen moleculesUp to $16 pN$Adhikari et al. [54,55]
Rat tail tendon type I collagenInternal strain regulated remodelingDittmore et al. [44]
Swiss-Webster mice33% strain in mouse anulus fibrosus tail intervertebral diskLotz et al. [49]
Mice tail tendon fasciclesStrain up to $∼5−8%$ and larger (in cell-free (i.e., protein synthesis free) tissue)↓ ($<∼5−8%$ strain)Saini et al. [59]
– ($>∼5−8%$ strain)
MMP-8
Bovine collagen I$≤50%$ network strainBhole et al. [42]
Bovine collagen IStretching to over 100%Flynn et al. [45]
MMP-9
Rat tail tendon type I collagenInternal strain regulated remodelingDittmore et al. [44]
Unspecified MMP
Human dentin collagen100,000 cycles, 2 Hz, 49 N (applied to dentin beams)Toledano et al. [185]
Trypsin
Collagen type III$0 pN$/trimer, $9 pN$/trimerKirkness et al. [117]

Some works find a strain-stabilization of collagen fibers to enzymatic degradation (“↓”), while others find that stretch may accelerate collagen degradation (“↑”), or it may be that no effect (“–”) was detected. In this table, Refs. [44] and [53] can be viewed as somewhat exceptional cases; please refer to Secs. 3.1 and 3.2 for some discussion on the findings that are presented in these works.

### 3.1 Strain Can Stabilize Collagen to Enzymatic Degradation.

Different experiments have shown that straining of fibrillar collagen may result in a decrease of enzymatic degradation, in which case the collagen remodeling process is denoted as strain-stabilized. The underlying molecular mechanisms that accompany the strain-stabilization are discussed in Sec. 5.

A reduced degradation of strained collagen to bacterial collagenases has been reported by Robitaille et al. [11], Bhole et al. [42], Camp et al. [43], Flynn et al. [46,47], Nabeshima et al. [48], Ruberti et al. [50], Wyatt et al. [51], Zareian et al. [52], and Saini et al. [59]. Several works report a strain-stabilization of collagen to degradation by MMPs, for example, to degradation by MMP-1 in Refs. [49] and [59] and by MMP-8 in Refs. [42] and [45]. Toledano et al. [185] explore the load-stabilization of dentin collagen against MMP degradation, without specifying the MMP type. Nabeshima et al. [48] discuss the modes of action in the way that bacterial and mammalian collagenases degrade collagen. It is found that different types of collagen react differently to mechanical stimuli, and that results from one type of collagenase do not extrapolate to other collagenase enzymes. The works [187,188] report an increasing ratio of collagen type III to type I when the cardiac tissue cells are mechanically stretched. Studies indicate that fibroblast cells “sense,” “integrate,” and “respond” to mechanical stimuli while in the act of depositing ECM components [189191]. Several structures have been proposed as sensors of mechanical stimuli including extracellular matrix molecules, the cytoskeleton, transmembrane proteins, proteins at the membrane-phospholipid interface, elements of the nuclear matrix, chromatin, and the lipid bilayer itself [192].

In contrast to the previously indicated studies that focus on effects of an externally applied tensile load, it can also be the case that material defects driven by internal strain may play a similar role. For example, Dittmore et al. [44] surmise that this could remove buckling within the collagen that in turn regulates enzymatic breakdown. More generally, internal strain may remove the spontaneous formation of defects, which are typically the preferred enzymatic cleavage locations, and this removal may in turn inhibit degradation of fibrillar collagen by MMP-1 and MMP-9.

### 3.2 Strain May Also Promote Enzymatic Collagen Degradation.

Other experiments find an opposite effect, one in which mechanical loading appears to promote the enzymatic degradation of collagen. For example, Adhikari et al. [54,55] have shown that in comparison to unstrained collagen I, strained collagen undergoes a stronger degradation in the presence of MMP-1. Kirkness and Forde [117] report different responses of collagen III triple helices to axial straining below $10 pN$: (a) the collagen helices are stabilized or tightened; (b) the collagen extends without any change in the degradation behavior; and (c) strain promotes the collagen degradation. It is found that the degradation of collagen type III will increase in the presence of trypsin, and that this increase becomes stronger with straining of the collagen triple helices. Willett et al. [186] report degradation increase of collagen in an in vitro tensile overload tendon in the presence of α-chymotrypsin and trypsin.

The question arises as to whether the strain-promotion of enzymatic degradation can be viewed as an overload effect, perhaps after an earlier regime of strain-stabilization (as discussed in Sec. 3.1) that remained undetected. Ellsmere et al. [53] study the enzymatic degradation of a bovine pericardium in the presence of bacterial collagenase. Their overall conclusions are stated in the form of a strain-promotion effect. Consider, however, the results that are presented in Fig. 5 of Ref. [53] which illustrates the extension of statically and dynamically loaded specimens. During the first hours, the specimens with the lowest static load application show a relatively large extension in comparison to the specimens that are subjected to higher static (and dynamic) loads. In the time-to-failure tests, the specimens that experience the larger loads fail before the specimens that are subjected to a lower load application.

### 3.3 Collagen Degradation Behavior That Varies According to Strain Range.

There are also studies that report a degradation behavior that depends on the amount of strain in the way that is specifically suggested by an overload effect, i.e., a small amount of fibrillar strain may stabilize collagen to degradation, but a larger amount of strain enhances the degradation. Huang and Yannas [56] report decreased collagen degradation to bacterial collagenase for relatively low strains, and at strains beyond the certain threshold value of 4%, the collagen degradation increases again. Similar effects are reported elsewhere. For example, Ghazanfari et al. [57] and Yi et al. [58] find that the degradation rate has its minimum at a static strain of around 20%. Section 7.1 presents a modeling approximation of the strain-dependent degradation behavior that changes with the amount of strain.

Fig. 3
Fig. 3

Key experimental findings are reported by Saini et al. [59], three of which are especially notable. First, mechanical strains suppress collagen degradation by bacterial collagenases and MMP-1 in cell-free (i.e., protein synthesis free) tissue for strain magnitudes up to $∼5−8%$. Although the results are described as noisy at larger strains, the authors conclude that, at larger strains, the degradation is nearly independent of the strain. Second, strain suppression of the collagen degradation rate is not related to the degradation mode, whereas there is a correlation between the local tissue strain, the degradation rate, and organization of collagen. Third, enzymatic degradation of collagen is correlated to the unfolded triple-helical collagen content.

### 3.4 Broader Biochemical and Physical Process Interactions.

While this article has its specific focus on the enzymatic degradation of collagen, it is important to note that further factors may be involved in such a degradation process. For example, a potential mechanism for rheumatoid arthritis is the nonenzymatic degradation of fibrillar collagen by the action of specific antibodies [193]. Dingal and Discher [194] remark that mechanical straining regulates how collagen breaks down as a function not only of enzymes but of other degradation agents as well. Lennox [195], Bass et al. [196], and Miles and Ghelashvili [197] report an increased thermal stability of strained collagen. At larger strains beyond a certain threshold that governs failure in the collagen structure, this stabilization effect is reversed whereupon the thermal stability of collagen decreases with increasing strain (see, e.g., Ref. [198]). If the strain is sufficiently large beyond the elastic region, then the collagen network will undergo damage processes [199].

The activation and expression of ECM degrading enzymes, such as MMP-1, MMP-2, and MMP-9, are sensitive to the type and form of any applied mechanical loading (uniaxial static stretch, biaxial static stretch, cyclic uniaxial stretch, cyclic biaxial stretch, cyclic hydrostatic pressure, etc.) [200205]. Such loading can mediate a change in the cell proliferation [206,207], inhibit or stimulate the proteoglycan synthesis [208], increase or decrease the fibronectin messenger ribonucleic acid [209], and reduce or elevate the collagen levels. Wang et al. [210,211] assess the impact of the cell source and mechanical stimuli on the regulation of ECM gene and protein expression.

The fiber remodeling process has been studied in the context of athletic training. Miller et al. [212] report an increased collagen and muscle protein synthesis in human patella tendon and quadriceps muscle after strenuous sports exercise. The research by Magnusson et al. [213] concludes that mechanical loading of collagenous tissue may result both in protein synthesis and collagen degradation. In the first $24−36 h$ after acute exercise in humans, the collagen degradation is larger than the collagen synthesis, so that a lack of rest between exercises may lead to increased risks for injury. This collagen net loss period is followed by a period of around $36−72 h$ after exercise with a collagen net synthesis.

The hormone mediated regulation of cervical tissue remodeling has been a longstanding topic of research [214]. Hormones 17β-estradiol and progesterone are reported to inhibit collagen degradation by corneal fibroblasts [215]. During pregnancy, hormones are essential for the regulation of soft-tissue remodeling within the female reproductive system. A decrease in the collagen content and an increasing extensibility of rat cervices that are subjected to relaxin are reported in Ref. [216]. It is also found that the temporal change in the progesterone–to–17β-estradiol hormone ratio during pregnancy regulates the turnover of elastin and collagen [217]. That work illustrates how the cervical stiffness decreases with increasing hormone ratio throughout pregnancy. Postpartum, the cervical stiffness increases as the hormone ratio decreases. The effect of progesterone on the mechanical properties of engineered cervical tissue is also significant [218]. The impact of β-estradiol and relaxin-1 on collagen production of the muscular fasciae is examined in Ref. [219], which finds that the production of type I and III collagen will decrease when exposed either to relaxin-1 or else to a combination of relaxin-1 and β-estradiol. Klug et al. [220], in a study of hormonal factors in cardiac tissue remodeling, report that the development of myocardial fibrosis depends on the circulating levels of both angiotensin II and aldosterone. The expression of several genes may be activated by angiotensin II. Aldosterone may lead to ventricular fibrosis, and it has been speculated that this may be a result of collagen secretion in cardiac fibroblasts [220].

## 4 Brief Recap of Basic Issues in the Mechanical Modeling of Fibrous Materials

Mathematical modeling of soft-tissue mechanical response provides a tool for estimating tissue health and tissue damage and for gauging the safety and efficacy of medical treatments. In the mechanical modeling of collagenous tissue, the fibers are often regarded as being embedded in a ground substance. In analogy to the inclusion-embedding methodological treatment of engineering composites, this ground substance is sometimes referred to as the ground substance matrix or just as the matrix in an applied mechanics sense. In using such terminology, care must be taken so as to avoid conflating this notion of a ground substance matrix with the ECM in biological soft tissue, as the ECM includes collagen fibers as a significant component.

The schematic Fig. 3 depicts the different relevant material configurations of a collagenous tissue and its individual constituents, showing its development at different instants of time. The mapping between these configurations is described by deformation gradients or deformation-gradient-like mapping tensors. The natural or stress-free configuration of the ground substance constituent may differ from the natural (stress-free) configuration of the various fibers. This produces residual stress when the tissue is free of external loads. For modeling purposes, the reference configuration κ0 for the constitutive theory is often taken to be the natural or stress-free configuration of the ground substance constituent. This choice can be motivated on the basis of the ground substance properties having significantly less time scale dependence compared to the more active changes taking place in the fibrous constituents. Each material point in κ0 is located by its position vector X. Each such location is regarded as a mixture of the various tissue constituents. The terminology representative volume element is sometimes employed to emphasize that each reference location X is a complex (and changing) mixture of fibers and ground substance.

Fig. 4
Fig. 4

Real-time tissue locations are denoted by position vectors $x=x(t)$ where t is the current time. The associated real-time (or actual) configurations are denoted by $κ(t)$. The local features of the deformation from X to x are described by deformation gradient tensor $F=∂x/∂X$, which is also time dependent, $F=F(t)$.

Biological soft tissues show a nonlinear mechanical behavior, and the fibrous constituent typically makes this behavior anisotropic. For all constituents, the elastic aspect of this behavior is described in terms of a strain energy density function W. This strain energy is a function of the right Cauchy–Green deformation tensor $C=FTF$ and the various fiber directions. The tensor $E=(1/2)[C−I]$ is a finite strain tensor which extends the classical small strain tensor so as to apply to nonlinear deformations. Specific material models typically specify the dependence of W on C in terms of scalar deformation measures such as the invariants of C or the eigenvalues of C, the latter of which have an interpretation in terms of material stretching. Because each material location is regarded as containing both ground substance material and fiber material, the modeling treatment proceeds in a homogenized sense. This motivates decomposing W, which describes the overall composite mixture, into constituent parts
$W=Wm+Wf$
(1)

where Wm describes the mechanical behavior of the ground substance material, and Wf describes the mechanical behavior of the embedded fiber constituent. It is then typical that Wm is isotropic and Wf is anisotropic. The use of numerous mathematical functions to describe the mechanical behavior of biological soft tissue is summarized by Chagnon et al. [221], which outlines the various statistical considerations, strain tensor aspects, and strain invariant approaches as they relate to specific polynomial, power law, exponential, and logarithmic forms for Wm and Wf. Wineman and Pence [222] summarize a number of constitutive models for fiber-reinforced materials that include features such as nonlinear viscoelasticity and an evolving microstructure due to a combination of mechanical and nonmechanical factors.

### 4.1 The Isotropic Ground Substance.

The isotropic part Wm of the strain energy density function in Eq. (1) is typically taken to depend upon the principal invariants of C. These invariants are $I1(C)=tr(C), I2(C)=(1/2){[tr(C)]2−tr(C2)}$, and $I3(C)=det(C)$. The third invariant of C is the squared magnitude of the material volume ratio of the deformed material to the undeformed one, $J=detF=I3(C)$. A material that changes its volume during the deformation is called compressible, and the isotropic strain energy density has the general form $Wm=Wm(comp)(I1(C),I2(C),I3(C))$.

Due to the large bound water content of soft tissue, the ground substance material often exhibits minimal compressibility. This motivates an incompressible modeling treatment, meaning that $J=I3(C)=1$, so that $Wm=Wm(inc)(I1(C),I2(C))$, i.e., the constant third invariant is not explicitly present in the argument list because it would serve no purpose. Conversely, it may be that changes to the tissue fluid content are significant over the time scale of the modeling treatment, with the resulting local tissue volume change determined by physiological and electrochemical considerations. In such a case, it is useful to treat this volume change in terms of a swelling ratio parameter, say, v, where v = 1 describes the unswollen state, and v > 1 denotes swelling. In this case, the previous condition is generalized to $J=det F=I3(C)=v$. The material can still be viewed as mechanically incompressible in the sense that mechanical loads (in contrast to chemical and osmotic effects) have negligible effect on the included liquid volume at each stage of the swelling process. Because of this volume constraint, the strain energy density function for the ground substance material is then modeled in terms of the first two invariants of C and the time-varying swelling ratio [72,223226], yielding $Wm=Wm(inc,swell)(I1(C),I2(C),v)$.

### 4.2 The “True” Fiber Stretch.

The mechanical properties of the fibers strongly differ from those of the ground substance. When swelling is present, the volume changes may be restricted to certain constituents, causing Wm and Wf in Eq. (1) to have very different dependence upon swelling [227229]. The fiber part of the strain energy Wf is also especially dependent on the fiber stretching. To determine this stretch, let N be a unit vector along a fiber direction in the reference configuration κ0. It is deformed into the vector $FN$ in the deformed configuration $κ(t)$. The vector $FN$ is not usually a unit vector, because F stretches N by the amount
$λ(κ0→κ)=FN·FN=N·CN$
(2)

which is not necessarily equal to one. The stretch (2) represents an intuitive conception of a true fiber stretch provided that the fiber is at its natural length in κ0. This requires, in turn, that both the ground substance and the fibers share a common natural or relaxed configuration. Even though such commonality is not always the case, it is a relatively common presumption in the modeling of collagenous tissue when the differences in the natural configuration of the constituents are sufficiently small. This subsection discusses this issue and in so doing outlines procedures for determining the true fiber stretch when the difference in the natural configurations of the constituents cannot be ignored.

The anisotropic part Wf of the strain energy density function (1) is often taken to depend only upon the material symmetry invariants (which are sometimes denoted as “pseudo-invariants”) which make reference to the individual fiber directions. The most basic of these is $I4=N·CN=[λ(κ0→κ)]2$. Thus, if κ0 also happens to be the fiber's natural configuration, then I4 is simply the squared magnitude of fiber stretch from its relaxed state. The invariant $I5=N·C2N$ does not have such a simple interpretation as it brings in localized shearing effects. If fibers are also present in a second direction, say, with unit vector $N′$ giving its directional orientation, then additional invariants are introduced (e.g., in the adventitia layer of arteries, see Sec. 4.5). The invariants $I6=N′·CN′$ and $I7=N′·C2N′$ for this second fiber family are analogous to the invariants I4 and I5 in the first fiber family. Interactions between the fiber families are described by invariants $I8=N·CN′ and I9=N·N′$ [230232]. Alternative choices for the invariants are also possible and sometimes used [233].

Fibers are formed at different times, and it may be that the natural (fully relaxed) configuration of the just-synthesized fiber is different from κ0, the natural configuration of the ground substance. This necessitates special care in the specification of Wf because the invariants I4, I5, I6, and I7 lose their intuitive interpretation if they are not referenced to their individual natural configurations (e.g., Refs. [234] and [235] for a single fiber family treatment and a multiple fiber family treatment, respecively). To address this issue, let $κf(τ)$ be the natural configuration of a collagen fiber that is synthesized at time τ. In general, the configuration $κf(τ)$ will differ from the natural configuration κ0 of the ground substance. The mapping from κ0 to $κf(τ)$ can be described locally by a mapping tensor $A=A(τ)$, which depends on τ. Sometimes, a notation such as $F(κ0→κf(τ))$ is used in place of A, since $A=F(κ0→κf(τ))$ can be interpreted as a deformation gradient. The use of A makes it possible to reserve the use of F for mappings from the reference configuration κ0 to a deformed configuration κ that was actually realized at some time during the physical process.

Figure 3 depicts different relevant material configurations for a collagenous tissue. The configuration κ0 is the reference configuration, which is chosen to be the natural or relaxed configuration of the ground substance in which the collagen fibers are embedded. The mapping from the configuration κ0 to the deformed one $κ(t)$ at current time t is described by the deformation gradient tensor $F(t)$. The configuration $κf(τ)$ is the relaxed or natural configuration of a basic fiber entity that has been synthesized at time τ. This figures shows three configurations $κf(τ1), κf(τ2)$, and $κf(τ3)$ for three instants τ1, τ2, and τ3 of fiber synthetization, where $τ1<τ2<τ3≤t$. The deformation-gradient-like tensor $A(τ)$ maps the location of the configuration $κf(τ)$ relative to κ0. Then, the deformation of a fiber from its natural configuration $κf(τ)$ into the deformed configuration $κ(t)$ is described by the tensor $F(t)[A(τ)]−1$, which depends on both the fiber creation time τ and the current time t. If the fibers are considered to be prestretched at the time of their creation, or if the ground substance is subjected to volume changes, then the fibers tend to contract the tissue. The special homeostatic configuration in which there is no external loading is denoted as $κ0d$. The tensor $Fκ0→κ0d$ describes the mapping from the reference configuration κ0 into the special homeostatic configuration $κ0d$.

A fiber unit vector N in the reference configuration (where the fiber is typically not relaxed) is deformed into its natural configuration $κf(τ)$ (where it is relaxed) via $AN$. The associated amount of relaxation stretch (which is measured from the configuration κ0 where the fiber is not relaxed) is $λ(κ0→κf)=AN·AN$. The intuitive notion of “true fiber stretch,” meaning the stretch from the fiber's natural stress-free configuration $κf(τ)$ to the current deformed configuration $κ(t)$, is formally denoted by $λ(κf(τ)→κ(t))$ and given by the ratio
$λ(κf→κ)=λ(κ0→κ)λ(κ0→κf)=FN·FNAN·AN$
(3)

The right-hand sides of Eqs. (2) and (3) coincide if the mapping tensor A is the identity tensor, because $A=I$ makes $AN·AN=1$. As indicated right after Eq. (2), this corresponds to a model where the fibers and the matrix share a common natural configuration, i.e., a model that posits $κf(τ)=κ0$, meaning that the fiber is relaxed if and only if the matrix is also relaxed.

The importance of the true stretch $λ(κf→κ)$ as given by Eq. (3) is that the condition $λ(κf→κ)=1$ denotes a fiber that is in a force-neutral state. In other words, if $λκf→κ=1$, then the fiber is neither in compression nor in tension, and so does not contribute to the overall state of stress. More generally, this $λ(κf→κ)$ naturally enters into the fiber's energy and stress constitutive laws in the mechanical modeling, both for a tension effect if $λ(κf→κ)>1$ and for a possible compression effect if $0<λ(κf→κ)<1$. Here, it is to be noted that collagen fibers are generally not capable of bearing significant compression loading, due to becoming crimped or wavy. In such cases, the fibers provide little or no compressive load bearing and have to be recruited in order to start bearing tensile load [236]. The length at which an initially wavy or crimped fiber starts to bear loading is sometimes denoted as slack length [237,238].

The treatment in terms of Eqs. (2) and (3) allows easy use of Eq. (1) in the mechanical analysis. The same ideas underly other treatments that seek to acknowledge the range of fibril natural states within the overall fibrous tissue which are due to fibril synthesis taking place while the tissue is deformed. For example, Ref. [238] presents a multiscale computational model that (among many other things) describes the variation in fibril mechanical response (which is due to a range of fibril stress-free states) in terms of a distribution in the stress-free fibril lengths.

### 4.3 Fibers That Are Deposited With Either Prestretch or Undulation.

If, as is commonly the case, the fiber deposition process is accompanied by non-negligible prestretch (e.g., Ref. [239]), then let $λ*$ denote the amount of stretch imparted to the just-created fiber as measured from its fully relaxed state. A process that could hypothetically release the just-created fiber, allowing it to revert back to its natural length, would then involve a relaxation stretch in the amount $1/λ*$ (undoing the prestretch). The prestretch $λ*$ corresponds to $λ(κf(τ)→κ(τ))$, where the time argument for both κf and κ is the creation instant τ. Conversely, $1/λ*$ corresponds to $λ(κ(τ)→κf(τ))$. In terms of this deposition prestretch, Eq. (3) becomes
$λ(κf→κ)=λ(κ0→κ)λ(κ0→κf)=λ*F(t)N·F(t)NF(τ)N·F(τ)N$
(4)

If $λ*>1$, then the prestretch gives fiber elongation with an associated immediate tensile contribution to the overall stress state. Conversely, if $0<λ*<1$, the prestretch involves fiber shortening (contraction) and hence a possible compressive contribution. When that happens, the tendency of fibers to buckle or otherwise crimp when compressed, as described in Sec. 2.2.1, may lead to wavy fiber morphologies at their creation time τ. Different structural models that account for such collagen morphologies are classified in Ref. [145] as the rigid corner model [240], the sinusoidal model [241,242], the helical model [243,244], the circular segment model [245], and the sequential loading model [246].

The above considerations leading from Eqs. (3) and (4) presume that the fiber prestretch $λ*$ is measured from the relaxed or force-neutral fiber state κf. If instead the prestretch $λ*$ is taken to be with respect to a configuration that differs from κf, then the relaxation stretch will generally not be given by $1/λ*$, and Eq. (3) will reduce to something other than Eq. (4). In principle, this can provide a more tractable expression compared to Eq. (4). While a well-intentioned modeling effort may elect to employ such alternative formulas for the sake of mathematical simplicity, great care must then be taken with respect to interpretation of the associated model predictions.

### 4.4 Fiber Remodeling in Homeostasis.

The homeostatic state (see Sec. 2.2) will typically adjust to current external and environmental factors. For the most basic analytical modeling treatments, these nonlocal factors are viewed as external input. In particular, the homeostatic state will generally depend on the external loading. The special homeostatic state associated with no external loading gives a stress-free configuration that shall be denoted as $κ0d$ (see Fig. 3). Although it is stress-free as a whole, the individual constituents are likely to be stressed. This occurs, for example, when prestretched fibers tend to contract the ground substance, or when a swollen ground substance material tends to stretch its embedded fibers. In both cases, this leads to elongated fibers in tension that are internally balanced by compressive stresses in the ground substance, giving an overall stress-free material at the tissue level (the stress response of the individual constituents is effectively averaged out).

The stress-free homeostatic configuration $κ0d$ should not be confused with κ0 which is the configuration in which the ground substance constituent by itself is stress-free. As a consequence, the stress-free homeostatic configuration $κ0d$ will generally differ from κ0. The configuration κ0 has already been chosen as the reference configuration for the constitutive relations in the modeling treatment, although, in some sense, $κ0d$ may be more intuitive since it is the configuration that the material would revert to if it was released from all external loadings. Standard transformation procedures allow the continuum mechanics treatment to be recast with a different configuration, including $κ0d$, becoming the reference configuration. While $κ0d$ is an intuitive choice, it does have the significant disadvantage that it would not be natural for any of the individual constituents (i.e., all of the constituents are prestressed in $κ0d$). By keeping κ0 as the reference state, everything is referred to a situation where the ground substance is stress-free [247]. In contrast, changing the reference state to $κ0d$ requires dealing with a prestressed ground substance, and this would generally require that an anisotropic constitutive law be used for Wm in Eq. (1). An alternative possibility is to use a different reference configuration for each constituent, typically that constituent's natural configuration, as invoked in constrained-mixture modeling (discussed in Sec. 4.6). For the fiber synthesis processes under consideration here, this leads to a notion of a separate reference configuration for fibers created at different times.

A thorough characterization of how the homeostatic stress-free state is determined by the constitutive theory often provides the most basic understanding of how the analytical model governs tissue reorganization. Contrary to a conventional elastic material with its time-independent mechanical behavior, in remodeling tissue the relation between the loading and deformation will likely depend on the past history of the material loading (rather than just on the current values, i.e., one must know the history of the boundary conditions). This history dependence, due to the underlying creation and degradation processes, is analogous to that which occurs in a viscoelastic material, even though the operative physical mechanism there is completely different (chain entanglement and relaxation).

Even though $κ0d$ is typically not the reference configuration for the constitutive relations, it is often taken as the starting configuration for the consideration of many loading or deformation scenarios, particularly in image-based modeling [248]. Such loading tends to disrupt any homeostatic balance that had been in place. Simulating the loss of homeostasis due to the disruption often necessitates numerical procedures that require significant computation. More readily computable is the hypothetical time sequence of homeostatic states associated with each time in the disruptive loading process. This sequence of homeostatic states may provide useful comparative values for the numerical treatment of the otherwise challenging fully time-dependent simulation. It may also serve as a useful starting state for various predictor–corrector type numerical algorithms that seek to simulate the complicated nonhomeostatic tissue response.

### 4.5 Complex Fiber Arrangements.

In biological soft tissue, the collagen fibers are generally organized in complicated multiscale arrangements that vary between tissue. The collagen distribution in cervical tissue offers a particularly striking example. The cervix can be divided into three zones, in which collagen fibers are arranged as follows: around the canal and also in the outermost zone, the fibers have a mean orientation parallel to the canal, whereas in the intermediate zone, the fibers instead have a preferred circumferential orientation [249251]. These three zones blend smoothly into each other, and the fibers are dispersed with respect to a preferred mean fiber direction [17]. In the simulation of cervical tissue response, this provides motivation for three-layer models for the collagen fiber arrangement [252,253].

Blood vessels are also broadly viewed as three-layer structures with: the tunica intima or innermost layer, the tunica media or middle layer, and the tunica adventitia or outermost layer [254,255]. Holzapfel [256] discusses the varying collagen arrangements in the different layers of arterial walls: in the intima layer, the collagen is organized in distinct families of dispersed fibers, whereas in the adventitia layer the collagen is organized in two helically arranged families, in which the orientation of the individual fibers deviate from the mean orientation of each fiber family.

A mathematical treatment in which the fibers are treated as either unidirectional or in terms of a small number of individual fiber directions provides the easiest and most straightforward modeling approach. More sophisticated approaches seek to account for the dispersion of collagen fibers. A variety of modeling treatments have been proposed, including those making use of distribution functions [257259] and angular integration methods [260,261]. The angular integration method takes the strain energy density Wf for the fibers in Eq. (1) to have the form
$Wf=∫ΩNwf(C,N) ρ̂(N) dΩ$
(5)

where wf is the energy density of the directed fibers per unit volume, and $ρ̂$ is a directional fiber density function that quantifies the fiber dispersion (see, e.g., Ref. [262]). The integration is over the different fiber orientations with $ΩN$ giving its range of solid angles. Since Eq. (5) is in terms of a specific C, and hence a single natural configuration, the mathematical form exhibited in Eq. (5) requires modification to address the type of time dependent prestretch issues described in Sec. 4.3. Computational complexity associated with Eq. (5) motivates alternative approaches for treating fiber dispersion. The generalized structure tensor method [263265] typically imposes less computation, because the numerical integration is carried out just once per element [266].

However, as pointed out in Ref. [267], care must be exercised in treating certain fiber dispersion specifications because the generalized structure tensor can potentially lead to undesirable modeling effects (e.g., the negative pressure values during tube inflation obtained in Fig. 12(b) of Ref. [267]). Correctly treating compressed fibers (the potentially nonload-bearing fibers) becomes a key consideration [268], either directly in the constitutive model [269] or in a numerical implementation [270]. These treatments and other methods are discussed in Ref. [271].

### 4.6 Growth, Remodeling, and Morphogenesis.

The issues under review here fall under the broader mechanobiological framework that considers tissue growth and ongoing tissue reorganization in general. Several recent books, significant portions of other texts, and major journal articles provide a comprehensive review of current analytical treatment methods within this broad subject area [6062,272]. This includes interesting discussion as to useful distinctions between the interrelated notions of growth, remodeling, and morphogenesis. With respect to growth, these frameworks motivate the introduction of a growth tensor, often denoted by $G=G(t)$, that is fundamental to the kinematic description. In the context of the type of mathematical description given in Sec. 4.2, this would typically cause the tensor F to be replaced by $FG$ in various places (possibly altering the role of the tensor A as described in Sec. 4.2). From a historical perspective, significant contributions in this field are the kinematic growth theory [64] and the concept of constraint mixture modeling [27]. Two hybrid growth and remodeling models, the evolving recruitment stretch model and the homogenized constrained-mixture model, are discussed in Ref. [65], which focuses on the merits of both of the original motivating models (kinematic growth theory and the constraint mixture modeling) along with possible means for overcoming some of their limitations.

Restricting attention here to these issues as they relate to a given fibrous component, this motivates the introduction of growth tensors $Gn$, one for each family of fibers. The volume or density change associated with that family is then captured by the time history of the determinant $Jn=detGn$. Evolution laws for this time history can then be posited, such as [61]
$Jn(t)=Jn(0)qn(t,0)+∫0tjn,+(τ)qn(t,τ) dτ$
(6)
where
$qn(t,τ)=exp(−kn−[t−τ])$
(7)

is a survival function that tracks the fiber produced at time τ that survives in undegraded form at current time t. The constant $kn−$ in Eq. (7) is a degradation rate coefficient, and $jn,+(τ)$ in Eq. (6) is a fiber production rate. These functions and coefficients can then be related to driving processes that incorporate even finer length scale considerations. Such treatments are not limited to integral forms of an evolution law for $Jn(t)$, with more detailed modeling treatments involving chemical reaction and diffusion processes being governed by ordinary and partial differential equations [72,73,273,274].

Mechanical models have been presented that specifically combine the remodeling of collagen at the microstructural level with tissue growth [78]. Using constrained-mixture hyperelasticity, soft-tissue wound healing has been simulated via an adaptive stress-driven approach for the synthesis of new collagen fibers in arrangements that are related to the principal stress directions [83]. The role of fiber growth and remodeling as it relates to the formation of aortic aneurysms is examined in Refs. [77,275], and [276]. All aspects of cardiac growth and remodeling theory, including myofiber remodeling, are active areas of current research [277]. These issues are very thoroughly reviewed in Ref. [63].

Similar to Eq. (5), the fiber strain energy function Wf for stretch-mediated enzymatic degradation of fibrillar collagen may also employ an integral with respect to the fiber creation times τ [131,234]
$Wf=∫t−Tmaxtwf(λfib(t,τ)) ζ(t,τ) dτ$
(8)

Here, Tmax is the physical upper bound to the mechanical lifespan of the fiber, and wf is again the energy density of the fiber per unit volume. The latter is modeled as a function of the fiber stretch, for example, $λfib(t,τ)=λ(κf→κ)$ with $λ(κf→κ)$ given in Eq. (2). Analogously to the memory kernel in viscoelastic constitutive laws and also the survival function $qn(t,τ)$ in Eqs. (6) and (7), the survival kernel $ζ(t,τ)$ in Eq. (8) accounts for the effect of the deformation history on the fiber density. Follow up details are provided in Sec. 7.1. Finally, it is to be remarked that integral forms for Wf that combine the type of angular integration in Eq. (5) with the time integration in Eq. (8) can be invoked for the consideration of history mediated fiber dispersions.

### 4.7 Modeling at Various Length Scales.

As indicated in Fig. 1, collagen fibers exhibit complex structure at many different length scales. Indeed, the figure does not do justice to its exquisite architecture. The synthesis and degradation processes reviewed in Secs. 2 and 3 are operative chemically and mechanically at the finest length scales [141,278,279]. Even so, its overall effect is manifested at all length scales, including the larger length scales that are appropriate for whole tissue and organ level modeling [280]. Mechanical processes that operate at the smaller length scales can be linked to the tissue scale by means of direct numerical multiscaling simulation procedures. However, the great disparity in length scales will tax even the most sophisticated extant computational resources. This motivates adapting a variety of modeling techniques, including volume averaging and homogenization approaches (e.g., Refs. [281] and [282]). These techniques, which were typically first developed for application to standard mechanical systems, may with care be adapted to the present biological process setting.

Figure 4 correlates specific modeling techniques to the collagen structure length scale at which they typically operate. Modeling at the molecular scale is often focused on molecular dynamics simulation. On the fibril and on the fiber scale, strain-mediated changes are typically accounted for by modeling a change in the fiber or fibril radius, or by modeling an overall collagen density change within a representative volume element. On the tissue scale, the material is often treated in an effective sense either by the application of homogenization strategies or by modeling in terms of a strain energy density function that has identifiable contributions from the different tissue constituents.

Fig. 5
Fig. 5

## 5 Molecular Scale Process Simulation

Enzymes act by means of their ability to access specific sites within the overall collagen complex. Continuum level concepts, such as load and strain, are agents of this interaction by virtue of the molecular conformational changes that they induce. In this regard, it has been suggested that strained single-molecules tend to degrade faster under tension [55], whereas strained “rope-like” collagen molecule arrangements tend to be stabilized to enzymatic degradation (“Like pulling on a wet rope to wring out water, tension squeezes out free volume or sterically shields binding sites via coiled-coil assembly to prevent enzyme access.” [194]). Both tension-degradation mechanisms can be represented by Michaelis–Menten kinetics [194] (also see the original article by Michaelis and Menten [283])
$δ(s)=δmaxsnKsn+sn$
(9)

In Eq. (9), the rate of degradation $δ(s)$ of a substrate s (in our case, the collagen fibril substrate) is presented in terms of the maximum degradation rate δmax at saturating concentrations of s, an enzyme affinity Ks for this substrate, and a cooperativity coefficient n. This enzyme affinity Ks is usually defined as a function of tension (or elongational strain). In the modeling [194] that seeks to explore potential mechanisms within the developing myocardium, the collagen is taken to be strain-stabilized to enzymatic degradation, applying the Michaelis–Menten kinetics in the form (9). The collagen messenger ribonucleic acid production rate is linked to the fibroblast crowding, which in turn depends on tension.

Molecular dynamics simulations provide insight into how forces affect molecular conformation, which in turn has a bearing on the ability of degrading enzymes to access key locations in the polymer chain [284,285]. Chang et al. [177] have conducted collagen molecule simulations of real sequences from mus musculus (wild type mouse) α-1 and α-2 chains in order to facilitate comparison with experimental data. They find that type I heterotrimers (two α-1 chains and one α-2 chain) will, upon mechanical loading, fold into a triple-helical structure that is potentially more protective against enzymatic degradation compared to the unloaded conformations. The regions that are naturally unfolded in the heterotrimers will naturally refold under mechanical loading and become less susceptible to enzymatic breakdown. The simulations also find that collagen homotrimers (three α-1 chains) can be more resistant to MMP cleavage.

The work by Teng and Hwang [178] addresses the debate as to whether mechanical loading of collagen fibrils increases or decreases the MMP cleavage rate, and their molecular dynamics work supports the former contention. Their simulations are conducted using the software chemistryatharvardmacromolecularmechanics. Even though some of the simulations found that stretching will result in increased unwinding, which may assist with cleavage by MMP, the findings also emphasize that the susceptibility of collagen to enzymatic cleavage depends on a multitude of factors. In addition to the magnitude of the loading and the general loading conditions (straining, torsion, etc.), the local arrangement of the collagen molecules has a significant role. While an increased collagen trimer unwinding may assist with enzyme cleavage, it is pointed out in Ref. [178] that, in general, unwinding does not necessarily destabilize a collagen trimer (the paper provides examples in which unwinding stabilizes the triple helix). Spontaneous periodic buckling of collagen has also been implicated in the remodeling of collagen. A mechanistic model at the molecular scale for this effect is provided in Ref. [44] with a focus on internal strain as a driving mechanism.

Elegant experiments that put individually anchored collagen trimers into tension, both by magnetic [43,54,55] and by centrifugal loading [117], find that such tension generally enhances enzymatic degradation.

Adhikari et al. [54] attach magnetic beads to collagen molecules, which have been exposed to MMP-1, and the fraction of beads f(t) still attached at time t is described by an exponential decay function $f(t)=a exp(−kdett)+c$. Here, kdet is a detachment rate, and a and c are two constants which are adjusted to fit the experimental data. For a fixed force, the dependence of the proteolysis rate k on the MMP-1 concentration $cMMP1$ is approximated via (also see Eq. (9))
$k=kcat cMMP1KD+cMMP1$
(10)

where KD is an effective dissociation parameter for MMP-1, and kcat is the maximal turnover rate that k approaches for increasing values of $cMMP1$. The parameter ratio $kcat/KD$ is observed to increase exponentially with the applied force.

Camp et al. [43] observe the opposite effect for bacterial collagenase, for which the collagen cleavage rate decreases with increased loading. Adhikari et al. [55] use a single-molecule magnetic tweezers assay to characterize collagen I cleavage by MMP-1 and collagenase from Clostridium histolyticum. The relation between the observed cleavage rate kobs and the applied load F is described by an exponential equation in the form
$kobs=k0 exp(FDkBT)$
(11)

where k0 corresponds to the cleavage rate for vanishing load, D is a parameter that describes the degree to which applied load alters the observed rate constant, and $kBT>0$ is the thermal energy. Note that parameter D will have units of distance. For MMP-1, the cleavage rate kobs is found to be described by Eq. (11) with distance parameter $D≈0.57 nm$.

Based upon tensile tests on single collagen I fibrils by Flynn et al. [46] and on monomers by Camp et al. [43] (see Sec. 3.1), Flynn et al. [47] have proposed the following fitting equation that relates the collagen cleavage rate kc to the force per monomer F:
$kc=0.059 s−1 exp(−0.82FpN)+0.0029 s−1$
(12)
Alves et al. [130] study how cells mediate the balancing of enzymatic digestion and repair of a collagen fiber. In that model, regenerative particles represent collagen monomers, and degrading particles represent enzymes that tend to cleave the collagen fibers. Two models are applied in Ref. [130]. The temporal control model simulates cells that actively measure the stiffness of their surrounding matrix. Based on these measurements, the cells either secrete degrading enzymes or upregulate enzyme inhibitors. The latter case stimulates the collagen synthetization. In the second model, the spatiotemporal control model, the degradation particle behavior is the same as in the temporal control model, but the probability of a regenerative particle to attach to a fiber is taken to be related to the local stiffness.
Kirkness and Forde [117] tether one side of collagen III trimers to glass and attach beads to the other side. Centrifugal forces acting on the beads strain the molecules. The change with time in the number of attached beads is described in Ref. [117] by a first-order decay model
$d[beads]dt=−kTr[beads]−kns[beads]$
(13)

where kTr is a trypsin-dependent bead detachment rate constant, and kns is an enzyme-independent bead detachment rate constant. They conclude that collagen proteolysis by trypsin is enhanced by force.

While these models capture the mechanosensitive aspect of enzymatic degradation on the molecular scale, typical simulations for biomechanics applications (including those that invoke multiscale approaches) are usually applied at length scales that exceed that of the individual polymer chain molecules. Examples of this are models that treat fibers as rod-like structures wherein remodeling is described in terms of ongoing change to the fibril radius, and models that track either the volume fraction or the density of collagen fibers (whose mechanical properties may change with time as remodeling proceeds). Such modeling approaches are discussed in Secs. 6 and 7. Also, because any particular model may incorporate elements of multiple approaches, the above useful classification scheme does not always have strict boundaries.

It is common to model collagen in terms of rod and beam structures when simulating the mechanical response of a collagen network at the fibrillar and microfibrillar scale. At the scale of collagen trimers, treatments in terms of homogeneous flexible rods that support bending, straining, and twist are reviewed in Ref. [286]. Collagen microfibrils have also been modeled in terms of wormlike chains [68,287], an approach that has been traditionally applied in the modeling of DNA helices [288].

This section reviews beam and rod type models for structural collagen in which the fibrillar radii is changing with time. The radius change is governed by biochemical processes under direct enzyme control whose efficacy is influenced by the mechanical loading. The mathematical governing laws are then typically posed in terms of mechanical variables. Such models often seek to provide a link with experiments that measure radius or diameter changes in stretched fibrils as a result of enzymatic dissolution [42,46]. Section 6.1 discusses models in which mechanical effects influence the degradation (subtractive) processes but not the augmentive depositional (additive) processes. Section 6.2 expands the discussion to models that seek to address the effect of load and deformation on both types of processes.

Different models have been developed that describe strain-stabilized fiber degradation in terms of an exponential governing function, i.e., the decline rate of the fiber radius is reduced when the fibers are strained. In a multiscale finite element method (FEM) treatment that relies on microstructural representative volume elements, Hadi et al. [289] model the relation between the axial force in the fiber and the stretch ratio λfib of the fiber. They adapt a relation from Ref. [259] in the form
$Ffib=Efib AfibB(exp(B[λfib2−12])−1)$
(14)
where Efib and Afib are the Young's modulus and the cross-sectional area of an individual collagen fiber, and B is a dimensionless multiplier. The cross-sectional area of the fiber is taken to be dependent on the amount of the fiber strain, and the following kinetic expression for the growth and decay of the fiber radius r is proposed in Ref. [289]:
$drdt=k3−r k1 exp(−k2[λfib−1])$
(15)
In Eq. (15), k1 and k2 are two kinetic parameters that define the sensitivity of the fiber decay to the amount of fiber stretch, and a factor k3 is added that represents a constant fiber growth rate. These parameters are positive so that increasing fiber stretch reduces the fiber decay. The kinetic expression in Eq. (15) can be brought into the form [289]
$r(t)=k3k4[1−e−k4t]+r0e−k4t$
(16)

where $r0=r(0)$ is the initial value for the radius at time t = 0, and we have applied the abbreviation $k4=k1 exp(−k2[λfib−1])$. In Eq. (16), the factor $k3/k4$ represents the response for $t→∞$ if the fiber stretch remains constant. This model (16) has some analogies to the fiber density function (29). Figure 5 gives a brief example on the development of the fiber radius with time. The multiscale model by Hadi et al. [289] then links the volume-averaged stress in the micrometer-scale representative volume elements to a finite element continuum model on a millimeter scale. This model in Eqs. (14) and (15) is then applied in Ref. [289] to describe experimental findings [42] on strain-stabilization of collagen fibers to enzymatic degradation. The work in Ref. [289] has also been applied in Ref. [290] in order to combine fiber remodeling with failure. Special cases of a fiber density model by Demirkoparan et al. [234] show analogies to the model in Eq. (15). The model [234] is discussed in Sec. 7.1.

Fig. 6
Fig. 6
Tonge et al. [291] in their modeling of strain-stabilized collagen degradation take into account that the collagen fibers may be crimped in their undeformed state. The collagen is regarded as dispersed, and this dispersion is modeled by means of the angular integration method (see Sec. 4.5). Building upon the elastica model for collagen fibers presented in Ref. [241], a micromechanical collagen beam of circular cross section with radius r is taken in Ref. [291] to have its radius governed by
$dDdt=−G1 exp(−WaxialG2)≡−kdeg$
(17)
in which $D=r/r0$ is the ratio of the radius r of the degraded collagen fiber to the radius r0 before the degradation has started, and G1 is the intrinsic degradation rate of the unstrained fiber. In the argument of the exponential function, G2 is the characteristic energy density for the mechanical inhibition of the degradation process, and
$Waxial=12E[λfib−1]2$
(18)

is the elongational part of the strain energy density (i.e., there is also a bending part, but it does not contribute to Eq. (17)). Here, λfib is the fibril stretch. The choice of a zero-order kinetic relation in Eq. (17) is motivated both from radius decay experimental observations [46,51,56,292294] and from molecular simulations [177]. On the tissue scale level, the hyperelastic behavior of the collagenous tissue is modeled in terms of a neo-Hookean isotropic ground substance and a strain energy density for the collagen fibers that combines Eq. (18) with an additional bending contribution. The strain-stabilization model (17) has subsequently been used in order to simulate degradation experiments of bovine corneal strips [52] and bovine pericardium strips under uniaxial tension [53]. The response of Eq. (17) to a fibril stretch λfib corresponds to the curve for $−kdeg$ in Fig. 6 using $G1=6.8×10−4 s−1, G2=700 J m−3$, and $E=50 MPa$ (parameter values taken from Ref. [295]).

Fig. 7
Fig. 7

### 6.2 Mechanical Regulation in Both Degradation and Deposition.

The models described in Sec. 6.1, with their focus on degradation, typically took the collagen synthesis to be independent of any mechanical loading. This section reviews mechanical models in which both the deposition of collagen and the degradation are mediated by mechanical stimuli.

The kinetic expression for a constant collagen fiber growth and a strain dependent collagen decay (15) has been modified by Gyoneva et al. [296] so that both the fiber decay and growths may become deformation dependent, i.e., change of fibrillar collagen radius r depends on growth and degradation with dependence on the strain of the fiber
$(drdt)=(drdt)rem+(drdt)add$
(19)
The first term in the right side of Eq. (19) is the fiber removal term, which describes the stretch-dependent fiber radius decay with time. This radius decay is taken in the form [296]
$(drdt)rem=−rr+rsmall k1 exp(−k2[λfib−1])$
(20)

in which $r/(r+rsmall)$ prevents negative values for the radius r (as might otherwise occur in the context of the model (15)).

The second term on the right side of Eq. (19) is the fiber addition term, which is taken in the form [296]
$(drdt)add=k3 exp(k4[λcell−1])︸f(λcell) exp(κ cos(2[θ−θ0]))I0(κ)︸g(θ)$
(21)

In Eq. (21), the factor $f(λcell)$ describes increase of collagen fiber growths with a cell stretch ratio λcell, which represents the stretch in a representative volume modeling cell, and k4 is a positive sensitivity parameter, i.e., the fiber growth increases with increasing values for λcell. The function $g(θ)$ in Eq. (21) is a von Mises distribution function, which quantifies the dispersion of the fibers. In this distribution function, κ is a shape or concentration parameter, θ is the orientation of a synthesized fiber, θ0 is the average orientation of the present fibers, and $I0(κ)$ is the zero-order modified Bessel function. Different modeling techniques that account for the dispersion of the fibers were briefly discussed in Sec. 4.5. Depending on the specifications for k4 and κ, the behavior of Eq. (21) can be strain dependent or independent and lead to an isotropic or anisotropic deposition of the fibers. For the specification $k4=κ=0$, the fiber radius growth term (21) reduces to the constant value k3, which corresponds to the modeling by Hadi et al. [289] in Eq. (15).

Argento et al. [297] model the impact of the scaffold architecture and fiber stretching on the collagen remodeling in the form
$dr(γ,t)dt=S−D$
(22)
which describes the change of the radius r of a fiber in the direction γ. The first term $S$ on the right side of Eq. (22) is the fiber growth rate, which is taken to depend upon all of the following: the mean synthetic activity of the cells, the scaffold fiber volume fraction, the collagen fiber volume fraction, and the collagen fiber geometry. The last term in Eq. (22) is the fiber degradation rate along the fiber direction γ using the strain-based Michaelis–Menten law (also see Eq. (9))
$D=k3−k4λfib(γ)−1k5+λfib(γ)−1$
(23)

for positive values of the collagen fiber radius $r(γ,t)$, whereas $D=0$ otherwise. The fiber degradation rate (23) reduces with the amount of fiber stretch $λfib(γ)$, and k3, k4, and k5 are three positive fitting parameters which provide experimental correlations. The finite element study of Ref. [297] then links the microscale modeling to the predicted mechanical behavior at the macroscale on the basis of different averaging theorems. Figure 7 depicts $dr/dt$ in Eq. (22) for $S=0$ (negligible fiber growth rate) and $D$ in Eq. (23) exhibiting its dependence on fiber stretch in the range $1≤λfib≤2$. The degradation rate is $dr/dt=−3.78×10−6 μm/s$ for $λfib=1$, whereas it approaches the limit $dr/dt=−0.69×10−6 μm/s$ as $λfib→∞$.

Fig. 8
Fig. 8
Jia and Nguyen [295] extend the model (17) in order to consider stretch-mediated collagen deposition
$dDdt=kdep−kdeg$
(24)
where $−kdeg$ is the previously indicated right side of Eq. (17), $D=r/r0$ is again the ratio of the radius r of the degraded collagen fiber to the initial radius r0, and the new deposition term is given by
$kdep=G3 exp(WaxialG4)$
(25)

The constant G3 is the collagen deposition rate of the unstrained fiber, and G4 is the characteristic energy density for mechanical inhibition of the deposition process (analogous to G1 and G2 in Eq. (17)). As in Ref. [291], the angular integration method is applied to account for the fiber dispersion, and the unloaded fibers are considered to be crimped. A key feature of the model in Ref. [295] is that newly deposited fiber is initially unstressed, so that fibers at different radii will generally have a different stress-free state. The resulting radially dependent stretch (see Eq. (11) of Ref. [295]) is thus described by a mathematical form similar to Eq. (3) in Sec. 4.2 of this review. This model is then applied to simulate the remodeling of collagen fibers under both static and cyclic loading. The intermediate curve in Fig. 6 shows how stretch-mediated collagen deposition $dD/dt$ in Eq. (24) is affected by the fibril stretch λfib. For comparison's sake, the uppermost curve shows the response with only the deposition term kdep on the right side of Eq. (24), and the lowermost curve shows the response when only $−kdeg$ is present on the right side of Eq. (24), thus recovering the simpler model Eq. (17). If $λfib=1$, meaning that the fibers are unstretched, then $dD/dt=G3−G1=−3.8×10−4 s−1<0$, whereupon the radius recedes. For $λfib=1.0037$, the deposition and degradation rates are in balance, and the fibril radius remains constant. For $λfib>1.0037$, the fibril radius increases exponentially.

## 7 Mechanosensitive Change in Collagen Volume and Density

The modeling in Secs. 5 and 6 treated the strain-mediated degradation of collagen by enzymes on the molecular scale and on the fibrillar scale. The associated kinematics was therefore cast in terms of changes in the molecular arrangement and in fibrillar geometry, respectively. This section reviews fiber and tissue scale modeling specifications that account for a change in the fibrillar collagen volume or density that are similarly mediated by mechanical stimuli. Density variables, here denoted by ρ, naturally enter these treatments. Certain aspects of these treatments can be harmonized with the radius-change models of Sec. 6 by putting the radius type variable of that section6 in correspondence with the density type variable which is described next.

To model experimental findings on collagen fibril strain-stabilization to enzymatic degradation as reported, for example, in Refs. [11] and [42], one may begin in terms of the local fiber density per unit volume in the reference configuration
$ρ(t)=∫t−Tmaxtζ(t,τ) dτ$
(26)

The fiber survival kernel ζ provides the contribution to the fiber density at time t due to basis fiber entities (“protofibers”) created at time τ that still survive at time t. Fibers have a range of possible lifespans with Tmax being the theoretical maximum. The kernel ζ accounts for the effect of the deformation history on each fiber's lifespan via an integral from the earliest relevant time instant $t−Tmax$ to the current time t in a manner that is reminiscent of the memory kernels used in viscoelasticity.

Using Eq. (26), the model in Refs. [234] and [247] posits a constant protofiber synthesis rate $χ>0$ (per unit volume in the reference configuration), and a protofiber dissolution rate that is both stretch dependent and proportional to the current fiber density. Letting $η=η(t,τ)≥0$ be the stretch-dependent dissolution rate, it follows that
$ζ(t,τ)=χ exp(−∫τtη(s,τ) ds)$
(27)

This equation permits the survival kernel function ζ to be expressed in terms of the more physically immediate dissolution rate function η. In the case of a constant value for η, an immediate integration puts Eq. (27) in correspondence with the survival function in Eq. (7).

As suggested by the previously indicated [11,42], the time dependence $η=η(t,τ)$ may arise from the effect of strain, suggesting that $η(t,τ)=η̂(λfib(t,τ))$. The constitutive function $η̂(·)$ directly relates dissolution rate to the current amount of each fiber's stretch $λfib(t,τ)$, where the stretch may depend on both the creation and current times τ and t in a remodeling tissue. A form for $η̂(λfib)$ that captures the stretch stabilization is
$η̂(λfib)={k1e−k2[λfib−1], if λfib≥1k1, if 0<λfib≤1$
(28)

The constant $k1>0$ defines the enzymatic protofiber degradation when the fiber is either unstretched or contracted (and hence likely to become crimped). The constant $k2>0$ provides an exponential decrease in dissolution rate with elongational fiber stretch λfib. In a continuum mechanics framework, the fiber stretch is treated on the basis of Sec. 4. In particular, λfib is determined from Eq. (3), including its specialization (4) for the case of fibers that are deposited with prestretch (see Sec. 4.3).

Models based on Eqs. (26)(28) yield stretch stabilization in the form of an enhanced fiber density ρ as λfib is increased. They are also able to replicate the type of degradation behavior provided by the fibril radius decay model of Ref. [289] that was reviewed in Sec. 6.1. That connection is most easily made by specializing the model (26)(28) to the case where both: (i) there is no theoretical upper bound to a fibril lifespan, and (ii) all fiber constituents have the same stress-free state which is also the stress-free state of the matrix. This amounts to taking $Tmax→∞$ in Eq. (26) and $A=I$ in Eq. (3). Then, the analogy between the model by Hadi et al. [289] as encapsulated in Eq. (15), and the models in Refs. [131,234], and [247] as governed by Eqs. (26)(28), can be obtained by considering two fiber stretch values: $λfib−$ for t < 0 and $λfib+$ for t > 0. Using Eq. (28), the integral in Eq. (26) can be evaluated, and the resulting fiber density function can be brought into the form
$ρ(t)=χη̂(λfib+)[1−e−η̂(λfib+)t]+χη̂(λfib−)e−η̂(λfib+)t$
(29)

Equation (29) describes the change in fiber density as the system transitions from the homeostatic fiber remodeling state associated with $λfib−$ to the homeostatic fiber remodeling state that is associated with $λfib+$. Note the correspondence with Eq. (15) after it has been brought into the form (16); in Eq. (29), the factor $χ/η̂(λfib−)$ is the initial density at t = 0, and $χ/η̂(λfib+)$ is the asymptotic density that is approached as t increases. This asymptotic approach is shown in panel (a) of Fig. 8 (blue curves) taking $k1=0.001086 s−1$ and $k2=1.289$ in Eq. (28) and taking $λfib−=1$ in Eq. (29). The two blue curves are for different values of $λfib+>1$. The chosen values for k1 and k2 provide a direct correspondence with results presented in Ref. [289].

Fig. 9
Fig. 9

The time and deformation history behavior of the model based on Eqs. (26)(28) is examined in Refs. [131] and [132] for uniaxial deformations, and then later for both biaxial deformation and inflation in Ref. [298]. In these studies, the fibers are taken to be dispersed in all directions by generalizing the fiber density function in Eq. (26) using the angular integration method. These works focus on two alternative ways in which the fiber natural configuration $κf(τ)$ may relate to the reference configuration as determined by the deformation gradient type mapping tensor $A(τ)$ (see again Fig. 3). For the first of these two alternatives, all constituents share a common natural configuration. This makes $A(τ)=I$ and corresponds to the just-discussed blue curves in panel (a) of Fig. 8. In the second alternative, the fibers are considered to be deposited in a stress-free condition. This makes $A(τ)=F(τ)$ and corresponds to taking $λ*=1$ in Eq. (4). The remaining two (red) curves in panel (a) of Fig. 8 show how the fiber density is affected under this second scenario. Unlike the more involved boundary value problems considered in Ref. [298], all of the curves in Fig. 8 correspond to a simple homogeneous stretching in the fiber direction with a jump from $λfib−=1$ to $λfib+>1$ at t = 0. Now the eventually attained new homeostatic state is the same as the original one, although the disruption provided by the abrupt change in stretch forces the system out of homeostasis for a significant period of time. Note, despite the equal mechanical properties of the fibers for all curves shown in Fig. 8(a), that the response of curves corresponding to Eqs. (3) and (4) is very different. This deviation in the remodeling behavior is solely due to the alternative specifications for the fiber natural configuration.

All of these processes are impacted if new fiber deposition involves prestretch ($λ*>1$). Panel (b) of Fig. 9 shows this effect, again on the basis of Eqs. (26)(28) with λfib given by Eq. (4). The remaining (green) curves of panel (b) show how prestretch $λ*>1$ further changes the fiber density when homeostasis is disrupted, and how this affects the time interval before homeostasis is restored.

Topol et al. [235] study the inflation behavior for a thick-walled hollow cylinder containing two embedded families of parallel fibers in a helical and symmetric arrangement. The fibers are considered to be strain-stabilized to enzymatic degradation and to undergo a continuous remodeling process. A follow-up [247] considers fibers of different lifespans (see Sec. 2.2) using different values for Tmax in the fiber density model (26), and its impact on homeostatic fiber remodeling versus dyshomeostatic fiber remodeling.

In certain cases, an ECM remodeling process with a decreased collagen synthetization may be desired for a specific medical application, for example, when a dense collagen network hinders or prevents therapeutic measures. In their thermodynamic modeling of tumor extracellurlar matrix subject to enzymatic degradation, Xue et al. [299] consider tumor ECM, consisting of proteoglycans and collagen network, that is subject to the poroelastic and polyelectroytic effect of interpenetrating fluid. This allows chemical and particle diffusion to the tumor site, and one focus of Ref. [299] is on the transport efficacy of nanomedicine sized carriers such as liposomes and viruses. In tumor cells with a high collagen density, the degradation of the collagen network may enhance the diffusion of both viruses and antibodies. Xue et al. obtain a mechanosensitive enzymatic degradation law in the form
$ddmdt=km0e exp(−γmDm)(1−dm)$
(30)

where dm is the degradation factor pertaining to tissue constituent m, $km0e$ is the enzymatic degradation rate without mechanical modulation, γm is a mechanosensitivity factor, and Dm is the driving force of degradation. The model (30) accounts for the enzymatic degradation of the ECM volume and also for strain-stabilization of collagen to such enzymatic degradation as reported in Refs. [43,46], and [50].

A multiscale computational model that also treats the strain-stabilization of collagen to enzymatic degradation is presented in Ref. [238]. Their model applies a fiber cleavage probability function in the exponential form
$Pfailproteo=exp(−ϕεmax)$
(31)

where the probability function provides results in the range $0≤Pfailproteo≤1$. The fitting constant $ϕ$ accounts for the decreasing probability of fiber cleavage with fiber peak strain $εmax$. This model (31) is motivated by the experiments of Flynn et al. [45] on the strain-stabilization of collagen fibrils against enzymatic degradation, for which Young et al. [238] determined $ϕ=300$.

Gaul et al. [300] remark that many modeling works on strain-mediated degradation in soft tissue focus on the collagenous part, while the complexity and heterogeneity of the material are often overlooked. The authors present a modeling framework for strain regulated enzymatic degradation of arterial tissue. In that work, the degradation behavior of the fiber–ground-substance unit (“fiber–matrix unit”) is modeled through the decrease of the fiber volume fraction $ϕf$ and matrix volume fractions $ϕm$ over time in the form
$dϕmdt=−Ψmϕm, dϕfdt=−Ψfϕf$
(32)
In Eq. (32), the ground substance degradation rate $Ψm=kmεf$ is considered to be a linear function of the strain $εf=(1/2)[λfib2−1]$, where km is a constant, and the fiber degradation rate $Ψf$ is a stepwise continuous function in the form [300]
$Ψf=Dc+{Dlεf,for εf<εt1DT1D(εf)2+T2Dεf+T3D,for εt1D≤εf≤εt2DDh(εf−εt2D)+T1D(εt2D)2+T2Dεt2D+T3D,for εf>εt2D$
(33)
The fiber stretch $λfib=λt/λp$, where λt is the aforementioned stretch ratio in the fiber–ground-substance unit and λp is a fiber prestretch (which plays a role akin to that of $λ*$ as used in Sec. 4.3). In Eq. (33), the parameters Dc, Dl, and Dh are the baseline constant degradation rate and the degradation moduli of the low and high strain regions, respectively. The three parameters
$T1D=Dl−Dh2[εt1D−εt2D]$
(34a)
$T2D=Dl−εt1DDl−Dhεt1D−εt2D$
(34b)
$T3D=Dlεt1D−T1D[εt1D]2−T2Dεt1D$
(34c)

ensure the continuous form of the fiber degradation rate function in Eq. (33). Motivated by the experiments reported in Refs. [56] and [57], the fiber degradation rate $Ψf$ decreases for $εf<εt1D$ and increases again for $εf>εt2D$. The interval $εt1D≤εf≤εt2D$ is the transition zone between the two degradation characteristics for small and large strains (Fig. 3). This model has been applied by Gaul et al. [301] in order to study a potential mechanism for degenerative arterial disease progression.

Remark. Here, it is interesting to point out that the type of models described in this section, with their mechanoregulation dependent on strain, can lead to qualitatively different behavior under strain control versus under stress control (isometric versus isotonic). To give an example, take the model that is given in Eqs. (26)(28) with its upper bound Tmax on a fiber lifespan and consider the response to the alternative loadings described next.

If a step deformation is applied to the material at time t0 and the material is held in this constant deformation state for time $t≥t0$, then at time $t=t0+Tmax$ all existing fibers have now been synthesized after the application of the deformation, and the fiber remodeling becomes homeostatic. For relatively simple deformations and idealized fiber arrangements, such a case may be analyzed analytically [302]. An example is shown in Fig. 10.

Fig. 10
Fig. 10

If instead a step force is applied at time t0 and the material remains subjected to this loading for time $t≥t0$, then the stretch must continually readjust as the remodeling proceeds in order for the load to remain fixed. Consequently, because the fiber degradation constitutive rule is explicitly stretch dependent, the remodeling process will likely not have achieved homeostasis at time $t=t0+Tmax$. Even such relatively simple force loading scenarios may therefore require a numerical treatment.

### 7.2 Mechanical Regulation in Deposition.

In some modeling works, the enzymatic collagen degradation is taken to be independent of strain, whereas the collagen deposition is mechanically regulated.

Boerboom et al. [303] present a numerical study of mechanically induced collagen fiber synthesis and degradation in the aortic valve, in which the change in the fiber volume fraction is presented in the form
$dθkdt=α(θtot)S(λk)−β(θk)D(λk)$
(35)

where the total fiber volume fraction θtot is the combined volume fraction of the individual fibers, $θtot=∑k=1Nθk$. The first term on the right side of Eq. (35) is the fiber creation term, in which the factor $α(θtot)$ is obtained from Michaelis–Menten kinetics (also see Eq. (9)) in order to keep the total fiber fraction below a threshold value. Here, $S(λk)=A(λk2−1)>0$ provides an upregulation of the fiber synthesis that depends on the amount of fiber stretch $λk≥1$. Conversely, for compressed fibers, there is no fiber synthesis whereupon $S(λk)=0$. The second term on the right side of Eq. (35) describes the fiber degradation. The factor $β(θk)$ depends upon the volume fraction of the individual fibers, and it prevents the degradation from decreasing below a certain threshold. The factor $D(λk)$ is taken to be a stretch-independent constant in Ref. [303], but it could more generally be taken as a function of the fiber stretch.

For tracking the evolution of the collagen density ρ, one may introduce a collagen degradation rate $ρ˙−$ and a collagen production rate $ρ˙+$ along with associated governing rules. For example, in the simulation of abdominal aortic aneurysms, Martufi and Gasser [304] take an equation equivalent to
$dρdt=−adeg ρ︸=ρ˙−+min[adeg ρ bmech(N),ρmax+]︸=ρ˙+$
(36)

The collagen degradation term $ρ˙−$ in Eq. (36) is a first-order kinetic expression in terms of the degradation time scale adeg. The collagen production term $ρ˙+$ in Eq. (36) is formulated in terms of a mechanical stimulus $bmech(N)$ that is associated with adeg, ρ, and with the orientation direction of the unit vector $N$. This collagen production term $ρ˙+$ is bounded above by a hypothetical maximum $ρ˙max+$. In Ref. [304], fibers are assumed to be synthesized with a specified amount of prestretch. This model is found to agree with experimental findings [45] on the strain-stabilization of collagen to enzymatic degradation. The collagen production term $ρ˙+$ registers the mechanical stimulus, whereas the $ρ˙−$ collagen degradation term provides an indirect dependence on strain through the collagen density ρ.

### 7.3 Mechanical Regulation in Both Degradation and Deposition.

This section turns to review collagen remodeling models in which both the collagen degradation rate and the collagen deposition rate depend on the straining of the collagen fibers. These works are often motivated by experiments which find that the collagen degradation decreases for relatively low amounts of strain whereas for relatively large strain the collagen degradation rate may increase (see Sec. 3.3).

Guided by experimental studies such as Refs. [42,50,51], and [56] on the strain-mediated enzymatic degradation of collagen fibers, Loerakker et al. [305] model the tissue as cells, collagenous fibers, and components of an isotropic ground substance. Their model seeks to describe strain-mediated change in the collagen volume fraction $φcfi$ in a specific fiber direction (hence the superscript index i). Production and degradation of this directional collagen in Ref. [305] are governed by
$dφcfidt=dφcf,prodidt−dφcf,degidt$
(37)
The first term on the right side of Eq. (37) describes the change in collagen volume fraction due to production of collagen. Alternative forms for this term are considered so as to emphasize different small scale effects. The second term on the right side of Eq. (37) describes the change in collagen volume fraction due to degradation of collagen, and two different models for the collagen degradation are proposed. The first degradation law in the form of a sigmoid function is motivated by experiments such as Wyatt et al. [51] which posit that the collagen degradation rate decreases with increasing strain
$dφcf,degidt=[Dmin+Dmax−Dmin1+10200[εe−εtrans]]φcfiτφ$
(38)
In Eq. (38), the parameters Dmin and Dmax serve to determine upper and lower thresholds for the amount of degradation, $εe$ is the fiber strain, $εtrans$ is the transition strain at which the change in the collagen degradation rate takes its local minimum, and $τφ$ is a characteristic time constant. In Eq. (38), the parameters Dmin and Dmax serve to determine upper and lower thresholds for the amount of degradation, $εe$ is the fiber strain, $εtrans$ is the transition strain at which the collagen degradation takes its minimum value (which corresponds to the maximum possible fiber stress), and $τφ$ is a time constant. The second degradation law is motivated by experimental observations (e.g., Ref. [56]) which report that the collagen fiber degradation rate decreases for smaller strains, but the degradation rate may increase when the strain exceeds a certain limit (denoted as $εe,p,max$). This second degradation law is a piecewise defined sigmoid function that is taken in the form
$dφcf,degidt={[Dmin+Dmax−Dmin1+10200[εe−εtrans]]φcfiτφfor εe≤εe,p,max[Dmin+Dmax−Dmin1+10200[2εe,p,max−εe−εtrans]]φcfiτφfor εe>εe,p,max$
(39)

The contribution of the fibers to the total stress on the macroscale then depends on the current collagen fiber orientation in terms of the individually considered fiber directions. The response of degradation rate model (39) to elastic strain is illustrated in Fig. 11 taking different values for the transition strain $εtrans$ which now serves to scale the size of the transition zone.

Fig. 11
Fig. 11
The model of Heck et al. [306] makes use of a collagen degradation law that is similar to Eq. (38), along with a collagen production rate that is taken to be the average of the minimum and the maximum in the degradation rate
$dφcf,prodidt=12[dφcf,deg,minidt+dφcf,deg,maxidt]$
(40)

This treatment presented in Ref. [306] is readily modified so as to incorporate a notion of fibril damage in the case of excessive strain.

Glaucoma is a group of eye disorders that may result in vision loss if it remains untreated. In order to describe mechanosensitive collagen turnover, the article by Grytz et al. [307] on lamina cribrosa thickening in early glaucoma introduce a growth Jacobian$Jgcol$ that describes the change in the relative volume that is occupied by the fibers. Making use of a stimulus function $ϕ=λfib−λhom$, which is the difference between the fiber stretch λfib and the stretch λhom that is maintained for homeostatic balance, the model proposed in Ref. [307] is governed by
$dJgcoldt=1τg[Jgcol,max−JgcolJgcol,max−1]2[Jgcol−Jgcol,min1−Jgcol,min]2︸kcol(Jgcol)ϕ$
(41)

Note that the change in the growth Jacobian $dJgcol/dt$ is taken to be the product of the stimulus function $ϕ$ and a weight function $kcol(Jgcol)$. The parameters $Jgcol,min$ and $Jgcol,max$ that appear in the weight function provide lower and upper bounds to the volume change, and τg is a characteristic time associated with the collagen turnover rate. The fibers are modeled as helically crimped, and the strain energy density of a fiber is dependent upon the overall fiber stretch previously obtained in Ref. [244]. The collagen contribution to the total strain energy density function of the overall tissue is then formulated in terms of the growth Jacobian of the fibers, the Jacobian of the surrounding bulk tissue, the volume fraction of the each constituent, and the strain energy density of the collagen.

While the primary focus of our review is on fibrillar collagen, similar modeling ideas have been applied to network forming collagen. Barocas et al. [308] discuss how strain-mediated enzymatic degradation of collagen is mainly demonstrated in fibrillar collagen. They then provide a network forming type IV collagen turnover model that is specifically applicable to the kidney glomerular basement membrane. In this model, the ongoing change of the glomerular basement membrane thickness
$γ=γd−γr$
(42)
involves both a tissue deposition rate γd and a tissue removal rate γr in the form
$γd=γd,maxεεd,501+εεd,50$
(43a)
$γr=γr,max11+εεr,50$
(43b)
where ε is the applied strain, $γd,max$ is the maximum deposition rate possible in the theoretical limit of an infinite strain, $εd,50$ is the strain at which the deposition rate is 50% of its maximum possible deposition rate, $γr,max$ is the maximum removal rate in the absence of any strain, and $εr,50$ is the strain at which the removal rate is 50% of its maximum possible removal rate. It is interesting to observe that the net deposition rate γ in Eq. (42) obeys
$limϵ→∞γ=γd,max, γ|ε=0=−γr,max$
(44)

Thus, as illustrated in Fig. 12, γ is negative in the absence of any strain and positive for large strains.

Fig. 12
Fig. 12

### 7.4 Swelling-Induced Retardation of Collagen Dissolution.

Modeling and simulation based on the different natural configurations of the ECM and the ground substance can be refined so as to address a variety of additional important process issues. Of particular significance here is the possibility of natural volume change in the ground substance, especially as a result of tissue swelling (see Sec. 4.1). As discussed in Sec. 2.2.1, a variety of different agents can provoke a swelling response. To this end, Topol et al. [309] model the mechanosensitive dissolution of fibrillar collagen as a response to volume changes in the ground substance. Figure 13 illustrates the predicted change in the fiber density ρ as given by Eq. (26) for an initially unswollen homeostatic state when the ground substance abruptly swells to twice its original volume. This puts the fibers in tension and the matrix in compression, and the analysis requires the determination of their separate stress contributions. This in turn requires specification of both Wm and Wf in Eq. (1)

Fig. 13
Fig. 13

Figure 13, which shows the predicted effect of deposition prestretch $λ*$, is obtained using the simulation algorithm presented in Ref. [309]. This makes use of a neo-Hookean Wm, with shear modulus μ, following the swelling treatment of Sec. 4.1, and strain energy contribution of the fibers Wf that is given by Eq. (8). The algorithm in Ref. [309] takes the standard model form $wf(λfib)=(1/2)γ(λfib2−1)2$ with fiber modulus γ. While Eq. (8) as employed in the examples in Fig. 13 considers a single parallel fiber arrangement, the treatment is readily generalized so as to address a network of dispersed remodeling fibers, for example, by means of the angular integration method (see Sec. 4.5).

This case of a swelling-induced fiber remodeling has several analogies to the strain-mediated treatment in Sec. 7.1, because the change in the ground substance volume stretches the embedded fibers. Nevertheless, while the strain induced case specifies all aspects of the deformation on the material, the swelling mediated case only stipulates the volume change of the material. Namely, the swelling case does not define the resulting specific shape of the material nor how this shape changes with the fiber remodeling process. This effect is demonstrated in Fig. 14, which provides additional detail on the step function free swelling treated in Fig. 13. The swelling is the sole mechanical driving agent which interrupts an original homeostatic state (there is no external loading). The single family of parallel fibers resists elongation in the fiber direction as determined by the ratio $μ/γ$. The graphs show six different cases (in terms of $μ/γ$ and $λ*$) of the time-dependent material contraction λc in the fiber direction, i.e., λc quantifies the ratio of material contraction in the fiber direction relative to the swollen matrix in the absence of fibers. After an initial relative contraction λc at time t = 0 due to the abrupt volume change, the material elongates in the direction of the parallel fibers until the fiber remodeling approaches homeostasis. This approach is slower when the fibers are synthesized with larger prestretch values $λ*$ or when lower values are employed for the matrix-to-fiber stiffness $μ/γ$. In other words, the approach is slower when the fibers exhibit a dominating mechanical behavior in comparison to the behavior of the ground substance.

Fig. 14
Fig. 14

The finite element simulations by Gou et al. [253] examine the effect of swelling on fiber remodeling in a study of maturing cervical soft tissue at different stages of pregnancy. The fiber remodeling process is mediated by combinations of swelling in the cervical ground substance in conjunction with an increasing intrauterine pressure. Particular emphasis in Ref. [253] is placed on homeostasis between a steady process of fiber creation and a strain-stabilized process of fiber breakdown.

## 8 Discussion

Table 6 summarizes the different mechanical models reviewed in this article, all of which incorporate a notion of strain regulated remodeling of fibrillar collagen. The remodeling processes are not isolated events, rather they are the result of myriad interrelated biochemical and physical processes. The different models each account for a limited number of phenomena. This is often sufficient for modeling collagen turnover within a certain time frame, or when the focus is on circumscribed features, such as a specific loading or a particular fiber arrangement.

Table 6

Mechanical modeling treatments for the strain and load mediated remodeling of fibrilar collagen in the presence of enzymes

Modeling workStrain impact on collagen degradationCollagen depositionComments on fiber dispersionPrestretched or crimped fibersGeneral comments on the modeling approach
Molecular scale process simulation as it relates to larger scale effects (Sec. 5)
Dingal and Discher [194]Strain dependentCollagen turnover model based on Michaelis–Menten kinetics [283]
Chang et al. [177]Study of mechanical force on collagen cleavage
Teng and Hwang [178]↑↓Cleavage depends on loading conditions and local molecular arrangement
Dittmore et al. [44]Internal strain driven collagen remodeling
Flynn et al. [46,47]Fitting based on fibril [46] and monomer tension experiments [43]
Alves et al. [130]Temporal control model and spatiotemporal control model are applied
Kirkness et al. [117]Modeling of centrifugal force loading of collagen III molecules
Hadi et al. [289]ConstantFEM continuum model: volume-averaged stress
Tonge et al. [291]ConstantAngular integrationaCrimpedTissue-level modeling: strain energy function
Mechanical regulation in both degradation and deposition (Sec. 6.2)
Gyoneva et al. [296]↑↓Strain dependentvon Mises distributionPrestressedFEM continuum model: volume-averaged stress
Argento et al. [297]Strain dependentbNormal distribution functionPrestressedFEM model: representative area element
Jia and Nguyen [295]Strain dependentAngular integrationaCrimpedTissue-level modeling: strain energy function
Mechanosensitive change in collagen volume and density (Sec. 7)
Demirkoparan et al. [131,234]ConstantAngular integrationPrestretched or crimpedFibers' strain energy density depends on fiber volume
Topol et al. [247,298]ConstantParallel [247]/angular integration [298]Prestretched, slackSimulation of inflation problems applicable to collagenous tissue
Xue et al. [299]Thermodynamic tumor ECM model
Young et al. [238]Cleavage probability degradation model
Gaul et al. [300,301]↑↓PrestretchedModeling of mechanosensitive collagen degradation in arterial tissue
Mechanical regulation in deposition (Sec. 7.2)
Boerboom et al. [303]Strain independentFEM aortic valve synthesis and degradation work
Martufi and Gasser [304]Strain independentcStrain dependentPrestretchedTurnover model with application to abdominal aortic aneurysms
Mechanical regulation in both degradation and deposition (Sec. 7.3)
Loerakker et al. [305,310]↓ [305,310], ↑↓ [305]Strain dependentdMacroscale stress depends on the current fiber orientations, stresses, and volume fractions
Heck et al. [306]Strain dependentdDegradation law similar to Ref. [305], fiber failure is considered
Grytz et al. [307]Strain dependentGeneralized structure tensorPrestretchedFEM growth and remodeling work on lamina cribrosa
Barocas et al. [308]Collagen IV turnover in the kidney glomerular basement membrane
Swelling-induced retardation of collagen dissolution (Sec. 7.4)
Gou et al. [253,309]ConstantPrestretchContinuum mechanics [309] and FEM study [253] of swelling soft tissue
Modeling workStrain impact on collagen degradationCollagen depositionComments on fiber dispersionPrestretched or crimped fibersGeneral comments on the modeling approach
Molecular scale process simulation as it relates to larger scale effects (Sec. 5)
Dingal and Discher [194]Strain dependentCollagen turnover model based on Michaelis–Menten kinetics [283]
Chang et al. [177]Study of mechanical force on collagen cleavage
Teng and Hwang [178]↑↓Cleavage depends on loading conditions and local molecular arrangement
Dittmore et al. [44]Internal strain driven collagen remodeling
Flynn et al. [46,47]Fitting based on fibril [46] and monomer tension experiments [43]
Alves et al. [130]Temporal control model and spatiotemporal control model are applied
Kirkness et al. [117]Modeling of centrifugal force loading of collagen III molecules
Hadi et al. [289]ConstantFEM continuum model: volume-averaged stress
Tonge et al. [291]ConstantAngular integrationaCrimpedTissue-level modeling: strain energy function
Mechanical regulation in both degradation and deposition (Sec. 6.2)
Gyoneva et al. [296]↑↓Strain dependentvon Mises distributionPrestressedFEM continuum model: volume-averaged stress
Argento et al. [297]Strain dependentbNormal distribution functionPrestressedFEM model: representative area element
Jia and Nguyen [295]Strain dependentAngular integrationaCrimpedTissue-level modeling: strain energy function
Mechanosensitive change in collagen volume and density (Sec. 7)
Demirkoparan et al. [131,234]ConstantAngular integrationPrestretched or crimpedFibers' strain energy density depends on fiber volume
Topol et al. [247,298]ConstantParallel [247]/angular integration [298]Prestretched, slackSimulation of inflation problems applicable to collagenous tissue
Xue et al. [299]Thermodynamic tumor ECM model
Young et al. [238]Cleavage probability degradation model
Gaul et al. [300,301]↑↓PrestretchedModeling of mechanosensitive collagen degradation in arterial tissue
Mechanical regulation in deposition (Sec. 7.2)
Boerboom et al. [303]Strain independentFEM aortic valve synthesis and degradation work
Martufi and Gasser [304]Strain independentcStrain dependentPrestretchedTurnover model with application to abdominal aortic aneurysms
Mechanical regulation in both degradation and deposition (Sec. 7.3)
Loerakker et al. [305,310]↓ [305,310], ↑↓ [305]Strain dependentdMacroscale stress depends on the current fiber orientations, stresses, and volume fractions
Heck et al. [306]Strain dependentdDegradation law similar to Ref. [305], fiber failure is considered
Grytz et al. [307]Strain dependentGeneralized structure tensorPrestretchedFEM growth and remodeling work on lamina cribrosa
Barocas et al. [308]Collagen IV turnover in the kidney glomerular basement membrane
Swelling-induced retardation of collagen dissolution (Sec. 7.4)
Gou et al. [253,309]ConstantPrestretchContinuum mechanics [309] and FEM study [253] of swelling soft tissue

↓ indicates that the collagen degradation (or deposition) decreases with increasing stretch, and ↑ indicates the opposite effect. ↑↓ indicates that the degradation behavior may depend on the amount of strain.

a

Fiber orientation distribution in the strain energy density function (continuum scale modeling).

b

Collagen fiber growth along a certain direction depends on the fiber radius growth rate.

c

Degradation has an indirect dependence on strain through the collagen density.

d

Depends on the degradation rates in all fibers.

As Fig. 1 implies, the hierarchical composition of collagen fibers motivates studies and modeling treatments at various lengths scales. Simulations at the molecular length scale seek to provide a fundamental understanding of how the collagen structure changes under mechanical loading, as well as the attendant consequences that these changes have for enzymatic degradation. On a larger length scale, several models describe how the fibril and fiber radii change as a result of loading and the related enzymatic degradation process. On the continuum length scale at which the material is often treated in a homogenized sense, certain models also account for a change in the collagen density with the deformation history. Different approaches have been applied in order to link the modeling on the different lengths scales. Some multiscale models then link the volume-averaged stress in the microscale representative volume elements to a continuum model on a much larger length scale [289,290,296]. Other approaches determine the strain energy density of the material on the continuum scale by integrating the contributions of the individual fibers over the different fiber orientations [131,234,291,295].

In many of the reviewed models, the mechanosensitivity of the collagen turnover dynamics is taken to be in either the creation or the degradation. Other models regard both processes as mechanosensitive, with mathematical dependence in terms of the relevant mechanical strain type variable that may lead to similar expressions for each effect (e.g., exponential forms for kdeg and kdep in Eqs. (17), (24), and (25)). This raises the question as to the extent in which, at least for computational modeling purposes, the mechanosensitivity can be relegated to only one of these domains, and whether treatments that pursue such one-process sensitivity are implicitly keying off such a consideration. In other words, from a computational perspective, under what circumstances is it a distinction without a difference? This also highlights the need for experimental refinements.

Experiments have found that a relatively small amount of fibrillar strain may stabilize collagen against enzymatic degradation. Larger amounts of strain are found to reverse this effect, causing collagen degradation to increase with any further increase in the amount of strain (as depicted in Figs. 9 and 11). However, excessive straining can lead to specific processes of damage initiation that require their own specialized modeling considerations. Peña et al. [38] consider both continuous and discontinuous damage in order to describe the softening phenomenon of fibrous soft biological tissues. Volokh [311,312] accounts for the failure of both the ground substance and the fibers at larger deformations. Holzapfel and Ogden [313] propose a progressive damage model for collagen fibers that is found to be capable of describing data from experiments on rat tail tendon tissue. Biological soft tisues are composed of solid and liquid constituents, and diffusion plays a key role in the collagen degradation and remodeling processes, which has been modeled in works such as Refs. [82] and [314]. The detailed role of chemical diffusion in collagen degradation is not covered in this review.

Injured, weak, or insufficient tissue may require some supporting materials for the healing process to take place or to be carried out efficiently. Such implant materials may be permanent or temporary ones. There is a growing interest for using biodegradable materials. The applicability of biodegradable materials in stents, which support arteries during their recovery process, has been studied in different works. Arterial stents are reported to be capable of inducing the collagen remodeling process [315]. Challenges lie in modeling how implant materials with either constant or changing mechanical properties affect, and are affected by, ongoing changes in the tissue's remodeling behavior as recovery proceeds. For example, Soares et al. [316] model biodegradable polymers with application to stents in a variational framework which accounts for degradation behavior that depends upon the ongoing deformation. Laubrie et al. [317] present a modeling work for arterial growth and remodeling after stent implantation.

Silk-based biomaterials have been produced in a variety of forms, and this also has given rise to new medical applications [318,319]. Silk-based biomaterials are degraded by enzymes, and there is evidence that this enzymatic degradation is also mediated by mechanical loading (see, e.g., Ref. [320]).

Modeling how mechanical stimuli related to biochemical process change will generally become more complex as an increasing number of the specific chemical reactions are taken into consideration. An example of such modeling complexity is exhibited in Fig. 1 of Ref. [321]. That model identifies key network regulators of cardiomyocyte mechanosignaling, and it comprises 125 activating or inhibitory reactions linking 94 process nodes.

With a better understanding of the interplay between the physical and biochemical processes, the simulation of fiber remodeling can be improved and refined. These models may then help to understand how healing progresses in collagenous tissue and to promote the development of therapeutic measures for both general applications and personalized medicine. Refined tissue models also help to predict how biological tissues grow and develop in alternative environments, which may aid in the planning and design of experiments that are often restricted due to costs, availability of biological materials, and ethical considerations.

## Acknowledgment

The statements made herein are solely the responsibility of the authors. We are grateful for focused technical observations and remarks from the reviewers, which led to key improvements in our article.

## Funding Data

• Qatar National Research Fund (a member of the Qatar Foundation) (NPRP Grant No. 8-2424-1-477; Funder ID: 10.13039/100008982).

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