Storset - Sideversjonshistorikk

Storset: Tømmer siden

2010-03-17T16:34:35Z

Tømmer siden

Storset på 12. mar. 2010 kl. 17:52

2010-03-12T17:52:59Z

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 17:52
Linje 16:		Linje 16:
	The plasma concentration of the drug is modeled by		The plasma concentration of the drug is modeled by

	(1)<math> {dc_1(t) \over dt} = -(k_1 + k_{12}) \cdot c_1(t) + {u(t) \over V_1} </math>		(1)<math>\begin{equation} {dc_1(t) \over dt} = -(k_1 + k_{12}) \cdot c_1(t) + {u(t) \over V_1} \end{equation}</math>

	where k<sub>1</sub> and k<sub>12</sub> are rate constants as define in figure 1 (k<sub>12</sub> is the link between plasma and active state of the drug and k<sub>1</sub> is general elimination of the drug from the plasma through other pathways), u(t) is the drug infusion rate function, and V<sub>1</sub> is the plasma volume of distribution (the total volume of plasma in the body). The active drug concentration is modeled by		where k<sub>1</sub> and k<sub>12</sub> are rate constants as define in figure 1 (k<sub>12</sub> is the link between plasma and active state of the drug and k<sub>1</sub> is general elimination of the drug from the plasma through other pathways), u(t) is the drug infusion rate function, and V<sub>1</sub> is the plasma volume of distribution (the total volume of plasma in the body). The active drug concentration is modeled by

	(2)<math>{dc_2(t) \over dt} = k_{12} \cdot {V_1 \over v_2} cdot c_1(t)- k_2 cdot c_2(t)</math>		(2)<math>\begin{equation}{dc_2(t) \over dt} = k_{12} \cdot {V_1 \over v_2} cdot c_1(t)- k_2 cdot c_2(t) \end{equation}</math>

	where k<sub>2</sub> is the elimination rate constant of active drug and V<sub>2</sub> is the volume of distribution of the cancer tumor. For both these equations it is important to remember that an interaction between the drug and a cancer or white blood cell does not lead to elimination of the drug. The drug affects the cell merely by its presence in the given volume of distribution. For both equations (2) and (3) the initial drug concentrations are set to zero:		where k<sub>2</sub> is the elimination rate constant of active drug and V<sub>2</sub> is the volume of distribution of the cancer tumor. For both these equations it is important to remember that an interaction between the drug and a cancer or white blood cell does not lead to elimination of the drug. The drug affects the cell merely by its presence in the given volume of distribution. For both equations (2) and (3) the initial drug concentrations are set to zero:

	(3) <math>c_1(0) = 0, c_2(0) = 0 </math>		(3) <math\>begin{equation}c_1(0) = 0, c_2(0) = 0 \end{equation}</math>

	The tumor count is modeled by the following differential equation:		The tumor count is modeled by the following differential equation:

	(4) <math>{dn(t) \over dt} = \lambda cdot n(t) cdot ln[\theta/n(t)]-k cdot [c_2(t) - c_{MIN}] cdot n(t) cdot H[c_2(t) - C_{MIN}, n(0) = n_0</math>		(4) <math>\begin{equation}{dn(t) \over dt} = \lambda cdot n(t) cdot ln[\theta/n(t)]-k cdot [c_2(t) - c_{MIN}] cdot n(t) cdot H[c_2(t) - C_{MIN}, n(0) = n_0 \end{equation}</math>

	The first term is the tumor growth term modeled using a [http://en.wikipedia.org/wiki/Gompertz_curve#Growth_of_tumors_and_Gompertz_curve Gompertz curve] growth type, which has previously been showed to fit cancer growth data quite accurate<ref></ref>. Here n(t) donates the number of cancer cells at time t and n<sub>0</sub> is the number of cancer cells at time 0. θ is the carrying capacity of the tumor, that is the maximal size the tumor can achieve at any time. When θ=n(t) we see that the growth term of equation (4) equals zero. λ is a constant modeling the proliferation ability of the cancer cells. The second term is the tumor cell-loss term, which is dependent of active drug concentration. H is the [http://en.wikipedia.org/wiki/Volume_of_distribution Heaviside step function], which leads to the cell-loss term equaling zero at all values of c<sub>2</sub>(t) < C<sub>MIN</sub>. C<sub>MIN</sub> is set as a threshold value of the active drug concentration. The drug will have no effect on the cancer cells under this active concentration. The cell-loss term is also proportional with the tumor cell count, n(t), and k is a proportionality constant which expresses the [http://en.wikipedia.org/wiki/Fractional_kill constant-cell-kill hypothesis] expected to be valid when cells are sensitive to the drug. The white blood cell count is modeled by		The first term is the tumor growth term modeled using a [http://en.wikipedia.org/wiki/Gompertz_curve#Growth_of_tumors_and_Gompertz_curve Gompertz curve] growth type, which has previously been showed to fit cancer growth data quite accurate<ref></ref>. Here n(t) donates the number of cancer cells at time t and n<sub>0</sub> is the number of cancer cells at time 0. θ is the carrying capacity of the tumor, that is the maximal size the tumor can achieve at any time. When θ=n(t) we see that the growth term of equation (4) equals zero. λ is a constant modeling the proliferation ability of the cancer cells. The second term is the tumor cell-loss term, which is dependent of active drug concentration. H is the [http://en.wikipedia.org/wiki/Volume_of_distribution Heaviside step function], which leads to the cell-loss term equaling zero at all values of c<sub>2</sub>(t) < C<sub>MIN</sub>. C<sub>MIN</sub> is set as a threshold value of the active drug concentration. The drug will have no effect on the cancer cells under this active concentration. The cell-loss term is also proportional with the tumor cell count, n(t), and k is a proportionality constant which expresses the [http://en.wikipedia.org/wiki/Fractional_kill constant-cell-kill hypothesis] expected to be valid when cells are sensitive to the drug. The white blood cell count is modeled by

	(5) <math>{dw(t) \over dt} = r_c - v cdot w(t) - \mu cdot w(t) cdot c_1(t-\tau)</math>		(5) <math>\begin{equation}{dw(t) \over dt} = r_c - v cdot w(t) - \mu cdot w(t) cdot c_1(t-\tau) \end{equation}</math>

	where r<sub>c</sub> is constant production rate of white blood cells (with dimension volume<sup>-1</sup>time<sup>-1</sup>), and v is the fraction of white blood cells which dies per time. The steady state of this system with no drug present gives the initial white blood cell count w(0) = w<sub>0</sub> = r<sub>c</sub>/v. The last term in equation (5) represents the loss of white blood cells because of drug present in the blood plasma. White blood cells are all produced from stem cells in the bone marrow and not by cell division during circulation. Because of this, it is reasonable to assume a constant production rate. This phenomenon will also introduce a delay from the white blood cells are produced in the bone marrow until they have matured and been transported into circulation in the blood system were they are susceptible for the drug. These considerations result in the last term of equation (5) where a delay, τ, is introduced into the plasma drug concentration. The interpretation of this would be that the white blood cells produced at time t, will not be susceptible for drug until a time t + τ in the future. Therefor is the delayed plasma drug concentration used. The total drug induced white blood cell-loss term is then proportional to the number of white blood cells, and the delayed drug concentration. μ is a proportionality constant with dimensions concentration<sup>-1</sup>time<sup>-1</sup>.		where r<sub>c</sub> is constant production rate of white blood cells (with dimension volume<sup>-1</sup>time<sup>-1</sup>), and v is the fraction of white blood cells which dies per time. The steady state of this system with no drug present gives the initial white blood cell count w(0) = w<sub>0</sub> = r<sub>c</sub>/v. The last term in equation (5) represents the loss of white blood cells because of drug present in the blood plasma. White blood cells are all produced from stem cells in the bone marrow and not by cell division during circulation. Because of this, it is reasonable to assume a constant production rate. This phenomenon will also introduce a delay from the white blood cells are produced in the bone marrow until they have matured and been transported into circulation in the blood system were they are susceptible for the drug. These considerations result in the last term of equation (5) where a delay, τ, is introduced into the plasma drug concentration. The interpretation of this would be that the white blood cells produced at time t, will not be susceptible for drug until a time t + τ in the future. Therefor is the delayed plasma drug concentration used. The total drug induced white blood cell-loss term is then proportional to the number of white blood cells, and the delayed drug concentration. μ is a proportionality constant with dimensions concentration<sup>-1</sup>time<sup>-1</sup>.
Linje 39:		Linje 39:
	Several constraints are put on the model to insure that the patient can handle the toxic side effects of the drug. Firstly, the maximum plasma drug concentration must never exceed a given value:		Several constraints are put on the model to insure that the patient can handle the toxic side effects of the drug. Firstly, the maximum plasma drug concentration must never exceed a given value:

	(6) <math>c_1(t_i) <= C_{MAX}</math>		(6) <math>\begin{equation}c_1(t_i) <= C_{MAX} \end{equation}</math>

	where t<sub>i</sub> represents the time interval of maximum plasma drug concentration in one given administration protocol. Also, a maximum drug exposure is introduced, meaning that the total amount of drug introduced to the patient over the whole treatment period must not exceed a given limit. This can be expressed by		where t<sub>i</sub> represents the time interval of maximum plasma drug concentration in one given administration protocol. Also, a maximum drug exposure is introduced, meaning that the total amount of drug introduced to the patient over the whole treatment period must not exceed a given limit. This can be expressed by

	(7) <math>{da(t) \over dt} = c_1(t), a(0) = 0</math>		(7) <math>\begin{equation}{da(t) \over dt} = c_1(t), a(0) = 0 \end{equation}</math>

	This is commonly known as area under the curve, AUC, because a(t) equals the integral from 0 to infinity over the concentration distribution of c<sub>1</sub>(t). Equation (7) leads to		This is commonly known as area under the curve, AUC, because a(t) equals the integral from 0 to infinity over the concentration distribution of c<sub>1</sub>(t). Equation (7) leads to

	(8) <math>a(T) <= A_{MAX}</math>		(8) <math>\begin{equation}a(T) <= A_{MAX} \end{equation}</math>

	where T is at time point where all drug is completely eliminated from the body (the integral is zero from point T until infinity, and therefore the integration can be stopped at this point). Also constraints on the white blood cell counts are introduced. First, a minimum value of w(t) is defined, under which the white blood cell count never can fall:		where T is at time point where all drug is completely eliminated from the body (the integral is zero from point T until infinity, and therefore the integration can be stopped at this point). Also constraints on the white blood cell counts are introduced. First, a minimum value of w(t) is defined, under which the white blood cell count never can fall:

	(9) <math>w(t_j) >= W_D</math>		(9) <math>\begin{equation} w(t_j) >= W_D \end{equation}</math>

	where t<sub>j</sub> represents the time interval of lowest white blood cell count. The second constraint puts a limit on how long the white blood cell count can stay under a given upper value, W<sub>U</sub>. The time t<sub>U</sub>(T) over which the white blood cell count is under W<sub>U</sub> must be less or equal to T<sub>U</sub>:		where t<sub>j</sub> represents the time interval of lowest white blood cell count. The second constraint puts a limit on how long the white blood cell count can stay under a given upper value, W<sub>U</sub>. The time t<sub>U</sub>(T) over which the white blood cell count is under W<sub>U</sub> must be less or equal to T<sub>U</sub>:

	(10) <math>t_u(T) <= T_u</math>		(10) <math>\begin{equation}t_u(T) <= T_u \end{equation}</math>

	==Drug delivery==		==Drug delivery==
	The optimization problem considers how to best infuse the drug over a period of time, t<sub>N</sub>, divided into N time treatment intervals, such that the tumor size is minimized without violating any of the given constraints. The infusion rates can vary between each of the time intervals such that		The optimization problem considers how to best infuse the drug over a period of time, t<sub>N</sub>, divided into N time treatment intervals, such that the tumor size is minimized without violating any of the given constraints. The infusion rates can vary between each of the time intervals such that

	(11) <math>u(t) = </math>		(11) <math>\begin{equation} u(t) = {d_i \over t_i} for 0 <= t < t_1, u(t) = {d_i \over t_i} for t_{i-1}<= t < t_i, i = 2,...,N, \end{equation}</math>

	is fulfilled. di donates the total amount of drug infused in the time interval from t<sub>i-1</sub> to t<sub>i</sub>. By examining different drug regimens, the optimal drug delivery scheme can be found using this given model.		is fulfilled. d<sub>i</sub> donates the total amount of drug infused in the time interval from t<sub>i-1</sub> to t<sub>i</sub>. By examining different drug regimens, the optimal drug delivery scheme can be found using this given model.

	=The model and results=		=The model and results=

Storset: /* Drug delivery */

2010-03-12T17:45:14Z

Drug delivery

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 17:45
Linje 58:		Linje 58:

	==Drug delivery==		==Drug delivery==
			The optimization problem considers how to best infuse the drug over a period of time, t<sub>N</sub>, divided into N time treatment intervals, such that the tumor size is minimized without violating any of the given constraints. The infusion rates can vary between each of the time intervals such that

			(11) <math>u(t) = </math>

			is fulfilled. di donates the total amount of drug infused in the time interval from t<sub>i-1</sub> to t<sub>i</sub>. By examining different drug regimens, the optimal drug delivery scheme can be found using this given model.

	=The model and results=		=The model and results=

Storset: /* Constraints */

2010-03-12T17:40:36Z

Constraints

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 17:40
Linje 37:		Linje 37:

	==Constraints==		==Constraints==
			Several constraints are put on the model to insure that the patient can handle the toxic side effects of the drug. Firstly, the maximum plasma drug concentration must never exceed a given value:

			(6) <math>c_1(t_i) <= C_{MAX}</math>

			where t<sub>i</sub> represents the time interval of maximum plasma drug concentration in one given administration protocol. Also, a maximum drug exposure is introduced, meaning that the total amount of drug introduced to the patient over the whole treatment period must not exceed a given limit. This can be expressed by

			(7) <math>{da(t) \over dt} = c_1(t), a(0) = 0</math>

			This is commonly known as area under the curve, AUC, because a(t) equals the integral from 0 to infinity over the concentration distribution of c<sub>1</sub>(t). Equation (7) leads to

			(8) <math>a(T) <= A_{MAX}</math>

			where T is at time point where all drug is completely eliminated from the body (the integral is zero from point T until infinity, and therefore the integration can be stopped at this point). Also constraints on the white blood cell counts are introduced. First, a minimum value of w(t) is defined, under which the white blood cell count never can fall:

			(9) <math>w(t_j) >= W_D</math>

			where t<sub>j</sub> represents the time interval of lowest white blood cell count. The second constraint puts a limit on how long the white blood cell count can stay under a given upper value, W<sub>U</sub>. The time t<sub>U</sub>(T) over which the white blood cell count is under W<sub>U</sub> must be less or equal to T<sub>U</sub>:

			(10) <math>t_u(T) <= T_u</math>

	==Drug delivery==		==Drug delivery==

Storset: /* Differential equations */

2010-03-12T17:23:19Z

Differential equations

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 17:23
Linje 30:		Linje 30:
	(4) <math>{dn(t) \over dt} = \lambda cdot n(t) cdot ln[\theta/n(t)]-k cdot [c_2(t) - c_{MIN}] cdot n(t) cdot H[c_2(t) - C_{MIN}, n(0) = n_0</math>		(4) <math>{dn(t) \over dt} = \lambda cdot n(t) cdot ln[\theta/n(t)]-k cdot [c_2(t) - c_{MIN}] cdot n(t) cdot H[c_2(t) - C_{MIN}, n(0) = n_0</math>

	The first term is the tumor growth term modeled using a [http://en.wikipedia.org/wiki/Gompertz_curve#Growth_of_tumors_and_Gompertz_curve Gompertz curve] growth type, which has previously been showed to fit cancer growth data quite accurate<ref></ref>. Here n(t) donates the number of cancer cells at time t and n<sub>0</sub> is the number of cancer cells at time 0. θ is the carrying capacity of the tumor, that is the maximal size the tumor can achieve at any time. When θ=n(t) we see that the growth term of equation (4) equals zero. λ is a constant modeling the proliferation ability of the cancer cells. The second term is the tumor cell-loss term, which is dependent of active drug concentration. H is the [http://en.wikipedia.org/wiki/Volume_of_distribution Heaviside step function], which leads to the cell-loss term equaling zero at all values of c<sub>2</sub>(t) < ~~CMIN~~. ~~CMIN~~ is set as a threshold value of the active drug concentration. The drug will have no effect on the cancer cells under this active concentration. The cell-loss term is also proportional with the tumor cell count, n(t), and k is a proportionality constant which expresses the [http://en.wikipedia.org/wiki/Fractional_kill constant-cell-kill hypothesis] expected to be valid when cells are sensitive to the drug. The white blood cell count is modeled by		The first term is the tumor growth term modeled using a [http://en.wikipedia.org/wiki/Gompertz_curve#Growth_of_tumors_and_Gompertz_curve Gompertz curve] growth type, which has previously been showed to fit cancer growth data quite accurate<ref></ref>. Here n(t) donates the number of cancer cells at time t and n<sub>0</sub> is the number of cancer cells at time 0. θ is the carrying capacity of the tumor, that is the maximal size the tumor can achieve at any time. When θ=n(t) we see that the growth term of equation (4) equals zero. λ is a constant modeling the proliferation ability of the cancer cells. The second term is the tumor cell-loss term, which is dependent of active drug concentration. H is the [http://en.wikipedia.org/wiki/Volume_of_distribution Heaviside step function], which leads to the cell-loss term equaling zero at all values of c<sub>2</sub>(t) < C<sub>MIN</sub>. C<sub>MIN</sub> is set as a threshold value of the active drug concentration. The drug will have no effect on the cancer cells under this active concentration. The cell-loss term is also proportional with the tumor cell count, n(t), and k is a proportionality constant which expresses the [http://en.wikipedia.org/wiki/Fractional_kill constant-cell-kill hypothesis] expected to be valid when cells are sensitive to the drug. The white blood cell count is modeled by

	(5) M4		(5) <math>{dw(t) \over dt} = r_c - v cdot w(t) - \mu cdot w(t) cdot c_1(t-\tau)</math>

	where rc is constant production rate of white blood cells (with dimension (volume)-1(time)-1), and v is the fraction of white blood cells which dies per time. The steady state of this system with no drug present gives the initial white blood cell count w(0) = w0 = rc/v. The last term in equation (5) represents the loss of white blood cells because of drug present in the blood plasma. White blood cells are all produced from stem cells in the bone marrow and not by cell division during circulation. Because of this, it is reasonable to assume a constant production rate. This phenomenon will also introduce a delay from the white blood cells are produced in the bone marrow until they have matured and been transported into circulation in the blood system were they are susceptible for the drug. These considerations result in the last term of equation (5) where a delay, τ, is introduced into the plasma drug concentration. The interpretation of this would be that the white blood cells produced at time t, will not be susceptible for drug until a time t + τ in the future. Therefor is the delayed plasma drug concentration used. The total drug induced white blood cell-loss term is then proportional to the number of white blood cells, and the delayed drug concentration. μ is a proportionality constant with dimensions (concentration)-1(time)-1.		where r<sub>c</sub> is constant production rate of white blood cells (with dimension volume<sup>-1</sup>time<sup>-1</sup>), and v is the fraction of white blood cells which dies per time. The steady state of this system with no drug present gives the initial white blood cell count w(0) = w<sub>0</sub> = r<sub>c</sub>/v. The last term in equation (5) represents the loss of white blood cells because of drug present in the blood plasma. White blood cells are all produced from stem cells in the bone marrow and not by cell division during circulation. Because of this, it is reasonable to assume a constant production rate. This phenomenon will also introduce a delay from the white blood cells are produced in the bone marrow until they have matured and been transported into circulation in the blood system were they are susceptible for the drug. These considerations result in the last term of equation (5) where a delay, τ, is introduced into the plasma drug concentration. The interpretation of this would be that the white blood cells produced at time t, will not be susceptible for drug until a time t + τ in the future. Therefor is the delayed plasma drug concentration used. The total drug induced white blood cell-loss term is then proportional to the number of white blood cells, and the delayed drug concentration. μ is a proportionality constant with dimensions concentration<sup>-1</sup>time<sup>-1</sup>.

	==Constraints==		==Constraints==

Storset: /* Differential equations */

2010-03-12T17:17:13Z

Differential equations

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 17:17
Linje 14:		Linje 14:

	==Differential equations==		==Differential equations==
	~~<nowiki>~~
	The plasma concentration of the drug is modeled by		The plasma concentration of the drug is modeled by

	(1) M1		(1)<math> {dc_1(t) \over dt} = -(k_1 + k_{12}) \cdot c_1(t) + {u(t) \over V_1} </math>

	where k1 and ~~k12~~ are rate constants as define in figure 1 (~~k12~~ is the link between plasma and active state of the drug and k1 is general elimination of the drug from the plasma through other pathways), u(t) is the drug infusion rate function, and V1 is the plasma volume of distribution (the total volume of plasma in the body). The active drug concentration is modeled by		where k<sub>1</sub> and k<sub>12</sub> are rate constants as define in figure 1 (k<sub>12</sub> is the link between plasma and active state of the drug and k<sub>1</sub> is general elimination of the drug from the plasma through other pathways), u(t) is the drug infusion rate function, and V<sub>1</sub> is the plasma volume of distribution (the total volume of plasma in the body). The active drug concentration is modeled by

	(2) M2		(2)<math>{dc_2(t) \over dt} = k_{12} \cdot {V_1 \over v_2} cdot c_1(t)- k_2 cdot c_2(t)</math>

	where k2 is the elimination rate constant of active drug and V2 is the volume of distribution of the cancer tumor. For both these equations it is important to remember that an interaction between the drug and a cancer or white blood cell does not lead to elimination of the drug. The drug affects the cell merely by its presence in the given volume of distribution. For both equations (2) and (3) the initial drug concentrations are set to zero:		where k<sub>2</sub> is the elimination rate constant of active drug and V<sub>2</sub> is the volume of distribution of the cancer tumor. For both these equations it is important to remember that an interaction between the drug and a cancer or white blood cell does not lead to elimination of the drug. The drug affects the cell merely by its presence in the given volume of distribution. For both equations (2) and (3) the initial drug concentrations are set to zero:

	(3) c1(0) = 0, c2(0) = 0		(3) <math>c_1(0) = 0, c_2(0) = 0 </math>

	The tumor count is modeled by the following differential equation:		The tumor count is modeled by the following differential equation:

	(4) M3		(4) <math>{dn(t) \over dt} = \lambda cdot n(t) cdot ln[\theta/n(t)]-k cdot [c_2(t) - c_{MIN}] cdot n(t) cdot H[c_2(t) - C_{MIN}, n(0) = n_0</math>

	The first term is the tumor growth term modeled using a ~~Gompertz curve(link to wikipedia,~~ http://en.wikipedia.org/wiki/Gompertz_curve#Growth_of_tumors_and_Gompertz_curve ) growth type, which has previously been showed to fit cancer growth data quite accurate~~[4]~~. Here n(t) donates the number of cancer cells at time t and n0 is the number of cancer cells at time 0. θ is the carrying capacity of the tumor, that is the maximal size the tumor can achieve at any time. When θ=n(t) we see that the growth term of equation (4) equals zero. λ is a constant modeling the proliferation ability of the cancer cells. The second term is the tumor cell-loss term, which is dependent of active drug concentration. H is the ~~Heaviside step function(~~http://en.wikipedia.org/wiki/Volume_of_distribution ), which leads to the cell-loss term equaling zero at all values of c2(t) < CMIN. CMIN is set as a threshold value of the active drug concentration. The drug will have no effect on the cancer cells under this active concentration. The cell-loss term is also proportional with the tumor cell count, n(t), and k is a proportionality constant which expresses the ~~constant-cell-kill hypothesis (~~http://en.wikipedia.org/wiki/Fractional_kill) expected to be valid when cells are sensitive to the drug. The white blood cell count is modeled by		The first term is the tumor growth term modeled using a [http://en.wikipedia.org/wiki/Gompertz_curve#Growth_of_tumors_and_Gompertz_curve Gompertz curve] growth type, which has previously been showed to fit cancer growth data quite accurate<ref></ref>. Here n(t) donates the number of cancer cells at time t and n<sub>0</sub> is the number of cancer cells at time 0. θ is the carrying capacity of the tumor, that is the maximal size the tumor can achieve at any time. When θ=n(t) we see that the growth term of equation (4) equals zero. λ is a constant modeling the proliferation ability of the cancer cells. The second term is the tumor cell-loss term, which is dependent of active drug concentration. H is the [http://en.wikipedia.org/wiki/Volume_of_distribution Heaviside step function], which leads to the cell-loss term equaling zero at all values of c<sub>2</sub>(t) < CMIN. CMIN is set as a threshold value of the active drug concentration. The drug will have no effect on the cancer cells under this active concentration. The cell-loss term is also proportional with the tumor cell count, n(t), and k is a proportionality constant which expresses the [http://en.wikipedia.org/wiki/Fractional_kill constant-cell-kill hypothesis] expected to be valid when cells are sensitive to the drug. The white blood cell count is modeled by

	(5) M4		(5) M4

	where rc is constant production rate of white blood cells (with dimension (volume)-1(time)-1), and v is the fraction of white blood cells which dies per time. The steady state of this system with no drug present gives the initial white blood cell count w(0) = w0 = rc/v. The last term in equation (5) represents the loss of white blood cells because of drug present in the blood plasma. White blood cells are all produced from stem cells in the bone marrow and not by cell division during circulation. Because of this, it is reasonable to assume a constant production rate. This phenomenon will also introduce a delay from the white blood cells are produced in the bone marrow until they have matured and been transported into circulation in the blood system were they are susceptible for the drug. These considerations result in the last term of equation (5) where a delay, τ, is introduced into the plasma drug concentration. The interpretation of this would be that the white blood cells produced at time t, will not be susceptible for drug until a time t + τ in the future. Therefor is the delayed plasma drug concentration used. The total drug induced white blood cell-loss term is then proportional to the number of white blood cells, and the delayed drug concentration. μ is a proportionality constant with dimensions (concentration)-1(time)-1.		where rc is constant production rate of white blood cells (with dimension (volume)-1(time)-1), and v is the fraction of white blood cells which dies per time. The steady state of this system with no drug present gives the initial white blood cell count w(0) = w0 = rc/v. The last term in equation (5) represents the loss of white blood cells because of drug present in the blood plasma. White blood cells are all produced from stem cells in the bone marrow and not by cell division during circulation. Because of this, it is reasonable to assume a constant production rate. This phenomenon will also introduce a delay from the white blood cells are produced in the bone marrow until they have matured and been transported into circulation in the blood system were they are susceptible for the drug. These considerations result in the last term of equation (5) where a delay, τ, is introduced into the plasma drug concentration. The interpretation of this would be that the white blood cells produced at time t, will not be susceptible for drug until a time t + τ in the future. Therefor is the delayed plasma drug concentration used. The total drug induced white blood cell-loss term is then proportional to the number of white blood cells, and the delayed drug concentration. μ is a proportionality constant with dimensions (concentration)-1(time)-1.
	~~</nowiki>~~

	==Constraints==		==Constraints==

Storset på 12. mar. 2010 kl. 16:48

2010-03-12T16:48:59Z

@@ Linje 4: / Linje 4: @@
 =Biological background=
 =History=
 =Mathematical model=
 ==Outline==
 ==Differential equations==
 ==Constraints==
 ==Drug delivery==

Storset: /* Parameter values */

2010-03-12T16:36:05Z

Parameter values

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 16:36
Linje 24:		Linje 24:
	! width="1000" \| Comment		! width="1000" \| Comment
	\|-		\|-
	! <~~math~~>~~V_1~~</~~math~~>		! V<sub>1</sub>
	\| 25		\| 25
	\| L		\| L
	\| Volume of plasma in adult human		\| Volume of plasma in adult human
	\|-		\|-
	! <~~math~~>~~V_2~~</~~math~~>		! V<sub>2</sub>
	\| 15		\| 15
	\| L		\| L
	\|		\|
	\|-		\|-
	! ~~<math>~~CL~~</math>~~		! CL
	\| 50		\| 50
	\| L/day		\| L/day
	\|		\|
	\|-		\|-
	! <~~math~~>~~k_1~~+~~k_12~~</~~math~~>		!_k<sub>1</sub>+k<sub>12</sub>
	\| 2		\| 2
	\| /day		\| /day
	\|		\|
	\|-		\|-
	! <~~math~~>~~k_1~~</~~math~~>		! k<sub>1</sub>
	\| 1.6		\| 1.6
	\| /day		\| /day
	\|		\|
	\|-		\|-
	! <~~math~~>~~k_12~~</~~math~~>		! k<sub>12</sub>
	\| 0.4		\| 0.4
	\| /day		\| /day
	\|		\|
	\|-		\|-
	! <~~math~~>~~k_2~~</~~math~~>		! k<sub>2</sub>
	\| 0.8		\| 0.8
	\| /day		\| /day
Linje 64:		Linje 64:
	\|		\|
	\|-		\|-
	! <~~math~~>~~n_0~~</~~math~~>		! n<sub>0</sub>
	\| 30*10^9		\| 30*10^9
	\| cells		\| cells
Linje 74:		Linje 74:
	\|		\|
	\|-		\|-
	! <~~math~~>~~C_MIN~~</~~math~~>		! C<sub>MIN</sub>
	\| 0.1		\| 0.1
	\| μg/mL		\| μg/mL
	\|		\|
	\|-		\|-
	! ~~<math>~~k~~</math>~~		! k
	\| 30		\| 30
	\| L/(g*day)		\| L/(g*day)
	\|		\|
	\|-		\|-
	! <~~math~~>~~r_c~~</~~math~~>		! r<sub>c</sub>
	\| 1.2*10^9		\| 1.2*10^9
	\| /(L*day)		\| /(L*day)
Linje 99:		Linje 99:
	\|		\|
	\|-		\|-
	! <~~math~~>~~W_0~~</~~math~~>		! W<sub>0</sub>
	\| 8*10^9		\| 8*10^9
	\| /L		\| /L
	\|		\|
	\|-		\|-
	! ~~<math>~~v~~</math>~~		! v
	\| 0.15		\| 0.15
	\| /day		\| /day
	\|		\|
	\|-		\|-
	! <~~math~~>~~C_MAX~~</~~math~~>		! C<sub>MAX</sub>
	\| 10		\| 10
	\| μg/mL		\| μg/mL
	\|		\|
	\|-		\|-
	! <~~math~~>~~W_D~~</~~math~~>		! W<sub>D</sub>
	\| 2*10^9		\| 2*10^9
	\| /L		\| /L
	\|		\|
	\|-		\|-
	! <~~math~~>~~T_U~~</~~math~~>		! T<sub>U</sub>
	\| 7		\| 7
	\| days		\| days
	\|		\|
	\|-		\|-
	! <math>\delta t</math>		! <math>\delta</math> t
	\| 0.52		\| 0.52
	\| days		\| days
	\|		\|
	\|-		\|-
	! <~~math~~>~~A_MAX~~</~~math~~>		! A<sub>MAX</sub>
	\| 30		\| 30
	\| μg*day/mL		\| μg*day/mL
	\|		\|
	\|-		\|-
	! <~~math~~>~~t_N~~</~~math~~>		! t<sub>N</sub>
	\| 21		\| 21
	\| days		\| days
	\|		\|
	\|-		\|-
	! ~~<math>~~T~~</math>~~		! T
	\| 31		\| 31
	\| days		\| days

Storset: /* Summary */

2010-03-12T16:28:09Z

Summary

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 16:28
Linje 2:		Linje 2:

	=Summary=		=Summary=


	=Biological background=		=Biological background=

Storset: /* Summary */

2010-03-12T16:24:20Z

Summary

← Eldre sideversjon		Sideversjonen fra 12. mar. 2010 kl. 16:24
Linje 2:		Linje 2:

	=Summary=		=Summary=
	~~<ref>~~

	=Biological background=		=Biological background=