### Predetermined conditions and notations about the head and cup center measurement

For the purpose of evaluating the accuracy of the wear calculations, a set of generalized data of the wear measurement of a single hip was prepared in this study. It was assumed that the head center penetrated postoperatively in accordance with the principle of polyethylene wear.

The horizontal and vertical lines were defined as the *x*- and *y*-axes, respectively, and the coordinates of the head center relative to the cup center immediately after surgery was defined as (0, 0). The medial and proximal directions were defined as positive, and the lateral and distal directions as negative. Bedding-in was defined as \(\overrightarrow{b}=\left({b}_{x},{b}_{y}\right)\) and presumed to occur until the first postoperative follow-up (*t*_{1}). Wear was defined to progress steadily after surgery at a rate of \(\overrightarrow{w}=\left({w}_{x},{w}_{y}\right)\) per year (\(\overrightarrow{b}\) and \(\overrightarrow{w}\) were parallel, and \({b}_{y}>0, {w}_{y}>0\)). According to these definitions, the coordinates of the head center at postoperative period *t*_{k} (year) could be calculated as

$$\left({x}_{k},{y}_{k}\right)=\overrightarrow{b}+{t}_{k}\overrightarrow{w}=\left({b}_{x}+{t}_{k}{w}_{x},{b}_{y}+{t}_{k}{w}_{y}\right),$$

(1)

where measurements of the head center positions are presumed to be performed *n* + 1 times in this study (*k* = 0, 1, 2, …, *n*, *t*_{0} = 0) (Fig. 2).

Errors in measuring \(\left({x}_{k},{y}_{k}\right)\) were defined as \(\left({x}_k^{\prime },{y}_k^{\prime}\right)\). Then, \(\left({x}_k+{x}_k^{\prime },{y}_k+{y}_k^{\prime}\right)\) could be used as the coordinates measured at postoperative period *t*_{k}. For accurate evaluation of the following wear calculations, the systematic error was presumed to be eliminated from \(\left({x}_k^{\prime },{y}_k^{\prime}\right)\). Then \(\left({x}_k^{\prime },{y}_k^{\prime}\right)\) represent random errors in measuring \(\left({x}_{k},{y}_{k}\right)\), and

$$E({{x}^{\prime}}_{k})=E({{y}^{\prime}}_{k})=0,$$

(2)

where *E* is the expected value. The bedding-in and steady-state wear rate were calculated using the conventional and novel methods based on these definitions. Both calculation methods start after the penetration vector, and the measurement errors at each follow-up period were provided (Fig. 3).

$$\left({X}_{k},{Y}_{k}\right)=\left({x}_{k}+{x\mathrm{^{\prime}}}_{k}-{x\mathrm{^{\prime}}}_{0},{y}_{k}+{y\mathrm{^{\prime}}}_{k}-{y\mathrm{^{\prime}}}_{0}\right)$$

(3)

### Accuracy evaluation of the wear calculations using the predetermined measurement values

The accuracy of the wear calculation method was evaluated by comparing the predetermined true values of the bedding-in (\(\overrightarrow{b}\)) and wear rate (\(\overrightarrow{w}\)) with their expected values of the calculated results using \(\left({X}_{k},{Y}_{k}\right)\) (*k* = 0, 1, 2, …, *n*). The calculation method could be claimed as accurate when they were consistent [22].

Meanwhile, the best-fit line for multiple points \(\left({u}_{k},{v}_{k}\right)\) (*k* = 1, 2, …, *n*) is \(y=ax+b\), *a* and *b* can be calculated as follows using the least-squares method [17, 23].

$$\begin{array}{l}a=\frac{n\sum_{k=1}^{n}{u}_{k}{v}_{k}-\sum_{k=1}^{n}{u}_{k}\sum_{k=1}^{n}{v}_{k}}{C}.\\ b=\frac{\sum_{k=1}^{n}{({u}_{k})}^{2}\sum_{k=1}^{n}{v}_{k}-\sum_{k=1}^{n}{u}_{k}{v}_{k}\sum_{k=1}^{n}{u}_{k}}{C}.\\ \left(C=n{\sum }_{k=1}^{n}{({u}_{k})}^{2}-\left({\sum }_{k=1}^{n}{{u}_{k}}\right)^{2}\right)\end{array}$$

These formulae can be represented more simply as follows:

$$a={\textstyle\sum_{k=1}^n}M_kv_k$$

(4)

$$b={\textstyle\sum_{k=1}^n}N_kv_k$$

(5)

where \({M}_{k}=\frac{n{u}_{k}-\sum_{l=1}^{n}{u}_{l}}{C}, {N}_{k}=\frac{\sum_{l=1}^{n}{\left({u}_{l}\right)}^{2}-{u}_{k}\sum_{l=1}^{n}{u}_{l}}{C}.\)

### Conventional method

Conventional wear calculations start by transforming the penetration vectors with measurement errors into scalar. Negative wear was used intact in this study because it seemed the most popular option in previous studies [18,19,20,21]. Therefore, the penetration at *t*_{k} years \(({P}_{k})\) was calculated as:

$${P}_{k}=sgn\left({Y}_{k}\right)\sqrt{{\left({X}_{k}\right)}^{2}+{\left({Y}_{k}\right)}^{2}},$$

(6)

where \(sgn\left({Y}_{k}\right)=1\) when \({Y}_{k}\geqq 0\), and \(sgn\left({Y}_{k}\right)=-1\) when \({Y}_{k}<0.\) Using the least-squares method for linear regression (Eqs. (4) and (5)), the steady-state wear rate and bedding-in (\({W}_{c}\) and \({B}_{c}\)) were calculated as follows (Fig. 4).

$$\begin{array}{c}W_c=\sum_{k=1}^nM_kP_k\\B_c=\sum_{k=1}^nN_kP_k\end{array}$$

$$(M_k=\frac{nt_k-\sum_{l=1}^nt_l}C,N_k=\frac{\sum_{l=1}^n\left(t_l\right)^2-t_k\sum_{l=1}^nt_l}C,C=n{\textstyle\sum_{k=1}^n}{(t_k)}^2-\left({\textstyle\sum_{k=1}^n}t_k\right)^2)$$

Therefore, we were able to conclude that the conventional method was accurate when the following equations were satisfied.

$$E\left({\textstyle\sum_{k=1}^n}M_kP_k\right)={\textstyle\sum_{k=1}^n}M_kE\left(P_k\right)=\left|\overrightarrow w\right|$$

(7)

$$E\left({\textstyle\sum_{k=1}^n}N_kP_k\right)={\textstyle\sum_{k=1}^n}N_kE\left(P_k\right)=\left|\overrightarrow b\right|$$

(8)

When all measurements were performed without an error, that is, \({P}_{k}=\left|\overrightarrow{b}+{t}_{k}\overrightarrow{w}\right|=\sqrt{{\left({x}_{k}\right)}^{2}+{\left({y}_{k}\right)}^{2}},\) the calculated results were consistent with the true values (\({W}_{c}=\left|\overrightarrow{w}\right|, {B}_{c}=\left|\overrightarrow{b}\right|\)). Thus,

$$\begin{array}{c}\sum_{k=1}^nM_k\sqrt{\left(x_k\right)^2+\left(y_k\right)^2}=\left|\overrightarrow w\right|\\{\textstyle\sum_{k=1}^n}N_k\sqrt{\left(x_k\right)^2+\left(y_k\right)^2}=\left|\overrightarrow b\right|\end{array}$$

Because \({M}_{k}\) and \({N}_{k}\) could take any real values depending on the measurement period

$$E\left({P}_{k}\right)=\sqrt{{\left({x}_{k}\right)}^{2}+{\left({y}_{k}\right)}^{2}}$$

(9)

was necessary and sufficient for Eqs. (7) and (8) to be satisfied.

### Novel method

We propose a novel calculation method in which penetration vectors are used without the transformation before the linear regression. In this method, the *x* and *y* components of the wear rate and bedding-in (*W*_{x}, *B*_{x}, *W*_{y}, and *B*_{y}) were calculated separately. Best-fit lines for \(\left({t}_{k},{X}_{k}\right)\) and \(\left({t}_{k},{Y}_{k}\right)\) (*k* = 1, 2, …, *n*) were respectively defined as

$$\begin{array}{c}y={W}_{x}x+{B}_{x}\\ y={W}_{y}x+{B}_{y},\end{array}$$

where \({W}_{x}\), \({B}_{x}\), \({W}_{y}\), and \({B}_{y}\) could be calculated using Eqs. (4) and (5), as follows (Fig. 5).

$$W_x={\textstyle\sum_{k=1}^n}M_kX_k$$

(10)

$$B_x={\textstyle\sum_{k=1}^n}N_kX_k$$

(11)

$$W_y={\textstyle\sum_{k=1}^n}M_kY_k$$

(12)

$$B_y={\textstyle\sum_{k=1}^n}N_kY_k$$

(13)

$$(M_k=\frac{nt_k-\sum_{l=1}^nt_l}C,N_k=\frac{\sum_{l=1}^n\left(t_l\right)^2-t_k\sum_{l=1}^nt_l}C,C=n{\textstyle\sum_{k=1}^n}\left(t_k\right)^2-\left({\textstyle\sum_{k=1}^n}t_k\right)^2)$$

Their expected values were as follows.

$$E\left(W_x\right)={\textstyle\sum_{k=1}^n}M_kE\left(X_k\right).$$

(14)

$$E\left(B_x\right)={\textstyle\sum_{k=1}^n}N_kE\left(X_k\right).$$

(15)

$$E\left(W_y\right)={\textstyle\sum_{k=1}^n}M_kE\left(Y_k\right).$$

(16)

$$E\left(B_y\right)={\textstyle\sum_{k=1}^n}N_kE\left(Y_k\right).$$

(17)

Therefore, we could conclude that the novel method was accurate when these expected values were consistent with the predetermined true values, *w*_{x}, *b*_{x}, *w*_{y}, and *b*_{y}, respectively.