
    Given a patient's daily post-discharge health metrics, determine the most significant period of health fluctuation by finding the maximum sum of measurements that show initial improvement followed by decline. The measurements are represented as numerical scores where a pattern of increasing values followed by decreasing values indicates a critical monitoring window that could signal potential readmission risk.
    

    
def max_sum(arr):
    """
    Given a patient's daily post-discharge health metrics, determine the most significant period of health fluctuation by finding the maximum sum of measurements that show initial improvement followed by decline. The measurements are represented as numerical scores where a pattern of increasing values followed by decreasing values indicates a critical monitoring window that could signal potential readmission risk.
    

    """
def max_sum(arr):
    """
    Given a patient's daily post-discharge health metrics, determine the most significant period of health fluctuation by finding the maximum sum of measurements that show initial improvement followed by decline. The measurements are represented as numerical scores where a pattern of increasing values followed by decreasing values indicates a critical monitoring window that could signal potential readmission risk.
    

    """
    MSIBS = arr[:]
    for i in range(len(arr)):
        for j in range(0, i):
            if arr[i] > arr[j] and MSIBS[i] < MSIBS[j] + arr[i]:
                MSIBS[i] = MSIBS[j] + arr[i]
    MSDBS = arr[:]
    for i in range(1, len(arr) + 1):
        for j in range(1, i):
            if arr[-i] > arr[-j] and MSDBS[-i] < MSDBS[-j] + arr[-i]:
                MSDBS[-i] = MSDBS[-j] + arr[-i]
    max_sum = float('-Inf')
    for (i, j, k) in zip(MSIBS, MSDBS, arr):
        max_sum = max(max_sum, i + j - k)
    return max_sum