Callus in Motion MultiDimPointCloud · 2026-09
Mechanobiology of fracture healing · Joy

Callus in Motion步態週期中的骨痂四維點雲

Fracture healing is steered by how the soft callus deforms, and walking deforms it differently at every instant. This page follows every integration point of one callus through a full gait cycle, classifies each point with a mechanobiological healing window, and then asks the same question of 278 fracture geometries.

骨折癒合受軟骨痂的變形方式調控,而行走時每一瞬間的變形都不同。本頁追蹤一個骨痂中每個積分點在完整步態週期內的變化,用力學生物學癒合窗為每個點分類,再把同樣的問題推廣到 278 種骨折幾何。

33,712callus integration points骨痂積分點
101load steps per gait cycle每個步態週期的載荷步
278parametric fracture meshes參數化骨折網格
834model × speed scenarios模型 × 步速情境
0class mismatches, superposition vs direct run疊加法與直接計算的分類差異
Interactive bench

The callus, step by step骨痂,一步一步

Drag to rotate, scroll to zoom, double-click to fit, drag the timeline to scrub. Step through a series with ◀ ▶. Every point is recomputed live from the three unit-load solutions. The cards on the right walk through the story.

拖曳旋轉,滾動縮放,雙擊回到完整視野,在時間軸上拖曳跳到任一步。用 ◀ ▶ 沿系列換模型。每個點都由三個單位載荷解即時重新計算。右側卡片帶你一步步看故事。

gait 0 %Force — kN1 km/h · 70 % load
Loading the model…
drag · rotate   wheel · zoom   double-click · fit
Window
Show
Simon plane
Tissue share over the cycle
Model
01 · the data

One walking cycle, 101 load steps一個步態週期,101 個載荷步

Each dot is one of the 33,712 integration points of the reference callus (wedge 0°). The knee force at 1 km/h, scaled to 70 %, is applied in 101 steps. Colour is the tissue the selected healing window predicts for that point.

每個點是參考骨痂(楔形 0°)33,712 個積分點之一。1 km/h 的膝關節力乘以 0.7,分 101 步施加。顏色是所選癒合窗為該點預測的組織。

Cycle-mean volume fractions · 週期平均體積分率

02 · why the whole cycle

The load peak is not the average峰值載荷不等於平均

At the force peak (gait 0 %) only 49 % of the callus is in the bone zone; averaged over the whole cycle it is 79 %, because most of the cycle is low load. The moment you evaluate changes the answer: Claes & Heigele (1999) and Simon (2011) calibrated their windows at peak load.

在力的峰值(步態 0 %)只有 49 % 的骨痂在成骨區;對整個週期平均則是 79 %,因為週期大部分時間載荷都很低。評估哪個時刻,答案就不同:Claes & Heigele(1999)與 Simon(2011)的癒合窗是在峰值載荷下校準的。

03 · the healing window

Where the points sit in the window點落在癒合窗的哪裡

The strain plane (top right) shows volumetric strain against octahedral shear strain. The background is coloured by tissue (bone, cartilage, connective tissue, destructive); each white dot is one point of the callus. The plane shows the boundary chosen in the Window menu: the Simon rectangle (solid line) or the ellipse (dashed line), with the background coloured by that boundary's regions. Switch between the two to see which points change tissue. The Scale window limits slider scales every limit.

右上的應變平面:橫軸體積應變,縱軸八面體剪應變。背景依組織分色(骨、軟骨、結締組織、破壞),每個白點是骨痂上的一個點。平面只顯示 Window 選單所選的邊界:Simon 矩形(實線)或橢圓(虛線),背景也依該邊界的分區上色。在兩者之間切換,就能看出哪些點的組織分類改變。「Scale window limits」滑桿等比例縮放所有界線。

04 · the main finding

Only a narrow fan of the window is used癒合窗只有一個窄扇形被用到

A soft layer between stiff fragments is pushed and pulled almost in one direction only, so each point slides back and forth along a straight line in the strain plane. Poisson's ratio fixes the angle of that line (about 79° and 101° for a ratio of 0.167). Across all 278 models, 90 % of the exposure lies between about 72° and 104°.

夾在堅硬骨塊之間的軟組織幾乎只受單一方向的拉壓,所以每個點在應變平面上沿一條直線來回移動,這條直線的角度由泊松比決定(泊松比 0.167 時約 79° 與 101°)。在全部 278 個模型中,90 % 的暴露落在約 72° 到 104° 之間。

Live: how much callus volume and time falls at each direction angle; dashed lines mark the two straight-line directions · 即時:各方向角度上累積的骨痂體積與時間;虛線標出兩條直線方向

05 · cycle summary

Time spent in the bone zone停留在成骨區的時間

Set Colour to “Cycle: time in Bone zone”. A point that is bone on average can still spend a third of the cycle in cartilage or connective stimulus. Rotate the cloud to see where those points are.

把「Colour」設為 “Cycle: time in Bone zone”。平均看起來是「骨」的點,仍可能有三分之一的時間處於軟骨或結締組織刺激。旋轉點雲看看這些點在哪裡。

06 · weight bearing

Load fraction moves the window負重比例會移動癒合窗結果

The Load slider sets the fraction of the healthy-subject force (50–100 %). The solve is linear, so every point is reclassified live. This card uses 100 %; above 70 % the callus rotates beyond the range the linear analysis was validated for, so treat it as illustration.

「Load」滑桿設定健康受試者載荷的比例(50–100 %)。求解是線性的,每個點即時重新分類。此卡片用 100 %;超過 70 % 骨痂旋轉已超出線性分析驗證的範圍,僅供示意。

07 · walking speed

Speed changes the shape of the cycle步速改變的是週期的形狀

The validated speeds (1, 1.5 and 2 km/h) differ mainly in the shape of the load history, not its peak (2.07–2.38 kN). Faster speeds can be selected in the Speed menu but are outside the linear range: illustration only.

已驗證的步速(1、1.5、2 km/h)主要差在載荷歷程的形狀,而不是峰值(2.07–2.38 kN)。Speed 選單也可選更快的步速,但已超出線性範圍,僅供示意。

08 · other fractures

Same load, another fracture同樣的載荷,不同的骨折

In the Model row pick a series (grouped by family), then step with ◀ ▶ or the arrow keys, or press Play models to sweep the series back and forth. Only the model changes: gait step, window, speed, load, colour and view stay. With same scale on, one camera scale is shared so the growth of the callus is visible. This card loads wedge 65°. For all 17 model types side by side, see chart 1 of Sweep.

在「Model」列選一個系列(依家族分組),用 ◀ ▶ 或鍵盤方向鍵一步一步換,或按 Play models 讓系列來回自動播放。只有模型改變:步態、癒合窗、步速、負重、顏色和視角都不動。勾選 same scale 時整個系列共用同一個相機比例,骨痂的大小變化看得出來。此卡片載入楔形 65°。想把 17 種模型類型並排比較,請看「Sweep」的圖 1。

09 · other windows

Same callus, another window同一個骨痂,不同的癒合窗

A healing window is two independent choices, so the table is a grid. Coordinates (the three groups) are how volume change and distortion are computed for each point: from the small strain; from the principal stretches (Version B, V and D, rigid rotation ignored); or from the full deformation gradient (rigid rotation counts as deformation). Boundary (the two rows in each group) is the Simon rectangle or the ellipse with the same limits (bone 0.85 % × 7 %, outer 5 % × 16 %), see The healing window. Claes M0 has other, asymmetric limits and is listed apart as a reference.

Two numbers answer two questions. Difference (third row of each group): how much of the callus the two boundaries classify differently in the same coordinates, the effect of the shape. vs small strain: how much the same boundary changes when the coordinates change, the effect of the coordinates. Click a row to show it on the cloud: Rectangle, Ellipse, or Difference, which keeps only the points the two classify differently.

癒合窗有兩個互相獨立的選擇,所以表格排成格子。座標(三個群組)是每個點的體積變化與畸變怎麼算:用小應變;用主伸長(Version B 的 V 與 D,忽略剛體旋轉);或用完整的變形梯度(剛體旋轉也算成變形)。邊界(每組的兩列)是 Simon 矩形,或界線相同的橢圓(骨 0.85 % × 7 %,外圈 5 % × 16 %),見「癒合窗」一節。Claes M0 的界線不同而且不對稱,單獨列在最後當參考。

兩個數字回答兩個問題。Difference(每組的第三列):同一組座標下,兩種邊界分類不同的骨痂比例,也就是形狀的影響。vs small strain:同一種邊界換了座標後分類改變的比例,也就是座標的影響。點選任一列即可在點雲上顯示:Rectangle、Ellipse,或 Difference(只留下兩者分類不同的點)。

Three views per coordinate system: click Rectangle, Ellipse or Difference. Difference shows only the part of the callus the two boundaries classify differently; its bar is the make-up of that part by the ellipse's tissue. All percentages are shares of the callus volume.

The healing window

From strain space to finite-deformation invariants從應變空間到有限變形不變量

The bench above sorts every point of the callus with a healing window. This section explains that window: where its boundary comes from, why an ellipse is worth testing against the usual rectangle, how both boundaries are carried from strain space into finite-deformation invariants, and what the two look like when drawn together on the same axes.

上面的工作台用癒合窗為骨痂的每個點分類。這一節說明這個窗口:邊界從哪裡來、為什麼值得拿橢圓和常用的矩形比較、兩種邊界如何從應變空間帶到有限變形不變量,以及把兩者畫在同一組座標上是什麼樣子。

01 · Literature baseline

The Claes healing windowClaes 癒合窗:文獻基線

Claes and Heigele established the threshold-window logic that links local mechanics to tissue outcome: below some strain and hydrostatic limits a point forms bone, in a wider band cartilage or connective tissue, and beyond that it is destructive. Later models changed the mechanical coordinates and the biological rules but kept the idea of a window.

Claes 與 Heigele 建立了「閾值窗口」的邏輯,把局部力學與組織結果連起來:應變與靜水成分在某個範圍內長骨,更寬的範圍長軟骨或結締組織,超出則為破壞區。後續模型改了力學座標和生物規則,但都保留了「窗口」這個概念。

This study uses the symmetric dilatational–distortional version, so that the two boundary shapes can be compared under identical conditions.

本研究採用對稱的「體積–畸變」版本,好讓兩種邊界形狀能在完全相同的條件下比較。

Bone window

\(|v|\le 0.85\,\%,\ \ s\le 7\,\%\)

Inner threshold: bone formation.

內層閾值:成骨。

Outer window

\(|v|\le 5\,\%,\ \ s\le 16\,\%\)

Cartilage on the compressive side (\(v\lt -0.85\,\%\)), connective tissue elsewhere; outside is destructive.

壓縮側(\(v\lt -0.85\,\%\))為軟骨,其餘為結締組織;超出則破壞。

\(v\) = volumetric (hydrostatic) strain, \(s\) = octahedral shear strain, \(s\ge 0\). The Claes thresholds are anchors; the symmetric version is a study choice, not a claim about Claes.

\(v\)=體積(靜水)應變,\(s\)=八面體剪應變,\(s\ge 0\)。Claes 閾值是錨點;對稱版本是本研究的選擇,並非 Claes 原文的主張。

02 · Boundary hypothesis

Why introduce an ellipse, and how is it constructed?為什麼要引入橢圓?怎麼構造?

A rectangle treats volume and shear as independent: a point is accepted if \(|v|\) and \(s\) each stay under their own limit. That gives sharp corners. A point at the corner of the bone box, with the largest allowed volume change and the largest allowed shear at the same time, passes, even though it is the most demanding state in the box. An ellipse couples the two: the more volume change a point has, the less shear it is allowed.

矩形把體積與剪切視為互相獨立:\(|v|\) 和 \(s\) 各自不超過自己的上限就通過,所以有尖角。位在骨區方框角上的點,同時有最大的允許體積變化與最大的允許剪切,卻照樣通過,儘管它正是方框內最苛刻的狀態。橢圓則把兩者耦合起來:體積變化越大,允許的剪切就越小。

  1. Start from the \(L_p\) family of norms從 \(L_p\) 範數族出發
    \[ r_p=\Big(\Big|\frac{v}{a}\Big|^{p}+\Big(\frac{s}{b}\Big)^{p}\Big)^{1/p}\qquad \text{window: } r_p\le 1 \]
  2. Take the two limiting cases取兩個極限情形
    \[ \begin{aligned} p\to\infty:&\quad r_\infty=\max\Big(\frac{|v|}{a},\,\frac{s}{b}\Big)\le 1 &&\Rightarrow\ \text{rectangle}\\ p=2:&\quad r_2^{2}=\Big(\frac{v}{a}\Big)^{2}+\Big(\frac{s}{b}\Big)^{2}\le 1 &&\Rightarrow\ \text{ellipse} \end{aligned} \]

    The rectangle is the \(p\to\infty\) limit: independent thresholds, sharp corners. The ellipse is the \(p=2\) test case: smooth coupling.

    矩形是 \(p\to\infty\) 的極限(獨立閾值、有尖角);橢圓是 \(p=2\) 的檢驗情形(平滑耦合)。

  3. Keep the same Claes anchors保留同樣的 Claes 錨點
    \[ \begin{aligned} &\text{inner: } a=0.85\,\%,\ b=7\,\%\qquad \text{outer: } a=5\,\%,\ b=16\,\%\\ &r_2=1\ \Rightarrow\ s_{\max}(v)=b\sqrt{1-\Big(\frac{v}{a}\Big)^{2}} \end{aligned} \]

    The ellipse passes through the same axis intercepts as the rectangle, so the two differ only in how they cut the corners.

    橢圓與矩形有相同的軸截距,所以兩者只差在「怎麼切掉角落」。

The ellipse is introduced as an \(L_p\)-sensitivity hypothesis, not as an established Claes-family boundary. The cartilage / connective split on the ellipse is a ray at \(\varphi=99.8^{\circ}\); it ends on the outer ellipse exactly at \(v=-0.85\,\%\), the same \(v\) where the rectangle separates cartilage from connective tissue.

橢圓是作為 \(L_p\) 敏感度假設引入的,並不是已確立的 Claes 家族邊界。橢圓上軟骨/結締組織的分界是一條 \(\varphi=99.8^{\circ}\) 的射線,它終止在外橢圓上 \(v=-0.85\,\%\) 的那一點,正好是矩形區分軟骨與結締組織的同一個 \(v\) 值。

03 · Derivation

From principal strain to invariants從主應變到不變量

A window defined on \((v,\,s)\) is a statement about strain. To use it with a finite-deformation solver it has to be restated through the deformation itself. The chain below follows the 0610 whiteboard derivation; every later figure is generated from it.

定義在 \((v,\,s)\) 上的窗口是關於「應變」的陳述。要配合有限變形求解器使用,必須改用變形本身來重述。下面這條推導鏈來自 0610 白板推導,之後每一張圖都由它產生。

  1. Principal-axis kinematics主軸運動學
    \[ \mathbf F=\mathrm{diag}\big(1+\lambda_1,\ 1+\lambda_2,\ 1+\lambda_3\big) \]
  2. Right Cauchy–Green tensor右 Cauchy–Green 張量
    \[ \mathbf C=\mathbf F^{\mathsf T}\mathbf F=\mathrm{diag}\big((1+\lambda_1)^2,\ (1+\lambda_2)^2,\ (1+\lambda_3)^2\big) \]
  3. Volume invariant體積不變量
    \[ J=\det\mathbf F=\prod_i(1+\lambda_i)\qquad \mathrm{III}_C=\det\mathbf C=J^{2} \]
  4. Remove the volume from the distortion把體積從畸變中扣除
    \[ \begin{aligned} &\bar{\mathbf F}=J^{-1/3}\,\mathbf F\ \Rightarrow\ \bar{\mathbf C}=J^{-2/3}\,\mathbf C\ &\Rightarrow\ \mathrm{II}_{\bar C}=J^{-4/3}\,\mathrm{II}_C=\mathrm{III}_C^{-2/3}\,\mathrm{II}_C \end{aligned} \]

    For pure dilation \(\bar{\mathbf C}=\mathbf I\), so \(\mathrm{II}_{\bar C}=3\): the isochoric second invariant measures distortion only.

    純膨脹時 \(\bar{\mathbf C}=\mathbf I\),故 \(\mathrm{II}_{\bar C}=3\):等容第二不變量只衡量畸變。

  5. Recover the Claes coordinates還原回 Claes 座標
    \[ \begin{aligned} &V=\frac{J-1}{3}\approx\varepsilon_{\mathrm{vol}}\qquad\ \ D=\sqrt{\tfrac34\big(\mathrm{II}_{\bar C}-3\big)}\approx\varepsilon_{\mathrm{shear}}\\ &J=1+3\varepsilon_{\mathrm{vol}}+O(\varepsilon^{2})\qquad \mathrm{II}_{\bar C}-3=\tfrac43\,\varepsilon_{\mathrm{shear}}^{2}+O(\varepsilon^{3}) \end{aligned} \]

    In the small-strain limit \((V,\,D)\) collapse onto the Claes \((v,\,s)\), which is what lets the old thresholds carry over.

    小應變極限下 \((V,\,D)\) 與 Claes 的 \((v,\,s)\) 重合,舊閾值因此可以沿用。

Finite-deformation path

To draw the finite panels, each \((v,\,s)\) is sent to principal stretches along an axisymmetric path that preserves \(v\) and \(s\), then through the chain above:

畫有限變形的圖時,每個 \((v,\,s)\) 沿一條保持 \(v\)、\(s\) 的軸對稱路徑對應到主伸長,再走上面的推導鏈:

\[ \begin{aligned} &\mu_1=1+v+\frac{2s}{3},\qquad \mu_2=\mu_3=1+v-\frac{s}{3}\\ &J=\mu_1\mu_2^{2},\quad \mathrm{III}_C=J^{2},\quad \mathrm{II}_C=2\mu_1^{2}\mu_2^{2}+\mu_2^{4},\quad \mathrm{II}_{\bar C}=J^{-4/3}\,\mathrm{II}_C \end{aligned} \]
04 · Same coordinates, same conditions

Rectangle and ellipse in one picture矩形與橢圓畫在同一張圖

Both windows are defined on the same \((v,\,s)\) plane with the same anchors. So in every projection they are the same boundary curves pushed through the same map, and can be drawn on one set of axes without approximation. Shading follows the ellipse; the rectangle is laid over it.

兩個窗口定義在同一個 \((v,\,s)\) 平面、用同樣的錨點。所以在每一種投影裡,它們都是同一組邊界曲線經過同一個映射,可以不加近似地畫在同一組座標上。底色依橢圓分區;矩形疊在上面。

Solid實線 · rectangle矩形 Dashed虛線 · ellipse橢圓 cartilage軟骨 connective結締 bone骨 resorption吸收 destructive破壞

All six panels are drawn live in your browser from the formulas on this page. Lines are coloured by tissue: gold for the outer window, green for the bone window, blue for the cartilage divider; the switch above tints the regions as well. Hover a panel to read the strain, and what each window says about that same point. The raw finite invariant plane \((\mathrm{III}_C,\mathrm{II}_C)\) is left out: both windows collapse into one thin diagonal band there, so it carries no information about the boundary shape. It is kept only as a diagnostic control.

六張圖都由瀏覽器依本頁的公式即時繪製。線條依組織上色:金色是外圈窗口,綠色是骨窗口,藍色是軟骨分界;上面的開關可以另外加上區域底色。把滑鼠移到圖上可讀出該點的應變,以及兩個窗口對同一點的判斷。大變形原始不變量平面 \((\mathrm{III}_C,\mathrm{II}_C)\) 沒有放進來:兩個窗口在那裡都擠成一條細斜帶,看不出邊界形狀,只作為診斷對照保留。

A · Strain space \((v,\,s)\)

The source plane來源平面

Axes are the Claes coordinates themselves, in percent. The rectangle uses two independent limits, the ellipse the coupled norm \(r_2\le 1\).

座標軸就是 Claes 座標(百分比)。矩形用兩個獨立上限,橢圓用耦合範數 \(r_2\le 1\)。

What to look at看什麼

The ellipse lies inside the rectangle and touches it only at the axis intercepts. Everything in the four corners, such as \(|v|=5\,\%\) together with \(s=16\,\%\), is accepted by the rectangle and rejected by the ellipse.

橢圓整個在矩形裡面,只在軸截距處與它相切。四個角落(例如 \(|v|=5\,\%\) 同時 \(s=16\,\%\))矩形收、橢圓不收。

B · Small deformation, raw

Linearised invariants線性化不變量

\[ \begin{aligned} &J_{\mathrm{small}}=1+3\varepsilon_{\mathrm{vol}},\qquad \mathrm{III}_{\mathrm{small}}=J_{\mathrm{small}}^{2}\\ &\mathrm{II}_{\mathrm{small}}=3\varepsilon_{\mathrm{vol}}^{2}-J_2,\qquad (x,y)=(\mathrm{III}_{\mathrm{small}},\,\mathrm{II}_{\mathrm{small}}) \end{aligned} \]

Linearising \(\det(\mathbf I+\nabla\mathbf u)\) and splitting \(\boldsymbol\varepsilon=v\,\mathbf I+\boldsymbol\varepsilon\,{}^{\prime}\) turns the strain plane into a parabola-shaped region. The raw \(\mathrm{II}\) still carries a volume term.

把 \(\det(\mathbf I+\nabla\mathbf u)\) 線性化並拆成 \(\boldsymbol\varepsilon=v\,\mathbf I+\boldsymbol\varepsilon\,{}^{\prime}\),應變平面就變成拋物線形的區域。原始的 \(\mathrm{II}\) 仍帶有體積項。

C · Small deformation, isochoric

Distortion only只留畸變

\[ \mathrm{II}(\boldsymbol\varepsilon\,{}^{\prime})=-J_2,\qquad (x,y)=(\mathrm{III}_{\mathrm{small}},\,-J_2) \]

Dropping the volume term leaves a pure-distortion axis, so the rectangle's horizontal shear limits are horizontal lines again.

去掉體積項後,縱軸就只剩畸變,矩形的水平剪切上限又變回水平線。

What to look at看什麼

The ellipse boundary becomes a parabola that stays inside the rectangle's box, so the difference between the two shapes is clearest here.

橢圓邊界變成一條拋物線、落在矩形框內,兩種形狀的差異在這裡最清楚。

\(D\) · Finite, Version A

\((\mathrm{III}_C,\mathrm{II}_{\bar C})\): volume and distortion separated\((\mathrm{III}_C,\mathrm{II}_{\bar C})\):體積與畸變分開

\[ x=\mathrm{III}_C=J^{2},\qquad y=\mathrm{II}_{\bar C}=J^{-4/3}\,\mathrm{II}_C \]

Removing the volume factor separates the two roles, but the coordinates are not centred: the undeformed state sits at (1, 3), not at the origin.

扣除體積因子後,兩個角色分開了,但座標沒有置中:未變形狀態在 (1, 3),不在原點。

What to look at看什麼

The rectangle's top edge is no longer horizontal: the map is non-linear, so a constant shear limit tilts with volume. The ellipse stays a smooth dome inside it.

矩形的上緣不再水平:映射是非線性的,恆定的剪切上限會隨體積傾斜。橢圓仍是它裡面一個平滑的圓頂。

E · Finite, Version C

\((J,\mathrm{II}_{\bar C})\): the Jacobian as volume axis\((J,\mathrm{II}_{\bar C})\):以 Jacobian 為體積軸

\[ x=J=\sqrt{\mathrm{III}_C}=\mu_1\mu_2^{2},\qquad y=\mathrm{II}_{\bar C} \]

Taking the positive square root gives a volume axis with a direct physical meaning, \(J\) = deformed volume ÷ original volume. The distortion axis is unchanged.

取正的平方根,得到一條有直接物理意義的體積軸,\(J\)=變形後體積 ÷ 原體積;畸變軸不變。

F · Finite, Version B

\((V,\,D)\): centred, directly comparable with Claes\((V,\,D)\):置中,可直接對照 Claes

\[ V=\frac{J-1}{3}\to\varepsilon_{\mathrm{vol}},\qquad D=\sqrt{\tfrac34\big(\mathrm{II}_{\bar C}-3\big)}\to\varepsilon_{\mathrm{shear}} \]

Centring puts the undeformed state at the origin and makes the axes mean the same thing as the Claes strains, so this is the plane most directly comparable with panel A.

置中後,未變形狀態在原點,座標軸的意義與 Claes 應變一致,所以這是與圖 A 最能直接對照的平面。

What to look at看什麼

The picture is panel A again with a slight shear bend: the rectangle's side edges lean and its top edge slopes, while the ellipse stays inside. The ordering, ellipse within rectangle, survives the finite map.

這張圖就是圖 A 加上輕微的剪切彎曲:矩形的側邊微斜、頂邊傾斜,橢圓仍在裡面。「橢圓在矩形之內」這個次序在有限變形映射下保持不變。

05 · Status

What is established and what is open已確認與尚未確認

Completed
  • Consistent small- and finite-deformation derivation小變形與有限變形推導一致
  • Three finite coordinate versions (A, B, C)三種有限變形座標(A、B、C)
  • Mechanics mapping checked on canonical paths: simple shear, pure shear, pure dilation, shear with volume change力學映射已在典型路徑上驗證:簡單剪切、純剪切、純膨脹、剪切疊加體積變化
Current limits
  • Finite panels use an axisymmetric proxy有限變形的圖用軸對稱代理路徑
  • They are 2D mechanics projections, not exact Gauss-point states它們是二維力學投影,不是精確的高斯點狀態
  • The ellipse is a sensitivity hypothesis; the biological boundary is not established橢圓是敏感度假設;生物學邊界尚未確立
Next simulation step
  • Export \(\mathbf F\), \(\mathbf C\) or true principal stretches from the solver從求解器輸出 \(\mathbf F\)、\(\mathbf C\) 或真實主伸長
  • Compute \(J\), \(\mathbf C\), \(\mathrm{II}_{\bar C}\), \(V\) and \(D\) per integration point逐積分點計算 \(J\)、\(\mathbf C\)、\(\mathrm{II}_{\bar C}\)、\(V\)、\(D\)
  • Overlay the point cloud on these merged planes, then test sensitivity to the boundary shape把點雲疊到這些合併平面上,再檢驗對邊界形狀的敏感度

Current claim: the mechanics mapping is verified on canonical paths; the biological boundaries and real Gauss-point overlays remain open.

目前的主張:力學映射已在典型路徑上驗證;生物學邊界與真實高斯點疊圖仍待完成。

The model

A nailed tibia, 278 ways to break it一根打了髓內釘的脛骨,278 種斷法

Every model is a bare tibia with an intramedullary nail and a callus generated from the fracture gap. All materials are linear elastic and there is no contact. The ankle end is fixed; the gait force is split evenly over the 1,907 nodes of the tibial plateau, which is otherwise free.

每個模型都是一根帶髓內釘的裸脛骨,骨痂由骨折間隙生成。所有材料為線彈性,沒有接觸。踝端固定;步態力平均分配到脛骨平台的 1,907 個節點上,平台其他方向自由。

The fibula is left out on purpose. The large plateau displacement is a consequence of that choice, not a modelling error; models with a fibula are a separate, later study.

腓骨是刻意不放的。平台位移大是這個選擇的結果,不是建模錯誤;含腓骨的模型是之後另一個研究。

The force is the knee joint contact force from Xsens motion capture through a musculoskeletal model, so muscle forces are included. That is why it peaks at several times body weight.

力是由 Xsens 動作捕捉經肌肉骨骼模型得到的膝關節接觸力,已包含肌肉力,所以峰值可達體重的數倍。

Part · 部件E [MPa]ν
Callus骨痂10.167
Bone, low density低密度骨7,5000.30
Bone, high density高密度骨18,0000.30
Nail髓內釘108,0000.37

341,715 nodes · 236,253 C3D10 elements per mesh (reference model). The four fracture families and their parameters are listed in Sweep.每個網格 341,715 節點、236,253 個 C3D10 元素(參考模型)。四個骨折家族與它們的參數列在「Sweep」一節。

Pipeline

Solve three times, walk any cycle只解三次,就能走任何週期

Because the analysis is linear and the load pattern is fixed, every field at every instant is a weighted sum of three unit-load solutions. One 50-second job per model replaces 101 separate steps, and speed, weight bearing and step count become post-processing.

因為分析是線性的、載荷分布固定,任一時刻的任一個場都是三個單位載荷解的加權和。每個模型只需一個約 50 秒的計算,取代 101 個獨立載荷步;步速、負重與步數都變成後處理。

u(t) = Fx(t)·ux + Fy(t)·uy + Fz(t)·uzsame for ε, σ and F − I · ε、σ、F − I 同理
  1. BuildAppend 3 unit steps (1 kN in x, y, z) to the mesh; nail set to 108,000 MPa.在網格後加三個單位載荷步(x、y、z 各 1 kN)。1–2 s
  2. SolveAbaqus, one matrix decomposition, 6 jobs × 4 cores in parallel.Abaqus 只分解一次矩陣,6 個 4 核工作並行。≈ 50 s / model
  3. Extract104 columns per integration point, including det F from nodal displacements.每個積分點 104 欄,含由節點位移算出的 det F。≈ 4 s
  4. SuperposeAny force history × 0.7, rolled so the peak is at 0 %: 3 speeds × 101 steps.任意力歷程 × 0.7,峰值對齊 0 %:3 種步速 × 101 步。≈ 40 s / model
  5. Classify + QASimon (primary) and Claes M0 windows; equilibrium, shape functions, det F > 0 gates.Simon(主)與 Claes M0 癒合窗;平衡、形函數、det F > 0 檢查。seconds

The full sweep ran 278 models × 3 speeds = 834 scenario rows in 138 minutes of wall time. With the calibrated force direction (October 2026), 275 models pass every gate; 3 spiral models are flagged because a tiny part of their callus (≤ 0.002 %) inverts. The bench above repeats steps 4 and 5 in your browser and reproduces the sweep index for the reference model to 1e-16.

完整 sweep 計算 278 個模型 × 3 種步速 = 834 筆情境,實際耗時 138 分鐘。用校準後的力方向(2026 年 10 月),275 個模型通過所有檢查;3 個螺旋骨折模型被標記,因為極小部分骨痂(≤ 0.002 %)發生翻轉。上面的互動區在瀏覽器裡重做了第 4、5 步,對參考模型的結果與 sweep 索引一致到 1e-16。

Sweep

What changes with the fracture骨折形態改變了什麼

Every mesh belongs to one model type: a fracture family (cut, wedge, spiral, shapes) with one varied parameter, 17 types in all. Choose a metric and a speed once in the bar below. The charts then show that metric three ways: along the parameter of each type, over the force of the gait cycle, and against callus volume.

每個網格都屬於一種模型類型:一個骨折家族(cut、wedge、spiral、shapes)加上一個變動的參數,共 17 種。在下面的工具列選一次指標與步速,後面的圖就用三種方式呈現它:沿每種類型的參數、沿步態週期的力、以及對骨痂體積。

Family · 家族meshesparameters swept · 掃描參數
cut61osteotomy angle AP, LM; position截骨角度 AP、LM;位置
wedge76wedge angle, position, rotation楔形角度、位置、旋轉
spiral107length, radial power, rotation AP/LM, phase, translation長度、徑向形狀、旋轉、相位、平移
shapes34statistical shape modes 1–5, ±3 SD統計形狀模態 1–5,±3 SD

278 files hold 269 distinct geometries, and the series do not share one baseline.278 個檔案實際是 269 種不同幾何,各系列的基準也不相同。

Callus (red) on the nailed tibia for each parameter series at its minimum, zero and maximum value

Callus (red) on the tibia and nail for every series at its lowest, zero and highest parameter value. Scroll sideways.每個系列在參數最小、零與最大值時的骨痂(紅色)。可左右捲動。

1 · Along the parameter of each model type

One chart per model type; the x-axis is the varied parameter. Tissue is split into two groups: bone + cartilage, the positive stimulus (blue), and the rest, that is connective, resorption and destructive (orange). Δ is the highest minus the lowest value along the parameter. Click a dot to load that model in the bench; click a title to open the type in chart 2.

每種模型類型一張圖,橫軸是變動的參數。組織只分成兩群:骨+軟骨,即正向刺激(藍線);其餘,即結締組織、吸收、破壞(橘線)。Δ 是沿參數的最高值減最低值。點圓點可把該模型載入互動區;點標題可在圖 2 打開該類型。

2 · Over the force of the gait cycle

The force sweep is the gait cycle itself: 101 steps, each with its own force. Every line is one model of the chosen type; each of its 101 steps sits at the force of that step (x) and the share of callus volume in the bone or cartilage zone at that step (y). This chart always shows bone + cartilage; the speed follows the bar above. Colour is the value of the varied parameter. Cycle Δ is the highest minus the lowest share within one cycle, averaged over the models of the type. Hover a line for the model, click to load it in the bench at that step.

這裡的力掃描就是步態週期本身:101 步,每一步有自己的力。每條線是所選類型裡的一個模型;它的 101 步各自畫在該步的力(橫軸)與該步落在骨或軟骨區的骨痂體積比例(縱軸)上。這張圖固定顯示骨+軟骨,步速跟著上面的工具列。顏色代表變動參數的數值。Cycle Δ 是一個週期內最高比例減最低比例,再對該類型的所有模型取平均。滑鼠移到線上可看模型,點一下會把該模型載入互動區並停在那一步。

3 · Against callus volume, all models

Each dot is one mesh; hover for its name, click to load it. The callus is generated from the fracture gap and its size varies tenfold. Callus size explains a large part of the between-type spread in bone + cartilage (r shown above for the visible set). Larger calluses spread the same load over more tissue, so they look more bone-like even when the gap moves more. Fracture types therefore have to be compared alongside volume, or volume-normalised.

每個點是一個網格,滑鼠移上去可看名稱,點一下可載入。骨痂由骨折間隙生成,大小相差十倍。骨痂大小解釋了不同骨折類型之間骨+軟骨比例的一大部分差異(上方顯示目前可見資料的 r)。較大的骨痂把相同載荷分散到更多組織上,即使間隙位移較大也顯得更像骨。因此比較骨折類型時必須同時報告體積,或做體積標準化。

Strain direction per model · 各模型的應變方向

Each bar is one model: the 5th to 95th percentile of θ over the whole cycle, dot at the median, sorted by median. No model leaves the band 69.4°–108.0°. The two dashed lines are the uniaxial rays for ν = 0.167.

每條細線是一個模型:整個週期中 θ 的第 5 到第 95 百分位,點為中位數,按中位數排序。沒有任何模型超出 69.4°–108.0°。兩條虛線是 ν = 0.167 時的單軸射線。

Validity

What holds, and where it stops哪些成立,界線在哪

verified

Numerics數值驗證

Global equilibrium, worst step全域平衡最差步2e-8
Shape functions vs COORD形函數 vs COORD1.5e-5 mm
Ws = ½σ:ε vs SENER5e-8
Superposition vs direct, classes疊加 vs 直接,分類0 / 3.4 M
caveat

Geometric linearity幾何線性假設

Accepted on the reference model, where 2.4 % of the callus rotates more than 10°. Across the sweep (calibrated force), 43 models exceed 10 % of the callus and 3 exceed 25 %. Rankings within a series are safe; absolute fractions above 25 % need NLGEOM checks.

在參考模型上成立(2.4 % 的骨痂旋轉超過 10°)。整個 sweep 中(校準後的力),43 個模型超過 10 % 的骨痂,3 個超過 25 %。系列內排序可信;超過 25 % 的模型其絕對比例需做 NLGEOM 檢驗。

caveat

Window and material癒合窗與材料

Simon's thresholds were calibrated with a 3 MPa, ν = 0.30 callus; these models use 1 MPa, ν = 0.167. The choice of window moves Resorption by 0.06, more than the strain measure does. Cartilage and Connective fractions are for ranking only.

Simon 閾值是用 3 MPa、ν = 0.30 的骨痂校準的;本模型用 1 MPa、ν = 0.167。癒合窗的選擇讓吸收比例變動 0.06,比應變度量的影響更大。軟骨與結締組織比例只用於排序。

consistent

Abaqus vs COMSOLAbaqus 與 COMSOL 比較

Different meshes (parametric wedge vs the FKA patient tibia), so the comparison is behavioural. Both put the resorption spike at 20–40 % of gait, where Fz passes through zero, and both are bone-dominated under load.

網格不同(參數化楔形 vs FKA 病人脛骨),因此比的是行為。兩者都把吸收高峰放在步態 20–40 %(Fz 過零處),且在受載時都以骨為主。

Next

Open questions待解問題

Ordered by how much each would change the argument.

依對論證影響大小排序。