A new one-DOF fully parallel mechanism for modelling passive motion at the human tibiotalar joint

Knowledge on how ligaments and articular surfaces guide passive motion at the human ankle joint complex is fundamental for the design of relevant surgical treatments. The paper presents a possible improvement of this knowledge by a new kinematic model of the tibiotalar articulation. Passive motion, i.e. in virtually unloaded conditions, was captured in vitro in four lower leg specimens by means of a surgical navigation system with cluster of active markers attached to the tibia and talus. The anatomical geometry of the passive structures, i.e. articular surfaces and attachment areas of the ligaments, were taken by digitisation with a pointer. An equivalent spatial mechanism for the passive motion simulation was defined by three sphere-to-sphere contact points and two rigid links. These contact points were identified at the lateral talo-fibular articulation and at the medial and lateral aspects of the articulation between tibial mortise and trochlea tali. The two rigid links were identified by the isometric fibres at the calcaneofibular and tibiocalcaneal ligaments. An optimisation algorithm was developed for the identification of the final geometrical parameters resulting from an iterative refining process, which targets best matching between model predictions and corresponding experimental measurements of the spatial motion. The specimen-specific equivalent spatial mechanisms replicated the original passive motion very well, with mean discrepancies in position smaller than 2.5 mm and in rotation smaller than 1°. The study demonstrates that the articular surfaces and the ligaments, acting together as a mechanism, control the passive kinematics of the ankle joint.     

Subject-specific knee joint geometry improves predictions of medial tibiofemoral contact forces

Estimating tibiofemoral joint contact forces is important for understanding the initiation and progression of knee osteoarthritis. However, tibiofemoral contact force predictions are influenced by many factors including muscle forces and anatomical representations of the knee joint. This study aimed to investigate the influence of subject-specific geometry and knee joint kinematics on the prediction of tibiofemoral contact forces using a calibrated EMG-driven neuromusculoskeletal model of the knee. One participant fitted with an instrumented total knee replacement walked at a self-selected speed while medial and lateral tibiofemoral contact forces, ground reaction forces, whole-body kinematics, and lower-limb muscle activity were simultaneously measured. The combination of generic and subject-specific knee joint geometry and kinematics resulted in four different OpenSim models used to estimate muscle–tendon lengths and moment arms. The subject-specific geometric model was created from CT scans and the subject-specific knee joint kinematics representing the translation of the tibia relative to the femur was obtained from fluoroscopy. The EMG-driven model was calibrated using one walking trial, but with three different cost functions that tracked the knee flexion/extension moments with and without constraint over the estimated joint contact forces. The calibrated models then predicted the medial and lateral tibiofemoral contact forces for five other different walking trials. The use of subject-specific models with minimization of the peak tibiofemoral contact forces improved the accuracy of medial contact forces by 47% and lateral contact forces by 7%, respectively compared with the use of generic musculoskeletal model.     

Muscle and external load contribution to knee joint contact loads during normal gait

Large knee adduction moments during gait have been implicated as a mechanical factor related to the progression and severity of tibiofemoral osteoarthritis and it has been proposed that these moments increase the load on the medial compartment of the knee joint. However, this mechanism cannot be validated without taking into account the internal forces and moments generated by the muscles and ligaments, which cannot be easily measured. Previous musculoskeletal models suggest that the medial compartment of the tibiofemoral joint bears the majority of the tibiofemoral load, with the lateral compartment unloaded at times during stance. Yet these models did not utilise explicitly measured muscle activation patterns and measurements from an instrumented prosthesis which do not portray lateral compartment unloading. This paper utilised an EMG-driven model to estimate muscle forces and knee joint contact forces during healthy gait. Results indicate that while the medial compartment does bear the majority of the load during stance, muscles provide sufficient stability to counter the tendency of the external adduction moment to unload the lateral compartment. This stability was predominantly provided by the quadriceps, hamstrings, and gastrocnemii muscles, although the contribution from the tensor fascia latae was also significant. Lateral compartment unloading was not predicted by the EMG-driven model, suggesting that muscle activity patterns provide useful input to estimate muscle and joint contact forces.     

In vivo tibiofemoral cartilage deformation during the stance phase of gait

The knowledge of articular cartilage contact biomechanics in the knee joint is important for understanding the joint function and cartilage pathology. However, the in vivo tibiofemoral articular cartilage contact biomechanics during gait remains unknown. The objective of this study was to determine the in vivo tibiofemoral cartilage contact biomechanics during the stance phase of treadmill gait. Eight healthy knees were magnetic resonance (MR) scanned and imaged with a dual fluoroscopic system during gait on a treadmill. The tibia, femur and associated cartilage were constructed from the MR images and combined with the dual fluoroscopic images to determine in vivo cartilage contact deformation during the stance phase of gait. Throughout the stance phase of gait, the magnitude of peak compartmental contact deformation ranged between 7% and 23% of the resting cartilage thickness and occurred at regions with thicker cartilage. Its excursions in the anteroposterior direction were greater in the medial tibiofemoral compartment as compared to those in the lateral compartment. The contact areas throughout the stance phase were greater in the medial compartment than in the lateral compartment. The information on in vivo tibiofemoral cartilage contact biomechanics during gait could be used to provide physiological boundaries for in vitro testing of cartilage. Also, the data on location and magnitude of deformation among non-diseased knees during gait could identify where loading and later injury might occur in diseased knees.     

Development and validation of a subject-specific moving-axis tibiofemoral joint model using MRI and EOS imaging during a quasi-static lunge

The aims of this study were to introduce and validate a novel computationally-efficient subject-specific tibiofemoral joint model. Subjects performed a quasi-static lunge while micro-dose radiation bi-planar X-rays (EOS Imaging, Paris, France) were captured at roughly 0°, 20°, 45°, 60°, and 90° of tibiofemoral flexion. Joint translations and rotations were extracted from this experimental data through 2D-to-3D bone reconstructions, using an iterative closest point optimization technique, and employed during model calibration and validation. Subject-specific moving-axis and hinge models for comparisons were constructed in the AnyBody Modeling System (AMS) from Magnetic Resonance Imaging (MRI)-extracted anatomical surfaces and compared against the experimental data. The tibiofemoral axis of the hinge model was defined between the epicondyles while the moving-axis model was defined based on two tibiofemoral flexion angles (0° and 90°) and the articulation modeled such that the tibiofemoral joint axis moved linearly between these two positions as a function of the tibiofemoral flexion. Outside this range, the joint axis was assumed to remain stationary. Overall, the secondary joint kinematics (ML: medial–lateral, AP: anterior-posterior, SI: superior-inferior, IE: internal-external, AA: adduction-abduction) were better approximated by the moving-axis model with mean differences and standard errors of (ML: −1.98 ± 0.37 mm, AP: 6.50 ± 0.82 mm, SI: 0.05 ± 0.20 mm, IE: 0.59 ± 0.36°, AA: 1.90 ± 0.79°) and higher coefficients of determination (R²) for each clinical measure. While the hinge model achieved mean differences and standard errors of (ML: −0.84 ± 0.45 mm, AP: 10.11 ± 0.88 mm, SI: 0.66 ± 0.62 mm, IE: −3.17 ± 0.86°, AA: 11.60 ± 1.51°).     

Multiscale ordination: a method for detecting pattern at several seales

Multivariate analyses of vegetation data have been restricted to a single scale of sampling, or multiscale sampling has been restricted to a single species. However, vegetation scientists need to be able to explore spatial relationships of many species over many scales. We present a modification of Noy-Meir & Anderson's (1971) method of multiscale ordination by summing two-term local covariance matrices and smoothing the component profiles. The advantages of our method are: 1) results are less subject to the starting position of the transect, 2) matrices may be added at any block size, and 3) plots of factor scores are smoothed by a moving weighted average to better reveal patterns at a prescribed scale.This procedure provides statistical associations of species over a range of scales. The scales which exhibit the association to the maximum extent are then determined from multiscale ordination. The relationships of different associations and their scales can then be examined. The application of the method to fabricated data proved successful in recovering the structure built into the data. When used on real vegetation data, from a community and a landscape, the method revealed the details of species associations over a range of scales, and of the relationships among associations.Abbreviations PCA = Principal Components Analysis - DCA = Detrended Correspondence Analysis - TTLC = Two-Term Local Covariance     

In vivo tibiofemoral contact analysis using 3D MRI-based knee models

This paper quantified the motion of the tibiofemoral contact points during in vivo weight bearing flexion using MRI- based 3D knee models and two orthogonal fluoroscopic images. The contact points on the medial and lateral tibial plateau were calculated by finding the centroid of the intersection of the tibial and femoral cartilage layers and by using the bony geometry alone. Our results indicate that the medial femoral condyle remains in the central portion of the tibial plateau and the lateral condyle translates posteriorly with increasing flexion. Using the bony contact model increased the total translation of the medial and lateral condyles by 250 and 55%, respectively, compared to the cartilage contact model. These results suggest that using the bony geometry alone may not accurately represent the articular surfaces of the knee. Articular cartilage geometry may have to be used to accurately quantify tibiofemoral contact.     

A three-dimensional study of the kinematics of the human knee

This paper represents a three-dimensional study of the human knee-joint and studies kinematic effects of the cruciate ligaments. Two methods were used for our studies, one method was preferred. This method used a time lapse photograph and strobe light to give us a plot of reference points to carry out our analysis using the method of Rouleaux applied to three dimensions. Five cadaver joints were used, each of which was used for three series of experiments, including the joint with capsule intact, with one of the ligaments cut and with the remaining ligament cut. Both lateral and medial studies were conducted to provide data for a three-dimensional study.

It was found that the cruciate ligaments had little effect on the kinematics of the knee, and that the knee motion remained unchanged after cutting one or both of the cruciate ligaments. It was concluded that the motion of the knee was due to the geometry of the bones and perhaps the collateral ligaments, and that the joint could be replaced with a prosthesis having a three dimensional axis of rotation with a fixed center.     

A multiaxial force-sensing implantable tibial prosthesis

Accurate in vivo measurement of tibiofemoral forces is important in total knee arthroplasty. These forces determine polyethylene stresses and cold-flow, stress distribution in the implant, and stress transfer to the underlying implant bone interface. Theoretic estimates of tibiofemoral forces have varied widely depending on the mathematical models used. The six degrees of freedom of motion, complex articular surface topography, changing joint-contact position, intra- and extra-articular ligaments, number of muscles crossing the knee joint, and the presence of the patellofemoral joint contribute to the difficulty in developing reliable models of the knee. A prototype instrumented total knee replacement tibial prosthesis was designed, manufactured, and tested. This prosthesis accurately measured all six components of tibial forces (R2>0.997). The prosthesis was also instrumented with an internal microtransmitter for wireless data transmission. Remote powering of the sealed implanted electronics was achieved using magnetic coil induction. This device can be used to validate existing models of the knee that estimate these forces or to develop more accurate models. In conjunction with kinematic data, accurate tibiofemoral force data may be used to design more effective knee-testing rigs and wear simulators. Additional uses are intraoperative measurement of forces to determine soft-tissue balancing and to evaluate the effects of rehabilitation, external bracing, and athletic activities, and activities of daily living.     

Ligament balancing in TKA: evaluation of a force-sensing device and the influence of patellar eversion and ligament release

Ligament balancing in total knee arthroplasty may have an important influence on joint stability and prosthesis lifetime. In order to provide quantitative information and assistance during ligament balancing, a device that intraoperatively measures knee joint forces and moments was developed. Its performance and surgical advantages were evaluated on six cadaver specimens mounted on a knee joint loading apparatus allowing unconstrained knee motion as well as compression and varus-valgus loading. Four different experiments were performed on each specimen. (1) Knee joints were axially loaded. Comparison between applied and measured compressive forces demonstrated the accuracy and reliability of in situ measurements (1.8N). (2) Assessment of knee stability based on condyle contact forces or varus-valgus moments were compared to the current surgical method (difference of varus-valgus loads causing condyle lift-off). The force-based approach was equivalent to the surgical method while the moment-based, which is considered optimal, showed a tendency of lateral imbalance. (3) To estimate the importance of keeping the patella in its anatomical position during imbalance assessment, the effect of patellar eversion on the mediolateral distribution of tibiofemoral contact forces was measured. One fourth of the contact force induced by the patellar load was shifted to the lateral compartment. (4) The effect of minor and major medial collateral ligament releases was biomechanically quantified. On average, the medial contact force was reduced by 20% and 46%, respectively. Large variation among specimens reflected the difficulty of ligament release and the need for intraoperative force monitoring. This series of experiments thus demonstrated the device's potential to improve ligament balancing and survivorship of total knee arthroplasty.     

A 3D lower limb musculoskeletal model for simultaneous estimation of musculo-tendon,joint contact,ligament and bone forces during gait

Musculo-tendon forces and joint reaction forces are typically estimated using a two-step method, computing first the musculo-tendon forces by a static optimization procedure and then deducing the joint reaction forces from the force equilibrium. However, this method does not allow studying the interactions between musculo-tendon forces and joint reaction forces in establishing this equilibrium and the joint reaction forces are usually overestimated. This study introduces a new 3D lower limb musculoskeletal model based on a one-step static optimization procedure allowing simultaneous musculo-tendon, joint contact, ligament and bone forces estimation during gait. It is postulated that this approach, by giving access to the forces transmitted by these musculoskeletal structures at hip, tibiofemoral, patellofemoral and ankle joints, modeled using anatomically consistent kinematic models, should ease the validation of the model using joint contact forces measured with instrumented prostheses. A blinded validation based on four datasets was made under two different minimization conditions (i.e., C1 – only musculo-tendon forces are minimized, and C2 – musculo-tendon, joint contact, ligament and bone forces are minimized while focusing more specifically on tibiofemoral joint contacts). The results show that the model is able to estimate in most cases the correct timing of musculo-tendon forces during normal gait (i.e., the mean coefficient of active/inactive state concordance between estimated musculo-tendon force and measured EMG envelopes was C1: 65.87% and C2: 60.46%). The results also showed that the model is potentially able to well estimate joint contact, ligament and bone forces and more specifically medial (i.e., the mean RMSE between estimated joint contact force and in vivo measurement was C1: 1.14BW and C2: 0.39BW) and lateral (i.e., C1: 0.65BW and C2: 0.28BW) tibiofemoral contact forces during normal gait. However, the results remain highly influenced by the optimization weights that can bring to somewhat aphysiological musculo-tendon forces.     

A comparative evaluation of the mechanical properties of the rabbit medial collateral and anterior cruciate ligaments.

The biomechanical properties of the medial collateral and anterior cruciate ligaments from 30 New Zealand White rabbits were measured. Because of its complex geometry, the ACL was divided into two portions (medial and lateral) to provide uniform loading. This allowed an examination of the intra-ligamentous properties. A laser micrometer system was used to measure the cross-sectional area for tensile stress and a video dimension analyzer was used to measure the strain. The mechanical properties (stress-strain curves) of the MCL and ACL were different, with the modulus (determined between 4 and 7% strain) in the MCL (1120 +/- 153 MPa) more than twice that of either portion of the ACL (516 +/- 64 and 516 +/- 69 MPa for the medial and lateral portions, respectively). This higher modulus correlated with the more uniform and dense appearance of the collagen fibrils examined with scanning electron microscopy (SEM).     

A comparison of the influence of global functional loads vs. local contact anatomy on articular cartilage thickness at the knee

Cartilage contact geometry, along with joint loading, can play an important role in determining local articular cartilage tissue stress. Thus individual variations in cartilage thickness can be associated with both individual variations in joint loading associated with activities of daily living as well as individual differences in the anatomy of the contacting surfaces of the joint. The purpose of this study was to isolate the relationship between cartilage thickness predicted by individual variations in contact surface geometry based on the radii of the femur and tibia vs. cartilage thickness predicted by individual variations in joint loading. Knee magnetic resonance (MR) images and the peak knee adduction moments during walking were obtained from 11 young healthy male subjects (age 30.5+/-5.1 years). The cartilage thicknesses and surface radii of the femoral and tibial cartilage were measured in the weight-bearing regions of the medial and lateral compartments of three-dimensional models from the MR images. The ratio of contact pressure between the medial and lateral compartments was calculated from the radii of tibiofemoral contact surface geometries. The results showed that the medial to lateral pressure ratios were not correlated with the medial to lateral cartilage thickness ratios. However, in general, pressure was higher in the lateral than medial compartments and cartilage was thicker in the lateral than medial compartments. The peak knee adduction moment showed a significant positive linear correlation with medial to lateral thickness ratio in both femur (R(2)=0.43,P<0.01) and tibia (R(2)=0.32,P<0.01). The results of this study suggest that the dynamics of walking is an important factor to describe individual differences in cartilage thickness for normal subjects.     

The effect of hamstring muscle compensation for anterior laxity in the ACL-deficient knee during gait

The hamstring muscles have been recognized as an important element in compensating for the loss of stability in the ACL-deficient knee, but it is still not clear whether the hamstring muscle force can completely compensate for the loss of ACL, and the consequences of increased hamstring muscle force. A two-dimensional anatomical knee model in the sagittal plane was developed to examine the effect of various levels of hamstring muscle activation on restraining anterior tibial translation in the ACL-deficient knee during level walking. The model included the tibiofemoral and patellofemoral joints, four major ligaments, the medial capsule, and five muscle units surrounding the knee. Simulations were conducted to determine anterior tibial translation and internal joint loading at a single selected position when the knee was under a peak external flexion moment during early stance phase of gait. Incremental hamstring muscle forces were applied to the modeled normal and the ACL-deficient knees. Results of simulations showed that the ACL injury increased the anterior tibial translation by 11.8mm, while 56% of the maximal hamstring muscle force could reduce the anterior translation of the tibia to a normal level during the stance phase of gait. The consequences of increased hamstring muscle force included increased quadriceps muscle force and joint contact force.     

Tensile properties, collagen content, and crosslinks in connective tissues of the immature knee joint

Background

The major connective tissues of the knee joint act in concert during locomotion to provide joint stability, smooth articulation, shock absorption, and distribution of mechanical stresses. These functions are largely conferred by the intrinsic material properties of the tissues, which are in turn determined by biochemical composition. A thorough understanding of the structure-function relationships of the connective tissues of the knee joint is needed to provide design parameters for efforts in tissue engineering.

Methodology/Principal Findings

The objective of this study was to perform a comprehensive characterization of the tensile properties, collagen content, and pyridinoline crosslink abundance of condylar cartilage, patellar cartilage, medial and lateral menisci, cranial and caudal cruciate ligaments (analogous to anterior and posterior cruciate ligaments in humans, respectively), medial and lateral collateral ligaments, and patellar ligament from immature bovine calves. Tensile stiffness and strength were greatest in the menisci and patellar ligament, and lowest in the hyaline cartilages and cruciate ligaments; these tensile results reflected trends in collagen content. Pyridinoline crosslinks were found in every tissue despite the relative immaturity of the joints, and significant differences were observed among tissues. Notably, for the cruciate ligaments and patellar ligament, crosslink density appeared more important in determining tensile stiffness than collagen content.

Conclusions/Significance

To our knowledge, this study is the first to examine tensile properties, collagen content, and pyridinoline crosslink abundance in a direct head-to-head comparison among all of the major connective tissues of the knee. This is also the first study to report results for pyridinoline crosslink density that suggest its preferential role over collagen in determining tensile stiffness for certain tissues.     

The effect of the frontal plane tibiofemoral angle and varus knee moment on the contact stress and strain at the knee cartilage

Subject-specific models were developed and finite element analysis was performed to observe the effect of the frontal plane tibiofemoral angle on the normal stress, Tresca shear stress and normal strain at the surface of the knee cartilage. Finite element models were created for three subjects with different tibiofemoral angle and physiological loading conditions were defined from motion analysis and muscle force mathematical models to simulate static single-leg stance. The results showed that the greatest magnitude of the normal stress, Tresca shear stress and normal strain at the medial compartment was for the varus aligned individual. Considering the lateral knee compartment, the individual with valgus alignment had the largest stress and strain at the cartilage. The present investigation is the first known attempt to analyze the effects of tibiofemoral alignment during single-leg support on the contact variables of the cartilage at the knee joint. The method could be potentially used to help identify individuals most susceptible to osteoarthritis and to prescribe preventive measures.     

城市绿色基础设施降温作用及其影响因素研究进展

城市绿色基础设施（UGI）作为一种多尺度、多类型和多功能的绿色网络,对缓解城市升温和提升城市气候韧性具有重要作用。尽管如此,当前仍缺乏关于UGI降温作用研究方法、降温机制、尺度效应和影响因素等方面的系统性综述。为此,从尺度、方法、降温机制、影响因素等多个方面详尽回顾了UGI降温作用研究进展,并讨论了其局限性、挑战和未来研究方向。研究发现：（1） UGI降温作用的研究尺度可归纳为站点、建筑、街道、绿地、街区和城市6类,其中站点、绿地和街区等微尺度和局部尺度研究较多,而城市尺度关注则较少。（2）研究方法可分为温度数据获取、降温作用测度和影响因素分析三部分。其中,温度数据获取主要采用实地观测、遥感观测和模型模拟三类方法;降温作用测度可分为基于空间参考、基于时间或情景参考和基于统计关系三类;影响因素分析多采用相关分析与多重线性回归等传统统计方法,此外,可探究非线性影响的机器学习方法也逐渐得到应用。（3） UGI降温作用主要依赖于遮荫、蒸散发与冷空气团平流三类机制;UGI植被物种构成与形态、UGI空间配置和外部环境是影响降温作用的三类主要因素;不同尺度降温作用的主控影响因素存在差异,随着尺度扩大UGI空间配置因素的重要性逐渐增加。最后,研究从时空尺度、降温机制、综合影响因素、生态系统服务视角等方面讨论了当前研究的局限和未来挑战,并强调了新型数据获取技术、组合降温策略以及城市尺度UGI系统降温作用等未来研究方向。     

Spatial reconstruction of the human motion based on images of a single camera

The inverse dynamic analysis procedures used in the study of the human gait require that the kinematics of the supporting biomechanical model is known beforehand. The first step to obtain the kinematic data is the reconstruction of human spatial motion, i.e., the evaluation of the anatomic points positions that enables to uniquely define the position of all anatomical segments. In photogrammetry, the projection of each anatomical point is described by two linear equations relating its three spatial coordinates with the two coordinates of the projected point. The need for the image of two cameras arises from the fact that three equations are necessary to find the original spatial position of the anatomical point. It is shown here that the kinematic constraint equations associated with a biomechanical model can be used as the extra set of equations required for the reconstruction process, instead of the equations associated with the second camera. With this methodology, the system of equations arising from the point projections and biomechanical model kinematic constraints are solved simultaneously. Since the system of equations has multiple solutions for each image, a strategy based on the minimization of the cost function associated to the smoothness of the reconstructed motion is devised, leading to an automated computer procedure enabling a unique reconstruction.     

A reciprocal connection factor for assessing knee-joint function

In the knee joint, interactions between instantaneous kinetics and kinematics associated with ligamentous and articular tissues are not fully understood. These structures may be represented by the instantaneous screw axis ($) (ISA) and static force vectors ($′). Geometric changes to the joint structure affecting motion have not been fully explained, especially after surgical reconstruction and replacement procedures. The ISA offers a joint-characterisation approach, which is dependent on the combined forces of ligaments, articular contacts and muscles. The standard four-bar linkage model in the sagittal plane demonstrates that the normal contact force and the lines of action of the cruciate ligaments always intersect at the centre of rotation of the joint. A kinematic knee model in which the articular surfaces in the lateral and medial compartments as well as the isometric fascicles in the engaged ligaments may be represented as five constraints in a one-degree-of-freedom parallel spatial mechanism. This study provides a theoretical foundation to elucidate the role of each of these elements in the control of the ISA. A recourse to the principle of virtual work explained through d'Alembert's principle for reducing a dynamics problem to an instantaneous static scenario allows screws to be applied to the biomechanics of human motion. The principle of reciprocity links these approaches together to explain the transmitting load between the tibia and the femur as well as the relative motion within the knee joint. A principal clinical implication of this study is the introduction of the reciprocal connection factor to evaluate knee kinematics and kinetics in one simple term, allowing the quantitative assessment of the outcome of knee-joint treatment and rehabilitation methods.     

An approach to multiscale modelling with graph grammars

Background and Aims

Functional–structural plant models (FSPMs) simulate biological processes at different spatial scales. Methods exist for multiscale data representation and modification, but the advantages of using multiple scales in the dynamic aspects of FSPMs remain unclear. Results from multiscale models in various other areas of science that share fundamental modelling issues with FSPMs suggest that potential advantages do exist, and this study therefore aims to introduce an approach to multiscale modelling in FSPMs.

Methods

A three-part graph data structure and grammar is revisited, and presented with a conceptual framework for multiscale modelling. The framework is used for identifying roles, categorizing and describing scale-to-scale interactions, thus allowing alternative approaches to model development as opposed to correlation-based modelling at a single scale. Reverse information flow (from macro- to micro-scale) is catered for in the framework. The methods are implemented within the programming language XL.

Key Results

Three example models are implemented using the proposed multiscale graph model and framework. The first illustrates the fundamental usage of the graph data structure and grammar, the second uses probabilistic modelling for organs at the fine scale in order to derive crown growth, and the third combines multiscale plant topology with ozone trends and metabolic network simulations in order to model juvenile beech stands under exposure to a toxic trace gas.

Conclusions

The graph data structure supports data representation and grammar operations at multiple scales. The results demonstrate that multiscale modelling is a viable method in FSPM and an alternative to correlation-based modelling. Advantages and disadvantages of multiscale modelling are illustrated by comparisons with single-scale implementations, leading to motivations for further research in sensitivity analysis and run-time efficiency for these models.