Fig 1.
This workflow sketch identifies key features and stages of our approach.
We have two concurrent long-term goals. 1) Build the case for plausible similarities—analogies—between Callus Subregion Analog final states (green boxes) and stained tissue sections in Fig 2 obtained from mouse 2 after 10 days of healing (shaded gray boxes). 2) Also build the case that strong analogies can exist between Analog Mechanisms and Processes occurring during execution and corresponding mouse callus healing mechanisms and processes at comparable granularities. An objective of this work is stage 6: demonstrate quantitative similarities between simulated final states and the day-10i Target Region.
Fig 2.
Shown are images of sagittal sections through mouse tibia calluses that were stained for tissue, bone, and cartilage using Hall-Brunt Quadruple.
(A) Mouse 1 on day-7; we decided to focus on the Target Region within yellow-boxed area. (B) Mouse 2 on day-10.
Fig 3.
Discretized, images that map to sagittal sections through mouse tibia calluses.
(A) Discretized, colorized counterpart to day-7 tissue section in Fig 2A. The yellow-box area corresponds to the one in Fig 2A. The white box is the Target Region initial state. (B) Discretized, colorized counterpart of the day-10i tissue section. The white box is the Target Region final state. (C & D) Focus is drawn to the right sides. Arrows indicate directions of apparent local tissue changes. Some of that directional change that was ongoing in day-7 (C) continued into day-10 (D). Change in the other areas, particularly around the top edge of the Target Region, would have started after day-7. These apparent directional changes were taken into consideration for Mechanisms 2–4.
Fig 4.
This is an idealized representation of key features of the space of tibia fracture healing in matched mice that are part of the same experiment.
The height of the sketch represents the large interindividual variability, which is a product of each fracture being unique. Consequently, each healing process path is necessarily partially customized. Changing colors represent evolving subprocesses; the colors are not related to those used to represent the nine tissue types. The five short white bars on days 7 and 10 represent different tissue sections taken at comparable locations within different tibia fractures. The dotted path illustrates a trace of the healing process path for one of the five day-7 tibia fractures. Moving from day-7 to day-10 along the dotted path, the unshaded bar on day-10 indicates the illustration that we created, the day-10i tissue section. The two additional unshaded bars labeled a and b illustrate future extensions of the simulated Healing Process forward to day-14 and backward to day 4 along the same healing path. Two dashed gray lines illustrate that in humans fracture nonunion can occur. Absent interventions, nonunions in mice do not occur. The circumstances leading to nonunions in humans are unknown. The transition from one color to another occurs later toward the bottom, which illustrates that the pace of healing can be different from one fracture to another.
Fig 5.
MASON displays of Target Region along with three influence grids.
(A) Target Region initial state. (B) Target Region final state. The partially obscured TUs in the outermost columns and rows have an incomplete Moore neighborhood. They remained inactive for the duration of all simulations, but they do provide type information for their interior neighbors. All other TUs have 8 Moore neighbors and are active. For Mechanisms 2–4, behaviors of TUs within the two green-bordered regions require logic that is different from that used elsewhere. (C-E) The logic followed by each Mechanism 2 TU used probability values from one of these three grids. Darkest blue represents the largest value. White cells have a probability of zero or are unused. (C) Probability values used by TUs within the rectangular region. (D) Probability values hued by TUs within the triangular region. (E) Probability values used by TUs elsewhere.
Fig 6.
Selected tissue unit (TU) activities.
(A) This sequence is an example of a sequence of events occurring within a single time step, and ending with a teal TU (outlined in yellow) transitioning into a burgundy TU. Step 1: the burgundy TU outlined in white is selected randomly. A probability determines if it is (or is not) given an opportunity to act, which it is given in this example. It will not have another opportunity to act until the next time step. Thereafter, probabilities determine whether events at steps 2–4, in sequence, occur or not. The logic for all transitions is diagrammed in Fig 7. Step 2: a burgundy TU can initiate change in only one of the three TU neighbors marked by asterisks. With probabilities specified in Fig 7A, the east location is selected. Step 3: two questions are asked. Can the TU at that location transition? If yes, is the precondition for transition met? A teal TU can transition. Had the burgundy TU located north been chosen, the answer to the question would be no, because a burgundy is not allowed to transition. Nothing further would happen during that time step. The transition precondition for a teal TU is that the number of gray and burgundy TUs in the Moore neighborhood of the TU outlined in white be ≥ 4. In this example, the precondition is met (marked by circles). Step 4: A teal TU can transition to burgundy or gray with equal probability. In this example transition to burgundy occurs. (B) An example of a non-biomimetic feature (asterisks) occasionally encountered using Mechanisms 3 and 4 that requires a specialized rule, described in the text, to correct.
Fig 7.
Tissue Unit (TU) Logic for Mechanisms 3 and 4.
(A) This panel shows an example transition, which occurs within a single time step. A selected TU, S, identifies nearby locations for transition, selects a site, L, and, if the conditions further described in Fig 6A are met, L transitions to a new final state F. (B) Allowed transitions for TUs are shown. A selected TU, S, may initiate a transition for one of its neighbor cells, as determined by the probabilities shown in Fig 8. The probability grid determines whether a transition occurs for TU S, as well as if the conditions shown are met. For example, in the upper left panel of (B), suppose a blue TU is selected. It will next determine the site for transition, L, according to the probabilities in Fig 8. If L is green or light blue, L will be allowed to transition to blue. (C) Detailed logic for whether a transition occurs is shown. Details here are intended to expand upon Fig 6. At each time step, every TU is selected exactly once. Each TU is given an opportunity to act each time step, according to a global probability. If selected, a TU, S, selects a site, L, for possible transition. Site L is selected based upon the probability grids in Fig 8 for the selected TU S. Whether L transitions is also governed by the probability in the center of the grid for S. If the probability is met and if L is one of the allowed types, shown in panel (B), L may transition.
Fig 8.
Probabilities governing TU transitions in mechanisms 3 and 4.
The center value is the probability each TU will have an opportunity to act each time step. The value at each shaded location specifies the probability that a transition event may occur at that location. The types of allowed transitions are illustrated in Fig 7. (A) The precondition for transition is that each of these four TU types have ≥ 4 Moore neighbors that are of the same type. (B) These are the three TU types that enter Target Region from the north. Their rules are different from counterparts in A. Gray* and burgundy* have no precondition for transition. They can transition into gray*, burgundy*, white, or green. The precondition for a blue* transition is that ≥ 4 Moore neighbors are blue* and/or are at the Target Region border. It can transition into gray, burgundy, teal, or blue.
Fig 9.
Examples of day-10 simulation results.
From the sets of 25 maximum Similarity values (one for each Monte Carlo execution) from which the Table 1 values were derived, we selected these eight examples. The first four have the largest maximum Similarity value for the specified set: (A) Mechanism 2, (B) Mechanism 3, (C) Mechanism 4 when calculating Similarity, and (D) Mechanism 4 when calculating UL-Similarity. (E-H) From all other Mechanism 4 maximum Similarity values, we selected these four as exhibiting good overall biomimicry to the day-10i image based simply on visual comparisons. E and F are examples that exhibit near maximum UL-Similarity.
Table 1.
Similarity criteria.
Fig 10.
Temporal profiles of Similarity values for selected executions.
Single executions were selected from those used to provide the summary results for Mechanism 4 in Table 1. (A) Profiles selected from the 25 Monte Carlo executions when calculating Similarity. The three profiles with the widest line widths exhibited the largest maximum Similarity values. The three profiles with narrowest line widths exhibited the three smallest maximum Similarity values. Arrows at bottom: those on the left mark the three earliest occurrences of a maximum Similarity value. Those on the right mark the three latest occurrences of a maximum Similarity value. (B) Profiles selected the 25 Monte Carlo executions when calculating UL-Similarity. The presentation is as described in A but y-axis values are different. (C) Examples are plotted of ascending portions of Similarity value profiles when calculating Similarity. Two of widest profiles exhibit the steepest increase. The wide profile at bottom exhibits the slowest increase. The three narrow profiles were selected randomly from the other 22.
Fig 11.
Characteristics of obstacles listed in the introduction.
Each spectrum provides a different perspective on research aimed at characterizing and explaining a phenomenon, fracture healing in this case. Relative to research in other biomedical domains, fracture-healing research can be characterized by a left-of-center location on each spectrum. Spectra locations considerably right-of-center are most supportive of the conventional inductive methods used by current multiscale modeling and simulation researchers. The green shaded curve illustrates that, as one moves leftward, the number and variety of equally possible explanations increases dramatically, which can be a serious barrier to progress.