JP2014008193A - Magnetic resonance apparatus and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To improve the accuracy of detecting the edge position of the liver.SOLUTION: Concerning each of navigator data, the edge position of the liver is detected using thresholds th (ef=0.25, 0.33 and 0.5) to create an edge position profile showing a temporal change of the detected edge position. Subsequently, an index showing the accuracy of detecting the edge position is calculated for each edge position profile, and the value of an edge coefficient used in main scanning is determined based on the value of the index.

Description

  The present invention relates to a magnetic resonance apparatus that images a body moving part by breathing, and a program applied to the magnetic life unit.

  A navigator gating technique is known as a technique for imaging a subject using a respiratory synchronization method (see Patent Document 1).

JP 2009-254392 A

  In the navigator gating technique, a navigator sequence for detecting the position of a moving part (for example, the position of the edge of the liver) immediately before (or immediately after) an imaging sequence for acquiring imaging region data of a subject. Is executed. Then, only the data of the imaging region acquired when the detection position is included in the predetermined range is adopted as the data for image reconstruction. Therefore, ghosts and the like can be reduced.

  As a technique for detecting the position of the edge of the liver, for example, a technique by Du is known (Reference: Du YP, Saranathan M, Foo TK. An accurate, robust, and computationally efficient navigator algorithm for measuring diaphragm positions. Cardiovasc Magn Reson. 2004; 6: 483-490). Although the detection method described in this reference has high calculation efficiency, an error in the edge detection position may increase depending on conditions for acquiring navigator echoes. When the edge position detection error increases, it becomes difficult to reduce body movement artifacts. Therefore, it is desired to provide a method with excellent edge position detection accuracy.

The first aspect of the present invention has a plurality of navigator sequences for obtaining navigator data including position information of the edge of the first part from a navigator area including the first part and the second part that move. Pre-scan,
A main scan having a navigator sequence for acquiring navigator data including position information of an edge of the first part from the navigator area, and an imaging sequence for acquiring data of an imaging area including a body moving part; ,
A magnetic resonance apparatus for performing
The signal value of the first part and the signal value of the second part in each navigator data acquired by the pre-scan is obtained, and the difference between the signal value of the first part and the second signal value And edge position detection means for detecting the position of the edge of the first part in each navigator data based on the coefficient multiplied by the difference, the value of the coefficient being changed, Edge position detecting means for detecting the position of the edge of the first part,
Determination means for determining the value of the coefficient used when detecting the position of the edge of the first part in the main scan based on the position of the edge of the first part detected for each value of the coefficient And
The edge position detecting means includes
The signal value of the first part and the signal value of the second part in the navigator data acquired by the main scan are obtained, and the difference between the signal value of the first part and the second signal value The magnetic resonance apparatus detects the position of the edge of the first part in the main scan based on the coefficient determined by the determining means.

The second aspect of the present invention has a plurality of navigator sequences for acquiring navigator data including position information of edges of the first part from a navigator area including the first part and the second part that move. Pre-scan,
A main scan having a navigator sequence for acquiring navigator data including position information of an edge of the first part from the navigator area, and an imaging sequence for acquiring data of an imaging area including a body moving part; ,
A magnetic resonance apparatus program for executing
The signal value of the first part and the signal value of the second part in each navigator data acquired by the pre-scan is obtained, and the difference between the signal value of the first part and the second signal value And edge coefficient detection processing for detecting the position of the edge of the first part in each navigator data based on the coefficient multiplied by the difference, the value of the coefficient being changed, Edge position detection processing for detecting the position of the edge of the first part,
Determination processing for determining the value of the coefficient used when detecting the position of the edge of the first part in the main scan based on the position of the edge of the first part detected for each value of the coefficient Is a program for causing a computer to execute
The edge position detection process includes
The signal value of the first part and the signal value of the second part in the navigator data acquired by the main scan are obtained, and the difference between the signal value of the first part and the second signal value A program for detecting the position of the edge of the first part in the main scan based on the coefficient determined by the determination process.

  Based on the edge position of the first part detected for each coefficient value, the value of the coefficient used when detecting the position of the edge of the first part in the main scan is determined. Therefore, the detection error of the position of the edge of the first part in the main scan can be reduced.

It is the schematic of the magnetic resonance apparatus of one form of this invention. It is explanatory drawing of the sequence performed with this form. It is a figure which shows an imaging | photography area | region schematically. It is explanatory drawing of the prescan A. FIG. The navigator data _D 1 to D _n obtained for each navigator sequence _P 1 to P _n is a diagram schematically showing. It is a figure which shows an edge position profile. 6 is an explanatory diagram of a main scan B. FIG. It is a figure which shows an example when the detection error of the edge position of a liver becomes large. It is a figure which shows the flow at the time of performing the prescan A and the main scan B, and acquiring the image data of the liver of the subject 11. FIG. The navigator data _D 1 to D _n obtained for each navigator sequence _P 1 to P _n is a diagram schematically showing. It is explanatory drawing of the calculation method of threshold value th. It is a figure which shows the edge position of the liver detected in the case of ef = 0.25. is a diagram showing an edge position profile _{E 2} in the case of ef = 0.33. is a diagram showing an edge position profile _{E 2} in the case of ef = 0.5. It is a figure which shows roughly an example of the range AW at the end of spitting. 6 is an explanatory diagram of a main scan B. FIG.

  Hereinafter, although the form for inventing is demonstrated, this invention is not limited to the following forms.

FIG. 1 is a schematic view of a magnetic resonance apparatus according to one embodiment of the present invention.
A magnetic resonance apparatus (hereinafter referred to as “MR apparatus”, MR: Magnetic Resonance) 100 includes a magnet 2, a table 3, a receiving coil 4, and the like.

  The magnet 2 includes a bore 21 in which the subject 11 is accommodated, a superconducting coil 22, a gradient coil 23, and an RF coil 24. The superconducting coil 22 applies a static magnetic field, the gradient coil 23 applies a gradient magnetic field, and the RF coil 24 transmits an RF pulse. In place of the superconducting coil 22, a permanent magnet may be used.

  The table 3 has a cradle 3a. The cradle 3a is configured to be able to move into the bore 21. The subject 11 is transported to the bore 21 by the cradle 3a.

  The reception coil 4 is attached to the abdomen of the subject 11. The receiving coil 4 receives a magnetic resonance signal from the subject 11.

  The MR apparatus 100 further includes a transmitter 5, a gradient magnetic field power source 6, a receiver 7, a control unit 8, an operation unit 9, a display unit 10, and the like.

The transmitter 5 supplies current to the RF coil 24, and the gradient magnetic field power supply 6 supplies current to the gradient coil 23.
The receiver 7 performs signal processing such as detection on the signal received from the receiving coil 4.

  The control unit 8 transmits necessary information to the display unit 10 and reconstructs an image based on data received from the receiver 7 so as to realize various operations of the MR device 100. Control the operation of each part. The control unit 8 is configured by, for example, a computer. The control unit 8 includes navigator data creation means 81 to edge coefficient determination means 83 and the like.

The navigator data creating means 81 creates navigator data including position information of the liver edge.
The edge position detection means 82 detects the edge position of the liver based on the navigator data.
The edge coefficient determining means 83 determines the value of an edge coefficient ef (see formula (1) described later) used when detecting the position of the edge of the liver in the main scan.

  The control unit 8 is an example constituting the navigator data creation means 81 to the edge coefficient determination means 83, and functions as these means by executing a predetermined program.

The operation unit 9 is operated by an operator and inputs various information to the control unit 8. The display unit 10 displays various information.
The MR apparatus 100 is configured as described above.

FIG. 2 is an explanatory diagram of a sequence executed in this embodiment, and FIG. 3 is a diagram schematically showing an imaging region.
In this embodiment, pre-scan A and main scan B are executed.
The pre-scan A is a scan for detecting the movement range in the SI direction of the upper edge of the liver, and the main scan B is a sequence for acquiring image data of a part including an organ such as the liver. Hereinafter, the pre-scan A and the main scan B will be described in order.

FIG. 4 is an explanatory diagram of the pre-scan A.
In prescan A, navigator sequence P ₁ to P _n for obtaining navigator echo from the navigator region R _n is performed. By the navigator echo obtained from the navigator region R _n, navigator data D including the position information of the liver edge is obtained. In this embodiment, a navigator sequence is used in which the lung side has a low signal and the liver side has a high signal, so that a step in signal value appears at the boundary between the lung and the liver, as shown in FIG. Therefore, by detecting the step of this signal value from the navigator data D, the edge position of the liver can be detected. In the pre-scan A, n navigator sequences P _{1 to} P _n are executed, so that navigator data representing the signal value in the SI direction of the navigator region R _n is obtained for each navigator sequence. FIG. 5 schematically shows navigator data D _{1 to} D _n obtained for each navigator sequence P _{1 to} P _n .

By detecting the edge position of the liver for each of the navigator data D _{1 to} D _n , it is possible to know how the edge position of the liver changes with time. Therefore, it is possible to obtain an edge position profile representing a temporal change in the edge position of the liver. FIG. 6 shows an edge position profile. The edge position profile is used to obtain a range AW of the breathing end of the subject 11. After executing pre-scan A, main scan B is executed.

FIG. 7 is an explanatory diagram of the main scan B.
In the main scan B, navigator sequences Q _{1 to} Q _m and imaging sequences IS _{1 to} IS _m are executed alternately. The navigator sequences Q _{1 to} Q _m are sequences for acquiring navigator echoes from the navigator region R _n as with the navigator sequences P _{1 to} P _n . The imaging sequences IS _{1 to} IS _m are sequences for acquiring k-space data from an imaging region including the liver.

In the main scan B, when the edge position of the liver detected in the navigator sequence is included in the range AW of the end of breathing exhalation, the k-space data acquired in the immediately subsequent imaging sequence is used for image reconstruction. Adopt as data. On the other hand, when the edge position of the liver detected by the navigator sequence is not included in the end-of-breathing end range AW, the k-space data acquired by the immediately following imaging sequence is the image reconstruction data. Not adopted. For example, in FIG. 7, since the liver edge position x ₁ detected by the navigator sequence Q ₁ is out of the range AW at the end of breathing exhalation, k-space data acquired by the imaging sequence IS ₁ immediately after that Is not adopted as image reconstruction data. Similarly, the edge position x ₂ of the liver detected by the navigator sequence Q ₂ also, since out of the range AW of end spitting breath data k-space acquired by the immediately following imaging sequence IS _2, the image It is not adopted as reconstruction data.

On the other hand, the edge position x ₃ of the liver detected by the navigator sequence Q ₃ are because they are included in the scope AW of end spitting breathing, data of the acquired k-space in the imaging sequence IS ₃ immediately thereafter, the image Adopted as reconstruction data.

  Accordingly, image reconstruction is performed using only k-space data acquired when the position of the liver edge is included in the end-of-breathing end range AW, so that body motion artifacts such as ghosts are reduced. Image data can be obtained.

  However, the navigator data may contain noise that adversely affects the detection accuracy of the liver edge. Such noise causes an increase in detection error of the edge position of the liver. FIG. 8 shows an example when the detection error of the edge position of the liver becomes large.

FIG. 8 shows a case where two signal intensity steps appear near the boundary between the lung and the liver due to the influence of noise. In Figure 8, an example right step x _a is the actual liver edge position is indicated. In this case, if the left step _xb is erroneously detected as the edge position of the liver, a detection error of Δx occurs. Therefore, even when the edge of the liver is actually out of the end-of-breathing end range AW, it may be determined that the liver is included in the end-of-breathing end range. In this case, since the k-space data acquired when the edge of the liver deviates from the end-of-breathing end range AW is used as image reconstruction data, it causes body motion artifacts.

In FIG. 8, although an example right step x _a is the actual liver edge position is indicated, in some cases, the left of the step x _b is the actual liver edge position. In this case, when the thus detected to the right of the step x _a as the edge position of the liver, again, cause motion artifacts.

  Therefore, in the present embodiment, even if the navigator data includes noise that adversely affects the detection accuracy of the liver edge, the detection method of the liver edge so that the detection error of the liver edge position is as small as possible. Is devised. Hereinafter, a liver edge detection method according to this embodiment will be described with reference to FIG.

  FIG. 9 is a diagram showing a flow when the pre-scan A and the main scan B are executed and the liver image data of the subject 11 is acquired.

In step ST1, pre-scan A (see FIG. 4) including navigator sequences P _{1 to} P _n is executed. The navigator echoes obtained by the navigator sequences P _{1 to} P _n are received by the receiving coil 4 (see FIG. 1), subjected to predetermined signal processing by the receiver 7, and output to the control unit 8. In the control unit 8, the navigator data creation unit 81 (see FIG. 1) creates navigator data including the position information of the liver edge for each navigator sequence P _{1 to} P _n based on the data received from the receiver 7. To do. FIG. 10 schematically shows navigator data D _{1 to} D _n obtained for each navigator sequence P _{1 to} P _n . After performing the pre-scan A, the process proceeds to step ST2.

In step ST2, the edge position detecting means 82 (see FIG. 1) detects the position of the edge of the liver for each navigator data D _{1 to} D _n . However, as described with reference to FIG. 8, the navigator data may include noise that adversely affects the detection accuracy of the liver edge. Therefore, in the present embodiment, even if such navigator data includes such noise, the edge of the liver is detected as follows so that the detection error of the edge position of the liver becomes as small as possible.

  First, a threshold value of a signal value for detecting the position of the liver edge is calculated. FIG. 11 shows a method for calculating the threshold th.

The threshold th is obtained by the following formula.
th = (max−min) * ef + min (1)

Where max: the maximum signal value of navigator data
min: Minimum signal value of navigator data
ef: edge coefficient (0 <ef <1)

  In this embodiment, since the edge coefficient ef is a value of 0 <ef <1, the threshold th is a value that falls within a range between the minimum value min and the maximum value max. Therefore, by detecting the position of the intersection point between the threshold value th and the navigator data, it is possible to detect the position where the step difference of the signal value of the navigator data appears. Since the edge position of the liver is considered to be the position where the step of the signal value appears, the edge position detecting means 82 detects the position of the intersection of the threshold th and the navigator data as the edge position of the liver. Note that the value of the threshold th can be changed according to the value of the edge coefficient ef. In this embodiment, three values ef = 0.25, ef = 0.33, and ef = 0.5 are considered as the value of the edge coefficient ef, and the edge position of the liver is detected for each edge coefficient ef. .

Hereinafter, the edge position of the liver detected for each edge coefficient ef will be described.
First, the edge position of the liver detected when ef = 0.25 will be described (see FIG. 12).

FIG. 12 is a diagram showing the edge position of the liver detected when ef = 0.25.
Edge position detecting means 82, the navigator data _D 1 to D _n, respectively, for detecting the edge positions _x 1 ~x _n liver using a threshold th (ef = 0.25) of formula (1). Then, as shown in the lower side of FIG. 12, an edge position profile E ₁ (ef = 0.25) representing the temporal change of the detected edge position is created.

Similarly, for each of the navigator data D _{1 to} D _n , the edge position of the liver when ef = 0.33 and the edge position of the liver when ef = 0.5 are detected, and an edge position profile is created. . Figure 13 shows the edge position profile _{E 2} in the case of ef = 0.33, indicating the edge position profile _{E 3} when the ef = 0.5 in FIG. 14.
After creating the edge position profiles E _{1 to} E ₃ , the process proceeds to step ST3.

In step ST3, the edge coefficient determining means 83 (see FIG. 1) uses an edge used when detecting the edge position of the liver in the main scan B (see FIG. 2) based on the edge position profiles E _{1 to} E ₃ . Determine the value of the coefficient ef. This determination is made as follows.

First, the edge coefficient determination unit 83 calculates an index representing the detection accuracy of the edge position when the edge coefficient ef = 0.25 is adopted based on the edge position profile E ₁ (ef = 0.25). Below, the calculation method of this parameter | index is demonstrated.

In this embodiment, the index F is obtained by the following formula.

Here, x _i : edge position obtained by i-th navigator sequence P _i x _{i + 1} : edge position obtained by i + 1-th navigator sequence P _{i + 1} N: number of navigator sequences executed in prescan A

In the formula _{(2), x i + 1} -x i is, i + 1-th and the navigator sequence _{P i +} edge location _{x i + 1} obtained by _1, i th navigator difference diff between the obtained edge position _{x i} by the sequence _{P i} ( = Xi _{+ 1- xi} )). Therefore, the index F represents the sum of the absolute values of the edge position difference diff (= x _{i + 1} −x _i ). When the detection error of the edge position is small, the detected edge position is sufficiently close to the actual edge position, so the difference diff (= x _{i + 1} −x _i ) of the edge position is a value of about several millimeters at most. Therefore, when the edge position detection error is small, the value of the index F is not so large. On the other hand, when the detection error of the edge position is large, the detected edge position greatly deviates from the actual edge position in the SI direction, and therefore the edge position difference diff (= x _{i + 1} −x _i ) tends to increase. Means that the value of the index F tends to increase.

  Therefore, it can be considered that the detection error of the edge position is small as the value of the index F is small, and the detection error of the edge position is large as the value of the index F is large.

In the above description, the case of calculating the index F with the edge coefficient ef = 0.25 has been described. However, when the edge coefficient ef = 0.33, the index F using the edge position profile E ₂ (see FIG. 13). When the edge coefficient ef = 0.5, the index F may be calculated using the edge position profile E ₃ (see FIG. 14).

  In this way, the index F representing the edge position detection accuracy can be calculated for each edge coefficient ef = 0.25, 0.33, 0.5. The edge coefficient determining means 83 uses the edge F to detect the edge position of the liver in the main scan B from the edge coefficients ef = 0.25, 0.33, 0.5. Determine the value of the coefficient ef. As described above, when the detection error of the edge position is small, the value of the index F tends to be small. Therefore, the edge coefficient determination means 83 has an edge coefficient ef = 0.25, 0.33, 0.55. Thus, the value of the edge coefficient ef that minimizes the value of the index F is obtained. Here, it is assumed that the index F with the edge coefficient ef = 0.5 is the minimum value. Therefore, the edge coefficient determining unit 83 determines the value of the edge coefficient ef used when detecting the edge position of the liver in the main scan B as ef = 0.5. After determining the value of the edge coefficient ef, the process proceeds to step ST4.

In step ST4, based on the edge position profile E ₃ obtained when the edge coefficient ef = 0.5, sets the range AW of spitting end of respiration of the subject 11. FIG. 15 schematically shows an example of the range AW at the end of spitting. In this embodiment, a certain range centering on the minimum value of the edge position is set as the end range AW of breathing. After setting the range AW at the end of breathing, the process proceeds to step ST5.

In step ST5, the main scan B is executed.
FIG. 16 is an explanatory diagram of the main scan B.
In the main scan B, navigator sequences Q _{1 to} Q _m and imaging sequences IS _{1 to} IS _m are executed alternately. In the scan B, and based on the navigator echo obtained by the navigator sequence _Q 1 to Q _m, creating a navigator data _D 1 to D _m. The edge position detection means 82 sets the value of the edge coefficient ef in the equation (1) to the value ef = 0.5 determined in step ST3, and uses the equation of the threshold th set to ef = 0.5. Thus, the positions x _{1 to} x _m of the liver edges in the navigator data D _{1 to} D _m are detected. When the detected liver edge position is out of the range AW at the end of breathing, it is determined that the k-space data acquired by the immediately following imaging sequence is not used as the data for image reconstruction. Is done. On the other hand, when the detected edge position of the liver is included in the range AW at the end of breathing, it is determined that the k-space data acquired by the immediately following imaging sequence is adopted as data for image reconstruction. The In this way, the main scan B is executed, and the flow ends.

  In this embodiment, the value of the edge coefficient ef is changed, an edge position profile is created for each value of the edge coefficient ef, and an index E representing the detection accuracy of the position of the liver edge is calculated. Therefore, the value of the edge coefficient ef when the edge position detection error is small can be determined. In the main scan B, the position of the edge of the liver is detected using the value of the edge coefficient ef determined in this way, so that image data with reduced body motion artifacts can be obtained.

  In this embodiment, ef used when executing the main scan B is determined from the three edge coefficients ef (0.25, 0.33, 0.5). However, the edge coefficient ef is not limited to three and may be four or more. Further, the edge coefficient ef is not limited to the combination of 0.25, 0.33, and 0.5, and other combinations may be used.

  In this embodiment, only the navigator sequence is executed in the pre-scan A, but an imaging sequence may be executed in addition to the navigator sequence.

  In the main scan B, the navigator sequence and the imaging sequence are executed alternately. However, the main scan B is not limited to the case where the navigator sequence and the imaging sequence are executed alternately. For example, the navigator sequence may be repeatedly executed, and the imaging sequence may be executed only when the edge position of the liver is included in the range AW at the end of breathing.

  In this embodiment, an example of detecting the edge position of the liver has been described. However, the present invention is not limited to the case of detecting the edge position of the liver. For example, the present invention can be applied when detecting the edge position of the kidney).

2 Magnet 3 Table 3a Cradle 4 Receiving coil 5 Transmitter 6 Gradient magnetic field power supply 7 Receiver 8 Control unit 9 Operation unit 10 Display unit 11 Subject 21 Bore 22 Superconducting coil 23 Gradient coil 24 RF coil 81 Navigator data creation means 82 Edge Position detecting means 83 Edge coefficient determining means 100 MR apparatus

Claims (8)

A pre-scan having a plurality of navigator sequences for obtaining navigator data including position information of edges of the first part from a navigator area including a first part and a second part that move;
A main scan having a navigator sequence for acquiring navigator data including position information of an edge of the first part from the navigator area, and an imaging sequence for acquiring data of an imaging area including a body moving part; ,
A magnetic resonance apparatus for performing
The signal value of the first part and the signal value of the second part in each navigator data acquired by the pre-scan is obtained, and the difference between the signal value of the first part and the second signal value And edge position detection means for detecting the position of the edge of the first part in each navigator data based on the coefficient multiplied by the difference, the value of the coefficient being changed, Edge position detecting means for detecting the position of the edge of the first part,
Determination means for determining the value of the coefficient used when detecting the position of the edge of the first part in the main scan based on the position of the edge of the first part detected for each value of the coefficient And
The edge position detecting means includes
The signal value of the first part and the signal value of the second part in the navigator data acquired by the main scan are obtained, and the difference between the signal value of the first part and the second signal value A magnetic resonance apparatus that detects the position of the edge of the first part in the main scan based on the coefficient determined by the determining means. The edge position detecting means includes
For each value of the coefficient, create an edge position profile that represents a temporal change in the position of the edge of the first part,
The determining means includes
2. The magnetic resonance apparatus according to claim 1, wherein an index representing detection accuracy of an edge position of the first part is calculated for each edge position profile, and the value of the coefficient is determined based on the index. The determining means includes
The magnetic resonance apparatus according to claim 2, wherein the index is calculated based on a difference between an edge position obtained by an i-th navigator sequence and an edge position obtained by an i + 1-th navigator sequence in the pre-scan. The magnetic resonance apparatus according to claim 3, wherein the determination unit calculates the index using the following equation.

Here, F: the index x _i: position of _i-th navigator edge obtained by the sequence x _{i + 1:} i _{+ 1} th navigator the edge position obtained by the sequence N: number of the navigator sequence executed by the prescanning
The edge position detecting means includes
The maximum value of the signal value of navigator data is obtained as the signal value of the first part, and the minimum value of the navigator signal value is obtained as the second signal value. The magnetic resonance apparatus described. The magnetic resonance apparatus according to claim 5, wherein the edge position detection means detects the position of the edge of the first part using a threshold value represented by the following expression.

th = (max−min) * ef + min

Where th: threshold
max: Maximum signal value of navigator data
min: Minimum signal value of navigator data
ef: the coefficient   The magnetic resonance apparatus according to claim 1, wherein the first part is a liver and the second part is a lung. A pre-scan having a plurality of navigator sequences for obtaining navigator data including position information of edges of the first part from a navigator area including a first part and a second part that move;
A main scan having a navigator sequence for acquiring navigator data including position information of an edge of the first part from the navigator area, and an imaging sequence for acquiring data of an imaging area including a body moving part; ,
A magnetic resonance apparatus program for executing
The signal value of the first part and the signal value of the second part in each navigator data acquired by the pre-scan is obtained, and the difference between the signal value of the first part and the second signal value And edge coefficient detection processing for detecting the position of the edge of the first part in each navigator data based on the coefficient multiplied by the difference, the value of the coefficient being changed, Edge position detection processing for detecting the position of the edge of the first part,
Determination processing for determining the value of the coefficient used when detecting the position of the edge of the first part in the main scan based on the position of the edge of the first part detected for each value of the coefficient Is a program for causing a computer to execute
The edge position detection process includes
The signal value of the first part and the signal value of the second part in the navigator data acquired by the main scan are obtained, and the difference between the signal value of the first part and the second signal value A program for detecting a position of an edge of the first part in the main scan based on the coefficient determined by the determination process.