# US 11,113,362 B2 · Multicomponent model parameterisation

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- Number: US 11,113,362 B2
- Document ID: US11113362B2
- Status: Granted
- Kind: B2
- Application number: 15/561,219
- Filed: 2016-03-24
- Published: 2018-03-08
- Granted: 2021-09-07
- Earliest claimed: 2015-03-24
- Assignees: Malvern Panalytical Limited
- Applicants: Malvern Panalytical Limited
- Inventors: Seyi Latunde-Dada; Oksana Iryna Leszczyszyn; Karl Hampton; Rachel Bott
- Web page: https://patentgrep.com/patent/US11113362B2

## CPC

- G06F 17/18: for evaluating statistical data {, e.g. average values, frequency distributions, probability functions, regression analysis (forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04)}
- G01N 13/00: Investigating surface or boundary effects, e.g. wetting power; Investigating diffusion effects; Analysing materials by determining surface, boundary, or diffusion effects (scanning-probe techniques or apparatus G01Q)
- G01N 15/0227: using imaging; using holography
- G06F 17/12: Simultaneous equations {, e.g. systems of linear equations}
- G01N 15/075: by optical means
- G01N 2013/003: {Diffusion; diffusivity between liquids}
- G01N 2015/0053: {in liquids, e.g. trouble}
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- G01N: INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES (measuring or testing apparatus or processes other than immunoassay, involving enzymes or microorganisms C12M, C12Q)

## Abstract

A method of estimating a parameter for fitting a multi-component Taylorgram model to Taylorgram data g(t) is disclosed. The data comprises a multi-component Taylorgram peak or front at t=tr. The method comprises: evaluating a value of an integration or differential of the data; determining the parameter, based on an analytical expression that includes the value of the integral or differential of the data, the parameter corresponding with a physical property of a sample from which the Taylorgram data was obtained.

## Claims

### Claim 1

A method of estimating a diffusion coefficient or hydrodynamic radius of a sample by:
performing a Taylor dispersion analysis, wherein performing the Taylor dispersion analysis comprises:
flowing a carrier solution through a capillary;
injecting a sample fluid into the capillary to form a sample plug or a sample front;
determining Taylorgram data by measuring absorbance over time at a detector as the sample plug or sample front flows past the detector; and
using a computer to fit a multi-component Taylorgram model to the Taylorgram data g(t) obtained from the sample, the Taylorgram data comprising a multi-component Taylorgram peak or front at t=tr; the method comprising:
evaluating a value of an integration or a differential of the data;
determining the diffusion coefficient or hydrodynamic radius of a component of the multi-component Taylorgram model, based on an analytical expression that includes the value of the integral or differential of the data, the diffusion coefficient or hydrodynamic radius corresponding with a physical property of a component of the sample from which the Taylorgram data was obtained;
determining a maximum value, w, of a differential of the data,
d
⁢
⁢
g
⁡
(
t
)
dt
,
and a value t′ corresponding with the maximum value, w; and
determining the diffusion coefficient or hydrodynamic radius from the values w, t′ and tr.

### Claim 2 (depends on 1)

The method of claim 1, wherein evaluating a value of an integration or a differential of the data comprises evaluating at least one of: a first differential of the data, a second differential of the data, a third differential of the data, a first integral of the data, a second integral of the data and a third integral of the data.

### Claim 3 (depends on 1)

The method of claim 1, comprising:
evaluating, based on the data, at least two values selected from:
x, a value of the data at t=tr;
y, a value of a second differential of the data,
d
2
⁢
g
⁡
(
t
)
dt
2
,
at t=tr;
z, a value derived from an integration of the data, ∫g(t)dt, starting or ending at t=tr; and
u, a value of an integration of the integration of the data, ∫∫g(t)dt2, starting or ending at t=tr; and then
determining an estimate for the diffusion coefficient or hydrodynamic radius, wherein determining the estimate comprises solving at least two simultaneous equations, the simultaneous equations each respectively comprising one of the evaluated values.

### Claim 4 (depends on 1)

The method of claim 1, wherein the simultaneous equations comprise at least two of:
x
=
∑
i
=
1
n
⁢
A
i
;
y
=
-
d
2
⁢
g
⁡
(
t
)
dt
2
=
∑
i
=
1
n
⁢
A
i
σ
i
2
;
z
=
2
π
⁢
I
=
∑
i
=
1
n
⁢
A
i
⁢
σ
i
,
where I is a value of the integration of the data, ∫g(t)dt, starting or ending at t=tr; and
u=Σi=1nAiσi2.

### Claim 5 (depends on 4)

The method of claim 4, wherein the model comprises a two-component Taylorgram model, and wherein at least one of the following conditions are met:
none of A1, A2, σ1, σ2 are known before the diffusion coefficient or hydrodynamic radius is estimated;
one of σ1 and σ2 are known, and none of the remaining parameters A1, A2, σ1, σ2 are known before the diffusion coefficient or hydrodynamic radius is estimated;
the second component has a negative amplitude A2, and the amplitude of the second component A2, is smaller than the amplitude of the first component A1.

### Claim 6 (depends on 4)

The method of claim 4, wherein:
i) the model comprises a two or three component Taylorgram model, and wherein a relationship between the values of σi for the sample is known, and wherein the absolute values of Ai and σi for the sample are not known, and the ratio between the values of Ai for the sample is not known; or
ii) the model comprises a two, three or four component Taylorgram model, and wherein the values σi for the sample are each known but the values of Ai for the biopharmaceutical formulation are not known.

### Claim 7 (depends on 1)

The method of claim 1, wherein the Taylorgram model is of the general form:
g
model
⁡
(
t
)
=
∑
i
=
1
2
⁢
A
i
·
e
-
0.5
·
(
t
-
t
r
σ
i
)
2
;
the diffusion coefficient or hydrodynamic radius is selected from Ai and αi; and
the second component has a negative amplitude A2, and the magnitude of the amplitude of the second component A2, is smaller than the magnitude of the amplitude of the first component A1.

### Claim 8 (depends on 7)

The method of claim 7, wherein the diffusion coefficient or hydrodynamic radius is determined based on at least one of the expressions:
w
=
A
1
σ
1
⁢
e
-
0.5
;
and
t
′
=
t
r
±
σ
1
.

### Claim 9 (depends on 8)

The method of claim 8, wherein a further parameter of the model is determined based on at least one of the expressions:
z
=
A
1
σ
1
2
+
A
2
σ
2
2
,
and
⁢
⁢
x
=
A
1
+
A
2
.

### Claim 10 (depends on 1)

An apparatus comprising a processor, configured to perform the method of claim 1.

### Claim 11 (depends on 10)

The apparatus of claim 10, further comprising an instrument for performing a Taylor dispersion analysis, so as to obtain the data.

### Claim 12 (depends on 1)

The method of claim 1, further comprising performing a measurement on the sample to obtain the Taylorgram data.

### Claim 13

A method of estimating a diffusion coefficient or hydrodynamic radius of a sample by:
performing a Taylor dispersion analysis, wherein performing the Taylor dispersion analysis comprises:
flowing a carrier solution through a capillary;
injecting a sample fluid into the capillary to form a sample plug or a sample front;
determining Taylorgram data by measuring absorbance over time at a detector as the sample plug or sample front flows past the detector; and
using a computer to fit a two-component Taylorgram model to the Taylorgram data g(t), the data being a Taylorgram obtained from the sample comprising a multi-component Taylorgram peak or front at t=tr, the method comprising:
performing a least squares fit of a single component Taylorgram model to the data g(t);
determining the diffusion coefficient or hydrodynamic radius by finding a root of a cubic equation, the cubic equation derived from: an analytical expression for the integrated residual squared error, R2, resulting from a fit of a one-component Taylorgram model to a two-component Taylorgram distribution; and a priori knowledge of the standard deviation, α, of a dominant Taylorgram component of the data g(t);
and,
wherein the two-component Taylorgram model is of the general form:
g
model
⁡
(
t
)
=
∑
i
=
1
2
⁢
A
i
·
e
-
0.5
·
(
t
-
t
r
σ
i
)
2
;
and,
the cubic equation is:
a·s23+b·s2+c=0, where
s
2
=
σ
2
σ
2
2
+
σ
s
2
;
a
=
2
⁢
(
A
peak
⁢
σ
1
σ
1
2
+
σ
s
2
-
A
s
)
;
b
=
A
s
-
2
⁢
A
peak
(
σ
1
σ
1
2
+
σ
s
2
)
3
;
and
c
=
A
s
(
2
⁢
(
σ
1
σ
1
2
+
σ
s
2
)
3
-
σ
1
σ
1
2
+
σ
s
2
)
.

### Claim 14 (depends on 13)

A non-transitory computer readable medium, containing a set of instructions that are operable to cause a computer to perform the method of claim 13.

### Claim 15 (depends on 13)

The method of claim 13, further comprising performing a measurement on the sample to obtain the Taylorgram data.