Fig 1.
Scale-dependent features informing the prediction of 10-year femur strength in a living human.
A cross-section of clinical quantitative computed tomography image set of an adult human femur at 0.625 mm resolution showing spatial heterogeneity in distribution of volumetric bone mineral density (vBMD). vBMD contour levels are shown in logarithmic scale to highlight regions of low vBMD. Details of cohort from which CT data is obtained and details of CT data processing are given in Ref. [31].
Fig 2.
Scale separation maps for predicting 10-year femur strength in a living human.
(A) Scale-dependent features identified using quantitative computed tomography (qCT) and ex vivo fall strength measurements. (B) Scale-dependent features identified using qCT only.
Fig 3.
Multiscale model orchestrations for predicting 10-year femur strength in a living human.
(A) Scale-dependent features identified using quantitative computed tomography (qCT) and ex vivo fall strength measurements. (B) Scale-dependent features identified using qCT only. Definitions of the mathematical symbols are given in the main text.
Fig 4.
Predicted volumetric bone mineral density (vBMD) distribution at a cross-section of an adult human femur.
The cross-section is identical to that in Fig 1 but after the application of models (R12′-1), (S2′) and (R12′-2) in sequence, resulting in a modified spatial distribution in vBMD at an instant when the subject is 70 years old. vBMD contour levels are shown in logarithmic scale to highlight regions of low vBMD, where the percentage change is highest.
Fig 5.
Scale-separation map and orchestration for alternative multiscale model.
(A) Scale-separation map and (B) multiscale model orchestration. Several instrumentation capabilities are employed in this alternative model to identify scale-dependent features in a large portion of reality. The closed-loop workflow around scale S1 in the multiscale model orchestration indicates an update following a time increment.
Fig 6.
Scale-separation map and orchestration for multiscale model to predict 8-week femur strength in a mouse.
(A) Scale-separation map and (B) multiscale model orchestration. Instrumentation capabilities employed are appropriate for the small size of the animal. The closed-loop workflow around scale S1 in the multiscale model orchestration indicates an update following a time increment.
Fig 7.
Scale-separation maps for different multiscale modelling approaches.
(A) Scale-separation map for a multiscale model using the approach presented in this paper. In the absence of experimental capability to realise the real scale denoted by the box with a sloping stripe pattern, the relation model (not shown) bridging the real scales denoted by boxes with grid and vertical stripe patterns is validated by considering the isolated hypothetical scale shown by the dashed outline box. When isolating this hypothetical scale, known effects from other scales, such as that denoted by the horizontal stripe pattern, must be controlled for. When the real scale denoted by the sloping stripe pattern can be experimentally realised, more direct validation of this relation model is possible. (B) Scale separation map for an applied mathematics and continuum (micro)mechanics based multiscale model. Dotted outline box denotes the fine-scale or full model. The other two boxes denote successively coarser-grain models. The edges of each box denote the assumed range of validity of the corresponding model with no explicit link to experimental capability. “Coarsening” always leads to a model with a range of validity that is a subset of the range of validity of the original model.