Figure 1.
Concept depiction – the structural constitution of the hidden complex tendinous networks (e.g., inset in a) can be inferred via the most informative force-motion data.
Steps (a)–(c) are performed cyclically until a termination criterion is met after which the best model (d) is chosen. (c) represents the evolved models from step (b) being actuated by a random set of input forces (or tests) shown using arrows. The length of each arrow represents the magnitude of the force applied. The actuation directions may or may not be held fixed (here, shown fixed). The actuation force set that generates the maximum discrepancy in the force-motion response between the evolved models is chosen to perform the experiment in (a).
Figure 2.
Models are evolved from the primordial mesh of strings in (a).
Length of each string behaves as a topological and parametric design variable. For the informative equilibrium configurations that the mesh observes, if some strings remain slack, they are eliminated from the topology. Only taut strings in some or all configurations get retained. Length and cross sections of strings are evolved as design variables. (b) Each taut string in the parent mesh is modeled as a large deformation truss finite element. Slack strings do not support the external loads. Analysis performed is geometrically (for synthetic targets) and materially (for the finger extensor mechanism) nonlinear and equilibrium is achieved through efficient, quadratically convergent Newton-Raphson iterations.
Figure 3.
Inference of the synthetic target networks using the informative force-motion data generated from the inference process.
The grounded nodes are shown using squares and the actuation forces are depicted using the arrows. Each model is evolved until they see 20 informative experimental data sets. (a) The ‘AFH’ target. (b) adapted Winslow's Rhombus or ‘aWR’ target. This is an adaptation of the Zancoli's representation [17] of the finger extensor mechanism (Fig. 4a). In the latter, the diagonal bands and lateral offshoots overlap while in this adaptation, the corresponding strings are fused. The top two grounded ports (through which the reaction forces are measured) are not interconnected. (c) Best eight models evolved through the informative data from the ‘AFH’ target. (d) Best eight models evolved through the informative data from the ‘aWR’ target. Strings colored yellow are slack in the shown equilibrium configuration. Those colored red are taut. Slack strings get taut for some other informative load set that they see during their evolution. Models (iii) in (c) and (ii) in (d) with the least cross validation errors (Table I) resemble in structural constitution with their respective targets.
Figure 4.
Structural inference of the finger extensor mechanism extracted from the middle finger of a human cadaver hand.
(a) The interpretation by Zancoli [17] and Garcia-Elias [18] as Winslow's Rhombus is widely accepted. A characteristic of this structure is the overlap between the lateral offshoot and the diagonal band on both symmetric sides. (b) The extensor tissue was carefully extracted during the day of the experiment. (c) the tissue mounted over the experimental bed for force-motion data extraction (d) magnified view of the extensor tissue (e) ten best inferred networks that are all functionally equivalent within the training set and cross validation errors of 7.9% and 7.2% respectively (Table 2). The models are structurally diverse. Strings colored yellow are not taut in the equilibrium configuration shown. Those colored red are taut. Slack strings do become taut when some other test, used to infer the model network, is employed. Models (i), (iv), (viii) and (ix) in (e) structurally resemble with the Winslow's rhombus in (a) with notable deviations besides the presence of additional strings. In model (i), all modules according to the classical description in (a) are captured except for the left lateral offshoot which is shorter in length. In (iv), a string connects the middle and terminal slips. In (viii), the central band is replaced by a diamond. In model (ix), the left lateral offshoot does not overlap with the left diagonal band but instead, is fused with it.
Table 1.
Training set and cross validation errors of various models inferred through the informative data obtained from synthetic ‘AFH’ and ‘aWR’ targets (Fig. 3).
Figure 5.
Topology matters and so does informative data generated from the inference process.
Training set (left column) and cross validation (right column) errors when (a) the ‘AFH’ target is inferred and (b) when the ‘aWR’ network is inferred. Solid lines represent errors with sequential informative experimental data when both topologic and parametric inference is performed. Dotted lines correspond to cases when random experimental data is sequentially employed for inference. Dashed lines show errors for only parametric inference with basic (Fig. S2 c–d) topologies. Error bars represent standard errors across three executions of the inference processes. Topologically and parametrically inferred networks with sequentially introduced informative force-motion data are more functionally proximal to the respective targets compared to (i) when random data is used and (ii) when module/string interconnectivity is ignored during inference.
Figure 6.
Topology matters and so does informative data generated from the inference process.
Errors incurred during the inference of the finger extensor mechanism. (a) Training set errors. (b) Cross validation errors. Progression of the error values are depicted as the number of data sets are introduced for model evolution. Error values are depicted for five executions for the topologic and parametric inference using informative tests. Bars represent standard errors. Solid lines correspond to the mean error when informative data is employed in topological and parametric inference of the target. Dotted lines show mean errors when sequential random data sets are used (error bars depicted for 20 executions). Dashed lines represent mean errors (error bars depicted for 20 executions) when only parametric evolution is performed using the basic topology where in only 6 strings connect all accessible (input and output) ports and no other interconnection is allowed (Fig. S2 d). At the end of the inference process, the mean cross validation error when models are topologically inferred using informative tests is 6.8%, better (by about 3%) than parametric-only inference with informative tests and improved (by about 7%) compared to the topological inference with random tests. The topology chosen for parametric-only inference (Fig. S2 d) comprises most components of the Winslow's Rhombus (Fig. 4a) except for the lateral offshoots and transverse bands. We observe in Fig. 4e that some inferred models do resemble the Winslow's Rhombus structurally. The low difference (3%) in the mean cross validation errors suggests that the disparity between the topologically and parametrically inferred models and the parametric-only inferred models (both using informative tests) is minor but essentially topologic (we reckon this could be because of the absence of the lateral offshoots in the all-in-all topology in Fig. S2d). The difference of 7% in (b) between models inferred using informative and random tests is however, relatively significant and is expected to be larger with further improvements in the inference process (see discussion). As observed in [56], the margin gained with the estimation-exploration algorithm grows with the complexity of the problem. Additionally, one may require more number of trials (which will add to the temporal and computational cost) to achieve a certain accuracy. With the estimation-exploration algorithm, a given accuracy can be achieved faster.
Table 2.
Training set and cross validation errors of the ten best models inferred through the informative force-motion data obtained from the finger extensor mechanism.
Figure 7.
Individual strings contribute significantly to the functionality of the chosen models.
(a) model (i) in Fig. 4e and (c) model (iv) in Fig. 4e with string members numbered. (b) Each string member is removed one at a time from model (i) and cross validation errors are computed. The intact model (with no string removed) exhibits the least cross validation error. (d) Each string member is removed one at a time from model (iv) and cross validation errors are computed. On removal of strings 22 or 24, the error decreases. This suggests that connection between middle and distal slips should be absent. Removing string 7 or 10 from the model in (c) does not alter the cross validation error from that of the intact model. This is because these strings do not get taut when cross validating load sets are used. Visual inspection further shows that these strings barely get taut when the model is actuated with the informative load sets used for its evolution. This suggests that strings 7 and 10 do not participate in the network which explains why the training set error for this model is larger (Table 2) compared to that of model (i) in (a).
Figure 8.
Additional information to accurately decipher the structural constitution of a finger extensor network is critical.
First and third columns: Best models are evolved with linear elastic tendon properties (E = 1 GPa). The models should emulate the informative experimental data from the cadaver hand. String numbers shown with the evolved networks. Second and fourth columns: Cross validation errors (×100 for percentage) for corresponding models on the left. The first bar represents the error when the corresponding network on the left is intact. The following bars depict the error when the strings are snapped one at a time. The errors for the intact models are equal to or greater than 30%. From each model, elimination of certain strings lowers the cross validation errors suggesting that the intact models do not depict the functionality of the actual extensor mechanism accurately. If erroneous postulations are employed, the experimental data obtained by the inference process may not be informative after all.